RU2134193C1 - Method of remote control of anthropomorphic walking and copying robot - Google Patents

Abstract

FIELD: robotics. SUBSTANCE: data on angular orientation of robot are transmitted to suspension follow-up mechanism over communication channel. Follow-up mechanism gives body of man-operator the same angular orientation as that of robot's corpus. It allows man to feel with his equilibrium organs all variations occurring in angular orientation of robot. EFFECT: higher efficiency of control. 6 cl, 141 dwg

Description

 The described method allows you to develop and manufacture the equipment necessary for remote control in copy mode walking anthropomorphic walking robot.

 Initially, we consider methods for manufacturing sensors that determine the angular orientation of an arbitrarily moving solid (in this case, the robot body).

 In the literature I reviewed regarding the problem of creating anthropomorphic walking robots, it says that "A walking robot must be equipped with a gyroscopic sensor to determine orientation in space." /cm. E.P. Popov, A. S. Yushchenko "Robots and Man", Moscow, 1984, p. 63 /.

 Such sensors that determine the angular orientation are necessary to ensure a stable gait of the robot.

 However, the mechanical “free” gyroscope has a number of obvious flaws, the combination of which, apparently, is one of the obstacles to creating full-fledged walking robots.

 The first, obvious, but not the main, drawback of the gyroscope is the mechanical rotation of the flywheel and the need to constantly maintain this fast rotation.

The second flaw of the gyroscope / it is less obvious, but much more significant /: the so-called. A “free” gyroscope mounted on a gimbal is not fully free of this word. Indeed, let the first axis of rotation of the "free" gyroscope be directed vertically / Fig. 1 /, where the gyro flywheel rotating on the axis OO _{1 is} depicted by an ellipse. The small / inner / gimbal frame is indicated by the letters A, B, C, D. It has a rotation axis PP ₁ . The large gimbal frame E, K, L, M rotates on the axis NN ₁ in bearings mounted in the robot body. According to the idea, the rotation axis of the flywheel OO ₁ should always maintain its direction in space, regardless of how the angular orientation of the robot body changes, but this is not so, in some cases a situation may arise when the axis OO ₁ will be tilted under the influence of external forces, changing the orientation of the robot body. Consider this situation.

жестко связанную с осью O-O₁ гироскопа. Орт

направим вдоль оси O-O₁. Маховик гироскопа крутится, но векторы

в этом кручении не участвуют. Вектор

направлен вдоль оси P-P₁, проходящей через подшипники внутренней рамки карданного подвеса. Вектор

будет направлен "к нам". Введем теперь правую тройку векторов

жестко связанную с корпусом робота. И пусть начальный момент, когда робот стоит вертикально, векторы

совпадают по своему направлению с ортами

гироскопа. Теперь допустим, что робот из вертикального положения принял положение "упор лежа" /проще говоря, лег на живот/, тогда рамки карданного подвеса займут по отношению к гироскопу положение, изображенное на фиг. 2. Если теперь робот начнет перекатываться из положения "упор лежа" на спину, то рамки карданного подвеса находятся в таком положении /фиг. 2/, что не дают возможности оси вращения "свободного" гироскопа сохранить свое направление в пространстве. При перекатывании робота с живота на спину ось вращения гироскопа может оказаться повернута на 180^o/я привожу здесь наиболее неблагоприятный случай/.We introduce the right triple of vectors

rigidly connected to the axis OO _{1 of the} gyroscope. Ort

direct along the axis OO ₁ . The gyro flywheel is spinning, but the vectors

do not participate in this torsion. Vector

directed along the axis of PP ₁ passing through the bearings of the inner frame of the gimbal. Vector

will be sent "to us." We introduce now the right triple of vectors

rigidly connected to the body of the robot. And let the initial moment when the robot is standing upright, vectors

coincide in their direction with the orts

gyroscope. Now suppose that the robot from the upright position has taken a “lying position” position (in other words, it lies on its stomach), then the frames of the gimbal will occupy the position shown in FIG. 2. If now the robot starts to roll from the "stop lying" position to the back, then the gimbal frames are in this position / Fig. 2 /, which does not allow the axis of rotation of the "free" gyroscope to maintain its direction in space. When the robot rolls from the abdomen to the back, the axis of rotation of the gyroscope may be rotated 180 ^o / I give here the most unfavorable case /.

After such a roll, the gyro sensor readings will already be incorrect. / An even worse situation may arise when the axis of the flywheel OO ₁ will be bent and the gyroscopic orientation sensor will fail. / In real technical devices / rockets, airplanes / the indicated flaw of the "free" gyroscope is eliminated by the introduction of "unloading engines" that do not give the gimbal frames go to the same plane. But in any case, a mechanical gyroscope is a complex device that is poorly suited for a walking anthropomorphic robot.

 It is possible to use interference laser gyroscopes as sensors that determine the orientation of a walking robot. But such interferometers are large, require complex alignment, roads and are not very reliable.

 The aim of the present invention is a method by which it is possible to produce compact, reliable, technological in the production of angular orientation sensors.

 For this purpose, it is supposed to use two devices different in their principles.

 The first of these is a liquid "plumb sensor". It allows you to determine the initial direction of the "top-bottom", and also allows you to measure the slow slopes of the body, changing the orientation.

 The second of them is a liquid inertial sensor that allows you to measure arbitrary rotational movements with significant acceleration.

 Plumb sensor.

 We consider the projection of a vertical line / top-to-bottom line / onto three “unit” circles inscribed on the sides of the cube / cm. FIG. 3 /. In FIG. 3 shows a cube with a side in two arbitrary units. Three circles of unit radius are inscribed on the sides of the cube, i.e. with diameters of two arbitrary units. The front walls of the cube are depicted by hatching, as if they are absent.

We introduce two coordinate systems into the cube: the Cartesian coordinates X, Y, Z and the polar coordinates θ, φ. The angles θ, φ vary between 0 ^o ≤ θ ≤ 180 ^o and 0 ^o ≤ φ ≤ 360 ^o / cm. FIG. 4/.

The angles θ and φ determine the orientation / slope / of the plumb line vector relative to the coordinate system X, Y, Z / cm. FIG. 4/. In FIG. 4, the vertical line vector is indicated by the abbreviation "OL", the projection of the vertical line vector is indicated by the abbreviation "FLOOR". The angle θ is measured from the Y axis of the coordinate system X, Y, Z, i.e., if the vector of the vertical line coincides / parallel to / the Y axis, then θ = 0 ^o . And if the vertical line vector is not parallel to the Y axis, then the angle θ ≠ 0 ^o / cm. FIG. 4/. The angle φ takes its origin from the X axis of the coordinate system X, Y, Z and increases if the projection of the vertical line vector is shifted / rotated / relative to the coordinate system from the X axis in the direction of the Z axis / cm. FIG. 4/.

 The sensor - plumb line is arranged as follows: each unit circle is a set of photocells arranged around a circle / cm. FIG. 5/. In front of the “single circle of photocells” a transparent disk-shaped “cell” is installed, filled with a transparent damping liquid / alcohol, kerosene, etc. /. This cuvette contains a small drop of mercury relative to the total volume of the cuvette. Radiation is incident on a cuvette from a light source, illuminating almost all photocells of a single circle. However, several photocells are closed from the light source by an opaque drop of mercury. These closed photocells characterize the lowest point of a unit circle of photocells / cm. FIG. 5/. In FIG. 5 depicts a “unit circle” consisting of only 18 photocells. In the real design of the plumb sensor, there should be much more, for example, a thousand.

 Each photocell belonging to a single circle is connected with its own output to its individual clocked "D" trigger. Further, all “D” triggers are connected to the priority encoder / cm. Fig.6 /. / "A single circle - a set of photocells," D "triggers, a priority encoder is best made in the form of one large integrated circuit, that is, on a single chip. It is possible to use a" CCD circle "as a measuring" unit circle ". That is charge-coupled device made in the form of a circle with sequential interrogation of elements.

 Inherently FIG. 6 is a “side view” of the so-called unit circumference of the sensor - plumb. This "side view" allows us to trace the connection of the sensor elements - plumb. In FIG. 6 numbers indicate: 1 - light source, 2 - cuvette, 3 - a drop of mercury, 4 - a set of photocells, 5 - a set of "D" triggers, 6 - priority encoder, 7 - electrical leads from the encoder, on which a binary code that characterizes the lowest point of this unit circumference of the sensor is a plumb line.

 Shielding a drop of mercury photocells causes the appearance of a priority binary code encoder on the output. This code, the position of the mercury drop, determines the projection of the plumb line onto the “unit circle”.

 The plumb sensor uses three “unit circles”. This enables us to obtain projections of a plumb line on three mutually perpendicular planes, see Fig. 7. As a result, we obtain three angles α, β, γ characterizing the slope of the coordinate system X, Y, Z with respect to the vertical line vector.

т. е. угол α возрастает, если в плоскости Z, О, Y вектор проекции отвесной линии смещается от оси Z в направлении к оси Y /см. фиг. 9 /.The angle α takes values in the interval 0 ^o ≤ α ≤ 360 ^o and characterizes the slope of the coordinate system X, Y, Z obtained as a result of rotation of the coordinate system X, Y, Z around the axis X. The angle α takes its origin / reference / from the Z axis , i.e., if the Z axis coincides with the projection vector of the plumb line on the plane Z, O, Y, then the angle α = 0 ^o / cm. FIG. eight /. The reference direction of the angle α is determined based on the condition that the right coordinate system X, Y, Z rotates around the X axis, which is determined by the vector product

i.e., the angle α increases if in the plane Z, O, Y the projection vector of the plumb line is shifted from the Z axis in the direction to the Y / cm axis. FIG. nine /.

т. е. угол β возрастает, если в плоскости X, Y, Z вектор проекции отвесной линии смещается от оси X в направлении к оси Z /см. фиг. 11/.The angle β takes values in the interval 0 ^o ≤ β≤360 ^o and characterizes the slope of the coordinate system X, Y, Z obtained as a result of rotation of the coordinate system X, Y, Z around the axis Y. The angle β takes its origin / reference / from the axis X , i.e. if the X axis coincides with the projection vector of the plumb line on the plane X, O, Z, then the angle β = 0 ^o / cm. FIG. ten/. The reference direction of the angle β is determined based on the condition that the right coordinate system X, Y, Z rotates around the Y axis, which is determined by the vector product

i.e., the angle β increases if, in the X, Y, Z plane, the projection vector of the plumb line is shifted from the X axis towards the Z / cm axis. FIG. eleven/.

т.е. угол γ возрастает, если в плоскости Y, O, X вектор проекции отвесной линии смещается от оси Y в направлении к оси X /см. фиг. 13 /.The angle γ takes values in the range 0 ^o ≤ γ ≤360 ^o , and characterizes the slope of the coordinate system X, Y, Z obtained as a result of rotation of the coordinate system X, Y, Z around the axis Z. The angle γ takes its origin / reference / from the axis Y, i.e., if the Y axis coincides with the projection vector of the plumb line on the Y, O, X plane, then the angle γ = 0 ^o / cm. FIG. 12 /. The reference direction of the angle γ is determined based on the condition that the right coordinate system X, Y, Z rotates around the Z axis, which is determined by the vector product

those. the angle γ increases if, in the Y, O, X plane, the projection vector of the plumb line is shifted from the Y axis towards the X / cm axis. FIG. 13 /.

Now the problem is as follows: it is necessary to determine the angles θ and φ, knowing the angles α, β, γ.
The following is a conclusion of the formulas connecting the angles α, β, γ with the angles θ and φ. These formulas are needed for programming ROM / read-only memory, see FIG. fourteen/. The angles α, β, γ, as well as the angles θ and φ, form a discrete finite set of values, which allows the use of ROM when recalculating the angles α, β, γ into angles θ and φ. In FIG. 14 digits 1, 2 and 3 are marked the so-called. "unit circles", and 4 - ROM. In this case, each unit circle measures its own angle: α, β, or γ, and represents a single electron-optical structure made on one substrate. / It is assumed that the topology of each such "unit circle" has a structure identical to the block diagram shown in FIG. 6. / The objective of FIG. 14 is to emphasize the use of ROMs for converting angles α, β, γ to angles θ, φ. It should be noted that in FIG. 14, several very important intermediate blocks are missing. / In more detail, this block diagram will be presented in FIG. 66, after the corresponding description. /
To derive mathematical formulas, we make some geometric constructions / cm. FIG. fifteen /. Where the sphere of the "unit" radius is inscribed in the cube ABCDD'A'B'C '. In the center of the sphere originates the coordinate system X ', Y', Z '. The relative orientation of the coordinate system and the cube is such that the axes X ', Y', Z 'are orthogonal to the corresponding faces of the cube. / Projections of a sphere and a cube on a plane perpendicular to the axes of the coordinate system give us three “unit circles” inscribed in squares, as can be seen from FIG. fifteen./
FIG. 15 allows you to divide the sphere into six angular sectors. Depending on which face of the cube the vertical line intersects, we define the angular sector.

Значением угла γ пренебрегаем, т. к. мы имеем возможность вычислить наклон /углы θ и φ / по двум углам α и β.
Привожу математические формулы, позволяющие вычислить для первого сектора θ и φ по значениям углов α и β . Для этого рассмотрим следующие геометрические построения /см. фиг. 17/. Где совмещены центры двух "единичных окружностей", по которым производится отсчет углов α и β, с реперной точкой O' координат X', Y', Z'. Согласно этому рисунку вектор отвесной линии принадлежит первому угловому сектору. Проекции отвесной линии на соответствующие "единичные окружности" дают углы α и β.
Из фиг. 17 видно, что
φ = β. (5)
Угол θ находим из прямоугольного треугольника O'PN, см. фиг. 18, которая является частью фиг. 17. Вершина P треугольника O'PN равна 90^o, поэтому угол θ можно найти как арктангенс отношения катетов NP и PO'
θ = arctg(NP:PO′). (6)
Но отрезок O'P, согласно фиг. 18, равен отрезку LM прямоугольного треугольника O'LM. А отрезок NP, согласно фиг. 18, равен гипотенузе KO' прямоугольного треугольника KLO'. Таким образом получаем
O′P = LM = |O′L|•tgα = 1•tgα; (7)
NP = KO′ = |O′L|:sinβ = 1:sinβ. (8)
Тогда формула 6 трансформируется в такой вид

Это и есть искомая формула, связывающая для первого углового сектора углы α и β с углом θ.
Второй угловой сектор получится, если отвесная линия пересекает сторону ADD'A' куба /см. фиг. 15 /. В этом случае углы проекций отвесной линии на единичные окружности заключены в пределах

Значением угла α пренебрегаем.The first angular sector will turn out if the plumb line intersects the side of the ABCD cube / cm. FIG. 16 /. In this case, the projection angles of the plumb line on unit circles are within

We neglect the value of the angle γ, since we are able to calculate the slope / angles θ and φ / from the two angles α and β.
I give mathematical formulas that allow us to calculate θ and φ for the first sector from the values of the angles α and β. To do this, consider the following geometric constructions / cm. FIG. 17 /. Where the centers of two "unit circles" are combined, along which the angles α and β are counted, with the reference point O 'of the coordinates X', Y ', Z'. According to this figure, the vertical line vector belongs to the first angular sector. The projection of the plumb line on the corresponding "unit circles" gives the angles α and β.
From FIG. 17 shows that
φ = β. (5)
The angle θ is found from the right triangle O'PN, see FIG. 18, which is part of FIG. 17. The vertex P of the triangle O'PN is 90 ^o , so the angle θ can be found as the arctangent of the ratio of the legs NP and PO '
θ = arctan (NP: PO ′). (6)
But the segment O'P, according to FIG. 18 is equal to the segment LM of the right triangle O'LM. And the segment NP, according to FIG. 18 is equal to the hypotenuse KO 'of the right triangle KLO'. Thus we obtain
O′P = LM = | O′L | • tgα = 1 • tgα; (7)
NP = KO ′ = | O′L |: sinβ = 1: sinβ. (eight)
Then formula 6 transforms into this form

This is the desired formula relating the angles α and β to the angle θ for the first angular sector.
The second angular sector will turn out if the plumb line intersects the ADD'A 'side of the cube / cm. FIG. fifteen /. In this case, the projection angles of the plumb line on unit circles are within

The angle α is neglected.

Третий угловой сектор получится, если отвесная линия пересекает сторону DCC'D' куба /см. фиг. 15/. В этом случае углы проекций отвесной линии на единичные окружности заключены в пределах

Значением угла β пренебрегаем.We produce the necessary geometric constructions / cm. FIG. 19 /. Where can I see
φ = β (11)
The angle θ is found from the right triangle O'PN, see FIG. 20. The vertex P of the triangle O'PN is 90 ^o , the angle θ, as in the previous case, is expressed through the arctangent of the ratio of the legs of NP and PO '
θ = arctan (NP: PO ′). (12)
Given the fact that from right triangles KLO 'and MPO' we have
NP = (cosβ) ^-1 and PO ′ = ctgγ. (13)
We obtain in the final form the formula for finding the angle θ:

The third angular sector will turn out if the plumb line intersects the DCC'D 'side of the cube / cm. FIG. fifteen/. In this case, the projection angles of the plumb line on unit circles are within

The angle β is neglected.

Учтем, что отрезок O'L равен единице, тогда формула 16 примет вид
θ = arctg LN, (17)
Отрезок LN находим с помощью теоремы Пифагора из прямоугольного треугольника LMN /см. фиг. 24/

Катет LM получаем из прямоугольного треугольника O'LM /см. фиг. 25/
LM = tgγ. (19)
А катет MN получаем из прямоугольного треугольника O'LK /см. фиг. 26/
MN = LK = tg(90^o-α) = ctgα (20)
Тогда формула 18 примет вид

Отсюда получим в окончательном виде формулу для угла θ

Формула для нахождения угла φ получается как арктангенс отношения катетов MN и LM прямоугольного треугольника LMN /см. фиг. 24/. С учетом формул 19 и 20 получим

Здесь необходимо сделать еще несколько замечаний, связанных с формулой 23
Если угол γ = 0^o и угол α < 90^o, то угол φ = 90^o.We produce the necessary geometric constructions / cm. FIG. 21 /. It differs significantly from that of FIG. 17 by the fact that an additional line LN is introduced into it. This can be seen in FIG. 22. The angle θ is found from the right triangle O'LN as the arctangent of the ratio of the legs of LN and O'L / cm. FIG. 23 /

We take into account that the segment O'L is equal to unity, then formula 16 takes the form
θ = arctan LN, (17)
The segment LN is found using the Pythagorean theorem from a right-angled triangle LMN / cm. FIG. 24 /

The LM leg is obtained from the right triangle O'LM / cm. FIG. 25 /
LM = tgγ. (19)
And the leg MN is obtained from the right triangle O'LK / cm. FIG. 26 /
MN = LK = tg (90 ^o -α) = ctgα (20)
Then formula 18 will take the form

From here we obtain in final form the formula for the angle θ

The formula for finding the angle φ is obtained as the arctangent of the ratio of the legs MN and LM of the right triangle LMN / cm. FIG. 24 /. Given formulas 19 and 20, we obtain

Here it is necessary to make a few more remarks related to formula 23
If the angle γ = 0 ^o and the angle α <90 ^o , then the angle φ = 90 ^o .

If the angle γ = 0 ^o and the angle α> 90 ^o , then the angle φ = 270 ^o .

If the angle γ = 0 ^o and the angle α = 0 ^o , then the angle φ has an indefinite value.

The expression "angle φ has an indefinite value" becomes clear if for the coordinate system X, Y, Z / cm. FIG. 4 / to build a "tilt globe" - a sphere of arbitrary radius, with a grid on its surface consisting of meridians and parallels / cm. FIG. 27 /. Meridians are lines on the surface of the slope globe along which the angle φ has a constant value. Parallels are lines along which the angle θ remains constant. But on the surface of the inclination globe there are two special ones, the so-called "polar" points. At these points, the angle φ has an indefinite value, and when these points intersect along the meridian, the angle φ experiences a jump of 180 ^o . At the polar points it is not necessary to know the values of the angle φ, these points are characterized by a different angle, the angle θ. Whenever θ = 0 ^o or θ = 180 ^o means that the vertical line vector passes through the polar point, i.e. this means that the coordinate system X, Y, Z is inclined in space so that the vertical line vector is parallel to the axis Y of the coordinate system.

Значением угла β пренебрегаем. Геометрические построения выполнены в фиг. 28 - это общий вид.Fourth corner sector:
A plumb line intersects the side of ABB'A 'cube / cm. FIG. fifteen /. In this case, the projection angles of the plumb line to unit circles are within

The angle β is neglected. Geometric constructions are made in FIG. 28 is a general view.

Т.е. получаем
θ = arctg(-|LN|). (25)
Отрезок LN находим с помощью теоремы Пифагора из прямоугольника LMNK /см. фиг. 31/

Отрезок LK находим с помощью фигуры 32, которая получается, если посмотреть на систему координат X', Y', Z', изображенную на фиг. 29 вдоль оси Z'. Имеем

Отрезок LM находим из фиг. 33

Формула 25 с учетом формул 26, 27 и 28 запишется так

Угол φ согласно фиг. 31 ищем в виде арктангенса отношения отрезков LM и LK

С учетом формул 27 и 28 получим

Здесь опять необходимо сделать несколько замечаний в связи с тем, что в секторе 4 находится полярная точка. Поэтому для формулы 31 имеем дополнения:
Если угол γ = 180^o и угол α > 270^o, то угол φ = 90^o.FIG. 29 highlights from FIG. 28 only those image elements that we need to derive formulas. From FIG. Figure 29 shows that to find the angle θ, it is necessary to use a right-angled triangle O'LN / cm. FIG. thirty /. The leg LN of the triangle O'LN is connected with the angle θ in this way

Those. we get
θ = arctan (- | LN |). (25)
The segment LN is found using the Pythagorean theorem from the rectangle LMNK / cm. FIG. 31 /

The line segment LK is found using figure 32, which is obtained by looking at the coordinate system X ', Y', Z 'shown in FIG. 29 along the z axis. We have

The segment LM is found from FIG. 33

Formula 25 taking into account formulas 26, 27 and 28 is written as

The angle φ according to FIG. 31 we are looking in the form of arctangent for the ratio of the segments LM and LK

Given formulas 27 and 28, we obtain

Here again, it is necessary to make a few comments in connection with the fact that sector 4 has a polar point. Therefore, for formula 31 we have the additions:
If the angle γ = 180 ^o and the angle α> 270 ^o , then the angle φ = 90 ^o .

If the angle γ = 180 ^o and the angle α <270 ^o , then the angle φ = 270 ^o .

If the angle γ = 180 ^o and the angle α = 270 ^o , then the angle φ has an indefinite value.

Значением угла α пренебрегаем. Геометрические построения /общий вид/ выполнены на фиг. 34. Фиг. 35 выделяет из фиг. 34 только необходимые элементы изображения, которые нужны для вывода формул. Мы видим из фиг. 35, что
φ = β. (33)
Угол θ находим из прямоугольного треугольника O'PN как арктангенс отношения катетов NP и O'P

Отрезок NP получаем из прямоугольного треугольника О'LK, это демонстрирует фиг. 36

Согласно фиг. 37;

Тогда формула 34 запишется в таком виде

Шестой угловой сектор:
Отвесная линия пересекает сторону A'B'C'D' куба /см. фиг. 15/. Углы проекции отвесной линии на единичные окружности заключены в пределах

Пренебрегаем значением угла γ. Необходимые геометрические построения выполнены на фиг. 38 и 39, откуда видно
φ = β. (39)
Угол θ находим как арктангенс отношения катетов TN и O'T прямоугольного треугольника O'TN /см. фиг. 39/

Отрезок TN получаем из треугольника O'LM /фиг. 40/

Отрезок O'T находим из треугольника O'LK /фиг. 41/;

В окончательном виде формула для угла θ запишется

Достоинством предлагаемого жидкостного датчика - отвеса является его технологичность. Необходимо только изготовить соответствующую микросхему, содержащую в себе единичную окружность фотоэлементов, набор триггеров и шифратор. При современном уровне развития микроэлектронной промышленности это не составит особого труда. А дальше при окончательной сборке датчика-отвеса не будет больших проблем с настройкой. Например: пусть на эталонном /горизонтальном/ стенде во время сборки датчика-отвеса выяснилось, что микросхема, определяющая угол α, дает ошибочные показания /см. фиг. 42/. В этой ситуации специалисту, осуществляющему сборку, достаточно будет, учитывая показания контрольных приборов, слегка повернуть микросхему, а затем закрепить ее в нужном положении /см. фиг. 43/.Fifth corner sector:
A plumb line intersects the side of the BCC'B 'cube / cm. FIG. fifteen /. The projection angles of the plumb line to unit circles are within

The angle α is neglected. Geometric constructions / general view / are made in FIG. 34. FIG. 35 highlights from FIG. 34 only the necessary image elements that are needed to derive the formulas. We see from FIG. 35 that
φ = β. (33)
The angle θ is found from the right triangle O'PN as the arctangent of the ratio of the legs of NP and O'P

The segment NP is obtained from the right triangle O'LK, as shown in FIG. 36

According to FIG. 37;

Then the formula 34 is written in this form

Sixth corner sector:
A plumb line intersects the A'B'C'D 'side of the cube / cm. FIG. fifteen/. The projection angles of the plumb line to unit circles are within

Neglect the angle γ. The necessary geometric constructions are made in FIG. 38 and 39, from where it can be seen
φ = β. (39)
We find the angle θ as the arctangent of the relation of the legs of TN and O'T of the right triangle O'TN / cm. FIG. 39 /

The segment TN is obtained from the triangle O'LM / Fig. 40 /

The segment O'T is found from the triangle O'LK / Fig. 41 /;

In the final form, the formula for the angle θ is written

The advantage of the proposed liquid sensor - plumb is its manufacturability. It is only necessary to make an appropriate microcircuit containing a single circle of photocells, a set of triggers and an encoder. At the current level of development of the microelectronic industry, this will not be difficult. And then, during the final assembly of the plumb sensor, there will be no big problems with tuning. For example: let it be found out on a standard / horizontal / stand during the assembly of the plumb sensor that the microcircuit that determines the angle α gives erroneous readings / cm. FIG. 42 /. In this situation, it will be enough for the specialist performing the assembly, taking into account the indications of the control devices, slightly turn the microcircuit, and then fix it in the desired position / cm. FIG. 43 /.

 Other sensors - plumb lines, which are based on other technical solutions, require more painstaking adjustment. So in the sensor - plumb, made on three spring accelerometers located along the axes of the coordinate system X, Y, Z / cm. FIG. 44 /, it is necessary to take into account the scatter of the parameters of three accelerometers: small differences in the stiffness of the springs, in friction, in the mass of inertial weights. All this makes tuning the sensor - a plumb line, consisting of spring accelerometers, more time-consuming.

 The disadvantage of a liquid plummet sensor is that it can give accurate readings only if the body on which the plummet sensor is mounted performs a translational / or almost translational / movement in space for a sufficiently long time, without strong shocks and rotations , i.e. liquid sensor - the plumb line gives the correct readings if it has been in the inertial / or almost inertial / reference system for a long time.

 / Here, under the expressions: “for a sufficiently long time” and “long enough”, we mean a time longer than the time of calming the natural vibrations / vibrations of mercury droplets in the cells /.

 Liquid inertial accelerated rotation sensor.

Operating principle:
Consider such a device, conventionally call it "measuring coil":
The tank filled with liquid is divided into two chambers by an elastic membrane / cm. FIG. 45 /, where V ₁ is the first chamber, V ₂ is the second chamber, M is an elastic membrane.

 From one chamber to another there is a tube T, also filled with liquid. The tube has a circle shape of radius R.

где ρ - плотность жидкости, l - длина окружности /длина трубки

- вращательное ускорение.With the accelerated rotation of such a device around the point "O" on different sides of the membrane creates a pressure drop

where ρ is the fluid density, l is the circumference / tube length

- rotational acceleration.

Под действием перепада давления ΔP упругая мембрана прогнется. Таким образом, следя за смещением мембраны, можно получить значение вращательного ускорения

Тогда угол α′, на который повернулся в пространстве измерительный виток под воздействием вращательного ускорения

за время t, вычисляется по формуле

Мы должны иметь возможность измерять вращательные ускорения, происходящие вокруг любого произвольного направления /вокруг любой произвольной оси/, поэтому в жидкостном инерционном датчике вращательных ускорений необходимо использовать три таких измерительных витка. Причем каждый из этих измерительных витков будет иметь свою отдельную емкость и свою отдельную упругую мембрану, т.е. измерительные витки будут работать независимо друг от друга. Располагаться они будут в трех взаимно ортогональных плоскостях /см. фиг. 46/.Formula 42 is obtained by analogy with the formula for calculating the pressure created by a column of liquid in the field of gravitational forces. Only in our case, the role of "gravity acceleration" is rotational acceleration

Under the action of a pressure drop ΔP, the elastic membrane bends. Thus, by monitoring the displacement of the membrane, we can obtain the value of rotational acceleration

Then the angle α ′, which turned the measuring coil in space under the influence of rotational acceleration

over time t, is calculated by the formula

We must be able to measure rotational accelerations occurring around any arbitrary direction / around any arbitrary axis /, therefore it is necessary to use three such measuring coils in a liquid inertial rotational acceleration sensor. Moreover, each of these measuring coils will have its own separate capacitance and its own separate elastic membrane, i.e. measuring coils will work independently of each other. They will be located in three mutually orthogonal planes / cm. FIG. 46 /.

 From formula 42 it is seen that to increase the sensitivity of the device, it is necessary to take a liquid with the largest possible specific density ρ / for example, mercury / and increase the tube length T / cm. FIG. 45 /. The length of the measuring tube can be increased several times, but in order for the entire device to remain compact, the tube will have to be rolled up in a special - "spiral" way. In this case, it is necessary to carefully ensure that additional / unplanned / loops lying in mutually orthogonal planes are not accidentally formed. This is especially important if the total number of measuring coils in the "spiral" is small. And if we fold the tube incorrectly, the device will give incorrect readings. In FIG. 47 shows how to roll a measuring tube. FIG. 48 gives the view of FIG. 47 on the left. In FIG. 49 is a perspective view of FIG. 47. Another possible variant of folding the “spiral” is shown in FIG. 50 - 52. As can be seen here, two tubercles formed on the measuring tube. The presence of such tubercles does not affect the correct functioning of the device, despite the fact that they do not lie in the same plane with the main measuring coils. The fact is that the effect of two such tubercles on the liquid filling the measuring tube is mutually compensated. It is possible to roll the measuring tube into a multi-turn coil - a solenoid, by wrapping it on a coil in several layers / similar to how a copper wire is wound in the manufacture of an electromagnet / see Fig. 53. This last design in the form of a coil - a solinoid has the greatest sensitivity, and the appearance of several additional turns in mutually orthogonal planes is no longer scary, because their influence on the accuracy of the device can be neglected. Of course, after the measuring tube is folded into a solenoid or into a “spiral”, formula 42 will no longer be absolutely accurate, but this is not a problem either, because the device will have to be empirically calibrated anyway.

 The liquid inertial accelerated rotation sensor proposed here responds only to rotational accelerations / to a change in the angular orientation of the body in space / and completely ignores the linear accelerations of the body / accelerations arising from the translational movement of the body /.

 The liquid inertial accelerated rotation sensor is designed to measure significant rotational accelerations, thereby it compensates for the disadvantages of the liquid sensor - plumb / cm. above /, which cannot work in non-inertial reference systems.

The displacement of the membrane in the liquid inertial accelerated rotation sensor can be determined using the change in electric capacitance, i.e. the membrane will have to play the role of one of the plates of a capacitor of variable capacity. This is illustrated in FIG. 54, which shows two capacitors of variable capacitance C ₁ and C ₂ connected in series with each other. Moreover, the change in capacitance of these capacitors is mutually opposite, i.e., if the capacitance of one capacitor increases, then the capacitance of the other at this time necessarily decreases, and vice versa. From the capacitor C _{2 there} are conclusions to which the measuring device is connected, here, in fact, a capacitive voltage divider is obtained. The capacitors are supplied with a reference sinusoidal voltage U _op having a fixed frequency and amplitude.

Depending on which liquid is used in the liquid sensor: dielectric / kerosene / or electric current conductor / mercury /, the following sensor designs are possible:
FIG. 55 .. As a fluid filling the sensor, we use a dielectric / kerosene /. Then only one membrane is used: M, it acts as a lining of an electric capacitor. The other two covers: K ₁ and K ₂ are located directly inside the chambers V ₁ and V ₂ filled with kerosene. Corresponding electrical conclusions are made from the membrane and from two other plates.

Иначе датчик ускоренных вращений окажется чувствительным к поступательным ускорениям, что крайне нежелательно./
Основным достоинством жидкостного инерционного датчика вращений является его простота по сравнению с другими устройствами, выполняющими аналогичную роль. Например, по сравнению с лазерными интерференционными гироскопами, изготовленными с использованием оптических волокон. Однако жидкостной инерционный датчик вращений имеет один достаточно серьезный недостаток: в нем используется упругий элемент - мембрана, которая может послужить источником т.н. возвращающей силы. Как следствие, в датчике возможно появление колебаний и, конечно, резонанса при определенной частоте. Надо сказать, что по своей сути жидкостной инерционный датчик вращательных ускорений представляет из себя модифицированный пружинный датчик - акселерометр, см. фиг. N 44. В данном случае в жидкостном инерционном датчике ускоренных вращений роль инерционной массы играет жидкость, а роль пружины - упругая мембрана. Так что вполне естественно, что жидкостному инерционному датчику ускоренных вращений, как и всякому датчику ускорений с инерционной массой присущ этот недостаток: опасность возникновения резонанса. Однако в технике сплошь и рядом применяются датчики-акселерометры, главным образом в навигационных системах кораблей, самолетов, баллистических ракетах. И это несмотря на опасность появления резонанса. Надеюсь, что и жидкостной инерционный датчик ускоренных вращений найдет свое применение в технике, несмотря на опасность проявления резонанса, тем более, что существуют способы, позволяющие существенно ослабить влияние резонанса на точность измерений:
Во - первых, сама жидкость, заключенная в трубке, может выполнить роль демпферирующего элемента и погасить резонансные колебания. Дело в том, что при колебаниях жидкость должна будет поочередно перетекать в небольших количествах по трубке Т то в одну камеру V₁, то в другую V₂. /см. фиг. N 45/, при этом жидкость будет тереться о стенки трубки Т, гася возникшие колебания.If an electric current conductor, for example, mercury, is used as the liquid filling the inertial accelerated rotation sensor, then a different sensor design / cm should be used. FIG. 56 /. Two elastic metal membranes have already been applied here: M ₁ and M ₂ . A metal spacer P is installed between them, abutting the membranes with convex / rounded / surfaces. The vertices / center points / rounded surfaces of the metal spacer are soldered to the corresponding membranes. Thus, the conductive liquid / mercury /, the membranes M ₁ , M ₂ and the spacer P are one solid conductor. The spacer is a capacitor plate. The voltage is removed from it with the help of terminal K ₃ , which at one end is immersed in mercury. Two other capacitor plates K ₁ and K _{2 are} made in the form of washers, mounted with their central holes on the outer surfaces of the chambers V ₁ and V ₂ made of a dielectric. / It is necessary to note here that the ratio of the total mass of the membranes M ₁ , M ₂ and the spacer P to the volume of an imaginary cylinder V _c , the bases of which are membranes / cm. FIG. 57 /, should be equal to the density of the liquid used in the sensor,

Otherwise, the accelerated rotation sensor will be sensitive to translational accelerations, which is extremely undesirable. /
The main advantage of a liquid inertial rotation sensor is its simplicity in comparison with other devices performing a similar role. For example, compared with laser interference gyroscopes made using optical fibers. However, the liquid inertial rotation sensor has one rather serious drawback: it uses an elastic element - a membrane, which can serve as a source of the so-called. restoring power. As a result, oscillations and, of course, resonance at a certain frequency are possible in the sensor. I must say that, in essence, the liquid inertial sensor of rotational accelerations is a modified spring sensor - accelerometer, see Fig. N 44. In this case, in the liquid inertial sensor of accelerated rotations, the role of the inertial mass is played by the liquid, and the role of the spring is the elastic membrane. So it is quite natural that the liquid inertial sensor of accelerated rotations, like any acceleration sensor with inertial mass, has this drawback: the risk of resonance. However, in technology, accelerometer sensors are often used, mainly in the navigation systems of ships, aircraft, and ballistic missiles. And this despite the danger of resonance. I hope that the liquid inertial accelerated rotation sensor will find its application in technology, despite the danger of resonance, especially since there are methods that can significantly reduce the effect of resonance on the measurement accuracy:
Firstly, the liquid itself, enclosed in a tube, can play the role of a damping element and damp out resonant vibrations. The fact is that during fluctuations, the liquid will have to alternate in small quantities alternately flowing through the T-tube into one chamber V ₁ , then into another V ₂ . /cm. FIG. N 45 /, while the liquid will rub against the walls of the tube T, damping the resulting vibrations.

 Secondly, it is possible to develop active methods of dealing with resonant oscillations. For this, it is necessary to transfer, by appropriate initial selection of the main characteristics of the sensor / selection of the membrane elasticity, selection of the length and cross section of the measuring tube / resonant frequency of the sensor to the high-frequency region, i.e. in a region that is absolutely not characteristic of a body whose changes in the angular orientation we measure. / For example, the rotation / vibration / body frequency of an anthropomorphic walking robot is likely to be less than 5 Hz. In this case, we must translate the natural vibration frequency of the liquid inertial accelerated rotation sensor to a region of the order of 10 Hz. / Next, in the electrical circuits intended for taking readings from the liquid inertial sensor, it will be necessary to introduce a band-suppression filter, and in the negative feedback circuit - narrow band filter. Moreover, the resonant frequency of these filters must coincide with the resonance frequency of the liquid inertial accelerated rotation sensor.

 The use of such filters in electronic circuits will allow us to highlight the low-frequency component of the rotations of the body / body of the robot / that we are interested in, even if the resonance in the sensor does occur. FIG. 58 and 59 explain this.

 In FIG. 58 is a graph of fluctuations in electrical voltage taken from a sensor to filters. And FIG. 59 shows that only the low-frequency component of the voltage passed through the filters.

 Electric circuits with which you can take readings from a liquid inertial accelerated rotation sensor can have the following structure: / cm. FIG. 60 /.

In FIG. 60 there are the following elements: capacitors C ₁ and C ₂ represent a membrane and plates K ₁ and K ₂ of the accelerated rotation sensor. These capacitors are connected to an alternating voltage generator U _op , a fixed frequency and amplitude. The voltage removed from the sensor membrane (in the diagram the membrane is marked with the letter "M") is applied to a band-suppression filter 1. Then the signal is amplified by an amplifier 2, which has a narrow-band filter 3 built in the negative feedback circuit. The use of filters eliminates the high-frequency component corresponding to resonant vibrations of the sensor. Next, a diode bridge 4, a rectifying voltage, is used. Behind it is an LC ₃ filter, which smooths out carrier frequency oscillations arising in the circuits from an alternating voltage generator U _op . Only the low-frequency component, characterizing the change in the angular orientation of the body / body of the robot /, enters the input of the sampling device - storage 5. Then comes the analog-to-digital Converter 6, followed by a programmable read-only memory 7, the output of which is a binary code. / EPROM 7 is required to configure the sensor. / The proposed scheme uses two independent lands "3 ₁ and З ₂ .

 There is another opportunity to significantly reduce the risk of resonant oscillations in the sensor. To do this, turn the spacer R / cm. FIG. 56 / in “The Fram device for reducing vibrations”, see the book: N.V. Roze “Dynamics of a rigid body” Leningrad 1932, pp. 285 - 289. However, I will not consider this possibility here, because I believe that the methods already considered for damping oscillations are sufficient.

а нам для дальнейших применений понадобится значение угла α′, на который произошел поворот корпуса робота за время t под воздействием вращательного ускорения

см. формулу (43). Поэтому рассмотрим метод численного интегрирования значений вращательных ускорений. Интегрирование будем производить с помощью цифровой /двоичной/ электроники.As can be seen from FIG. 60, we get the rotational acceleration value at the outputs of the ROM

and for further applications, we need the value of the angle α ′, by which the robot body rotated in time t under the influence of rotational acceleration

see formula (43). Therefore, we consider the method of numerical integration of rotational acceleration values. We will integrate using digital / binary / electronics.

 It is assumed that in the first seconds after turning on the power, the robot body is stationary or inactive. This gives us the opportunity to get the initial conditions for the "task of determining the angular orientation of the robot body", i.e. we can define the initial downward direction. In those first seconds, while the robot remains stationary, the electronics of the liquid orientation sensors have time to give readings thousands of times. Some of these indications will need to be remembered - they are needed for calculations.

через равные промежутки времени Δt.
Вначале найдем скорость поворота корпуса робота, которая получается в результате вращательных ускорений

Используем аппроксимацию: заменим интегрирование суммированием площадей трапеций, заключенных под графиком функции /см. фиг. 61/. Здесь надо особо подчеркнуть один принципиальный момент: из графика, изображенного на фиг. 61, видно, что в первые мгновения значения функции

равны нулю. Это крайне необходимое условие, а вовсе не случайность! Только при соблюдении этого обстоятельства можно выполнить интегрирование.The change in the angular orientation of the robot casing "from an electronic point of view" is slow, i.e. Electronics works much faster than rotations of the robot body in space. We select the values of rotational accelerations

at regular intervals Δt.
First, we find the rotation speed of the robot body, which is obtained as a result of rotational accelerations

We use the approximation: we replace the integration by summing the areas of the trapezoids enclosed under the graph of the function / cm. FIG. 61 /. One fundamental point should be emphasized here: from the graph depicted in FIG. 61, it can be seen that in the first instants of the function

equal to zero. This is an essential condition, and not an accident at all! Only under this circumstance can integration be performed.

за конечное время t_i, выражается формулой

-
площадь элементарной трапеции, заключенной под графиком функции

Это есть приращение угловой скорости вращения за "бесконечно малое" время Δt, прошедшее между моментами времени t_n-1 и t_n.Then the rotation speed of the robot body, obtained under the influence of rotational accelerations

for a finite time t _i , is expressed by the formula

-
area of the elementary trapezoid enclosed under the function graph

This is the increment of the angular velocity of rotation for the "infinitesimal" time Δt elapsed between the times t _n-1 and t _n .

Полная скорость вращения корпуса робота получается с учетом начальных условий

где

начальная малая скорость, полученная из показаний датчика - отвеса. /Напомню, что датчиком - отвесом можно пользоваться, когда робот малоподвижен, например, сразу после включения электропитания./ В общем случае α₀, хотя и малая, но все-таки отличная от нуля величина, поэтому ее надо учитывать. Вычисляется она как разность угла α₀, измеренного в момент времени t_o, и угла α_-1, измеренного в момент - t_-1 и запомненного электроникой.Then formula (6) can be rewritten as follows:

The full speed of rotation of the robot body is obtained taking into account the initial conditions

Where

initial low speed obtained from the readings of the sensor - plumb. / I remind you that the plumb sensor can be used when the robot is inactive, for example, immediately after turning on the power. / In the general case, α ₀ , although small, but still non-zero, so it must be taken into account. It is calculated as the difference between the angle α ₀ measured at time t _o and the angle α _-1 measured at time t _-1 and stored by the electronics.

где первый член описывает поворот, полученный роботом, под воздействием вращательных ускорений за промежуток времени, прошедший между моментами t_o и t_i.A complete rotation of the robot body that occurred during time t _i is determined by the formula

where the first term describes the rotation obtained by the robot under the influence of rotational accelerations for the period of time elapsed between the moments t _o and t _i .

Третий член - это начальный угол α₀, т. е. угол, на который был повернут корпус робота в момент времени t_o, его значение получено от датчика - отвеса.The second term describes the angle at which the robot body rotates under the influence of a low initial velocity

The third term is the initial angle α ₀ , that is, the angle by which the robot body was rotated at time t _o , its value is obtained from the plumb sensor.

Перепишем это в более наглядной форме:

Из формулы (52) видно, что полный угол, на который повернулся в пространстве корпус робота под воздействием произвольных ускорений, можно найти, если представить его так:

где

Эти две последнии формулы (53) и (54) более удобны по сравнению с формулой (52), т. к., основываясь на них, легче организовать "бесконечный вычислительный цикл" /см. фиг. 63 и 64/. К формулам (53) и (54) необходимо еще добавить начальные условия:
начальную скорость

и начальное ускорение

а также добавим еще одну формулу, которая получается из формулы (48),

Комментарии:
В соответствии с формулой (53) угол

на который повернулся в пространстве корпус робота к моменту времени t_i, вычисляется как сумма двух слагаемых:
Первое слагаемое:

это значение угла, на который был повернут корпус робота к моменту t_i-1. К моменту t_i угол

должен быть вычислен и запомнен электроникой.We use a similar technique: we replace integration by summing over areas / cm. FIG. 62 /. Then formula (50) will be written as follows:

We rewrite this in a more visual form:

From formula (52) it can be seen that the full angle at which the robot body rotated in space under the influence of arbitrary accelerations can be found if it is represented as follows:

Where

These last two formulas (53) and (54) are more convenient in comparison with formula (52), since, based on them, it is easier to organize an “infinite computational cycle” / cm. FIG. 63 and 64 /. To formulas (53) and (54), it is necessary to add the initial conditions:
initial speed

and initial acceleration

and also add another formula, which is obtained from formula (48),

Comments:
According to formula (53), the angle

on which the robot body turned in space at time t _i , is calculated as the sum of two terms:
The first term:

this is the value of the angle by which the robot casing was rotated by the time t _i-1 . To the moment t _{i the} angle

must be calculated and memorized by electronics.

это причащение угла поворота корпуса робота, возникшее за время Δt, прошедшее между моментами t_i-1 и t_i. Значение

вычисляется электроникой робота с учетом формул (55), (56) и (57), на основе показаний датчика - отвеса.The second term:

this is the communion of the angle of rotation of the robot body that arose during the time Δt elapsed between the moments t _i-1 and t _i . Value

calculated by the robot electronics, taking into account formulas (55), (56) and (57), based on the readings of the sensor - plumb.

 I will explain how the algorithm depicted in FIG. 63 and 64.

 Immediately after turning on the power to the robot, the computational process specified by the specified algorithm begins.

In block 2, the initial values are assigned to some parameters. The integer parameter J is necessary for branching in an "infinite loop", depending on whether the condition is satisfied. / Block 2 plays another very important role, which is not fully reflected in this algorithm. The fact is that the execution of the “infinite loop” will be interrupted from time to time, and the computational process will be returned to block 2. The reasons for the occurrence of such interruptions will be explained later in the section where the issue of the dominance of a sensor of one type over a sensor of another type is discussed. e. depending on the dynamics of changes in the angular orientation of the robot body. /
Entering current readings from a liquid sensor - plumb line and from a liquid inertial accelerated rotation sensor on the presented algorithm is shown in block 3.

 Block 4 provides the same interrupt mode, which was mentioned above.

 Block 5 - verification of the condition, if J = 0, then go to block 6.

на который повернут корпус робота, присваивается текущее значение угла α_i, измеренное датчиком - отвесом. А также в блоке 6 происходит присвоение переменной

текущего значения угла α_i, это - "задел на будущее" - в следующем периоде выполнения цикла нынешнее значение α_i будет уже считаться предыдущим.In block 6 we make some assignments. Parameter J is assigned the value one. The corner

on which the robot body is rotated, the current value of the angle α _i , measured by the plumb sensor, is assigned. And also in block 6, the variable is assigned

the current value of the angle α _i , this is a "reserve for the future" - in the next period of the cycle, the current value of α _i will already be considered the previous one.

на который повернулся корпус робота. После этого мы вновь возвращаемся к блоку 3, это показано стрелкой. Теперь начинается новый период цикла.Block 7 denotes the output of the value of the full angle

which turned the body of the robot. After that, we again return to block 3, this is indicated by the arrow. Now begins a new period of the cycle.

 Again, in block 3, the current values are inputted by the sensor - a plumb line and an inertial accelerated rotation sensor. After that, we go down to block 5, if we managed to go through block 4. Now the value of the parameter is J = 1, so we go to block 8, where the condition is checked.

 After that, we get to block 9, where, based on the previous and current readings of the plumb sensor, the initial angular velocity of rotation of the robot body is calculated.

на который повернулся в пространстве корпус робота, получается присвоением текущего значения угла α_i, измеренного датчиком - отвесом. Переменная

получает значение

/Как уже ранее указывалось

это приращение угла поворота корпуса робота, возникшее за промежуток времени Δt. /. Все остальные присвоения, осуществленные в блоке 10, необходимы, чтобы обеспечить работу алгоритма в следующем периоде, т.е. "нынешние значения становятся предыдущими" и запоминаются - записываются в соответствующие ячейки памяти. Так нынешняя скорость

для следующего периода цикла становится предыдущей

нынешний угол α_i станет предыдущим

Далее переходим к блоку 11, который символизирует собой вывод информации: полного угла

на который повернут корпус робота, и приращения угла поворота корпуса робота

После чего, мы вновь возвращаемся к блоку 3.In block 10, assignments are made. The J parameter is assigned the value "two", this is necessary for us so that in the next period it is possible to switch to another branch of the algorithm. Full angle

on which the robot body turned in space, is obtained by assigning the current value of the angle α _i , measured by a plumb sensor. Variable

gets value

/ As already indicated

this is the increment of the angle of rotation of the robot body, which occurred over a period of time Δt. /. All other assignments made in block 10 are necessary to ensure the operation of the algorithm in the next period, i.e. "current values become previous" and memorized - are recorded in the corresponding memory cells. So current speed

for the next cycle period becomes the previous

the current angle α _i will be the previous one

Next, go to block 11, which symbolizes the output of information: full angle

on which the robot body is rotated, and increments of the angle of rotation of the robot body

After which, we again return to block 3.

по времени. Этим самым мы получаем полный угол

на который повернут корпус робота к моменту t_i /блок 14/. В блоке 15 делаются необходимые присвоения для следующего периода цикла.Now, after the first two periods of the cycle, the robot body can undergo strong arbitrary rotational accelerations. We measure these accelerations with a liquid inertial accelerated rotation sensor / block 3 /. Previously, the parameter in block 10 was assigned the value 2, so now, after several checks for fulfilling the conditions / blocks 4, 5, 8 / we get to block 12, where double integration is carried out

by time. Thus we get the full angle

on which the robot body is rotated by the time t _i / block 14 /. In block 15, the necessary assignments are made for the next cycle period.

 Block 16 symbolizes the output of information, after which we again return to block 3, and the cycle becomes "infinite".

 Now the main role in determining the angular orientation of the robot body has passed entirely to the liquid inertial accelerated rotation sensor. This role will belong to him until the angular orientation of the robot body begins to change uniformly for a long time / under these conditions there will be an interruption depicted on the algorithm by block 4, parameter J will be set to zero in block 2, etc. / .

 To find the total angles β ′ and γ ′, one can imagine algorithms and formulas similar in structure, similar to formulas (53), (54), (55), (56), (57).

 So, we have at our disposal three angles α ′, β ′, γ ′, which are obtained taking into account the readings of two types of sensors, both a plumb sensor and a liquid inertial accelerated rotation sensor.

 By changing the values of these three angles α ′, β ′, γ ′, we can now determine the change in the angular orientations of the robot body. To do this, we introduce the right coordinate system XYZ, rigidly connecting it with the body of an anthropomorphic walking robot.

Here, for our theoretical constructions, it is necessary to use the Euler theorem, which states:
“If, with respect to some reference frame S, a rigid body has one fixed point, then the movement of the solid from any position to any other position can be accomplished by one rotation at a certain angle around a certain axis passing through the fixed point of the body.”
/ See book: I. I. Olkhovsky "Course in Theoretical Mechanics for Physicists". M., 1970, p. 148./
Suppose that for a sufficiently long period of time, the robot body, and with it the coordinate system XYZ, did not change its angular orientation in space / such a situation, as I have already indicated, arises in the first moments immediately after turning on the power of the anthropomorphic walking robot. The robot is still motionless, and the angular orientation sensors are already giving out their readings. /
Then the robot / together with the coordinate system XYZ. / begins to make its arbitrary movements in space, changing its angular orientation in an unpredictable way.

т. е. система координат XYZ совершила поворот в пространстве, перейдя в состояние, которое можно пометить индексом "k": X_k, Y_k, Z_k.At time t _i , the coordinate system XYZ had a slope θ _i , φ _i . We mark the coordinate system XYZ observed at time t _{i with the} index "i", so we get: X _i , Y _i , Z _i . After one period Δt of polling sensors at time t _k = t _i + Δt, the orientation sensors gave readings:

that is, the coordinate system XYZ made a rotation in space, passing into a state that can be marked with the index "k": X _k , Y _k , Z _k .

/это "бесконечно малые величины с точки зрения механики", но отнюдь не "с точки зрения" электроники/.In our case, the period of polling sensors Δt “from the point of view” of mechanics can be considered an “infinitesimal” time interval, i.e. the change in the angular orientation of the robot casing is slow compared to the speed of the electronics. Therefore, for the "infinitesimal" time Δt / for the period of the survey of orientation sensors / the robot body will have time to change / its orientation only by "infinitesimal" values:

/ these are "infinitesimal values from the point of view of mechanics", but by no means "from the point of view" of electronics /.

 As you know, infinitesimal rotations commute, therefore they can be identified with a vector, see the book: I. I. Olkhovsky, “A Course in Theoretical Mechanics for Physicists”. Moscow, 1970, p. 151, where it says something like this: an infinitesimal rotation can be specified by a vector, the modulus of this vector is equal to the angle of rotation, and the straight line on which the vector is located is the instantaneous axis of rotation.

характеризующий бесконечно малое изменение ориентации корпуса робота, произошедшее за время Δt, будет иметь в системе координат X_i, Y_i, Z_i компоненты:

и длину /модуль/:

Теперь рассмотрим очень важный принципиальный момент, играющий фундаментальное значение в теории изготовления механизма подвеса - устройства, с помощью которого мы будем управлять ходьбой антропоморфного шагающего робота.In our case, the vector

characterizing the infinitesimal change in the orientation of the robot body that occurred during the time Δt will have the following components in the coordinate system X _i , Y _i , Z _i :

and length / module /:

Now consider a very important point of principle that plays a fundamental role in the theory of manufacturing the suspension mechanism - a device with which we will control the walking of an anthropomorphic walking robot.

корпуса робота, совершаемый в пространстве вокруг мгновенной оси, иным /эквивалентным/ образом, а именно: как наклон корпуса робота, происходящий относительно вектора отвесной линии, и как поворот вокруг вектора отвесной линии, осуществляемый корпусом робота одновременно с наклоном.We have every right to decompose an arbitrary infinitesimal turn

the robot body, made in space around the instantaneous axis, in a different / equivalent / way, namely: as the inclination of the robot body relative to the vector of the vertical line, and as the rotation around the vector of the vertical line, carried out by the robot body simultaneously with the inclination.

Поворот корпуса робота вокруг вектора отвесной линии определяется еще одним бесконечно малым углом dξ′, который вычисляется как скалярное произведение двух векторов: вектора отвесной линии - вектора единичной длины, с направлением, взятым момент t_i, вектора

По сути мы здесь находим проекцию вектора

на единичный вектор отвесной линии /см. фиг. 65/.Thus, any arbitrary change in the orientation of the robot body is described by three independent parameters. The inclination of the robot body relative to the vertical line vector is determined by two infinitely small angles

The rotation of the robot body around the vertical line vector is determined by another infinitesimal angle dξ ′, which is calculated as the scalar product of two vectors: vertical line vector - unit length vector, with the direction taken moment t _i , of the vector

In fact, we find here the projection of the vector

per unit vector of a vertical line / cm. FIG. 65 /.

Таким образом угол dξ′ вычисляется по формуле

От дистанционного антропоморфного шагающего робота мы будем передавать по каналам связи на устройство управления три угла:

/см. фиг. 66/.The unit vector of the vertical line has at the time t _i in the coordinate system Z _i , Y _i , Z _i components / cm. FIG. 4/:

Thus, the angle dξ ′ is calculated by the formula

From a remote anthropomorphic walking robot, we will transmit three angles via communication channels to the control device:

/cm. FIG. 66 /.

In the block diagram shown in FIG. 66, the corresponding blocks depict: a liquid sensor - a plumb line and a liquid inertial sensor of accelerated rotation. / The device of these sensors was described above. /
Data from the sensors enter the computing unit and the control unit.

 Computation block: its purpose is to calculate the total angles: α ′, β ′, γ ′ and the angle dξ ′ based on the data from the liquid sensors. The operation of the computing unit is based on the algorithm shown in FIG. 63 and 64.

In addition, in the calculation block, the angles θ ′ and φ ′ / cm are used to calculate the angle dξ ′. formula (63) / obtained in the previous period. / This is shown in the diagram by arrows going from the ROM to the computation unit. /
The control unit, based on the readings of liquid sensors and from the dynamics of changes in these readings, sets various modes of operation of the calculation unit. / In the circuit of FIG. 66, the “steering” action of the control unit on the calculation unit is indicated by a thick arrow. /
The read-only memory / ROM / converts the angles: α ′, β ′, γ ′ into the angles θ ′ and φ ′. / This has already been mentioned, see fig. 14 and comments on it. /
Consider the principles of operation of the control unit.

 The operation of the control unit is based on the following: based on the data received from the liquid sensors in the control unit and based on the dynamics of changes in these data, it is necessary to determine the mode of changing the angular orientation of the robot body and, in accordance with a certain mode, change the calculation mode of the calculation unit.

 To solve this problem, it will be necessary at first to consider / more correctly say: "peep" / how a person determines his angular orientation in space, i.e. you need to see how the human organs of balance function.

 A person has liquid inertial "accelerated rotation sensors" in his head, this is the so-called otolithic organs. The person also has a “sensor - plumb line”, but it is not localized in any particular place in the body, but is “smeared” over virtually the entire body. Those. a person determines the direction of external accelerations / including the direction of gravitational forces / with the help of his whole body. Under the influence of external accelerations, individual parts of the human body are displaced relative to each other, in a certain way, a person feels this and subconsciously determines the direction and intensity of the action of external accelerations.

 The use by the human body of such a "plumb sensor" allows it to determine the initial downward direction. And the otolithic organs monitor subsequent changes in the angular orientation of the human body / including the rotations that occur around the vertical line vector, and the "plumb sensor" does not notice such rotations /.

I will give specific examples that explain the effect of a particular “sensor” in a given situation:
The first, very important case. It is often found in practice in us: the human body remains motionless or almost motionless for a long time, i.e. the human body does not experience large accelerations and rapid rotational movements. From this situation, a person subconsciously using his "sensor - plumb" receives information about the initial direction "down". In the future, this information "about the initial downward direction" will be very necessary for a person to determine his angular orientation when his body begins to move unevenly, that is, when rotational accelerations appear / see the second case /.

 Using mathematical terminology, we can comment on this first case as follows: "finding the initial direction" down "is the finding of the initial conditions in the problem of" determining the angular orientation of an arbitrarily moving human body ".

 The second, very common case: a person moves on foot over heavily rough terrain. In this case, the human body will certainly experience tremors and rotational accelerations. In this situation, the main role in determining the angular orientation in humans is played by the otolith organ.

 These two cases just examined are the most important for us, because they are most typical - nature created man "on the basis of the assumption" that he / man / will move first of all on his own, on foot, on the move.

 In the future, the person was very "spoiled" by inventing various technical means of transportation unplanned by the primitive nature: boats, carts, cars, etc. .... One more mechanism must be included in this list - a centrifuge, which, of course, does not apply to vehicles, however, it is the most "insidious" technical device for the human organs of balance.

 As a result of the use of technical means, the human balance organs often find themselves in situations that are “atypical from the point of view of pristine nature”. I will demonstrate this by the example of a centrifuge / cm. FIG. 67 /.

 Schematically, the centrifuge is arranged as follows: a vertical axis of rotation 3 is installed in bearings 1 and 2. A perpendicular rod 4 is fixed on it, at one end of which there is a counterweight 5, and a “cradle” 6 is suspended at the other.

 The person / in the figure it is indicated by the number 7 / lies in the cradle 6 when the centrifuge drive is turned off. After that, the centrifuge is rotated.

After some time, the cradle 6 together with the man 7 deviates from its initial position under the influence of centrifugal force, changing its angle by about 90 ^o relative to the rod 4 / cm FIG. 68 /.

 In this situation, the readings of various "equilibrium sensors" of the human body are extremely contradictory. The otolithic organ gives the following indications / cm. FIG. 69 /: "a person from a horizontal position" stood on his head "and spins around its axis with an angular velocity equal to the angular velocity of the centrifuge."

 And the "sensor is a plumb line", "smeared" over the human body, at this time gives completely different readings / cm. FIG. 70 /: "the human body lies horizontally in an unusually strong gravitational field."

 The human balance organs successfully cope with this contradiction: the person recognizes the correct readings of the "sensor - plumb".

 To create a remote, anthropomorphic robot controlled by a person in copying mode, it is necessary to use algorithms in the control unit that would, if possible, repeat the operating modes of the human balance organs.

 In FIG. 71 and 72 are given such control unit algorithms.

 The algorithm depicted in FIG. 71 is an "endless cycle". The purpose of this algorithm is to determine situations when the robot body for a long time moves in space without strong rotational accelerations. Then, an interrupt signal / cm should be sent from the control unit to the calculation unit. FIG. 63, block 4 /. Those. the algorithm shown in FIG. 71 identifies situations suitable for measuring “initial conditions”.

 Block 2, the algorithm depicted in FIG. 71, indicates the assignment of the initial value to the parameter S. This parameter will be necessary for us to measure a certain period of time.

вводятся в алгоритм от соответствующих измерительных витков жидкостного инерционного датчика ускоренных вращений. /Здесь в рассмотрение введены вращательные ускорения с верхними индексами α, β, γ. Если

то это означает, что корпус робота вращается в пространстве c переменной угловой скоростью вокруг оси X системы координат XYZ, жестко связанной с корпусом робота /см. фиг. 7 /, т.е. угол α изменяется неравномерно. Аналогично, если

то угол β изменяется неравномерно. И, наконец, если

то γ изменяется неравномерно./
В блоке 4 производятся вычисления. Электроника рассчитывает: квадрат абсолютного значения угловой скорости вращения корпуса робота M, полученного на основе измерений датчика - отвеса, и квадрат абсолютного значения вращательного ускорения, F, полученного на основе данных, измеренных инерционным датчиком ускоренных вращений.Block 3 denotes the input of data from liquid sensors. The angles α, β and γ are introduced from the corresponding unit circles of the sensor - plumb. Rotational Acceleration:

are introduced into the algorithm from the corresponding measuring coils of the liquid inertial accelerated rotation sensor. / Here, rotational accelerations with superscripts α, β, γ are introduced. If

this means that the robot body rotates in space with a variable angular velocity around the X axis of the XYZ coordinate system, rigidly connected to the robot body / cm. FIG. 7 /, i.e. the angle α varies unevenly. Similarly, if

then the angle β varies unevenly. And finally, if

then γ varies non-uniformly. /
In block 4, calculations are performed. Electronics calculates: the square of the absolute value of the angular velocity of rotation of the robot body M, obtained on the basis of the measurements of the plumb sensor, and the square of the absolute value of rotational acceleration, F, obtained on the basis of the data measured by the inertial sensor of accelerated rotations.

Next comes block 5 - this is a verification of the condition. If the robot body does not experience strong rotational accelerations: F≤C ₁ , then go to block 6. And if there are strong rotational accelerations: F> C ₁ , then go to block 10, where the “stopwatch” S = 0 is zeroed and return to start of cycle in block 3.

Block 6 is the same verification of the condition. If the condition: M <C ₂ is fulfilled, then this means that the robot body does not perform fast rotations in the space.

Blocks 7 and 8 organize a "stopwatch": if for a long time the robot body changes its angular orientation slowly and evenly, then the "infinite loop" of the algorithm manages to scroll many times, passing all the time through the same blocks with numbers: 3, 4 , 5, 6, 7, 8. And each time, when passing block 7, the value of the parameter S will increase by one - "the stopwatch is ticking." In block 8, the time measured by the "stopwatch" is compared with a specific time interval specified by the constant C ₃ . / The value of the constant C _{3 is} selected taking into account the speed of the robot electronics performing all the necessary computational operations of the algorithm. / As soon as the "stopwatch sticks" for more than 2.5 - 3 seconds / S> C ₃ /, the transition to block 9 occurs, i.e. the electronic control unit will issue an interrupt command / cm. FIG. 66 and 63 /. After that, the "stopwatch" will be reset: S = 0 and the execution of the cycle will start again.

 Presented by FIG. 71 algorithm of the electronic control unit allows an anthropomorphic walking robot to walk on very rough terrain on foot, and also use virtually any type of vehicle.

 In case the robot enters the centrifuge, another computational algorithm is developed for the control unit, as shown in FIG. 72. This algorithm will constantly “spin” in the control unit, analyzing the situation, in parallel with the algorithm presented in FIG. 71.

 For a robot trapped in a centrifuge, the inclination of the casing θ and φ should be determined using a plumb sensor, and the liquid inertial accelerated rotation sensor will only know: dξ is the rotation of the robot casing around the vertical line vector.

 / Earlier, I used similar letter designations, but with a stroke: α ′, β ′, γ ′, θ ′, φ ′, dξ ′ this was explained by the fact that the liquid inertial accelerated rotation sensor played the main role in determining the angular orientation of the robot body. And for him I chose hatched variables.

Т. е. наличие или отсутствие у переменной штриха показывает - какой датчик играл доминирующую роль при ее получении./
Физические принципы работы алгоритма, изображенного на фиг. 72, заключаются в следующем: если в течение 2 - 3 секунд наклон корпуса робота, измеренный датчиком - отвесом, существенно /более чем на 15^o/ отличается от наклона, вычисленного с помощью жидкостного инерционного датчика ускоренных вращений, то наклон корпуса робота необходимо определять по датчику - отвесу.In the situation when the robot enters the centrifuge, the main / dominant / role in determining the angular orientation goes to the sensor - a plumb line, for which variables without strokes were originally taken: α, β, γ, θ, φ.
Now, to these five unshaded variables, one more unshaded variable is added: dξ, whose value is calculated by the formula

That is, the presence or absence of a variable in a stroke indicates which sensor played a dominant role in its receipt. /
The physical principles of operation of the algorithm depicted in FIG. 72 are as follows: if, within 2 to 3 seconds, the inclination of the robot casing, measured by the plumb sensor, is significantly / more than 15 ^o / different from the inclination calculated using the accelerated liquid inertia sensor, then the inclination of the robot casing must be determined by sensor - plumb.

 In a normal situation, when an anthropomorphic robot moves around the terrain on its own / on foot / move, the mercury droplets in the plumb sensor will make small vibrations near the lowest points of the sensor’s individual circles. In this case, the downward direction, calculated on the basis of the readings of the liquid inertial accelerated rotation sensor, will not differ much from the true downward direction obtained by the plumb sensor. For a long time (for example, for three seconds), the readings of various types of sensors will coincide several times or will be very close to each other.

The situation when the robot enters the centrifuge is different: here for a long time / for example, for three seconds / the readings of the sensor - the plumb line and the readings of the liquid inertial sensor of accelerated rotations are very different, for example, differ by more than 15 ^o . / As an example, see FIG. 67-70 and related comments thereto. /
Such a long-term strong difference in the readings of the sensors will allow the electronics of the control unit to conclude that the robot is in a centrifuge.

 I took here as a "long time" three seconds. We can say that these three seconds is “adaptation time”, i.e. time of the robot getting used to the centrifuge mode / or time of weaning off from the centrifuge mode /.

 So we have two plumb line vectors. One vector of the plumb line is measured by a plumb sensor; the other vector of the plumb line is measured by a liquid inertial accelerated rotation sensor. You can find the angle between these two vectors using the scalar product of vectors.

где переменной R_i обозначено значение косинуса угла, заключенного между двумя векторами отвесных линий, полученных в момент времени t_i датчиками двух различных типов.By agreement, the vertical line vector has a unit length, its components in the XYZ coordinate system. such / see FIG. 4 /:
(sinθ) (cosφ); (cosθ); (sinθ) (sinφ). (64)
/ These were the components of the plumb line vector measured by the plumb sensor. /
For a vertical line vector, determined using a liquid inertial accelerated rotation sensor, in the component record, the dashed variables should be used:
(sinθ ′) (cosφ ′); (cosθ ′); (sinθ ′) (sinφ ′). (65)
Then the scalar product between two unit vectors of plumb lines will give us the cosine of the angle between them and will be written:

where the variable R _i denotes the cosine of the angle enclosed between two vectors of vertical lines obtained at time t _{i by} sensors of two different types.

 To obtain the angles θ and φ, as well as to obtain the angles θ ′, φ ′, you can use the same ROM / cm. FIG. 66 /, transforming the angles α, β, γ into the angles θ, φ. The fact is that the liquid sensor - plumb responds much faster compared to the liquid inertial sensor of accelerated rotation. Indeed, in order to take readings from a liquid plummet sensor, it is sufficient to apply a requesting gating pulse to the sensor trigger D - plummet “D”, after which the values of angles α, β, γ in the binary code will appear on the outputs of the priority encoders. Those. response time of the sensor - plumb is determined by the rate of transient processes in priority encoders / cm. FIG. 6 /. A liquid inertial accelerated rotation sensor is much slower. This is because in the beginning it is necessary to convert the analog signal / voltage / using an ADC to a binary code, which is then converted to another binary code using an EPROM / cm. FIG. 60 /. Then this converted binary code is then used to calculate the angles α ′, β ′, γ ′ according to the algorithm presented in FIG. 63 and 64.

 So, while in the block of calculations / cm. FIG. 66 / the angles α ′, β ′, γ ′ will be calculated, the values of the angles α, β, γ will be fed to the inputs of the ROM, and then the binary values of the angles θ and φ will appear at the outputs of the ROM. These values will be transferred to the control unit / cm. FIG. 66./, where they are subject to memorization and are used for further calculations.

/ An even more efficient use of the ROM is possible if, in addition to converting the angles α, β, γ into angles θ and φ, we add the values of the components of the plumb line vector in binary code. Then it is not necessary in the control unit to recalculate the angles into the components of the vector. /
Let's see how the algorithm works.

 Immediately after turning on the power to the robot, the algorithm shown in FIG. 72, starts to work. This moment is indicated by block 1.

 In block 2, the initial values are assigned to the parameters: U and T. Parameter U can take only two values, "zero" or "one." The initial value of the parameter U is set equal to zero / U = 0 /. Further, during the operation of the robot, the value of the parameter U as a result of the execution of the algorithm may change. If U becomes equal to unity, then this means that the robot fell into the centrifuge and now the sensor, the plumb line, dominates in determining its angular orientation.

 Parameter T is needed to count "adaptation time". This parameter is simultaneously used both to determine the duration of the robot’s “getting used to” mode to the centrifuge, and to determine the duration of the robot’s “weaning” mode from the centrifuge. The initial value of the parameter T is set equal to -C / T = -C /, where C is a positive integer. Its value is selected so that the "adaptation" time is approximately 2 to 3 seconds.

 Data input from liquid sensors of two different types is carried out in block 3 algorithm.

Block 4 of the presented algorithm symbolizes the operation of calculating R _i - the cosine of the angle enclosed between two vectors of steep lines whose directions were measured at time t _{i by} liquid sensors of various types. / I did not enter the calculation formula R _i directly into the algorithm because of its bulkiness / cm. formula (66) /./
Block 5 - verification of the condition: if R _{i is} less than the cosine of 15 ^o , then this means that the angle between the two vectors of steep lines measured by different sensors is more than 15 ^o . And if, the condition in block 5 for R _{i is} not fulfilled, that is, if R _{i is} greater than or equal to the cosine of 15 ^o , then the angle between two vectors of plumb lines is less than 15 ^o .

We make the assumption: let the robot not be exposed to centrifugal forces in the first minutes after turning on the power, i.e. It is in the usual, normal environment, for example, it moves on the ground on foot. In this situation, the algorithm shown in FIG. 72, works as follows: during the first period we go through the blocks: 1, 2, 3, 4, 5. Since the robot is not in a centrifuge, the condition recorded in block 5 is not fulfilled - the angle between two vectors of plumb lines is less than 15 ^o . Then we proceed to verify that the condition is fulfilled - block 6. According to block 2: T = -C, therefore, the condition in block 6 is not fulfilled, go to block 7, where the value of the parameter T decreases by one, now: T = -C-1. Next, we go on to verify that the condition is fulfilled in block 8 — the condition is fulfilled, go to block 9 — set the parameter: U to “zero,” and return to block 3.

 Thus, as a result of the first period of the algorithm, the value of the parameter T became: T = -C-1., And the value U is assigned the value "zero".

 Consider the second period of the cycle: sequentially go through the blocks with numbers: 3, 4, 5 and go to block 6. Now we have T = -C-1, so the condition in block 6 is satisfied and we go to block 3. Here we must say that block 6 / like block 10 of the presented algorithm / is intended to limit the absolute value of parameter T.

 Now no matter how many periods of the presented algorithm have passed, the parameter T has a value equal to: -С-1, parameter U = 0. It should be so, because the robot is not in the centrifuge.

We make another assumption:
Let the robot move for a long time on its own / on foot / move, and then jump into a spinning centrifuge.

So, from the very moment when the robot is in a rotating centrifuge, the angle formed by two vectors of steep lines measured by liquid sensors of various types will be more than 15 ^o . And now, when passing through each period of the cycle, the condition recorded in block 5 will be fulfilled, and the calculations will go to another branch of the algorithm - blocks 5, 10, 11, 12, 13. The processes in this new branch of the algorithm are largely similar to the processes flowing in blocks 6, 7, 8, 9, except that the parameter T will now increase, and the parameter U will take the value "one" if T becomes greater than C.

 Presented in FIG. 72 algorithm allows an anthropomorphic walking robot to adapt independently to a centrifuge, the axis of rotation of which can have any angular orientation in space.

 This concludes the description of the "vestibular apparatus" of the anthropomorphic walking robot / liquid sensors and the mathematical support they need /.

 I will now proceed to describe the mechanism of suspension of the human body - the operator, that is, to the description of the device that allows you to control the movements of the robot.

 The suspension mechanism of the human operator.

 We are faced with the task of creating a remote anthropomorphic walking robot. Such a robot will have increased maneuverability, i.e., the robot will be able to walk freely on two legs over very rough terrain, crouch, lie down, roll side to side, get up, climb stairs and even do mountain climbing / in the literal sense of the word /.

 The robot is controlled in copy mode using a tracking system, one part of which is represented by the robot itself, the other is made in the form of a control suit with rigid elements, worn on a human operator. Two types of sensors are installed in each joint of the anthropomorphic robot and in each joint of the control suit: one sensor measures the angle of the bend of the joint, the second measures the rotational moment of force applied to the rigid elements of the joint. Data from these sensors installed in the body of a remote-controlled robot is transmitted via a communication channel to the electronic logic circuits of the control suit. At the same time, the data from the sensors installed in the joints of the control suit also comes to the same logical circuits. Electronics analyzes this data and issues the necessary control commands to the power drives of the limbs of the robot and the control suit. Control commands are sent to a remotely controlled robot through the appropriate communication channels.

 The forceful effects exerted by the human operator on the rigid elements of some joint of the control suit are perceived by the corresponding sensor, which measures the moment of force. If, at the same time, a similar moment of forces does not act on a similar joint of an anthropomorphic robot, then the joint of the control suit begins to bend in the direction of the efforts exerted by the operator. At the same time, a similar joint of an anthropomorphic robot bends in the same direction. At the same time, the electronics ensures that the similar joints of the robot and the control suit bend at the same angular speed and at the same angle, i.e. the principle of copying must be followed. In turn, if the force impact of the external bodies surrounding the robot will not be compensated by force from the operator, then the joint of the robot and the joint of the control suit similar to it will begin to bend in the direction of the force exerted on the robot by external bodies. All this allows the human operator to subconsciously feel directly with their propreoceptors the spatial position of the limbs of the robot and the efforts exerted by the robot on the objects, as well as determine the shape of the objects and their weight by touch. Thus, the motor and sensory systems of the human operator are articulated with the mechanical limbs of the robot so that the latter become, as it were, a natural extension of the human limbs. The operator receives in general the same information from the limbs of the robot, as if a person directly manipulated the bodies surrounding the robot with his own hands.

In addition, electronic logic circuits, based on data received from the sensors of the angular orientation of the robot casing, should make appropriate amendments to the force effects exerted by the control suit on the human operator. This is necessary in order to compensate for the gravitational gravity of the structural elements of the robot and the control suit, i.e. it is necessary to ensure that the robot and the control suit do not hang with their weight on the hands and feet of the human operator, regardless of its orientation in space. In other words, the corresponding “electronic balances” must be made. "
At this stage, the main attention in the development of such a robot must be given to the design of the suspension mechanism, with which we must hang in the space of a human operator dressed in a control active exoskeleton.

The suspension mechanism must satisfy the following two requirements:
1 / Do not allow the human operator to touch extraneous objects with his limbs: the human operator, for example, should not reach the floor with his feet, otherwise this will interfere with the robot's walking control.

 The suspension mechanism should give the body of the human operator the same orientation in space as the robot body, the same angular velocities and the same rotational accelerations - all this will ensure vertical walking of the robot.

 “This will allow a person to feel with his balance organs all the changes occurring in the angular orientation of the robot body. At the same time, the human operator, controlling the vertical walking of the robot, will unconsciously strive to maintain vertical stability by involuntarily changing the position of his limbs accordingly, which will the stability of the vertical gait of the robot is preserved, as it will copy all these involuntary movements of the operator.

 The human operator when managing the robot’s walking will experience about the same sensations as when walking in a spacesuit on a treadmill mechanical simulator.

Thus, the following telemetric data are transmitted from the remote robot via communication channels to the logical electronic circuits of the control suit and suspension mechanism:
1 / data on the angular orientation of the robot body,
2 / data from sensors installed in all joints of the robot,
3 / video signal of the environment surrounding the robot,
4 / data characterizing the sound environment around the robot. / The latter is relevant for robotic soldiers and robotic firefighters. /
In turn, through the appropriate communication channels to the remote-controlled robot are transmitted:
1 / commands to control the power drives of the joints of the robot,
2 / signals allowing the robot to reproduce the spoken language of the human operator. / The latter is relevant for robotic soldiers and robotic firefighters. / "
Such robots can be used as divers' robots, robots - soldiers, robots - miners, robots - firefighters, etc.

 The use of robots as soldiers is discussed later, as a separate paragraph. And now I will demonstrate the use of robots as divers / cm. FIG. N 73 - 78 /, where the odd figures depict a human operator, dressed in a control suit and suspended from a suspension mechanism. Even figures having numbers: 74, 76, 78, depict a robot - diver.

 It must be said here that we cannot reproduce, with the help of the suspension mechanism, the translational accelerations experienced by the robot. Otherwise, the human operator would have to follow in space the same path that a robot follows. The path of this path would coincide with the path of the robot. For example, in the case of a robot-diver, we simply cannot physically allow ourselves to allow a human operator to walk around the ship in a control suit, freely, like a ghost, passing through all ship bulkheads.

 The control suit must exert a forceful effect on the human operator. This is necessary so that the human operator has the same proprioceptive sensations as the robot, that is, if the robot is holding an object in its hands, then the operator, under the force of the control suit, should feel the weight of the object, its the shape and, most importantly, the position of the robot’s arms in space. The control suit creates similar proprioceptive sensations for all joints of the human body - the operator.

 In FIG. 73, 75, 77 in a human operator, a satchel is shown behind his back. This is due to the fact that the “muscles” are located in the knapsack, responsible for the shoulder, cervical, waist and pelvic joints of the control suit. Placing these “muscles” behind a person’s back will free up more space in front and make controlling the robot more convenient.

 Let's move on to a more thorough acquaintance with the device of the suspension mechanism.

 It was previously indicated that the orientation of the body of the anthropomorphic robot is specified by three angles θ, φ, and dξ / cm. above/. The first two angles determine the inclination of the robot body relative to the vertical line vector. The third angle sets the rotation of the robot body around the vertical line vector.

In the suspension mechanism, in order to give the human body the operator the same orientation as that of the robot body, it is necessary to reproduce the same angles θ, φ, dξ.
We will rigidly connect the coordinate system XYZ / cm to the body of the anthropomorphic walking robot. FIG. 79 /. Now changing the angular orientation of the robot body means that the angular orientation of the XYZ coordinate system will also change.

 The angles θ and φ of the plumb line vector in the coordinate system XYZ are defined in the same way as shown in FIG. 4.

 If the robot makes a “forward tilt”, then the angle φ will increase / cm. FIG. 80 and 81 /.

If the robot is lying on its left side, then the value of the angle θ is equal to zero / cm. FIG. 82 /. And if the robot lies on its right side, then the value of the angle θ is 180 ^o / cm. FIG. 83 /.

 We will rigidly connect the analogous coordinate system XYZ with the body of a person - an operator suspended on a suspension mechanism.

The suspension mechanism has the following kinematic scheme:
These are six rings embedded in each other. We number them so that the smallest / inner / ring has N 1 and the largest / outer / ring has N 6, see FIG. 84.

 At the center of this system of rings nested into each other is a man - an operator dressed in his control suit. The vest of the control suit is rigidly connected to the ring 1 with the help of a radial rod, which originates on the inner surface of the ring 1 and ends on the back of a person - the operator / cm. FIG. 85 /.

Each ring has two external protrusions lying on a straight line passing through the center of the ring, i.e. lying on the continuation of the diameter / cm. FIG. 84 and 85 /. Rings of smaller diameter abut their protrusions against rings of larger diameter. Ring 6 lies with its protrusions on the base K ₁ -K _{2 of} the suspension mechanism / cm. FIG. 84 /.

/ In case we are dealing with a robot-diver, the rigid frame of the ship from which the robot is controlled will be such a basis for the suspension mechanism. And in cases when the robot will be used on land, for example, as a robot - a soldier, the basis for the suspension mechanism will be the frame of the machine with which the robot is controlled. /
If you look closely at FIG. 84, you can see that the protrusions of the rings with even numbers / 2, 4, 6 / lie on one straight line, which is directed along the X axis of the coordinate system XYZ rigidly connected with the human body - the operator. And in relation to the projections of the track with odd numbers, you can notice that they all lie on a straight, collinear Y axis of the XYZ coordinate system associated with the human body - the operator.

 FIG. 84 is drawn here to give an overview of the set of parts that make up the suspension mechanism. It shows the suspension mechanism in a situation where all the rings lie strictly in the same plane, but in a working environment, the real design of the suspension mechanism will very rarely look like this. The fact is that the protrusions on the rings are mechanical axes around which the rings rotate relative to each other, this is necessary for us in order to give the body of the human operator a certain angular orientation.

We proceed to consider the functions that perform the presented details of the suspension mechanism;
The role of rings 5 and 6 consists solely in that the axes, the protrusions of the ring 4, are always located along the vertical line vector / cm. FIG. 86./
/ It will be shown below that one of the purposes of ring 4 is to rotate the human body - the operator by an angle dξ around the vertical line vector and that is why the axis - the protrusions of ring 4 should always lie on a vertical line. /
Essentially, rings 5 and 6 form here a conventional gimbal, similar to that used for a mechanical gyro / cm. FIG. 1/.

 As an example, demonstrating the need for the introduction of such a gimbal, consider two situations.

 Let's say that from a ship under the conditions of sea rolling, a robot is controlled by a diver. The sensors installed on the ship will monitor the change in the angular roll of the ship and give the necessary commands to the gimbal formed by rings 5 and 6, while maintaining the vertical orientation of the axes - the protrusions of the ring 4.

 Or another example. A machine controlled by a robot soldier is not horizontal. In this situation, the gimbal will again help us out - the tabs of the ring 4 will be installed vertically.

I believe that here I have fully disclosed the purpose of the gimbal formed by rings 5 and 6. I will not return to this issue anymore. / On all subsequent figures, rings 5 and 6 will no longer be depicted, this is done in order to simplify the drawings, for their better understanding. Those. from now on, we agree to assume that the protrusions - the axes of the ring 4 are always located vertically.
By rotating the ring 1 relative to the ring 2 on the protrusions - axes, we will be able to specify the angle φ / cm in the body of the human operator, the operator. FIG. 85, 80 and 81 /.

 In the suspension mechanism, the angle θ for the human body - the operator in the general case is set by the simultaneous rotation of two rings 2 and 3.

 Here you will have to make very detailed explanations so that it becomes clear to you why rotations of two rings are used to set the angle θ.

 It would seem that to specify the angle θ to the body of a human operator, one can use the rotation of only one ring 2 / cm. FIG. 87 and 88 /, and generally discard the ring 3 and thereby significantly simplify the design of the suspension mechanism. In fact, such a "simplification" will very negatively affect the operation of the suspension mechanism.

The fact is that if you "throw out" the ring 3, then in situations where the value of the angle θ is close to 0 ^o or 180 ^o , in the simplified design of the suspension mechanism, significant singularities will arise. For instance:
Let the robot lie on the left side for a long time so that its slope relative to the vertical line vector is determined by two angles: θ = 1 ^o φ = 270 ^o i.e. the spatial orientation of the robot resembles FIG. 82. In a simplified design of the suspension mechanism, a similar inclination to the body of a human operator is specified by the corresponding rotation of two rings having numbers 1 and 2 / cm. FIG. 88 /.

Suppose that at some point the robot slightly changes the slope of its body, so that the angles θ and φ acquire the values: θ = 1 ^o and φ = 0 ^o . At the same time, for an outsider, the change in the orientation of the robot body seems so insignificant that its position in space can still be characterized with great success by FIG. 82. In the XYZ coordinate system, rigidly connected with the robot body, the angle between two vectors of plumb lines taken at different points in time is less than 1.5 ^o in absolute value, so that “by eye” the change in the orientation of the robot body is practically imperceptible.

The situation with giving the slope θ = 1 ^o , φ = 0 ^o to the human body as an operator in a simplified suspension / cm mechanism is completely different. FIG. 89 /.

If we now pay attention to the figures with numbers: 82, 88 and 89, we can see a clear discrepancy: the robot only slightly changed the inclination of its body, and in the simplified suspension mechanism, in order to give the human body the operator the same inclination, I had to turn the ring very quickly 1 at 90 ^o , this is the singularity that I spoke about.

 Further, we will consider only the "complicated" design of the suspension mechanism, which contains the ring 3, designed to eliminate singularities of this kind.

 How ring 3 copes with this task I will tell later, but now it is necessary to agree on the direction of reference and on the reference point of the angles that determine the mutual rotation of the rings in the suspension mechanism.

In FIG. 84 rings 1, 2, 3 and 4 have a “black mark” on one of their protrusions. Such "black marks" allow us to determine the positive direction of the reference angles that specify the mutual orientation of the rings of the suspension mechanism:
"The angle between the two rings increases if the ring of smaller diameter is rotated counterclockwise relative to the ring of larger diameter, provided that the" black mark "of the small ring is directed towards us."
Here was given the most general rule that determines the positive direction of the turns made relative to each other by the rings of the suspension mechanism. This rule is developed on the basis of the contract.

 We proceed to consider specific angles.

 The angle μ determines the rotation of the ring relative to ring 2. It increases if the body of the person - the operator, together with ring 1, performs "forward somersaults" with respect to ring 2. FIG. 90 and 91 /. The relative position of the rings 1 and 2 is taken as the reference point of the angle μ, when the X axis of the XYZ coordinate system, rigidly connected with the body of the operator, is perpendicular to the plane of ring 2, while the Z axis of the XYZ coordinate system is directed toward the black mark of the ring 2 / cm. FIG. 90 /.

 The angle λ determines the rotation of the ring 2 relative to the ring 3. The angle λ increases if the ring 2 rotates relative to the ring 3 in the direction specified by the general rule determining the positive direction of mutual rotation of the rings of the suspension mechanism. The origin of the angle λ is determined by the following mutual position of the rings 2 and 3 / cm. FIG. 92./: Ring 2 with its "black mark" is directed at us and perpendicular to the plane of ring 3. The "black mark" of ring 3 is directed to the right towards us, and the "black mark" of ring 1 is directed downward.

 The angle κ defines in the suspension mechanism the amount of rotation of the ring 3 relative to the ring 4. It increases if the ring 3 is rotated relative to the ring 4 in the direction specified by the general rule determining the positive direction of the mutual rotations of the rings of the suspension mechanism. By starting the angle κ, we choose the following relative position of the rings 3 and 4 / cm. FIG. 93 /: protrusions - the axis of the ring 4 is located strictly vertically, the "black mark" of the ring 4 is located at the bottom. The plane of ring 3 is perpendicular to the plane of ring 4, and the black mark is directed to the right. In this case, the "black mark" of the ring 2 is directed to us.

/ In general, FIG. 93 depicts a situation where all angles are equal to zero: μ = 0 ^o , λ = 0 ^o , κ = 0 ^o .
By changing the values of the angles μ, λ, κ in the suspension mechanism, we can set the human operator the same slope as that of the robot body.

 / It should be noted that turns at angles μ, λ, κ around the corresponding protrusions - the axes of the suspension mechanism commute - the result of the turns does not depend on the sequence of their execution! Those. the body of a human operator will take the same slope θ, φ in space at the same angles μ, λ, κ, regardless of the sequence of these turns. It would seem that this contradicts the well-known assertion that: “the result of two / or more / rotations by finite angles in the most general case depends on the sequence of performing these rotations” / cm. I.I. Olkhovsky "The Course of Theoretical Mechanics for Physicists", 1970. M. p. 150 /. However, this statement just quoted is valid only under the condition that all turns, both intermediate and resulting, are counted relative to absolutely motionless space.

In the case with the suspension mechanism, only the resulting rotation / tilt of the human body - the operator / is considered relative to the absolute space, and all intermediate rotations μ, λ, κ are counted relative to the rigid structural elements of the suspension mechanism, therefore, rotations by the angles μ, λ, κ and commute. If you wish, you can verify this using the appropriate geometric constructions. /
We show that a particular pair of angles θ and φ that determine the inclination of the human body - the operator / I emphasize: the inclination, and not the orientation in space / in the general case can be realized using a very large set of angles κ, λ, μ, if between these angles certain ratios.

 Consider FIG. 94, which shows rings 1, 2 and 3.

 Earlier, I pointed out that in order to give the body of the human operator the same angular orientation as that of the robot body, it is necessary to tilt the body of the human operator in a certain way and simultaneously turn it in a certain way around the vertical line vector. Rotations of the human body - the operator around the vertical line vector are carried out in the suspension mechanism due to the rotation of the ring 4 around its protrusions - axes. But now, at the moment, we are primarily interested in the slopes of the human body - the operator relative to the vertical line vector, and not the angular orientation in general. Therefore, we can now assume that the position of ring 4 is fixed in space and consider only the rotations of rings 1, 2, 3.

Rigidly connect with rings 2, 3 and 4 coordinate systems: X ₂ , Y ₂ , Z ₂ , X ₃ , Y ₃ , Z ₃ and X ₄ , Y ₄ , Z ₄ / in FIG. 94 in order to simplify the image, ring 4 is absent, however, there is an associated coordinate system X ₄ , Y ₄ , Z ₄ /.

Now our task is to obtain expressions / in the form of formulas / for the components of the vertical line vector in the coordinate system: XYZ, rigidly connected with the body of the human operator, sequentially performing the turns κ, λ, μ using the suspension mechanism.
We use matrices that describe the rotations of coordinate systems relative to each other.

The matrix A describing the rotation of the coordinate system X ₃ , Y ₃ , Z ₃ relative to the coordinate system X ₄ , Y ₄ , Z _{4 is} obtained if we look at the system X ₄ , Y ₄ , Z ₄ from the coordinate system X ₃ , Y ₃ , Z ₃ after the ring 3 rotates through an angle κ relative to ring 4, i.e. we consider the situation as if the coordinate system X ₄ , Y ₄ , Z ₄ together with the vertical line vector would rotate with respect to the coordinate system X ₃ , Y ₃ , Z ₃ as a whole.

/ In reality, of course, the opposite is true: the coordinate system X ₄ , Y ₄ , Z _{4 is} stationary, and the coordinate system X ₃ , Y ₃ , Z ₃ makes rotation relative to it.
When finding the matrix A, it is also necessary to take into account the mutual position of the coordinate systems X ₄ , Y ₄ , Z ₄ and X ₃ , Y ₃ , Z ₃ / rings 4 and 3 /, when κ = 0 ^o , i.e. origin of the angle κ / cm. FIG. 95 /. And also it is necessary to take into account the mutual position of the coordinate systems X ₃ , Y ₃ , Z ₃ and X ₄ , Y ₄ , Z ₄ after ring 3 has rotated relative to ring 4 in the positive direction of reading the angle κ / cm. FIG. 96 /.

Аналогичные методы используются для получения других матриц:
Матрица В получается с помощью фиг. 97 и 98, она описывает поворот на угол λ системы координат X₂, Y₂, Z₂ относительно системы координат X₃, Y₃, Z₃:

Матрица С получается с помощью фиг. 99 и 100, она описывает поворот на угол μ системы координат XYZ относительно системы координат X₂, Y₂, Z₂

В соответствии с фиг. 94 компоненты вектора отвесной линии в системе координат X₄, Y₄, Z₄ имеют постоянные значения
(0,0-1). (70)
Используя матрицы A, B, C найдем значение компонент вектора отвесной линии в системе координат XYZ. Для этого надо использовать матричное умножение: вначале умножим компоненты вектора отвесной линии /формула (70)/ на матрицу A, полученный результат затем умножим на матрицу B, а потом и этот результат умножаем на матрицу С. Таким образом мы получим компоненты вектора отвесной линии в системе координат XYZ в виде формул

/Здесь надо обязательно заметить, что матрицы A, B и C между собой не коммутируют, несмотря на то, что повороты на углы κ, λ, μ коммутируют. Т.е. порядок выполнения матричного умножения в данном случае надо строго соблюдать./
С другой стороны мы, используя фиг. 4, можем для компонент вектора отвесной линии записать такие выражения

Напомню: сейчас я доказываю, что какой-то конкретный наклон θ, φ тела человека-оператора в механизме подвеса может быть получен в общем случае очень большим набором значений углов κ, λ, μ, если они удовлетворяют формулам 71.Then, in accordance with FIG. 95 and 96 matrix A is written

Similar methods are used to obtain other matrices:
Matrix B is obtained using FIG. 97 and 98, it describes a rotation through an angle λ of the coordinate system X ₂ , Y ₂ , Z ₂ relative to the coordinate system X ₃ , Y ₃ , Z ₃ :

Matrix C is obtained using FIG. 99 and 100, it describes the rotation by the angle μ of the coordinate system XYZ relative to the coordinate system X ₂ , Y ₂ , Z ₂

In accordance with FIG. 94 components of the plumb line vector in the coordinate system X ₄ , Y ₄ , Z ₄ have constant values
(0.0-1). (70)
Using the matrices A, B, C, we find the value of the components of the vertical line vector in the XYZ coordinate system. To do this, you need to use matrix multiplication: first, we multiply the components of the vertical line vector / formula (70) / by matrix A, then we multiply the result by matrix B, and then we multiply this result by matrix C. Thus, we obtain the components of the vertical line vector in XYZ coordinate system in the form of formulas

/ It should be noted here that the matrices A, B and C do not commute with each other, despite the fact that the rotations by the angles κ, λ, μ commute. Those. the order of matrix multiplication in this case must be strictly observed. /
On the other hand, using FIG. 4, we can write such expressions for the components of the vertical line vector

Let me remind you: now I prove that a specific slope θ, φ of the body of a human operator in the suspension mechanism can be obtained in the general case by a very large set of values of the angles κ, λ, μ, if they satisfy formulas 71.

As an example, consider this situation: the vertical line vector has an orientation in the coordinate system defined by the angles θ = 30 ^o , φ = 0 ^o . Then from formulas (72) we obtain
X _ol = 0.5; Y _ol = 0.866; Z _ol = 0. (73)
In the suspension mechanism, this inclination of the human body - the operator relative to the vertical line vector can be / in principle / realized if the angles κ, λ, μ take the corresponding numerical values indicated in some / any / row of the table shown in FIG. 101. This is just clear evidence that, in the suspension mechanism, any tilt of the human operator’s body could, in principle, be obtained by an infinite set of appropriate combinations of the three angles κ, λ, μ, satisfying expressions 71 and 73.

 In this regard, a question arises before us; “How to choose the optimal, providing the most effective way to give the necessary tilt to the human operator’s body in the suspension mechanism from such an endless set of combinations of angles κ, λ, μ?” - this is necessary for the rational organization of the suspension mechanism electronics.

 To answer the question posed it is necessary to return to the concept of "globe tilt" / cm. above and FIG. 27 /.

 Let us depict globules of inclinations around the robot body and around the body of a human operator, as shown in FIG. 102 and 103.

 The robot is free, it can move / bend / as it pleases. The concept of “tilt globe” does not play a special role for him.

 The body of the human operator is not free, its orientation in space is set using the suspension mechanism, by rotating the corresponding rings around the protrusions of the axes. And it is precisely here, in the suspension mechanism, that the concept of a “tilt globe” plays an extremely important demonstration role.

The tilt globe, rigidly connected with the body of the human operator, is mentally divided into three sectors, highlighting on its surface two "polar circles" and the "equatorial part" / cm. FIG. 104, where the polar circles are darkened. Points on the surface of the slope globe belonging to the same polar circle satisfy the condition
θ ≤ 30 ^o . (74)
For the other polar circle
θ ≤ 150 ^o (75)
So, if we need the vector of the vertical line to pass for the human operator through the equatorial part of the globe, then we will set such inclinations of the human body as the operator only using two rings of the suspension mechanism: using the rotations of ring 1 and ring 2. Angle while κ must be equal to zero, i.e.

if: 30 ^o <θ <150 ^o , then κ = 0 ^o ! (76)
Then, in the equatorial sector, the task of tilting the body of a human operator will be extremely simple: by rotating the ring 1 we will set the angle φ, and by rotating the ring 2 the angle θ.
Thus, if we need to give the body of a human operator a slope characterized by angles θ and φ, where the angle θ is in the range 30 ^o <θ <150 ^o , then we can now do this in only two ways, depending on which the range of values belongs to the angle λ: so, if we choose 0 ^o ≤ λ ₁ ≤ 180 ^o , then to set the angles θ and φ it is necessary that the angles λ ₁ and μ ₁ take the following values:
λ ₁ = θ; μ ₁ = φ, (77)
and if we have chosen for the angle λ the interval of values 180 ^o <λ ₂ <360 ^o , then to set the same angles θ and φ it is necessary that the equalities
λ ₂ = 360-θ ^o ; μ ₂ = φ + 180 ^o . (78)
/ The question of which interval of values of the angle λ to choose for setting the slopes of the human body - the operator is solved simply: we agree that the electronics that provide the suspension mechanism should always set the mode 0 ^o ≤ λ ₁ ≤ 180 ^o in the first moments after turning on the device’s power . Further, during the operation of the suspension mechanism, the interval of values of the angle λ can change and become 180 ^o <λ ₂ <360 ^o , but for this, in the suspension mechanism, the vertical line vector must pass through the territory of the Arctic Circle. /
Now, instead of an infinite set of combinations of angles κ, λ, and μ, with certain relationships among themselves, we already have only two possibilities, which, of course, is a significant simplification of the situation.

/ Suppose, for example, at the moment the suspension mechanism sets the slope of the human operator’s body, characterized by angles θ and φ, at the same time the information comes from the robot that the slope of the robot body relative to the vertical line vector is determined by other angles θ ′ and φ ′. The electronics of the suspension mechanism calculates the quantities dθ = θ′-θ ≠ 0 ^o and dφ = φ′-φ ≠ 0 ^o , which act as mismatch signals. The task of the suspension mechanism is to give an inclination to the body of the human operator such that dθ = 0 ^o and dφ = 0 ^o . And that's all, you don’t have to do any particularly complicated calculations anymore, which allows us to greatly simplify the electronic circuits of the suspension mechanism - in fact, we have here a fairly simple, traditional tracking system. /
The equatorial sector 30 ^o <θ <150 ^o occupies a large part of the surface on the tilt globe, approximately 86.6% of the area of the sphere, so in most cases it will be possible to use relatively simple electronics in the suspension mechanism.

From FIG. 105 and 106, it becomes clear why there are only two possibilities that realize the same slope of the human operator’s body relative to the vertical line vector, provided that 30 ^o <θ <150 ^o .
In FIG. 105 and 106 depict the central details of the suspension mechanism - rings with numbers 1, 2, 3 and a human operator in a control suit. In both cases, we / external observers / look at the suspension mechanism from the same fixed point, ring 3 is motionless relative to us, but rings 2 and 1 make turns.

In the first case / Fig. 105 / we have such angles of rotation of rings 1 and 2: λ ₁ = 45 ^o and μ ₁ = 300 ^o . In the second case / FIG. 106 /: λ ₂ = 315 ^o and μ ₂ = 120 ^o .

To demonstrate the fact that the slopes of the human operator’s body in the case of λ ₁ = 45 ^o , μ ₁ = 300 ^o and in the case of λ ₂ = 315 ^o , μ ₂ = 120 ^{o are the} same, one can use formulas (71). Subsequently, we substitute the values of the angles λ ₁ = 45 ^o , μ ₁ = 300 ^o , κ = 0 ^o and λ ₂ = 315 ^o , μ ₂ = 120 ^o κ = 0 ^o in formulas (71). In both cases, we get the same values of the components of the plumb line vector in the XYZ coordinate system:
X _ol = 0.353; Y _ol = 0.707; Z _ol = 0.612, (79)
and this is indisputable evidence that the slopes of the human body - the operator in both cases are the same.

 Now we will consider such tilts of the human operator’s body, carried out by the suspension mechanism, in which the vertical line vector falls on the territory of the polar circle of the slope globe.

Let me remind you that I defined the Arctic Circle as the set of points on the surface of the slope globe having an angle θ ≤ 30 ^o , / the other polar circle is the set of points with: θ ≥ 150 ^o /.

Earlier, I pointed out that any point θ, φ on the surface of the globe can, in principle, be realized by a very large set of angles κ, λ, μ. In this regard, we had the problem of choosing the optimal combination from a large set of angles κ, λ, μ when setting the necessary slope for the human operator. For points of the equatorial part of the globe (30 ^o <θ <150 ^o ), this problem was solved by fixing the value of the angle κ: κ = 0 ^o .
Then the slopes of the human body - the operator in the suspension mechanism are set by rotating the rings 1 and 2 by the angles λ and μ, the values of which are calculated by formulas (77) or (78).

For the polar circles, this problem is eliminated in a different way;
We introduce for the globe points belonging to the polar circle the concept of "neighboring points" - these are two close points on the surface of the globe that are spaced apart by one division of the angle θ or / and / of the angle φ / cm. FIG. 107 /.

In FIG. 107 points: B, C, D are adjacent points to point A. Points: B, C are no longer adjacent to point D, because separated by two divisions of the angle φ.
The challenge facing us is to reduce, if possible, or even eliminate / the singularities near the polar points of the slope globe. Mathematically, this is expressed as
| Δμ | + | Δλ | + | Δκ | _ → min, (80)
where Δμ = μ′-μ is the difference in angles between the current μ rotation of ring 1 and the future μ ′ rotation of ring 1, which must be set to obtain the required inclination of the human operator’s body.

Similarly for Δλ and Δκ.
To solve this problem, we introduce two fairly obvious boundary conditions:
1: Angle κ ≠ 0 only in the Arctic Circle. When approaching from the inner regions of the polar circle to its periphery, the value of the angle κ smoothly tends to zero. Outside the polar circle κ = 0 ^o .
2: If the vertical line vector has a polar coordinate θ equal to zero for any φ, then κ = 0 ^o , λ = 0 ^o , the angle μ of rotation of the first ring of the suspension mechanism can have an arbitrary value.

которое может принимать угол κ для данной θ, φ точки полярного круга, определяется углом θ и задается кривой, изображенной на фиг. 108 - параболой, плавно переходящей в прямые. /Эта кривая выражается в аналитическом виде с помощью алгоритма, представленного на фиг. 109. / Полученное таким образом значение

выступает в дальнейшем как "радиус-вектор

" или как "амплитуда

".Next, we introduce some more conditions:
3: The angle κ in the Arctic Circle is generally non-zero and can take values in the range from -12 to +12 degrees. In this case, the absolute maximum value

which can take the angle κ for a given θ, φ point of the polar circle, is determined by the angle θ and is given by the curve depicted in FIG. 108 - a parabola, smoothly turning into straight lines. / This curve is expressed in analytical form using the algorithm presented in FIG. 109. / Value thus obtained

appears in what follows as a "radius vector

"or how" amplitude

"

" с помощью формулы

где

- "радиус-вектор", упоминавшийся в условии (3),
Т - "число секторов", на которые мы разбили воображаемую окружность, образованную радиус-вектором
t - целочисленный параметр: t ⊂ [0,T].
5: Для различных θ, φ точек полярного круга значение Т определяется в зависимости от значения θ:
если θ ⊂ ]0^o,9^o], то Т = 88; (82)
если θ ⊂ [10^o,29^o], то T = 4•(31 -θ ); (83)
если θ ≥ 30^o, то T = 0. (84)
6: Для каждой θ, φ, точки полярного круга определяем набор значений угла λ_N, используя набор значений угла κ_t, полученный для данной точки полярного, и формулу

Здесь необходимо следить за знаком, который появляется перед λ_i. Договоримся, что если

данные элементы "набора λ_N " имеют отрицательные значения. А если

то такие элементы "набора λ_N " имеют положительные значения.4: Each θ, φ of the polar circle point has a set of values of the angle κ _t , which is obtained "as a projection of the radius vector

"using the formula

Where

- "radius vector" referred to in condition (3),
T is the "number of sectors" into which we divided the imaginary circle formed by the radius vector
t is an integer parameter: t ⊂ [0, T].
5: For different θ, φ points of the polar circle, the value of T is determined depending on the value of θ:
if θ ⊂] 0 ^o , 9 ^o ], then T = 88; (82)
if θ ⊂ [10 ^o , 29 ^o ], then T = 4 • (31 -θ); (83)
if θ ≥ 30 ^o , then T = 0. (84)
6: For each θ, φ, polar circle point, we determine the set of values of the angle λ _N , using the set of values of the angle κ _t obtained for this polar point, and the formula

Here it is necessary to follow the sign that appears before λ _i . We agree that if

these elements of the “set λ _N " have negative values. What if

then such elements of the “set λ _N " have positive values.

и при t=T/2

дадут в "набор λ_N " еще по одному элементу, таким образом и получается: N=Т+ 2, N ⊂ [2, 90].
7: Набор значений угла μ для данной θ, φ точки полярного круга найдем с помощью формул
если λ_i > 0^o, то μ_i = φ-S_i; (86)
если λ_i < 0^o, то μ_i = φ+180-S_i; (87)
где S_i поправка, вычисляемая по формуле

Таким образом, каждой θ, φ точке полярного круга ставится в соответствие набор троек углов κ_n, λ_n, μ_n, где n ⊂ [1,N]. Каждая такая тройка углов κ_i, λ_i, μ_i может реализовать в механизме подвеса наклон θ, φ тела человека-оператора.Moreover, the number of elements in the “set of λ _N " will be equal to: N + T + 2 / Elements of the “set of λ _N ” formed at t = 0

and at t = T / 2

give to the “set λ _N " one more element, and thus it turns out: N = T + 2, N ⊂ [2, 90].
7: The set of values of the angle μ for a given θ, φ point of the polar circle will be found using the formulas
if λ _i > 0 ^o , then μ _i = φ-S _i ; (86)
if λ _i <0 ^o , then μ _i = φ + 180-S _i ; (87)
where S _{i is the} correction calculated by the formula

Thus, each θ, φ point of the polar circle is associated with a set of triples of angles κ _n , λ _n , μ _n , where n ⊂ [1, N]. Each such triple of angles κ _i , λ _i , μ _i can realize the slope θ, φ of the human operator’s body in the suspension mechanism.

содержащий N элементов /N троек углов/, и определяемый в соответствии с вышеперечисленными семью условиями.Now suppose that at some point in time the vector of the vertical line is located on the territory of the Arctic Circle at the point with polar coordinates θ, φ, while the inner rings of the suspension mechanism, defining the slope of the human operator’s body, are rotated relative to each other at certain angles κ, λ, μ. Suppose that we then had the need to change the tilt of the body of a human operator so that the vector of the vertical line was at the θ ′, φ ′ point of the polar circle. This new θ ′, φ ′ point of the polar circle corresponds to a set of triples of angles

containing N elements / N triples of angles /, and determined in accordance with the above seven conditions.

 It remains to choose from this set of triples of angles one that would satisfy formula (80) as much as possible, and turn the inner rings of the suspension mechanism by these angles. Thus, the problem of singularity near the polar points will be solved.

 The presented conditions allow us to develop an algorithm that is implemented by the author of the application in the form of a demo program DATAPK. BAS.

To obtain formula (85), it is necessary to equate the second lines of formulas (71) and (72), we write:
cosθ = cosλ • cosκ, (89)
whence formula (85) is easily obtained.

 We justify formulas (86) and (88).

 Due to the fact that in the polar circle the angle κ is not generally zero, the expression μ = φ / cm. formula (77) / does not hold, and we must now use formula (86) to calculate the angle μ. Where S plays the role of amendment.

 To justify formula 86, we use the following method: suppose that we look at the coordinate system XYZ so that for any of its inclinations, the Y axis is always directed strictly at us. Those. as if we were all the time rigidly planted with our eyes on the Y axis of the XYZ coordinate system and along with it we tilt about the vertical line vector.

Suppose that at the initial moment of time the coordinate system XYZ has such an orientation that the vertical line vector passes through the polar point of the inclination globe θ = 0 ^o , i.e. directed along the y-axis of the XYZ coordinate system. According to the second condition stated above, a similar orientation of the coordinate system is realized in the suspension mechanism by angles: κ = 0 ^o , λ = 0 ^o , and the angle μ can be equal to zero without loss of generality. Above, we agreed that we would look at the XYZ coordinate system so that the Y axis would be directed exactly at us all the time. Then this situation is depicted in FIG. 110.

Suppose that at the next moment, the XYZ coordinate system has undergone a slope in space. So that λ> 0 ^o . At the same time, we / external observers / also underwent a similar inclination and the coordinate system XYZ is still directed with its Y axis strictly “towards us”. However, the plumb line vector is already noncollinear to the Y axis and gives a projection onto the XOZ plane perpendicular to the Y axis of the XYZ coordinate system. From FIG. 111 it is seen that the projection of the vertical line vector lies on the X axis of the XYZ coordinate system, which means that φ = 0 ^o .
Now suppose that the XYZ coordinate system was rotated by the suspension mechanism around the Y axis, so that the angle μ received a positive increment. Then for us / observers / the relative orientation of the XYZ coordinate system and the vertical line vector will be presented in FIG. 112, which shows that φ ≠ 0 ^o and φ = μ.
In conclusion, let the XYZ coordinate system obtain, using the suspension mechanism, a positive increment of the angle κ: κ> 0 ^o / cm. FIG. 113 /. This shows that φ = μ + S.
Thus, formula (86) has been substantiated.

 Now we prove formula (88). To do this, we perform a number of geometric constructions.

 First, let the orientation of the XYZ coordinate system in space be such that the vertical line vector coincides in direction with the Y axis of the coordinate system / cm. FIG. 114 and 93 /.

This situation is realized in the suspension mechanism, when the angles μ, λ, κ are equal to zero, according to the second condition described above
Suppose that at the next moment the suspension mechanism changes the slope of the coordinate system XYZ so that λ> 0 ^o / cm. FIG. 115 /, while the vector of the vertical line remains on the territory of the Arctic Circle, i.e. θ <30 ^o . The angle between the axis X of the coordinate system and the projection of the vertical line vector determines the value of the angle φ, in this case φ = 0 ^o .

Now he admitted that after this the coordinate system XYZ was once again tilted by the suspension mechanism, so that the angle κ received a positive increment and became nonzero: κ> 0 ^o .

 The coordinate system XYZ is rotated by an angle κ by rotating it around a straight line perpendicular to the vector of the vertical line / in most situations this line does not coincide with any particular axis of the coordinate system XY2 /. Therefore, setting the suspension mechanism to a positive increment of the angle κ at which the angle κ becomes nonzero appears in the XYZ coordinate system as a change in the direction of the vertical line vector, and the angle between the previous direction of the vertical line vector and the new direction is equal to the increment of the angle κ. This is shown in FIG. 116.

/ Fig. 114, 115 and 116 give axonometric images of the XYZ coordinate system at various stages of its evolution, starting from the first moment when λ = 0 ^o , κ = 0 ^o , and ending with the moment when λ = 10 ^o , κ = 5 ^o . All three of these figures are mathematically accurate, i.e. Before drawing, I calculated the parameters of their elements: the direction and length of all vectors. But for further logical constructions of FIG. 114, 115 and 116 are inconvenient to use. /
I will draw another FIG. 117. It carries the same physical activity as FIG. 116. However, in this case, the relations between the image elements are greatly changed / in particular, the angles λ and κ / are significantly increased. Such distortions are quite admissible here; they are made for purely demonstrative purposes and will not affect the truth of the logical constructions given below.

So: in FIG. 117 depicts three non-coplanar rays originating at one common point "O". The ray [O, A] represents the Y axis of the XYZ coordinate system. /cm. FIG. 116 /. The ray [O, B] depicts the direction of the plumb line vector in a situation where the coordinate system XYZ had a slope: λ ≠ 0 ^o , κ = 0 ^o / cm. FIG. 115 /. The angle between the rays [O, A] and [O, B] is equal to λ. The ray [O, F] coincides in direction with the vector of the vertical line, when the inclination of the coordinate system is specified in the suspension mechanism by angles: λ ≠ 0 ^o , κ ≠ 0 ^o / cm. FIG. 116 /. The angle between the rays [O, B] and [O, F] is κ.
We cut the ray [O, A] perpendicular to the plane / cm. FIG. 118 /. Then the projections of the rays [O, B] and [O, F] on this plane will give us the segments [A, B] and [A, F]. Connect the points B and F by a straight line, we get the segment [B, F].

 So we got four triangles: Δ OAB, Δ OAF, Δ OBF and Δ ABF. The triangles Δ OAB and Δ OAF are rectangular, because legs: [A, B] and [A, F] lie in a plane perpendicular to the leg [O, A]. The plane of the triangle Δ OAB is perpendicular to the plane of the triangle Δ OBF, because in the suspension mechanism, the axes, the rotation around which the angles λ and κ are set, are perpendicular to each other.

 As a result of the fact that two mutually perpendicular planes: OBF and OAB intersect with the third plane ABF, perpendicular to the plane OAB, we get that the angles: ABF and OBF are straight.

Thus, all four triangles shown in FIG. 118, rectangular. Folded together they form a spatial figure: a triangular slant pyramid / cm. FIG. 118 /. A flat scan of this pyramid is shown in FIG. 119 - here all the triangles lie in the same plane. Let me remind you that the leg [A, B] of the triangle OAB was obtained as a result of the projection of the plumb line vector onto the BAF plane parallel to the XOZ plane, in a situation where the inclination of the XYZ coordinate system is determined by the angles λ = 10 ^o , κ = 0 ^o .

The hypotenuse [A, B] of the triangle ABF was obtained as a result of the projection of the vertical line vector on the plane ABF parallel to the XOZ plane of the XYZ coordinate system, in the situation when λ = 10 ^o , κ = 5 ^o .

Those. the value of the angle BAF is the value of the desired angle S / correction /!
Of the triangles shown in FIG. 119, we find the mathematical formula necessary to calculate the correction S.

т.е. формула (88) доказана.We make the assumption:
Let be
| OB | = 1, (90)
Then
| BF | = tgκ and | AB | = sinλ. (91)
From here it is already easy to get:

those. formula (88) is proved.

Finally, we justify the formula (87). Why will we look at the XYZ coordinate system so that the axis in all cases is directed exactly at us. Suppose that at the first moment the coordinate system had such a slope that the vertical line vector was parallel to the Y / cm axis. FIG. 120 /. Such a situation, according to the second condition described above, in the suspension mechanism is obtained if μ = 0 ^o , λ = 0 ^o , κ = 0 ^o .

At the next moment, the suspension mechanism changes the slope of the coordinate system so that λ <0 ^o , μ = 0 ^o , κ = 0 ^o / cm. FIG. 121 /. The projection of the plumb line vector onto the XOZ plane of the XYZ coordinate system was in the half-plane where X <0.

Then the suspension mechanism once again changed the inclination of the coordinate system, making κ> 0 ^o , / cm. FIG. 122 / now the projection of the vertical line vector is in that quarter of the coordinate plane where: Z> 0, X <0. Then for FIG. 122 can be written
φ = μ + 180 ^o + S, (93)
where μ without loss of generality is equal to zero. From this we obtain formula (87).

 In the real design of the suspension mechanism, specialized electronic circuits carrying out parallel information processing should be used to search for the necessary triple of angles κ, λ, μ. The structure of these circuits may be as shown in FIG. 123.

A set of ROMs / read-only memory / 90 pieces form the first row of electronic elements. / In FIG. 123 of these ROMs are shown only three. /
I will explain why in the first row 90 ROMs are used. The fact is that the maximum length N = T + 2 of a set of triples of angles κ, λ, μ can be 90. And in each ROM of the first row only one triple of angles is stored, which corresponds to a specific θ, φ point of the polar circle. From here comes a set of 90 ROMs.

The values of θ ′, φ ′, characterizing the slope that we are going to give to the body of a human operator, are fed to the inputs of the ROM of the first row through a common bus. After that, the values of triples of angles corresponding to the slope θ ′, φ ′ appear at the outputs of the ROM of the first row. Moreover, at the output of each ROM only the values of the angles of one triple κ ′, λ ′, μ ′ appear.
Further, these values of the triples of angles as operands are received on the second row of electronic elements, which consists of 90 PLM / programmable logic matrices /. Each triple of angles received from some ROM of the first row is fed to its "personal" second-row PLM. As another operand, the values of the triple of angles κ, λ, μ, which currently implements the slope θ, φ of the human operator’s body in the suspension mechanism, are received at the second-row PLC inputs from the corresponding common bus. At the outputs of the PLM, calibrated values of the absolute differences of the triples of angles appear - the values κ, λ, μ of the outer triple of angles are compared and all sorts of values of the triples of the angles κ ′, λ ′, μ ′ that will determine the next θ ′, φ ′ tilt of the human operator’s body. Those. in the second-row PLM, such expressions are calculated
K _n = | Δμ | + | Δλ | + | Δκ |, (94)
where n takes a value from 1 to 90.

At the same time, each second-line PLM calibrates its “personal” expression (94), supplementing it with the smallest integer that is greater than or equal to the value of expression (94). / For example, if in some i-th PLM of the second row it is calculated that
K _i = | Δμ | + | Δλ | + | Δκ | = 2.56,
then the calibrated value of this expression will appear at the output of the i-th PLM: K _i = 3, etc. /
Then, the calibrated values of K _n obtained at the PLM go to the inputs of the decoders / Дш /, where they are converted from a binary bit code to a bit code with the base unit. Then the values converted in the decoders are fed to the common bus. Moreover, this bit bus with a base unit, i.e. if at the output of some PLCs of the second row a calibrated value appears, equal to zero, then after the decoders on the discharge wire of the bus corresponding to zero / this, for example, may be the first wire of the "discharge bus, with the base unit" /, a positive potential appears. And if at the same time calibrated values equal to five appear at the outputs of some other second-order PLCs, then the positive potential also appears on the discharge wire of the bus corresponding to the five / sixth bus wire /, thus positive on the first and sixth bus wires potentials etc. Those. this bus in its discharge wires carries information simultaneously about all 90 K _n .

Further, electronic elements of the fourth row — ninety ROMs — are connected to this bus with their inputs. At the same time, the other inputs of the ROM as operands from the second-row PLCs individually receive calibrated values of K _n .

The purpose of the fourth row ROM is to compare the individual calibrated values of K _i received from the second row PLC, simultaneously with all K _n calibrated values available in the set / in the discharge bus with the base unit /.

In this case, the fourth-row ROMs are selected from the entire set of triples of angles κ ′, λ ′, μ ′, which determine the future tilt of the human operator’s body in the suspension mechanism, only those triple angles that satisfy expression (80). A positive potential appears at the output of the i-th ROM of the fourth row if its individual K _{1 is} less than or equal to the minimum value of the entire set of triples of angles.

 Then in the circuit there is a series of logical elements OR and NAND, the application of which allows to eliminate the ambiguity - to select from one of the three triples of angles that satisfy expression (80) one. This is due to the fact that the appearance of an i-th ROM of a positive potential at the output blocks the outputs of all (90-i) ROMs located to the right of it.

 In the last row of the electronic circuit under consideration, there is a set of 90 registers that open access to the output bus to the only three angles κ ′, λ ′, μ ′ satisfying all seven conditions described above.

 By turning the inner rings of the suspension mechanism at these angles, a new slope θ ′, φ ′ of the human operator’s body will be set.

 The speed of parallel processing of information in such an electronic structure is determined only by transients.

 Here I described only the idea of creating electronic circuits that quickly select from the set N = 90 the desired three angles, and I deliberately omitted certain nuances.

 Thus, here we fully examined the issues related to giving the body of a human operator the same tilt as that of the robot body.

But in order for the body of a human operator to receive a certain angular orientation in space, just tilting it is not enough, it is still necessary to make a rotation by a certain angle around the vector of the vertical line / cm. the text above, which says that any change in the orientation of the robot body, we can consider as the inclination of the robot body relative to the vector of the vertical line and as a rotation around the vector of the vertical line, which occurs simultaneously with the inclination. /
Earlier, I pointed out that in the suspension mechanism the rotation of the human body - the operator around the vertical line vector is carried out by rotating the ring 4 relative to rings 5 and 6. But besides this, the rotation of ring 4 has another very important "neutralizing" role.

 Consider these issues in more detail.

 Let the robot stand vertically / cm at the first moment. FIG. 79 /.

Now suppose a human operator decides to turn the robot "to the right." Then the operator begins to exert a corresponding forceful effect on the "pants" of the control suit with his feet, which in turn will cause the appearance of the corresponding efforts on the legs of the robot, after which the robot will begin to turn its body to the right. Such a small change in the orientation of the robot body will be detected by a liquid inertial accelerated rotation sensor, which will send information via the communication channel to the electronics of the suspension mechanism that the orientation of the robot body has changed by Δγ ≠ 0.
After that, the suspension mechanism will rotate the ring 4 / and with it the body of the person - the operator / by the angle Δγ, etc., until the robot / and the person-operator / controlling it / in full - by 90 ^o - complete space turn right, conceived by the human operator.

 The process of turning the robot and the human body - the operator "to the right" is shown in FIG. 79, 124, 125, and 126.

 At first, the robot stood upright, see FIG. 79, this position in the suspension mechanism corresponded to the orientation of the human body - the operator, depicted in FIG. 124. FIG. 125 depicts a robot already after it has completely completed the "right" turn, this position corresponds to a new orientation of the human body - the operator, defined in the suspension mechanism by turning the ring 4 / cm. FIG. 126 /.

 Now let's talk about the "neutralizing" role performed by ring 4.

 To do this, consider four poses that the robot sequentially takes in space.

Let first the slope of the robot body is set by the angles: θ = 40 ^o , φ = 270 ^o / cm. FIG. 127 /. At the next moment, the robot begins to lower the upper part of its body and at the same time rolls slightly onto the stomach, so that the angles θ = 40 ^o , φ = 0 ^o / cm. FIG. 128 /. Then the robot again rolls onto the left shoulder and at the same time raises the lower part of its body: θ = 40 ^o , φ = 90 ^o / cm. FIG. 129 /. And finally, the robot lowers its lower part of the body and rolls slightly onto its back: θ = 40 ^o , φ = 180 ^o / cm. FIG. 130 /. Then the robot again takes the pose depicted by figure 127, and the whole cycle repeats.

On the globe of the tilts of the human body - the operator, rotated in the suspension mechanism, with such tilts of the robot body, the end of the vertical line vector will describe circles around the polar point: θ = 0 ^o / cm. FIG. 131 /. And here in the suspension mechanism exactly that undesirable effect arises that needs to be neutralized by turning the ring 4.

 Note that in FIG. 127 - 130 the Z axis of the XYZ coordinate system, rigidly connected with the robot body, is always directed in one direction, along the same azimuth. Those. the robot does not make a real rotation in space around the vertical line vector, but only tilts relative to it.

While in the suspension mechanism, we will be forced to rotate the body of the human operator to give him the necessary slopes / cm. FIG. 132-135 /:
FIG. 132 corresponds to the pose of the robot depicted in FIG. 127,
FIG. 133 corresponds to the pose of the robot depicted in FIG. 128,
FIG. 134 corresponds to the pose of the robot depicted in FIG. 129,
FIG. 135 corresponds to the pose of the robot depicted in FIG. 130.

 Thus, in the suspension mechanism, when the body of the operator, the operator, is given the necessary tilt / by rotation of the rings 1, 2 and 3 /, an undesirable rotation around the vertical line vector occurs simultaneously.

 In order to neutralize this undesirable rotation of the human body - the operator around the vertical line vector, we will turn the ring 4 towards him. Then the human body - the operator will not make an unplanned turn in space around the vertical line vector. It is precisely in this that the neutralizing role of ring 4 lies.

So, in the suspension mechanism, the ring 4 for the "infinitesimal" period of time: dt should generally rotate around its protrusions - axes / cm Fig. 86, 93 / angle:
dψ = dξ-dN, (95)
where the angle dξ is calculated by formula (62). The physical meaning of the rotation through this angle dξ was disclosed earlier.

По своей сути величина dN - это угол нежелательного вращения, на который повернется тело человека - оператора вокруг вектора отвесной линии за время dt при задании ему необходимого наклона: θ_i+1, φ_i+1.
Бесконечно малый промежуток: dt берется между двумя близкими моментами времени t_i и t_i+1:
dt = t_i+1 - t_i. (97)
Значения дифференциалов dλ и dμ вычисляются исходя из текущего и будущего наклонов тела человека - оператора. Т.е. в момент времени t_i тело человека- оператора имеет наклон: θ_i, φ_i, который в механизме подвеса задан углами μ_i, λ_i, κ_i. Одновременно от робота по каналу связи к нам пришла информация, что наклон корпуса робота определяется иными углами θ_i+1, φ_i+1, близкими по своим значениям к углам θ_i, φ_i.
Для придания телу человека - оператора подобной угловой ориентации /как и у корпуса робота/ мы будем должны повернуть за время dt кольца 1, 2, 3, 4 на углы dμ, dλ, dκ и dψ.
Перед тем как выполнить повороты на эти углы электроника механизма подвеса должна вычислить их значения.The value of the neutralized rotation dN is calculated by the formula

In essence, the value of dN is the angle of unwanted rotation, which will turn the body of the person - the operator around the vector of the vertical line in time dt when he sets the necessary slope: θ _{i + 1} , φ _{i + 1} .
Infinitely small gap: dt is taken between two close points in time t _i and t _{i + 1} :
dt = t _{i + 1} - t _i . (97)
The values of the differentials dλ and dμ are calculated based on the current and future slopes of the human body - the operator. Those. at time t _{i, the} body of the human operator has a slope: θ _i , φ _i , which in the suspension mechanism is given by the angles μ _i , λ _i , κ _i . At the same time, information came to us from the robot via the communication channel that the tilt of the robot body is determined by other angles θ _{i + 1} , φ _{i + 1} , which are close in their values to the angles θ _i , φ _i .
To give the human body the operator a similar angular orientation / like the robot body / we will have to rotate the rings 1, 2, 3, 4 over the time dt by the angles dμ, dλ, dκ and dψ.
Before making turns at these angles, the electronics of the suspension mechanism must calculate their values.

The differential dμ is calculated as the difference between the future value of the angle μ _{i + 1} and the current value of the angle μ _i :
dμ = μ _{i + 1} -μ _i , (98)
where the value of the future angle μ _{i + 1 the} electronics of the suspension mechanism extracts from the set of ROMs, where the values of the rotation angles of the rings μ, λ, κ are “sewn” as a function of the angles θ, φ that determine the slope / cm. text above.

The differentials dλ and dκ are calculated in a similar way:
dλ = λ _{i + 1} -λ _i ; (99)
dκ = κ _{i + 1} -κ _i . (100)
In formula (96), the differentials dλ and dμ are faced by the coefficients [sinκ (t _i )] and [cosκ (t _i ) • cosλ (t _i )], the values of which are taken at the current time t _i .

 The justification formula (96).

To do this, consider the following example:
Assume that we have a solid body that rotates on the mechanical axis AB, see fig. 136 /. The axis of rotation AB is directed at an angle θ to the plumb line vector. We introduce two coordinate systems: the XYZ system and the X _o Y _o Z _{o system} , such that the Z _o axis of the X _o Y _o Z _{o system is} directed opposite the vertical line vector, the Y _o and Y axes coincide / are aligned / and are angled 90 ^o to the Z _o axis, the X axis is aligned with the mechanical axis of rotation AB, and the X _o axis is coplanar to the X / cm axis. FIG. 137./.

Despite the fact that the solid rotates around the axis AB, the coordinate systems X _o Y _o Z _o and XYZ are stationary in space, i.e. a solid does not involve a coordinate system in rotation.

от оси вращения A-B. /см. фиг. 138/.The angle N, at which the rigid body rotates around the Z _o axis / around the vertical line vector /, is defined as the arc described in space by a unit radius - a vector perpendicular to the Z _o axis. The rotation of this unit radius - vector around the axis Z _o draws in the plane X _o OY _{o a} circle of unit radius, which in FIG. 138 looks like an ellipse. / Unit radius - the vector is not yet connected with any particular material point of the solid. /
Now, in the solid, select the material point "A" located at a distance

from the axis of rotation AB. /cm. FIG. 138 /.

In FIG. 138 shows a general view of the coordinate systems X _o Y _o Z _o and XYZ / axonometric projections are used here /, the large ellipse represents the unit circle described in space around the axis Z _{o by the} unit radius - vector, and the small ellipse represents the circle described by the material point "A "during the rotation of a solid material around the mechanical axis AB.

пересекает окружность. Отсюда мы уже легко можем найти угловую скорость, которую имеет материальная точка "A" по отношению к оси Z_o в момент прохождения точкой "A" плоскости X_oOY_o.We find the linear velocity vector of the material point "A" at the moment when it intersects the unit circle when the solid rotates around the X axis. Next, we find the projection of this linear speed on the tangent to the unit circle drawn at the point where the vector

crosses a circle. From here we can already easily find the angular velocity that the material point "A" has with respect to the axis Z _o at the moment the point "A" passes through the plane X _o OY _o .

The point "A" was chosen by us in a solid quite arbitrarily, the concept: "unit radius vector" also has a fairly arbitrary interpretation. From here we get the projection of the angular velocity vector on the Z _o axis for all points of the solid when they intersect the X _o OY _o plane in their rotation.

FIG. 139 gives us an image of FIG. 138 above, so that the axis Z _o of the coordinate system X _o Y _o Z _{o is} directed towards us.

FIG. 140 gives the view of FIG. 138 on the side, so that the axis Y _o of the coordinate system X _o Y _o Z _{o is} directed towards us.

лежащий в плоскости X_oOY_o. Этот вектор берет свое начало в точке "О" и заканчивается в той точке единичной окружности, через которую материальная точка "A" пересекает плоскость X_oOY_o.In FIG. 138 and 139 the unit radius vector was introduced

lying in the plane X _o OY _o . This vector originates at the point "O" and ends at that point of the unit circle through which the material point "A" intersects the plane X _o OY _o .

имеет в системе координат X_oY_oZ_o следующие компоненты:
(cosρ, sinρ, 0). (101)
Теперь с помощью матрицы, описывающей переход от системы координат X_oY_oZ_o к системе XYZ /см. фиг. 140/:

можно записать компоненты вектора

в системе координат XYZ:
(cosρ•sinθ, sinρ, cosρ•cosθ). (103)
Отсюда уже легко получить компоненты вектора

в тот момент, когда материальная точка "A" пересекает при своем вращении единичную окружность, лежащую в плоскости X_oOY_o:
(0, sinρ, cosρ•cosθ). (104)
Найдем с помощью векторного произведения компоненты вектора линейной скорости материальной точки "A" в системе координат XYZ. Для этого рассмотрим векторное произведение двух векторов: вектора

и вектора

задающего вращение твердого тела. Вектор

имеет в системе координат XYZ следующие компоненты:

Тогда вектор линейной скорости материальной точки "A" :

Т.е. через компоненты линейная скорость материальной точки "A" в системе XYZ запишется так:

Теперь перейдем от системы координат XYZ к системе координат X_oY_oZ_o. Для этого транспонируем матрицу, записанную в выражении (102)

Тогда вектор линейной скорости материальной точки "A" в системе координат X_oY_oZ_o имеет следующие компоненты:

Найдем проекцию этого вектора линейной скорости на касательную к единичной окружности в точке, где заканчивается вектор

/см. фиг. 141/. Для этого необходимо вначале определить единичный вектор, перпендикулярный вектору

и лежащий в плоскости X_oOY_o. Такой вектор в системе координат X_oY_oZ_o имеет компоненты /см. фиг. 141 /:
(sinρ, -cosρ, 0). (110)
Искомая нами проекция вектора линейной скорости материальной точки "A" на касательную определяется скалярным произведением вектора линейной скорости /выражение (109)/ и единичного вектора, касательного к окружности /выражение (110)./:

Итак, мы получили

/О том, что связь между

и

будет именно такой, можно было догадаться. Собственно говоря, я так и поступил, а здесь уже просто подогнал доказательство под интуитивно полученный ответ./
Теперь уже не составит труда объяснить появление формулы (96).As can be seen from FIG. 139, vector

has the following components in the coordinate system X _o Y _o Z _o :
(cosρ, sinρ, 0). (101)
Now, using the matrix describing the transition from the coordinate system X _o Y _o Z _o to the XYZ / cm system. FIG. 140 /:

you can write the components of the vector

in the XYZ coordinate system:
(cosρ • sinθ, sinρ, cosρ • cosθ). (103)
From here it is already easy to obtain the components of the vector

at the moment when the material point "A" intersects during its rotation a unit circle lying in the plane X _o OY _o :
(0, sinρ, cosρ • cosθ). (104)
Using the vector product, we find the components of the linear velocity vector of the material point “A” in the coordinate system XYZ. To do this, consider the vector product of two vectors: a vector

and vector

specifying the rotation of a rigid body. Vector

has the following components in the XYZ coordinate system:

Then the linear velocity vector of the material point "A":

Those. through the components, the linear velocity of the material point "A" in the XYZ system can be written as follows:

Now we move from the coordinate system XYZ to the coordinate system X _o Y _o Z _o . To do this, we transpose the matrix written in the expression (102)

Then the linear velocity vector of the material point "A" in the coordinate system X _o Y _o Z _o has the following components:

We find the projection of this linear velocity vector onto the tangent to the unit circle at the point where the vector ends

/cm. FIG. 141 /. To do this, you must first determine the unit vector perpendicular to the vector

and lying in the plane X _o OY _o . Such a vector in the coordinate system X _o Y _o Z _o has components / cm. FIG. 141 /:
(sinρ, -cosρ, 0). (110)
The projection of the linear velocity vector of the material point “A” onto the tangent that we are looking for is determined by the scalar product of the linear velocity vector / expression (109) / and the unit vector tangent to the circle / expression (110) ./:

So we got

/ That connection between

and

it will be just that, one could guess. As a matter of fact, I did just that, and here I simply adapted the proof to the intuitively received answer. /
Now it will not be difficult to explain the appearance of formula (96).

на вектор отвесной линии. /Появление перед dλ "sin" а не "cos", как это казалось бы следует из формулы (112), объясняется тем, что в механизме подвеса угол κ берет свое начало /начинает свой отсчет/ в положении, когда плоскость кольца 3 перпендикулярна вектору отвесной линии /см. фиг. 93 //. Второй член суммы (96) определяет величину поворота тела человека - оператора вокруг вектора отвесной линии, вызванного вращением dμ кольца 1 механизма подвеса. Перед дифференциалом dμ имеется коэффициент, состоящий из произведения косинусов, его появление объясняется формулой (89).The first term [sinκ (t _i ) • dλ, written in formula (96), determines the amount of rotation of the human body, the operator, around the vertical line vector, when ring 2 rotates in the suspension mechanism by an angle dλ. At its core, the expression [sinκ (t _i ) • dλ is the projection of the "rotation vector"

on a sheer line vector. / The appearance before dλ of "sin" and not of "cos", as it would seem from formula (112), is explained by the fact that the angle κ originates in the suspension mechanism / starts its countdown / in the position when the plane of ring 3 is perpendicular to the vector plumb line / cm. FIG. 93 //. The second term of the sum (96) determines the amount of rotation of the human body - the operator around the vertical line vector caused by rotation dμ of ring 1 of the suspension mechanism. In front of the differential dμ there is a coefficient consisting of the product of cosines; its appearance is explained by formula (89).

на вектор отвесной линии всегда равна нулю. Однако повороты кольца 3 изменяют наклоны выступов - осей колец 1 и 2, поэтому-то значение угла κ и входит через "sin " и "cos" в коэффициенты, стоящие перед dλ и dμ.
Предложенная здесь конструкция механизма подвеса обладает еще одним достоинством, заключающимся в том, что при задании наклонов тела человека-оператора сингулярные эффекты могут проявляться в механизме подвеса только тогда, когда антропоморфный робот находится в максимально устойчивом положении - лежит на правом или левом боку, поэтому сингулярные эффекты не отразятся на устойчивости вертикальной походки робота.Formula (96) does not include the differential dκ, because axis - the protrusions of the ring 3 always remain perpendicular to the vector of the vertical line and the projection of the vector

per plumb line vector is always zero. However, the rotations of the ring 3 change the inclination of the protrusions - the axes of the rings 1 and 2, therefore the value of the angle κ also enters the coefficients facing dλ and dμ through "sin" and "cos".
The suspension mechanism design proposed here has one more advantage, namely, that when defining the body of a human operator, singular effects can appear in the suspension mechanism only when the anthropomorphic robot is in the most stable position — it lies on the right or left side, therefore, singular effects will not affect the stability of the vertical gait of the robot.

 The method proposed here (physical theory of the suspension mechanism and sensors of angular orientation) makes it possible to manufacture a remotely controlled, replicating, anthropomorphic walking robot with increased maneuverability and increased cross-country ability / compared to robots moving on a wheeled or tracked track /.

 A method of protecting the electronics of an anthropomorphic robot from water.

 In this method, the outer coating of the robot consists of two layers. The first layer is prefabricated shields made of stainless alloys. One of the purposes of these "knightly armor" is to protect the inside of the robot from the piercing and cutting effects of surrounding objects. To do this, the metal shields must completely cover the surface of the robot. In those places where the robot has joints, the shields should, if necessary, go behind each other so as not to interfere with the robot bending the joint.

Under the metal shields is a continuous polymer shell, like the one that is worn on the body of a human diver working in three-bolt diving equipment. / Probably, this shell will be better made not of rubber, but of nylon, which, as I know, is more resistant to the effects of hydrocarbon compounds. /
In those places where the robot has the most movable joints / shoulder, pelvic, knee, ... / between the metal shields and the polymer shell should be located chain-link gasket - a structure woven from metal rings.

 The purpose of this chain-link gasket is to protect the polymer shell from being accidentally pinched by metal shields and from tearing when the robot bends the joint and the shields go one after another. / The chain mail rings should be large enough not to fall between the shields, and the shields themselves should be carefully adjusted to each other so that the gap between them is as small as possible.

 When a robot diver is immersed to a depth, water, of course, will easily penetrate the gaps between the metal shields and through the chain mail rings, but it will not be able to go further inside the robot through an airtight, continuous, polymer shell.

 With an increase in the immersion depth of the robot-diver, the water pressure on the polymer shell will also increase. However, the polymer shell will retain the volume enclosed within it, as it is filled with an "incompressible" liquid / kerosene or diesel fuel /. Therefore, we will be able to prevent compression and rupture of the polymer shell under the influence of increasing external water pressure.

 Thus, the electronics of the robot-diver will be in a non-aggressive, dielectric environment / in kerosene or in diesel fuel / all the time, and we can protect it from the effects of sea water.

Note:
The dielectric fluid filling the polymer shell plays the role of a lubricant for the internal mechanisms of the robot - the diver / while it can even be forced to pump between the rubbing surfaces of the joints of the robot /. On land, the "knightly armor" of a robot-diver prevents the "shell" of the polymer shell in the lower part of the robot from being inflated by the weight of the dielectric fluid enclosed in the polymer shell.

Claims (6)

 1. A method for remote control of an anthropomorphic walking copying robot, characterized in that the data measured by the angular orientation sensors installed in the robot body is sent from the robot via a communication channel to the tracking suspension mechanism, which gives the body of the human operator the same angular orientation as near the robot casing, which allows a person to feel with their balance organs all the changes occurring in the angular orientation of the robot, this gives the human operator the ability to control the vertical stroke Fight of the robot, since the robot repeats all the operator’s movements, the control of the moving parts of the robot is carried out in copy mode using a tracking system, one part of which is represented by the robot itself, the other is made in the form of a control suit with rigid elements worn on a human operator, while the effects exerted on the elements of the control suit are transmitted via the communication channel to the power drives of the robot for the corresponding movements of its limbs, while similar movements in space are made and the limbs of the human operator, in turn, when the surrounding bodies act on the robot, their effects are transmitted via a communication channel to the power drives of the joints of the control suit to transfer the corresponding efforts to the operator, which allows the human operator to feel what effort he exerts on the bodies surrounding the robot .  2. The method according to claim 1, characterized in that the determination of the angular orientation of an arbitrarily moving robot is carried out by installing two types of sensors in the robot housing and organizing the computational interaction between the readings of these sensors.  3. The method according to claim 2, characterized in that as one of the sensors use a plummet sensor, consisting of three mutually orthogonal measuring circles consisting of photocells, each of which is covered by a disk-shaped transparent container filled with a damping liquid, where under the influence of gravitational forces moves a small ball of another substance with a different specific gravity and optical transparency with respect to the damping fluid, which allows you to modulate the light flux incident on the photo ementy measuring the circumference and determine the direction of gravitational forces.  4. The method according to claim 2, characterized in that as the second type of sensors use a liquid inertial sensor that allows to measure arbitrary accelerated rotational movements, consisting of three spiral-shaped tubes filled with liquid, each of which lies in an orthogonal plane along relative to two other tubes and connected at its ends to a differential pressure gauge.  5. The method according to claim 1, characterized in that to protect the electronics of the anthropomorphic robot from the aggressive effects of water, the robot is placed in a sealed polymer shell, on top of which prefabricated protective metal shields are installed, after which the sealed shell is completely filled with a dielectric fluid non-aggressive with respect to electronics the robot. 6. The method according to claim 1, characterized in that to give the space to the body of the human operator a certain angular orientation simultaneously produce a tilt of the human body and its rotation around the vertical axis at certain angles, which use the suspension mechanism, consisting of six nested in each other rings with increasing radii, and the first ring is made with a minimum radius, and the sixth ring is with a maximum radius, each inner ring abuts with a pair of axle-protrusions existing on it for the continuation of the diameter in the ring about a larger size, and the diameter of the ring, on the continuation of which the axis-protrusions are located, is perpendicular to the line that is formed by the axis-protrusions of the smaller radius ring resting on it, the sixth ring abuts with its protrusions-axes at the base of the suspension mechanism, while the rings are made with the possibility of forced rotation on the protrusions-axes relative to each other, the human operator, on whose body a control suit is worn, is located in a suspended state in the center of a system of rings embedded in each other on a radial a rod perpendicular to the line passing through the axis-protrusions of the first ring, one end of this rod originates on the inner surface of the first ring, and the other joins the rigid elements of the control suit in the area of the back of the human operator, in the suspension mechanism rotation of the human operator’s body around a vertical the spatial line is carried out by rotating the fourth ring around its axis-protrusions, which, thanks to the rotation of the rings five and six, take orientation in the Earth’s gravitational field along vert In addition, rotation of the fourth ring eliminates unwanted spatial rotations of the human operator’s body around the vertical axis, which occur in the suspension mechanism in the process of giving the human body the necessary tilt using rotations of the first, second and third rings, an imaginary system of polar coordinates is rigidly connected with body of a human operator and has an orientation with respect to a person in which the polar axis of the coordinate system always passes through the axis-protrusions of the first ring, in most In more cases, the inclination of the human operator’s body is determined by turning only two, the first and second rings, and the plane of the third ring remains perpendicular to the vertical line, in situations where the human operator lies in the suspension mechanism on the right or left side and the acute angle between the vertical the line and the straight line passing through the protrusions-axes of the first ring turn out to be less than 30 ^o , the third ring also takes part in setting the slopes of the human operator’s body; the angles at which the third ring can turn in this case lie in the interval from -12 to +12 ^o , while the same tilt of the human operator’s body can be set by turning at different angles of the first, second and third rings, the values of the angles at which the rings can be rotated in a similar situation must meet certain to mutual relations, thus, each tilt of the human operator’s body is characterized by a whole set of triples of angles of rotation of the rings, specialized combination chains perform parallel processing of information and select the optimal triple of angles from this set, poses wanting to obtain the necessary tilt of the human operator’s body with a minimal change in the angular orientation of the inner rings of the suspension mechanism, a significant reduction in the singularity near the polar points of the polar coordinate system rigidly connected with the human operator’s body is achieved, in addition, the general design of the suspension mechanism is such that the residual singular effects arise in the suspension mechanism when defining the slopes of the human operator’s body only when the human operator is in the most stable position - ezhit on the right or left side.

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