CN107802268A - A kind of outer instantaneous spiral shell rotating shaft measuring method of human elbow anterior flexion and rear stretching and forearm medial rotation rotation - Google Patents

A kind of outer instantaneous spiral shell rotating shaft measuring method of human elbow anterior flexion and rear stretching and forearm medial rotation rotation Download PDF

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CN107802268A
CN107802268A CN201711253617.4A CN201711253617A CN107802268A CN 107802268 A CN107802268 A CN 107802268A CN 201711253617 A CN201711253617 A CN 201711253617A CN 107802268 A CN107802268 A CN 107802268A
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CN107802268B (en
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李剑锋
张春召
周帅锋
高亚楠
张凯
曹强
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Beijing University of Technology
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    • AHUMAN NECESSITIES
    • A61MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61BDIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00Measuring for diagnostic purposes; Identification of persons
    • A61B5/103Detecting, measuring or recording devices for testing the shape, pattern, colour, size or movement of the body or parts thereof, for diagnostic purposes
    • A61B5/11Measuring movement of the entire body or parts thereof, e.g. head or hand tremor, mobility of a limb
    • A61B5/1121Determining geometric values, e.g. centre of rotation or angular range of movement
    • AHUMAN NECESSITIES
    • A61MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61BDIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00Measuring for diagnostic purposes; Identification of persons
    • A61B5/103Detecting, measuring or recording devices for testing the shape, pattern, colour, size or movement of the body or parts thereof, for diagnostic purposes
    • A61B5/11Measuring movement of the entire body or parts thereof, e.g. head or hand tremor, mobility of a limb
    • A61B5/1126Measuring movement of the entire body or parts thereof, e.g. head or hand tremor, mobility of a limb using a particular sensing technique
    • A61B5/1127Measuring movement of the entire body or parts thereof, e.g. head or hand tremor, mobility of a limb using a particular sensing technique using markers

Abstract

The invention discloses a kind of human elbow anterior flexion and rear stretching and the outer instantaneous spiral shell rotating shaft measuring method of forearm medial rotation rotation, belong to field of measuring technique, the present invention includes following three steps:(1) collection of human upper limb locomotion data:Two index point moving mass are fixed on operator's upper arm, forearm appropriate location, the foundation for obtaining human body upper arm, forearm movable information (2) forearm motion model first:Using human body upper arm, forearm as a rigid body, elbow joint broad sense single-degree-of-freedom and forearm broad sense single-degree-of-freedom kinematics model are established.(3) rotation instantaneous screw axis in joint solves:By collecting upper arm, forearm exercise data, structure forearm rigid body obtains instantaneous position of the elbow joint helical axis in motion process with respect to the space displacement matrix of upper arm rigid motion.Easy realization of the invention, measurement point are calculated simple and measured accurately by being strapped on skeleton.

Description

A kind of outer instantaneous spiral shell rotating shaft measurement of human elbow anterior flexion and rear stretching and forearm medial rotation rotation Method
Technical field
The invention belongs to skeleton field of measuring technique, specifically a kind of human elbow anterior flexion and rear stretching and forearm medial rotation The outer instantaneous spiral shell rotating shaft measuring method of rotation.
Background technology
The instantaneous rotation that human elbow anterior flexion and rear stretching moves around instantaneous aroused in interest rotary motion rule and human body ancon around dynamic point Rotating shaft is the basic data of skeletal system guiding movement, the bionical elbow joint mechanism of design and ancon motor function, is obtained relative Accurate ancon joint motions information is for ensureing that bone training effect, the bionical elbow joint mechanism of design and ancon functional rehabilitation have It is of practical significance.
By consulting domestic and foreign literature, human body ancon joint motions infomation detection is said from method of testing species:Greatly It is by ultraviolet light camera system, multidimensional camera system, ectoskeleton donning system, electromagnetic tracking system, flexible sensor system mostly System etc. carries out the relative motion data that detection obtains ancon joint, inherently divides, can be divided into two kinds of intrusive mood and non-intruding Measuring method, intrusive mood measuring method are exactly to be directed through skin, the corresponding sports of tissue measurement shoulders of human body bone, such as purple Outer smooth camera system, the autopsy experiment of human body ancon, this measuring method harm to the human body is big, general inadvisable, another It is non-intrusion type measuring method:By taking index point in skeleton skin surface patch, closed by camera acquisition human body ancon The moving image of section, by index point extraction and motion analysis, and based on the phase between the short disjunctor coordinate system of different limbs Pose is analyzed to obtain human body ancon joint motions information, in foregoing detection method, using elbow joint motion analysis model For single-degree-of-freedom rotary pair.
It is intricate shown in human body ancon joint Fig. 1 it can be seen from human anatomy, it by humerus, radius, ulna and its Between humeroradial joint, humeroulnar joint, articulatio radioulnaris proximalis and distal radioulnar jojnt be formed by connecting, moved according to human skeleton model Credit analyse, ancon joint (traditional elbow joint) complete it is anteflexion/after stretch and medial rotation/rotation of forearm outside motion process In, rotary shaft is transient change, and the instantaneou axis of rotation is not about the fixed-point motion of ancon humerus one, but rotates fortune around dynamic point It is dynamic, though and the instantaneou axis of rotation characteristics of motion is broad sense single-degree-of-freedom kinematic pair, extremely complex, not clear at present, this makes It is more complicated to study the motion in ancon joint, in order to obtain ancon joint motions information, the present invention proposes a kind of human elbow Anterior flexion and rear stretching and the outer instantaneous spiral shell rotating shaft measuring method of forearm medial rotation rotation, measuring method measurement are accurate.
The content of the invention
It is outer instantaneous that the technical problem to be solved in the present invention is to provide a kind of human elbow anterior flexion and rear stretching and forearm medial rotation rotation Spiral shell rotating shaft, and a kind of simple in construction, easy to use, the concise measuring method of algorithm.Its inventive concept is:In human body ancon In the research of structure, modelling processing is the beginning of all work, is gained knowledge according to human dissection, and with reference to skeleton Extract upper arm (being equivalent to determine rigid body) on the basis of, skeleton are kinematic out from human body, forearm (being equivalent to dynamic rigid body) is done Corresponding kinematics analysis, as shown in Figure 2,3.The kinematical properties are equivalent to the kinematics mould of general rigid body in rational mechanics Type, gained knowledge and learnt according to the general motion of rigid body in rational mechanics, forearm does anterior flexion and rear stretching relative to upper arm and medial rotation rotation is outer Motion process spin matrix, you can it is determined that the instantaneou axis of rotation now.
Scheme is used by the present invention solves technical problem:
A kind of outer instantaneous spiral shell rotating shaft measuring method of human elbow anterior flexion and rear stretching and forearm medial rotation rotation, ancon exercise data Collection:Four index points not in one plane are fixed on moving mass first, moving mass is fixed on motion flange, fortune Dynamic flange is fixed on close to the anterior appropriate location of upper arm top, forearm, using VICON measuring systems, obtains above index point Real time position.
The foundation of ancon motion model:To solve human elbow anterior flexion and rear stretching and the outer instantaneous spiral shell rotating shaft of forearm medial rotation rotation, Using human body upper arm as fixed rigid body, forearm a broad sense single-degree-of-freedom ancon joint motions are established as real time kinematics rigid body Model is learned, similarly, a broad sense single-degree-of-freedom forearm medial rotation is established and revolves outer kinematics model.
The solution of the joint instantaneou axis of rotation:It is constant with respect to the motion of rigid body according to the index point on rigid body during rigid motion Property, the coordinate data when index point of upper arm and forearm is moved by VICON automatic Optic Motion Capture Systems is collected, led to Structure space displacement matrix equation is crossed, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being transported relatively with upper arm The instantaneou axis of rotation during dynamic, the instantaneou axis of rotation during the outer motion of joint forearm medial rotation rotation is obtained, in order to quantify joint Center t at any time position, is updated to it with VICON detection frequency, is calculated using the matrix equation provided To any time ancon joint instantaneou axis of rotation, all parameters in matrix equation be all by the upper arm that above measures to obtain, The surface data of forearm motion.
The present invention compared with prior art, has advantages below and high-lighting effect:
The outer instantaneous spiral shell rotating shaft measuring method of a kind of human elbow anterior flexion and rear stretching and forearm medial rotation rotation of the present invention, Neng Gougen According to human body ancon bone and the anatomical structure in joint, a broad sense single-degree-of-freedom ancon joint and forearm kinematics model are established. According to the kinematic data of index point on VICON system detectio attribute blocks, the influence that measurement process is come due to Zona transformans is reduced, And by building space displacement matrix equation, joint screw parameter is extracted, accurate human body ancon joint is obtained and instantaneously revolves Rotating shaft.Specific checking computations are learnt:Of the invention easily to realize, measurement point is calculated simple and measured by being strapped on skeleton Accurately.The outer instantaneous spiral shell rotating shaft measuring method of a kind of human elbow anterior flexion and rear stretching and forearm medial rotation rotation, it is characterised in that including with Lower three steps;
(1) collection of ancon exercise data:Four index points not in one plane are fixed on moving mass first, moved It is fixed on soon on motion flange, motion flange is fixed on close to the anterior appropriate location of upper arm top, forearm, is surveyed using VICON Amount system, obtain the real time position of above index point.
(2) foundation of ancon motion model:In order to solve the instantaneous spiral shell rotating shaft of human elbow anterior flexion and rear stretching, using human body upper arm as Fixed rigid body, forearm are established a broad sense single-degree-of-freedom ancon articular kinesiology model, can similarly built as real time kinematics rigid body The broad sense single-degree-of-freedom kinematics model of the vertical outer motion of forearm medial rotation rotation.
(3) solution of the joint instantaneou axis of rotation:According on rigid body during rigid motion mark point with respect to rigid body motion invariance, The coordinate data when mark point of upper arm and forearm is moved by VICON automatic Optic Motion Capture Systems collects, and passes through Space displacement matrix equation is built, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being moved relatively with upper arm During the instantaneou axis of rotation.
Each step to more than is described further below:
The collection of ancon exercise data:Tester's upper arm is fixed on test platform, and upper arm flange is fixed on upper arm, on Cushion is added between arm flange and upper arm;Forearm flange is fixed on forearm, and bullet is equally added between forearm flange and hand Property pad.In order to allow VICON automatic Optic Motion Capture Systems more accurately to collect index point, the situation for avoiding losing a little occurs, and has Beneficial to elbow joint motion amplitude maximization, upper arm flange is with forearm flange as far as possible away from elbow joint.
In measurement process, when bending and stretching collection exercise data, forearm keeps nature, preceding after fixing upper arm and measuring table Arm is moved by Qu Xiangshen, reciprocal three cycles.Outside the rotation of forearm medial rotation during motion gathered data, adjustment forearm flange is fixed on forearm Appropriate location, VICON systems are made to be able to detect that all data of the index point in a period of motion, fixed upper arm and measurement are flat After platform, forearm is moved from medial rotation to outside rotation, reciprocal three periods of motion.
Using VICON measuring systems, the real time position data of above index point can be accurately obtained.By 8 marks of upper arm and forearm Point is designated as Pi, the position of each index point is (xi yi zi), wherein i=1,2,8, these real time position datas are all Being described under VICON coordinate systems, upper arm fixed coordinate system origin represents that x, y, z axle is represented with X1, Y1, Z1 respectively with O1, Being calculated to simplify, its direction is consistent with VICON system-based coordinate systems direction, wherein:
O1=(P1+P2+P3+P4)/4 (1)
P1、P2、P3、P4For upper arm index point.
Forearm index point is expressed as P in upper arm coordinate systemii, now i=5,6,7,8, wherein:
Pii=Pi-O1 (2)
The foundation of ancon motion model:To solve human elbow anterior flexion and rear stretching and the outer instantaneous spiral shell rotating shaft of forearm medial rotation rotation, Using human body upper arm as fixed rigid body, forearm establishes a broad sense single-degree-of-freedom ancon joint and preceding as real time kinematics rigid body Arm medial rotation revolves outer kinematics model.
The solution of the joint instantaneou axis of rotation:It is constant with respect to the motion of rigid body according to the index point on rigid body during rigid motion Property, the coordinate data when index point of upper arm and forearm is moved by VICON automatic Optic Motion Capture Systems is collected, led to Structure space displacement matrix equation is crossed, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being transported relatively with upper arm The instantaneou axis of rotation during dynamic.Joint helical axis Relative motility parameters are extracted, joint forearm is obtained and is being moved relatively with upper arm During the instantaneou axis of rotation.
S1 builds space displacement matrix equation:Forearm index point represents in fixed coordinate system O1, at 1, the j moment, the table of moment 1 Show that measurement starts, moment j represents measurement process, and be named as k, q, r, j on forearm index point, and Equation is expressed as at 4 points:
D in formula 31jImplication for the forearm j moment relative to the relative displacement matrix at 1 moment, (k1x,k1y,k1z) indicate for k Position o'clock at 1 moment relative to upper arm fixed coordinate system, (kjx,kjy,kjz) fixed for k index points at the j moment relative to upper arm The position of coordinate system.
Write as matrix equation:
Obtain transposed matrix:
S2 extracts joint helical axis Relative motility parameters, according to equation below, obtains helical axis kinematic parameter:
φ is spiral Shaft angle, and u is the direction vector of helical axis,For in the forearm at the 1 moment three-dimensional coordinate vector of a bit, s For the amount of movement in helical axis directions,Relative rotation matrices for the forearm j moment relative to 1 moment, as shown in Figure 5.
S3 obtains the relative instantaneou axis of rotation with upper arm in motion process of joint forearm according to the screw parameter tried to achieve.
Brief description of the drawings
Fig. 1 is human body ancon skeletal structure schematic diagram.
Fig. 2 is human body ancon articular kinesiology model schematic.
Fig. 3 is that human body ancon arthritis adhesive plaster index point moves schematic diagram.
Fig. 4 is attribute block, attribute block adpting flange, attribute block binding structure schematic diagram.
Fig. 5 is rigid motion helical axis kinematics parameters.
Fig. 6 is forearm anterior flexion and rear stretching and certain outer moment helical axis schematic diagram of medial rotation rotation.
Embodiment
Present invention specific implementation example is further illustrated below in conjunction with accompanying drawing
The collection of ancon exercise data:As shown in Figure 2,3, 4,8 index points are placed with the upper arm and forearm of tester, led to Cross the position that vicon automatic Optic Motion Capture Systems gather 8 index points.By taking flexion and extension as an example, in experimentation, collection During flexion/extension exercise data, forearm is moved by Qu Xiangshen, reciprocal three cycles.
Inscribed at 1, the position of 8 index points is:
1P1(88.45031,273.0711,837.2647),1P2(166.1756,274.8842,841.8455),
1P3(166.1756,274.8842,841.8455),1P4(132.6032,227.2049,818.8338),
1P5(439.7643,208.5741,757.6406),1P6(513.9912,196.7798,766.3503),
1P7(470.8855,197.1223,800.4714),1P8(479.417,202.7582,726.2701)
Under t, the position of 8 index points is:
tP1(88.22546,273.0368,837.277),tP2(166.0312,274.8918,841.9113),
tP3(125.8689,324.4791,855.2818),tP4(132.4993,227.2034,818.8992),
tP5(440.4059,209.7388,759.9684),tP6(515.2634,198.1683,770.8627),
tP7(471.2239,197.6752,803.7907),tP8(481.3215,204.1584,729.6448)
8 index points of upper arm and forearm are designated as Pi, the position of each index point is (xi yi zi), wherein i=1, 2,8, obtain the upper arm fixed coordinate system O1 origins of coordinates by (1) formula is at 1 moment:
1O=(477.0537,202.4352,766.0667)
The upper arm fixed coordinate system O1 origins of coordinates are in t:
tO=(477.0537 202.4352 766.0667)
Obtained by (2), the forearm index point data that 1 moment represented in upper arm coordinate system are:
1P55=(- 36.6478 7.3036-6.0982)1P66=(38.2097, -4.2669,4.7961)
1P77=(- 5.8298, -4.7600,37.7240)1P88=(4.2678,1.7233, -36.4218)
The data of t expression are in upper arm coordinate system:
tP55=(- 36.0082,8.4908, -2.6307)tP66=(39.4255, -2.6591,10.7712)
tP77=(- 5.6559, -3.7558,42.3359)tP88=(5.9133,2.9865, -31.8120)
The solution of the joint instantaneou axis of rotation:It is constant with respect to the motion of rigid body according to the index point on rigid body during rigid motion Property, the coordinate data when index point of upper arm and forearm is moved by VICON automatic Optic Motion Capture Systems is collected, led to Structure space displacement matrix equation is crossed, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being transported relatively with upper arm The instantaneou axis of rotation during dynamic.Specially:There are above-mentioned 1 moment and t data value, bring above-mentioned (3), (4), (5) equation into, Obtaining space displacement matrix is:
It is equal by both members corresponding amount according to equation (6), obtain helical axis kinematic parameter:Its x, y, z direction vectors are: 0.0201, -0.0374, -0.0119, the coordinate of any is on helical axis:(1.37,0.24978,1.521), according to what is tried to achieve Screw parameter u direction vector and the coordinate of a bit, it is relative instantaneous in motion process with upper arm to obtain joint forearm Rotary shaft.The coordinate of two moment points is similarly chosen, elbow joint medial rotation can be drawn and revolve outer instantaneous screw axis, before the specific a certain moment Instantaneous screw axis is stretched after bending and the outer instantaneous screw axis of medial rotation rotation is as shown in Figure 6.
In order to quantify articulation center t at any time position, it is updated with VICON detection frequency, utilized The simple matrix equation provided is calculated any time ancon joint instantaneou axis of rotation, and all parameters in equation are all logical Cross and above measure obtained upper arm, the surface data of forearm motion.
It is to be used in the described outer instantaneous spiral shell rotating shaft measuring method of a kind of human elbow anterior flexion and rear stretching and forearm medial rotation rotation Ancon joint motions information on the right side of human body is detected, is equally applicable to detect ancon upper arm joint motions information on the left of human body, is not limited to This example.
Above-mentioned embodiment is explained and illustrated to the essence of the present invention, but is not construed as limitation of the present invention, Any simple modifications made based on essence of the invention, as long as its kinematics and method of testing are based on the principle, should all fall into this Within the scope of invention right is claimed.

Claims (5)

1. a kind of human elbow anterior flexion and rear stretching and the outer instantaneous spiral shell rotating shaft measuring method of forearm medial rotation rotation, it is characterised in that including Three steps below;
(1) collection of ancon exercise data:Four index points not in one plane are fixed on moving mass first, moved Block is fixed on motion flange, and motion flange is fixed on close to the anterior appropriate location of upper arm top, forearm, is surveyed using VICON Amount system, obtain the real time position of above index point;
(2) foundation of ancon motion model:In order to solve the instantaneous spiral shell rotating shaft of human elbow anterior flexion and rear stretching, using human body upper arm as Fixed rigid body, forearm are established a broad sense single-degree-of-freedom ancon articular kinesiology model, can similarly built as real time kinematics rigid body The broad sense single-degree-of-freedom kinematics model of the vertical outer motion of forearm medial rotation rotation;
(3) solution of the joint instantaneou axis of rotation:According on rigid body during rigid motion mark point with respect to rigid body motion invariance, The coordinate data when mark point of upper arm and forearm is moved by VICON automatic Optic Motion Capture Systems collects, and passes through Space displacement matrix equation is built, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being moved relatively with upper arm During the instantaneou axis of rotation.
2. a kind of human elbow anterior flexion and rear stretching according to claim 1 and the outer instantaneous spiral shell rotating shaft measurement side of forearm medial rotation rotation Method, it is characterised in that the collection of ancon exercise data:Tester's upper arm is fixed on test platform, and upper arm flange is fixed on On arm, cushion is added between upper arm flange and upper arm;Forearm flange is fixed on forearm, same between forearm flange and hand Add cushion;In order to allow VICON automatic Optic Motion Capture Systems more accurately to collect index point, the situation for avoiding losing a little goes out It is existing, and be advantageous to elbow joint motion amplitude maximization, upper arm flange is with forearm flange as far as possible away from elbow joint.
3. a kind of human elbow anterior flexion and rear stretching according to claim 1 and the outer instantaneous spiral shell rotating shaft measurement side of forearm medial rotation rotation Method, it is characterised in that in measurement process, when bending and stretching collection exercise data, forearm keeps nature, fixed upper arm and measurement After platform, forearm is moved by Qu Xiangshen, reciprocal three cycles;Outside the rotation of forearm medial rotation during motion gathered data, forearm flange is adjusted Forearm appropriate location is fixed on, VICON systems is able to detect that all data of the index point in a period of motion, in fixation After arm and measuring table, forearm is moved from medial rotation to outside rotation, reciprocal three periods of motion.
4. a kind of human elbow anterior flexion and rear stretching according to claim 1 and the outer instantaneous spiral shell rotating shaft measurement side of forearm medial rotation rotation Method, it is characterised in that using VICON measuring systems, can accurately obtain the real time position data of above index point;By upper arm with before 8 index points of arm are designated as Pi, the position of each index point is (xi yi zi), wherein i=1,2 ..., 8, these real time position numbers According to all being described under VICON coordinate systems, upper arm fixed coordinate system origin is represented with O1, and x, y, z axle uses X1, Y1, Z1 respectively Represent, calculated to simplify, its direction is consistent with VICON system-based coordinate systems direction, wherein:
O1=(P1+P2+P3+P4)/4 (1)
P1、P2、P3、P4For upper arm index point;
Forearm index point is expressed as P in upper arm coordinate systemii, now i=5,6,7,8, wherein:
Pii=Pi-O1 (2)。
5. a kind of human elbow anterior flexion and rear stretching according to claim 1 and the outer instantaneous spiral shell rotating shaft measurement side of forearm medial rotation rotation Method, it is characterised in that the foundation of ancon motion model:To solve human elbow anterior flexion and rear stretching and the outer instantaneous spiral shell of forearm medial rotation rotation Rotating shaft, a broad sense single-degree-of-freedom ancon joint is established as real time kinematics rigid body using human body upper arm as fixed rigid body, forearm And forearm medial rotation revolves outer kinematics model;
The solution of the joint instantaneou axis of rotation:According to motion invariance of the index point on rigid body during rigid motion with respect to rigid body, lead to Cross coordinate data when VICON automatic Optic Motion Capture Systems move the index point of upper arm and forearm to collect, pass through structure Space displacement matrix equation is built, extracts joint helical axis Relative motility parameters, joint forearm is obtained and is being moved through relatively with upper arm The instantaneou axis of rotation in journey;Joint helical axis Relative motility parameters are extracted, obtain joint forearm relatively with upper arm in motion process In the instantaneou axis of rotation;
S1 builds space displacement matrix equation:Forearm index point represents that, at 1, the j moment, the moment 1 represents in fixed coordinate system O1 Measurement starts, moment j expression measurement process, and 4 points are named as k, q, r, j on forearm index point, and Equation is expressed as:
(kjx,kjy,kjz,1)T=[D1j](k1x,k1y,k1z,1)T
(qjx,qjy,qjz,1)T=[D1j](q1x,q1y,q1z,1)T
(rjx,rjy,rjz,1)T=[D1j](r1x,r1y,r1z,1)T (3)
(jjx,jjy,jjz,1)T=[D1j](j1x,j1y,j1z,1)T
D in formula 31jImplication for the forearm j moment relative to the relative displacement matrix at 1 moment, (k1x,k1y,k1z) indicate for k Position o'clock at 1 moment relative to upper arm fixed coordinate system, (kjx,kjy,kjz) fixed for k index points at the j moment relative to upper arm The position of coordinate system;
Write as matrix equation:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>D</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Obtain transposed matrix:
<mrow> <mo>&amp;lsqb;</mo> <msub> <mi>D</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mi>j</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
S2 extracts joint helical axis Relative motility parameters, according to equation below, obtains helical axis kinematic parameter:
φ is spiral Shaft angle, and u is the direction vector of helical axis,To be in the forearm at the 1 moment three-dimensional coordinate vector of a bit, s Amount of movement in helical axis directions,Relative rotation matrices for the forearm j moment relative to 1 moment;
S3 obtains the relative instantaneou axis of rotation with upper arm in motion process of joint forearm according to the screw parameter tried to achieve.
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