CN107791248A

CN107791248A - Control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions

Info

Publication number: CN107791248A
Application number: CN201710902038.1A
Authority: CN
Inventors: 杨亮亮; 王杰; 史伟民; 王飞; 胡斌; 王刘奎; 李翔; 钱良珠
Original assignee: Zhejiang University of Technology ZJUT
Current assignee: Zhejiang University of Technology ZJUT
Priority date: 2017-09-28
Filing date: 2017-09-28
Publication date: 2018-03-13
Anticipated expiration: 2037-09-28
Also published as: CN107791248B

Abstract

The invention discloses the control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, belongs to motion planning and robot control method and technology field.The motion control method of six axle six degree of freedom serial manipulators of prior art, it is only applicable to the six axle six degree of freedom serial manipulators for meeting pipper criterions, not high to the control accuracy that is unsatisfactory for the axle six degree of freedom serial manipulator of pipper criterions six, control process is cumbersome.The present invention proposes a kind of parsing control method of new foundation in identical joint coordinate system, when establishing coordinate system to six axle six degree of freedom serial manipulators, all joint coordinate systems are consistent with robot basis coordinates system coordinate direction, and the six axle six degree of freedom serial manipulators for being unsatisfactory for pipper criterions are handled by the way of " standardization " and " transformation approach ", control process is simply orderly, amount of calculation can effectively be reduced, and the six axle six degree of freedom serial manipulators suitable for being unsatisfactory for pipper criterions, control accuracy are high.

Description

Control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions

Technical field

The present invention relates to the control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, belong to machine People's motion control method technical field.

Background technology

Conventional robot kinematics' method for solving mainly has 3 kinds：DH parametric methods, geometric method and analytic method.DH parametric methods It is a kind of wide variety of method, its foundation to joint coordinate system requires very harsh, it is necessary to performed according to specified principle, And cumbersome matrix operation, be unfavorable for embody algorithm real-time.Geometric method typically solution is more difficult, and versatility is bad, only Suitable for the less mechanical mechanism of the free degree.Conventional analytic method is adapted to robot that is simple in construction, being easy to analysis, but computing The solution of arcsine and anticosine often occurs in journey, more solutions just occur in such Robotic inverse kinematics when calculating.

But traditional parsing control method is only applicable to the six axle six degree of freedom serial machines for meeting pipper criterions People-i.e. its be used for the joints axes in last three joints of control machine people's terminal angle and intersect at a point.If these three are closed The joints axes of section are not if intersecting at a point, then the shifted relative on some position is just certainly existed between them. Also, the position and posture of this biasing can also change and change with the motion state in last three joints.Because the biasing is grown The location status of degree has certain uncertainty, and its position and posture to robot end also has significant impact. So we can not be connected by the parsing control method of routine to this six axle six degree of freedoms for being unsatisfactory for pipper criterions Robot carries out motion control, causes that the robot control process to the type is cumbersome, and computationally intensive, control accuracy is not high.

The content of the invention

The defects of for prior art, it can accurately be controlled it is an object of the invention to provide one kind and be unsatisfactory for pipper standards The control process of six degree of freedom serial manipulator motion then is simple, the control method of the few robot of amount of calculation.

To achieve the above object, the technical scheme is that：

Based on the control method for the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, comprise the following steps：

Step 1：The six axle six degree of freedom serial manipulators for being unsatisfactory for pipper criterions are changed on an equal basis, make its change Change to and meet on six axle six degree of freedom serial manipulator models of pipper criterions；Step 2：To the six axles six degree of freedom When serial manipulator carries out Forward Kinematics Analysis, the direction vector parameter of biasing is reserved；Step 3：According to trying to achieve before Biasing direction vector parameter, to the six axles six degree of freedom serial manipulator carry out changes in coordinates, transformed to satisfaction Six axle six degree of freedom serial manipulators of pipper criterions；Step 4：Analysis of Inverse Kinematics and meter are carried out to it based on analytic method Calculate；Step 5：The problem of progress during Analysis of Inverse Kinematics is carried out be present to the six axles six degree of freedom serial manipulator Related supplemental and explanation；Step 6：The kinematic parameter solved, motion control card is dealt into, to being unsatisfactory for pipper criterions six Axle six degree of freedom serial manipulator is controlled.The present invention proposes that a kind of new foundation controls in the parsing of identical joint coordinate system Method, control process is simply orderly, can effectively reduce amount of calculation, and suitable for being unsatisfactory for six axles six of pipper criterions certainly It is high by degree serial manipulator, control accuracy.

As optimization technique measure, step 1：The six axles six degree of freedom serial manipulator is changed on an equal basis, makes it It is transformed into the six axle six degree of freedom serial manipulator models for meeting pipper criterions.

Although the type serial manipulator is unsatisfactory for pieper criterions, by after corresponding deformation, can still pass through DH parameter models calculate, and the connect end effector arm of mechanical robot of six axles have been carried out into conversion to a certain extent so that end The extended line of last three cradle head axis is held to intersect at a point.

As optimization technique measure, step 2：Positive kinematics are being carried out to the axle six degree of freedom serial manipulator of the type six When analysis, the direction vector parameter of biasing is reserved.

Robot Forward Kinematics Analysis is exactly known rod member geometric parameter and joint angle vector, seeks end effector of robot Position and posture relative to fixed reference coordinate.

Before robot Forward Kinematics Analysis is carried out, it is desirable to establish change of the coordinate system { i } relative to coordinate system { i-1 } Change.This general conversion is made up of four parameters.To any given robot, this conversion is an only variable Function, the other three parameter are determined by mechanical system.Ginseng is converted between the connecting rod of the six axles six degree of freedom serial manipulator It is as shown in the table for number.

Parameter list is converted between the connecting rod of the axle six degree of freedom serial manipulator of table six

Wherein：

a_iFor along the X under the i-th coordinate system_iAxle, the Z under the i-th coordinate system_iDirection of principal axis is moved to Z under i+1 coordinate system_i+1 The distance of axle；

α_iFor the X under the i-th coordinate system_iAxle, the Z under the i-th coordinate system_iThe Z that axle is rotated under i+1 coordinate system_i+1Axle Angle；

d_iFor along the Z under the i-th coordinate system_iAxle, the X under the i-th -1 coordinate system_i-1The X that axle is moved under the i-th coordinate system_iAxle Distance；

θ_iFor the Z under the i-th coordinate system_iAxle, the X under the i-th -1 coordinate system_i-1The X that axle is rotated under the i-th coordinate system_iAxle Angle.

With homogeneous coordinate transformation matrix come represent the normal solution computing of robot (Represent it must is it from i coordinates to j coordinate systems Between 4 × 4 transfer matrix；In formula：By s θ_iWhat is represented is sin θ_i；cθ_iThat represent is cos θ₁.And i=1,2,3...6)：

Wherein：

θ₂₃=θ₂+θ₃

n_x=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁sθ₂₃sθ₅

n_y=s θ₁sθ₂₃sθ₅+sθ₁cθ₂₃cθ₄cθ₅-cθ₁sθ₄cθ₅

n_z=c θ₂₃Sθ₅-sθ₂₃cθ₄cθ₅

o_x=s θ₁cθ₄-cθ₁cθ₂₃sθ₄

o_y=-c θ₁cθ₄-sθ₁cθ₂₃sθ₄

o_z=s θ₂₃sθ₄

a_x=c θ₁cθ₂₃cθ₄sθ₅-cθ₁sθ₂₃cθ₅+sθ₁sθ4Sθ₅

a_y=s θ₁cθ₂₃cθ₄sθ₅-sθ₁sθ₂₃cθ₅-cθ₁sθ₄sθ₅

a_z=-s θ₂₃cθ₄sθ₅-cθ₂₃cθ₅

p_x=-d₄cθ₁sθ₂₃+l₂cθ₁cθ₂+d₃sθ₁-d₂sθ₁+d₆a_x+l₆o_x

p_y=-d₄sθ₁sθ₂₃+l₂sθ₁cθ₂+d₂cθ₁-d₃cθ₁+d₆a_y+l₆o_y

p_z=-d₄cθ₂₃-l₂sθ₂+d₆a_z+l₆o_z

It is worth noting that：Here the o out calculated_x、o_y、o_zWith the o hereinafter quoted_x5、o_y5、o_z5Parameter values are Consistent.By θ₆=0 brings into above-mentioned equation, it is seen that：The direction vector at the exactly bigoted place of these three parameters description.

As optimization technique measure, step 3：According to the direction vector parameter for the biasing tried to achieve before, to six axle six Free degree serial manipulator carries out coordinate transform, is transformed to the six axle six degree of freedom serial manipulators for meeting pipper criterions On.

Analysis of Inverse Kinematics is the geometric parameter of known machine people's rod member, and given end effector is sat relative to fixed reference Mark the desired locations and posture (wherein, p of system_x′、p_y′、p_z' represent be end effector coordinate position, n_x、n_y、n_z、o_x、 o_y、o_z、a_x、a_y、a_zWhat is represented is the coordinate posture of end effector), can analysis robot make its end effector reach this Individual expected pose.

For Analysis of Inverse Kinematics, the coordinate of ending coordinates system posture is known, that is to say, thatParameters It is known.But due to θ₆Uncertainty, can not pass through, it is known that byTo drawTransformation matrix. Herein, " standardization " can be taken such to solve the problems, such as.

Above-mentioned Forward Kinematics Analysis is solved what is comeThe o drawn_x5、o_y5、o_z5It is added in whole Analysis of Inverse Kinematics Come, and given data is pre-processed：

Wherein：p_x′、p_y′、p_z' it is known parameters, by the p after change_x、p_y、p_zBring into matrix.

Why given data is pre-processed, be because the serial manipulator is unsatisfactory for pieper criterions.With biography The multiaxis serial manipulator of system is compared, and the joints axes in three joints in end of the type multiaxis serial manipulator are prolonged Long line fails to intersect at a point.One relative is had more so between them on its position of some on end effector Lateral offset.Also, the position and posture of this biasing can also change and change with the motion state in last three joints.Due to The location status of the offset lengths has certain uncertainty, and its position and posture to robot end also has significantly Influence.Only take to its ending coordinates parameter p_x′、p_y′、p_z' pretreated method is carried out, it could eliminate by above-mentioned inclined Put and do not know to the influence caused by Solve problems, derived so as to carry out further Analysis of Inverse Kinematics and process.In fact, This processing mode be by it is original, be unsatisfactory for the series connection that pieper criterion serial manipulators are converted into meeting pieper criterions Robot.

As optimization technique measure, step 4：Analysis of Inverse Kinematics and calculating are carried out to it based on analytic method.

By formulaIt can obtain：

It is equal by the element (2,4) on above-mentioned equation both sides for first axis joint angle, obtain：

In formula：

Pay attention to：Sign, θ in above formula be present₁There can be multiple solutions.Now, θ₁It is known.

It is equal by the element (Isosorbide-5-Nitrae) on above-mentioned equation both sides, (3,4) for the 3rd joint angles, obtain：

Wherein：

It is equal by the element (Isosorbide-5-Nitrae) (2,4) on above-mentioned equation both sides for second joint angles, obtain：

So, by θ₂₃=θ₂+θ₃It can obtain：

According to θ₁And θ₃Four kinds of solution may combine, and can obtain corresponding four kinds of probable values by above formula, then can obtain θ₂ Four kinds may solution.

For the 4th joint angles, by the element (1,3) on above-mentioned equation both sides, (3,3) are equal obtains：

θ₄(the s θ of=A tan 2₁a_x-cθ₁a_y, c θ₁cθ₂₃a_x+sθ₁cθ₂₃a_y-sθ₂₃a_z)

As s θ₅When=0, robot is in Singularity, meets degenerative conditions.Now, joint shaft 4 and 6 overlaps, and can only solve Go out θ₆With θ₄And with difference.In Singularity, θ can be arbitrarily chosen₄Angle, calculating corresponding θ₆Value.But this feelings Condition is existed only among the serial manipulator for meeting Pepeer criterions.This problem will be begged in detail in next operation By.

For the 5th axis joint angle, by the element (1,3) on above-mentioned equation both sides, (3,3) are equal obtains：

θ₅(± the M of=A tan 2₁, M₂)

Wherein：

M₁=(c θ₁cθ₂₃cθ₄+sθ₁sθ₄)a_x+(sθ₁cθ₂₃cθ₄-cθ₁sθ₄)a_y-sθ₂₃cθ₄a_z

M₂=-c θ₁sθ₂₃a_x-sθ₁sθ₂₃a_y-cθ₂₃a_z

For the 6th joint angles, it can obtain：

Wherein：

N=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁sθ₂₃sθ₅

O=s θ₁cθ₄-cθ₁cθ₂₃sθ₄

As optimization technique measure, step 5：1) to the analysis of generally robot degenerate problem

As s θ₅When=0, robot is in Singularity, meets degenerative conditions.Now, joint shaft 4 and 6 overlaps, and can only solve Go out θ₆With θ₄And with difference.In Singularity, θ can be arbitrarily chosen₄Angle, calculating corresponding θ₆Value.

But such case is existed only among the serial manipulator for meeting Pepeer criterions, in the six axles six degree of freedom It is that degenerate problem is not present in serial manipulator.Because under the type serial manipulator, the robot being connected with each other is not Conllinear problem can occur.

When Robotic inverse kinematics are analyzed, first the given data of Analysis of Inverse Kinematics is pre-processed.And And this processing mode be exactly be by it is original, be unsatisfactory for pieper criterion serial manipulators and be converted into meeting pieper criterions Serial manipulator.Therefore, during actually solving, or exist and work as s θ₅=0 robot meets the situation of degenerative conditions. Below, this degenerate case is carried out to detailed analysis.

Based on joint shaft 4 for coordinate, by s θ₅=0, c θ₅=± 1 tries to achieveTransition matrix：

Make (1,2), (3,2) are equal obtains：

-o_xsθ₁+o_ycθ₁=c θ₄sθ₆±sθ₄cθ₆

θ₄±θ₆(the o of=A tan 2_xsθ₁-o_ycθ₁,-o_xcθ₁cθ₂₃-o_ysθ₁cθ₂₃+o_zsθ₂₃)

By previously mentioned o_x5、o_y5、o_z5Parameter, θ can be obtained₆：

θ₆(the o of=A tan 2_x, o_x5)

It can to sum up obtain：

Wherein：The selection of the positive negativity of above formula is by c θ₅Positive and negative judge.

2) analysis that special circumstances are under degenerate state

Above the Singularity of the six axles six degree of freedom serial manipulator is generally analyzed.It is actual On, when the robot motion is to a certain particular point, unusual situation also occurs.Due to such case only exist and certain The state of a kind of location point.Therefore it is referred to as Singularity in particular cases.

Situation 1：Work as d₂=d₃, and the axis of joint shaft 1 and 4 (or can not solve θ when coincidence, under this state₁ And θ₄.Such case there is more than among the robot model after change, in original robot model and exist strange Dystopy shape problem.

Such case can occur in both cases：First, such location point is the transit point of movement locus.Second, such Location point is movement locus starting point.

If such location point is the transit point of movement locus, can solve this problem using " inheritance act ". That is, when there is such case, θ₁Value acquiescence inherit θ under neighbouring position dotted state₁Value, it is and then right θ₄Solved.

If such location point is the starting point of movement locus, can solve this problem using " Reconstruction Method ". That is, when there is such case, θ₁Value be defaulted as 0, so as to afterwards to θ₄Solved.

Situation 2：Work as d₂=d₃, and when the axis coincidence of joint shaft 1 and 6, θ can not be solved under this state₁And θ₆。 Such case is existed only among the robot model after conversion, in original robot model is asked in the absence of Singularity Topic.But in above-mentioned Analysis of Inverse Kinematics, the state of this kind of Singularity still occurs.

In this case, can solve this problem using " sciagraphy ".There is such case that is, working as When, θ can be obtained by the projection that robot is formed in xoy planes₁Numerical value.

Compared with prior art, the invention has the advantages that：

The present invention is according to the six axle six degree of freedom serial manipulators control feature for being unsatisfactory for pipper criterions, to six axles six When free degree serial manipulator establishes coordinate system, all joint coordinate systems are sat with robot basis coordinates system (global coordinate system) It is consistent to mark direction, and come the six axle six degree of freedom strings to being unsatisfactory for pipper criterions by the way of " standardization " and " transformation approach " Connection robot is handled；Bivariate tan is used when solving joint angles, so passes through the symbol can of independent variable The quadrant where joint angles is determined, avoids the possibility of Lou solution；Occurs the situation of multigroup solution for some joints, using " just Approximately principle ", a solution nearest from current angular is chosen, using the method rejected step by step, is greatly reduced because certain joint occurs The huge amount of calculation that more solutions are brought.

Solution hardly possible, redundancy for being unsatisfactory for the appearance of the axle six degree of freedom serial manipulator kinematics solution of pipper criterions six The problems such as solution, propose that a kind of new foundation in the Analytic Method method of identical joint coordinate system, overcomes six axle six degree of freedoms Serial manipulator is because mass motion caused by itself being unsatisfactory for pipper criterions (inverse solution) solves failure the shortcomings that.In redundancy During solution selection, the method rejected using nearby principle and step by step greatly reduces amount of calculation, raising arithmetic speed, while also avoid Driving phenomenon caused by choosing solution is improper, and then can accurately control and be unsatisfactory for the axle six degree of freedom cascade machine of pipper criterions six Device people.

The present invention proposes a kind of parsing control method of new foundation in identical joint coordinate system, and control process simply has Sequence, amount of calculation can be effectively reduced, and suitable for being unsatisfactory for six axle six degree of freedom serial manipulators of pipper criterions, control Precision is high.

Brief description of the drawings

Fig. 1 is the mechanical arm, three-D model schematic of such six degree of freedom series connection of the present invention；

Fig. 2 is such six axles six degree of freedom serial manipulator machinery local deformation of the invention and establishment of coordinate system schematic diagram；

Fig. 3 is the robot that the present invention is under degenerate state to such six axles six degree of freedom serial manipulator ordinary circumstance Schematic diagram；

Fig. 4 is in Singularity robot model's schematic diagram 1 time for the present invention to special circumstances；

Fig. 5 is in Singularity robot model's schematic diagram 2 times for the present invention to special circumstances.

Embodiment

In order to make the purpose , technical scheme and advantage of the present invention be clearer, it is right below in conjunction with drawings and Examples The present invention is further elaborated.It should be appreciated that specific embodiment described herein is only to explain the present invention, not For limiting the present invention.

On the contrary, the present invention covers any replacement done in the spirit and scope of the present invention being defined by the claims, repaiied Change, equivalent method and scheme.Further, in order that the public has a better understanding to the present invention, below to the thin of the present invention It is detailed to describe some specific detail sections in section description.Part without these details for a person skilled in the art Description can also understand the present invention completely.

The threedimensional model of six degree of freedom series connection mechanical arm includes pedestal, large arm, forearm and the part of end palm four composition, such as Shown in Fig. 1.Large arm can be around pedestal axial-rotation in model, and forearm can be around forearm axial direction around large arm axial-rotation, end's platform Rotation.

As shown in left in Figure 2.Coordinate system { 0 } is located at pedestal lower surface position, and coordinate system { 1 } is located at pedestal and connected with large arm Place centre position is connect, coordinate system { 2 } is located at the centre position of large arm rear end, and coordinate system { 3 } is located at the centre position of forearm rear end, Coordinate system { 4 } and { 5 } are respectively positioned on the centre position of forearm front end, and coordinate system { 6* } is located at end's platform surface parallel and coordinate It is the focal position on { 4 } z-axis extended line, coordinate system { 6 } is located at the surface location in end's platform.D in Fig. 2₁For coordinate system { 0 } distance between coordinate system { 1 } along the z-axis direction, d₂For coordinate system { 1 } and the distance of coordinate system { 2 } along the z-axis direction, d₃To sit The distance that mark system { 2 } moves with coordinate system { 3 } along z3 direction of principal axis, d₄Moved for coordinate system { 5 } and coordinate system { 6 } along z5 direction of principal axis Dynamic distance；l₁For big arm lengths, l₂For forearm lengths, l₃For end's platform length.

The present invention relates to the control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, described side Method comprises the following steps：

Step 1：The axle six degree of freedom serial manipulator of the type six is changed on an equal basis, it is transformed into satisfaction On six axle six degree of freedom serial manipulator models of pipper criterions.

This six axles six degree of freedom serial manipulator structure still has closing solution.Although the serial manipulator is unsatisfactory for Pieper criterions, but by after corresponding deformation, can still be calculated by DH parameter models.As shown in Fig. 2 by left figure The end effector arm of six axles series connection mechanical robot has carried out conversion to a certain extent, and result of variations is as shown in Fig. 2 right figures.Make The rotary shaft for obtaining last three mechanical arms in end intersects at a point.6* coordinate systems are can be passed through to translate by 6 coordinate systems in right figure Conversion obtains.

Step 2：When Forward Kinematics Analysis is carried out to the axle six degree of freedom serial manipulator of the type six, reserve The direction vector parameter of biasing.

Before robot Forward Kinematics Analysis is carried out, it is desirable to establish change of the coordinate system { i } relative to coordinate system { i-1 } Change.This general conversion is made up of four parameters.To any given robot, this conversion is an only variable Function, the other three parameter are determined by mechanical system.As shown in above-mentioned Fig. 2, the six axles six degree of freedom serial manipulator Connecting rod between transformation parameter it is as shown in the table.

Wherein：

θ₂₃=θ₂+θ₃

n_x=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁Sθ₂₃Sθ₅

n_y=s θ₁sθ₂₃Sθ₅+sθ₁cθ₂₃cθ₄cθ₅-cθ₁sθ₄cθ₅

n_z=c θ₂₃Sθ₅-Sθ₂₃cθ₄cθ₅

o_x=S θ₁cθ₄-cθ₁cθ₂₃sθ₄

o_y=-c θ₁cθ₄-sθ₁cθ₂₃sθ₄

o_z=S θ₂₃Sθ₄

a_x=c θ₁cθ₂₃cθ₄sθ₅-cθ₁sθ₂₃cθ₅+sθ₁sθ₄Sθ₅

a_y=s θ₁cθ₂₃cθ₄sθ₅-sθ₁sθ₂₃cθ₅-cθ₁sθ₄sθ₅

a_z=-s θ₂₃cθ₄Sθ₅-cθ₂₃cθ₅

p_x=-d₄cθ₁sθ₂₃+l₂cθ₁cθ₂+d₃sθ₁-d₂sθ₁+d₆a_x+l₆o_x

p_y=-d₄Sθ₁sθ₂₃+l₂sθ₁cθ₂+d₂cθ₁-d₃cθ₁+d₆a_y+l₆o_y

p_z=-d₄cθ₂₃-l₂sθ₂+d₆a_z+l₆o_z+h

It is worth noting that：Here the o out calculated_x、o_y、o_zWith the o hereinafter quoted_x5、o_y5、o_z5Parameter values are Consistent.By 0₆=0 brings into above-mentioned equation, it is seen that：The vector side at the exactly bigoted place of these three parameters description To.

Step 3：According to the direction vector parameter for the biasing tried to achieve before, to the axle six degree of freedom serial machine of the type six People carries out changes in coordinates, is transformed to the six axle six degree of freedom serial manipulators for meeting pipper criterions.

Analysis of Inverse Kinematics is the geometric parameter of known machine people's rod member, and given end effector is sat relative to fixed reference Mark the desired locations and posture (wherein, p of system_x’、p_y’、p_z' represent be end effector coordinate position, n_x、n_y、n_z、o_x、 o_y、o_z、a_x、a_y、a_zWhat is represented is the coordinate posture of end effector), can analysis robot make its end effector reach this Individual expected pose.

For Analysis of Inverse Kinematics, the coordinate of ending coordinates system posture is known, that is to say, thatParameters It is known.But due to θ₆Uncertainty, can not pass through, it is known that byTo drawTransformation matrix. Here it is possible to take " standardization " such to solve the problems, such as.

Why given data is pre-processed, be because the serial manipulator is unsatisfactory for pieper criterions.With biography The multiaxis serial manipulator of system is compared, and the joints axes in three joints in end of the type multiaxis serial manipulator are prolonged Long line fails to intersect at a point.One relative is had more so between them on its position of some on end effector Lateral offset.Also, the position and posture of this biasing can also change and change with the motion state in last three joints.Due to The location status of the offset lengths has certain uncertainty, and its position and posture to robot end also has significantly Influence.Only take to its ending coordinates parameter p_x′、p_y′、p_z' pretreated method is carried out, it could eliminate by above-mentioned inclined Put and do not know to the influence caused by Solve problems, so as to carry out further inverse kinematics derivation and analysis.It is in fact, this Processing mode be by it is original, be unsatisfactory for the serial machine that pieper criterion serial manipulators are converted into meeting pieper criterions People, as shown in Figure 2.The serial manipulator model transformed to after pretreatment is basic herein as shown in the right figure in Fig. 2 The coordinate system of upper foundation is as shown in the right figure in Fig. 2.

Step 4：Analysis of Inverse Kinematics and calculating are carried out to it based on analytic method.

By formulaIt can obtain：

It is equal by the element (2,4) on above-mentioned equation both sides for first joint angles, obtain：

sθ₁p_x-cθ₁p_y=d₂-d₃

To sum up, θ₁Solution can be write as：

It is equal by the element (Isosorbide-5-Nitrae) on above-mentioned equation both sides, (3,4) for the 3rd axis joint angle, obtain：

p_xcθ₁+p_ysθ₁=-d₄sθ₂₃+l₂cθ₂

p_z=-d₄cθ₂₃-l₂sθ₂

To sum up, θ₃Solution can be write as：

Wherein：

It is equal by the element (Isosorbide-5-Nitrae) (2,4) on above-mentioned equation both sides for second axis joint angle, obtain：

cθ₁cθ₂₃p_x+sθ₁cθ₂₃p_y-sθ₂₃p_z-l₂cθ₃=0

-cθ₁sθ₂₃p_x-sθ₁sθ₂₃p_y-cθ₂₃p_z+l₂sθ₃=d₄

To sum up, θ₂Solution can be write as：

So, by θ₂₃=θ₂+θ₃It can obtain：

cθ₁cθ₂₃a_x+sθ₁cθ₂₃a_y-sθ₂₃a_z=c θ₄sθ₅

-sθ₁a_x+cθ₁a_y=-s θ₄sθ₅

：θ₄(the s θ of=A tan 2₁a_x-cθ₁a_y, c θ₁cθ₂₃a_x+sθ₁cθ₂₃a_y-sθ₂₃a_z)

For the 5th, the 6th joint angles, by the element (1,3) on above-mentioned equation both sides, (3,3) are equal obtains：

(cθ₁cθ₂₃cθ₄+sθ₁sθ₄)a_x+(sθ₁cθ₂₃cθ₄-cθ₁sθ₄)a_y-sθ₂₃cθ₄a_z=s θ₅

-cθ₁sθ₂₃a_x-sθ₁Sθ₂₃a_y-cθ₂₃a_z=c θ₅

：

θ₅(± the M of=A tan 2₁, M₂)

Wherein：

M₂=-c θ₁sθ₂₃a_x-sθ₁sθ₂₃a_y-cθ₂₃a_z

Now, θ₁、θ₂、θ₃、θ₄、θ₅It is known.In formula before substitution, it can obtain：

N=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁sθ₂₃sθ₅

O=s θ₁cθ₄-cθ₁cθ₂₃sθ₄

And then：

Wherein：

Step 5：There is robot degeneration in above-mentioned Analysis of Inverse Kinematics to the six axles six degree of freedom serial manipulator to ask Topic carries out related supplemental and explanation.

Because the kinematics analysis mode of the serial manipulator to the type is different from conventional Calculation Method, and in method In also related to the conversion between two kinds of different type serial manipulators, conventional robot kinematics' Singularity Analysis The problem of being not enough to tackle in the presence of above-mentioned analysis.Therefore, to using the analysis to degenerate case present in motion process To solve the multiresolution issue occurred in Robotic inverse kinematics analysis.

When robot loses 1 free degree, and therefore do not moved back by Ji Cheng robots during desired state motion Change.Robot can degenerate under two conditions：

(1) joint of robot reaches its physics limit and can not further moved；

(2) if the z-axis in two similar joints becomes conllinear, robot may be changed into degeneration shape in its working space State.

This means now whichever joint motions will all produce same motion, controller, which can not also determine bottom, is Which moving in joint.No matter any situation, the available number of degrees of freedom, of robot is less than 6, thus the equation of robot without Solution.

1. pair generally analysis of robot degenerate problem

But such case is existed only among the serial manipulator for meeting Pepeer criterions, in the six axles six degree of freedom It is that degenerate problem is not present in serial manipulator, as shown in Figure 2.Because under the type serial manipulator, it is connected with each other Conllinear problem will not occur for robot.

Based on axle 4 for coordinate, by s θ₅=0, c θ₅=± 1 tries to achieveTransition matrix：

Make (1,2), (3,2) are equal obtains：

By previously mentioned o_x5、o_y5、o_z5Parameter, θ can be obtained₆：

θ₆(the o of=A tan 2_x, o_x5)

It can to sum up obtain：

2. the analysis that pair special circumstances are under degenerate state

Above the Singularity of the axle six degree of freedom serial manipulator of the type six is generally analyzed.It is real On border, when the robot motion is to a certain particular point, unusual situation also occurs.Due to such case only exist with The state of certain a kind of location point.Therefore it is referred to as Singularity in particular cases.

Situation 1：As shown in figure 4, work as d₂=d₃, and the axis of joint shaft 1 and 4 is (or when coincidence, under this state θ can not be solved₁And θ₄.Such case there is more than among the robot model after change, in original robot model Singularity be present.

Such case can occur in both cases：First, such location point is the transit point of movement locus；Second, such Location point is movement locus starting point.

Situation 2：As shown in figure 5, work as d₂=d₃, and when the axis coincidence of joint shaft 1 and 6, can not be solved under this state Go out θ₁And θ₆.Such case is existed only among the robot model after conversion, is to be not present very in original robot model Dystopy shape problem.But in above-mentioned Analysis of Inverse Kinematics, the state of this kind of Singularity still occurs.

It is worth it is to be noted that：It is this to ask method to draw two different results.

Claims

1. the control method based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, it is characterised in that：Including following Step：

Step 1：The six axle six degree of freedom serial manipulators for being unsatisfactory for pipper criterions are changed on an equal basis, transform to it Meet on six axle six degree of freedom serial manipulator models of pipper criterions；Step 2：Connected to the six axles six degree of freedom When robot carries out Forward Kinematics Analysis, the direction vector parameter of biasing is reserved；Step 3：It is inclined according to what is tried to achieve before The direction vector parameter put, changes in coordinates is carried out to the six axles six degree of freedom serial manipulator, is transformed to satisfaction Six axle six degree of freedom serial manipulators of pipper criterions；Step 4：Analysis of Inverse Kinematics and meter are carried out to it based on analytic method Calculate；Step 5：The problem of progress during Analysis of Inverse Kinematics is carried out be present to the six axles six degree of freedom serial manipulator Related supplemental and explanation；Step 6：The kinematic parameter solved, motion control card is dealt into, to being unsatisfactory for pipper criterions six Axle six degree of freedom serial manipulator is controlled.

2. the control method as claimed in claim 1 based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, its It is characterised by：

Step 1：The six axles six degree of freedom serial manipulator is changed on an equal basis, it is transformed into and meets pipper criterions Six axle six degree of freedom serial manipulator models；

Although the type serial manipulator is unsatisfactory for pieper criterions, by after corresponding deformation, can still be joined by DH Exponential model calculates, and the connect end effector arm of mechanical robot of six axles has been carried out into conversion to a certain extent so that end is most The extended line of three cradle head axis intersects at a point afterwards.

3. the control method as claimed in claim 2 based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, its It is characterised by：Step 2：Robot Forward Kinematics Analysis is exactly known rod member geometric parameter and joint angle vector, asks robot last Hold position and posture of the actuator relative to fixed reference coordinate；

Before robot Forward Kinematics Analysis is carried out, conversion of the coordinate system { i } relative to coordinate system { i-1 } is established；It is general this Individual conversion is made up of four parameters；To any given robot, this conversion is the function of only one variable, in addition Three parameters are determined by mechanical system；Transformation parameter such as following table institute between the connecting rod of the six axles six degree of freedom serial manipulator Show；

Wherein：

a_iFor along the X under the i-th coordinate system_iAxle, the Z under the i-th coordinate system_iDirection of principal axis is moved to Z under i+1 coordinate system_i+1Axle Distance；

α_iFor the X under the i-th coordinate system_iAxle, the Z under the i-th coordinate system_iThe Z that axle is rotated under i+1 coordinate system_i+1The angle of axle Degree；

d_iFor along the Z under the i-th coordinate system_iAxle, the X under the i-th -1 coordinate system_i-1The X that axle is moved under the i-th coordinate system_iAxle away from From；

θ_iFor the Z under the i-th coordinate system_iAxle, the X under the i-th -1 coordinate system_i-1The X that axle is rotated under the i-th coordinate system_iThe angle of axle Degree；

The normal solution computing of robot is represented with homogeneous coordinate transformation matrix：(Represent it must is from i coordinates to j coordinate systems 4 × 4 transfer matrix；In formula：By s θ_iWhat is represented is sin θ_i；cθ_iThat represent is cos θ₁.And i=1,2,3...6)

<mrow> <mmultiscripts> <mi>T</mi> <mn>6</mn> <mn>0</mn> </mmultiscripts> <mo>=</mo> <mmultiscripts> <mi>T</mi> <mn>3</mn> <mn>0</mn> </mmultiscripts> <mmultiscripts> <mi>T</mi> <mn>5</mn> <mn>3</mn> </mmultiscripts> <mmultiscripts> <mi>T</mi> <mrow> <mn>6</mn> <mo>*</mo> </mrow> <mn>5</mn> </mmultiscripts> <mmultiscripts> <mi>T</mi> <mn>6</mn> <mrow> <mn>6</mn> <mo>*</mo> </mrow> </mmultiscripts> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>o</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>-</mo> <msub> <mi>n</mi> <mi>x</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>+</mo> <msub> <mi>o</mi> <mi>x</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>o</mi> <mi>y</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>-</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>+</mo> <msub> <mi>o</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>o</mi> <mi>z</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>-</mo> <msub> <mi>n</mi> <mi>z</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mi>z</mi> </msub> <msub> <mi>c&theta;</mi> <mn>6</mn> </msub> <mo>+</mo> <msub> <mi>o</mi> <mi>z</mi> </msub> <msub> <mi>s&theta;</mi> <mn>6</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>z</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>

Wherein：

θ₂₃=θ₂+θ₃

n_x=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁sθ₂₃sθ₅

n_y=s θ₁sθ₂₃sθ₅+sθ₁cθ₂₃cθ₄cθ₅-cθ₁sθ₄cθ₅

n_z=c θ₂₃sθ₅-sθ₂₃cθ₄cθ₅

o_x=s θ₁cθ₄-cθ₁cθ₂₃sθ₄

o_y=-c θ₁cθ₄-sθ₁cθ₂₃sθ₄

o_z=s θ₂₃sθ₄

a_x=c θ₁cθ₂₃cθ₄sθ₅-cθ₁sθ₂₃cθ₅+sθ₁sθ₄sθ₅

a_y=s θ₁cθ₂₃cθ₄sθ₅-sθ₁sθ₂₃cθ₅-cθ₁sθ₄sθ₅

a_z=-s θ₂₃cθ₄sθ₅-cθ₂₃cθ₅

p_x=-d₄cθ₁sθ₂₃+l₂cθ₁cθ₂+d₃sθ₁-d₂sθ₁+d₆a_x+l₆o_x

p_y=-d₄sθ₁sθ₂₃+l₂sθ₁cθ₂+d₂cθ₁-d₃cθ₁+d₆a_y+l₆o_y

p_z=-d₄cθ₂₃-l₂sθ₂+d₆a_z+l₆o_z

It is worth noting that：Here the o out calculated_x、o_y、o_zWith the o hereinafter quoted_x5、o_y5、o_z5Parameter values are consistent 's；By θ₆=0 brings into above-mentioned equation, it is seen that：The direction vector at the exactly bigoted place of these three parameters description；Its In, p_x’、p_y’、p_z' represent be end effector coordinate position, n_x、n_y、n_z、o_x、o_y、o_z、a_x、a_y、a_zWhat is represented is end The coordinate posture of actuator.

4. the control method as claimed in claim 3 based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, its It is characterised by：

Step 3：According to the direction vector parameter for the biasing tried to achieve before, the six axles six degree of freedom serial manipulator is carried out Coordinate transform, transformed to and met on six axle six degree of freedom serial manipulators of pipper criterions；

Analysis of Inverse Kinematics is the geometric parameter of known machine people's rod member, gives end effector relative to fixed reference frame Desired locations and posture (wherein, p_x’、p_y’、p_z' represent be end effector coordinate position, n_x、n_y、n_z、o_x、o_y、o_z、 a_x、a_y、a_zWhat is represented is the coordinate posture of end effector), can analysis robot make its end effector reach this expection Pose；

For Analysis of Inverse Kinematics, the coordinate of ending coordinates system posture is known, that is to say, thatKnown to parameters； But due to θ₆Uncertainty, can not pass through, it is known that byTo drawTransformation matrix；Herein, " standardization " can be taken such to solve the problems, such as；

Above-mentioned Forward Kinematics Analysis is solved what is comeThe o drawn_x5、o_y5、o_z5Whole Analysis of Inverse Kinematics is added to, And given data is pre-processed：

<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mi>x</mi> </msub> <mo>=</mo> <msup> <msub> <mi>p</mi> <mi>x</mi> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>d</mi> <mn>6</mn> </msub> <msub> <mi>a</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>6</mn> </msub> <msub> <mi>o</mi> <mrow> <mi>x</mi> <mn>5</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mi>y</mi> </msub> <mo>=</mo> <msup> <msub> <mi>p</mi> <mi>y</mi> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>d</mi> <mn>6</mn> </msub> <msub> <mi>a</mi> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>6</mn> </msub> <msub> <mi>o</mi> <mrow> <mi>y</mi> <mn>5</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mi>z</mi> </msub> <mo>=</mo> <msup> <msub> <mi>p</mi> <mi>z</mi> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>d</mi> <mn>6</mn> </msub> <msub> <mi>a</mi> <mi>z</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>6</mn> </msub> <msub> <mi>o</mi> <mrow> <mi>z</mi> <mn>5</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>

Wherein：p_x′、p_y′、p_z' it is known parameters, by the p after change_x、p_y、p_zBring into matrix；

Why given data is pre-processed, be because the serial manipulator is unsatisfactory for pieper criterions；With it is traditional Multiaxis serial manipulator is compared, the extended line of the joints axes in three joints in end of the type multiaxis serial manipulator Fail to intersect at a point；The relative transverse direction of one is had more so between them on its position of some on end effector Biasing；Also, the position and posture of this biasing can also change and change with the motion state in last three joints；Because this is inclined Putting the location status of length has certain uncertainty, and its position and posture to robot end also has significant shadow Ring；Only take to its ending coordinates parameter p_x′、p_y′、p_z' pretreated method is carried out, it could eliminate by above-mentioned biasing not It is determined that to the influence caused by Solve problems, derived so as to carry out further Analysis of Inverse Kinematics and process；It is in fact, this Processing mode be by it is original, be unsatisfactory for the serial machine that pieper criterion serial manipulators are converted into meeting pieper criterions People.

5. the control method as claimed in claim 4 based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, its It is characterised by：Step 4：Analysis of Inverse Kinematics and calculating are carried out to it based on analytic method；

By formulaIt can obtain：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>n</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>z</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mmultiscripts> <mi>T</mi> <mn>6</mn> <mn>1</mn> </mmultiscripts> </mrow>

<mrow> <msub> <mi>&theta;</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <mo>,</mo> <msub> <mi>p</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> </mrow> <mi>&rho;</mi> </mfrac> <mo>,</mo> <mo>&PlusMinus;</mo> <mfrac> <msqrt> <mrow> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mi>&rho;</mi> </mfrac> <mo>)</mo> </mrow> </mrow>

In formula：

Pay attention to：Sign, θ in above formula be present₁There can be multiple solutions；Now, θ₁It is known；

For the 3rd joint angles, by the element (Isosorbide-5-Nitrae) on above-mentioned equation both sides, (3,4), equal (wherein element (Isosorbide-5-Nitrae) refers to The row parameter of equation both sides the 1st row of matrix the 4th), obtain：

<mrow> <msub> <mi>&theta;</mi> <mn>3</mn> </msub> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mo>&PlusMinus;</mo> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> </mrow>

Wherein：

<mrow> <mi>k</mi> <mo>=</mo> <mfrac> <mrow> <msup> <msub> <mi>d</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>l</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>p</mi> <mi>x</mi> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>p</mi> <mi>y</mi> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>p</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>4</mn> </msub> </mrow> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>c&theta;</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>n</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>z</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mmultiscripts> <mi>T</mi> <mn>6</mn> <mn>3</mn> </mmultiscripts> </mrow>

<mrow> <msub> <mi>&theta;</mi> <mn>23</mn> </msub> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mn>2</mn> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>d</mi> <mn>4</mn> </msub> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>p</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>,</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>c&theta;</mi> <mn>3</mn> </msub> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>p</mi> <mi>x</mi> <mi>c</mi> <mi>&theta;</mi> <mn>1</mn> <mo>+</mo> <mi>p</mi> <mi>y</mi> <mi>s</mi> <mi>&theta;</mi> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>pz</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>z</mi> </msub> <mo>,</mo> <msub> <mi>p</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow>

So, by θ₂₃=θ₂+θ₃It can obtain：

<mrow> <msub> <mi>&theta;</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>d</mi> <mn>4</mn> </msub> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>p</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>,</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>c&theta;</mi> <mn>3</mn> </msub> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>p</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mi>z</mi> </msub> <mo>,</mo> <msub> <mi>p</mi> <mi>x</mi> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>y</mi> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mo>&PlusMinus;</mo> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow>

According to θ₁And θ₃Four kinds of solution may combine, and can obtain corresponding four kinds of probable values by above formula, then can obtain θ₂Four Kind may solve；

θ₄=Atan2 (s θ₁a_x-cθ₁a_y, c θ₁cθ₂₃a_x+sθ₁cθ₂₃a_y-sθ₂₃a_z)

As s θ₅When=0, robot is in Singularity, meets degenerative conditions；Now, joint shaft 4 and 6 overlaps, and can only solve θ₆ With θ₄And with difference；In Singularity, θ can be arbitrarily chosen₄Angle, calculating corresponding θ₆Value；But such case is only It is present among the serial manipulator for meeting Pepeer criterions；This problem will be discussed in detail in next operation：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> <msub> <mi>c&theta;</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> <msub> <mi>c&theta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> <msub> <mi>c&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>d</mi> <mn>4</mn> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> <msub> <mi>s&theta;</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> <msub> <mi>s&theta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> <msub> <mi>s&theta;</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>d</mi> <mn>4</mn> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>d</mi> <mn>4</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>n</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>z</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mmultiscripts> <mi>T</mi> <mn>5</mn> <mn>4</mn> </mmultiscripts> </mrow>

For the 5th axis joint angle, by the element (1,3) at above-mentioned equation both ends, (3,3) are equal obtains：

θ₅=Atan2 (± M₁, M₂)

Wherein：

M₂=-c θ₁sθ₂₃a_x-sθ₁sθ₂₃a_y-cθ₂₃a_z

For the 6th joint angles, it can obtain：

<mrow> <msub> <mi>&theta;</mi> <mn>6</mn> </msub> <mo>=</mo> <mi>A</mi> <mi> </mi> <mi>tan</mi> <mn>2</mn> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>-</mo> <msub> <mi>o</mi> <mi>x</mi> </msub> </mrow> <msqrt> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>o</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>,</mo> <mfrac> <msub> <mi>n</mi> <mi>x</mi> </msub> <msqrt> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>o</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mi> </mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mn>2</mn> <mrow> <mo>(</mo> <mfrac> <mi>n</mi> <msqrt> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>o</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>,</mo> <mfrac> <mi>o</mi> <msqrt> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>o</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> </mrow>

Wherein：

N=c θ₁cθ₂₃cθ₄cθ₅+sθ₁sθ₄cθ₅+cθ₁sθ₂₃sθ₅

O=s θ₁cθ₄-cθ₁cθ₂₃sθ₄

6. the control method as claimed in claim 5 based on the six degree of freedom serial manipulator for being unsatisfactory for pipper criterions, its It is characterised by：

Step 5：1) to the analysis of generally robot degenerate problem

As s θ₅When=0, robot is in Singularity, meets degenerative conditions；Now, joint shaft 4 and 6 overlaps, and can only solve θ₆ With θ₄And with difference；In Singularity, θ can be arbitrarily chosen₄Angle, calculating corresponding θ₆Value；

But such case is existed only among the serial manipulator for meeting Pepeer criterions, connected in the six axles six degree of freedom It is that degenerate problem is not present in robot；Because under the type serial manipulator, the robot being connected with each other is to send out Raw conllinear problem；

When Robotic inverse kinematics are analyzed, first the given data of Analysis of Inverse Kinematics is pre-processed；And this Kind of processing mode be exactly be by it is original, be unsatisfactory for the series connection that pieper criterion serial manipulators are converted into meeting pieper criterions Robot；Therefore, during actually solving, or exist and work as s θ₅=0 robot meets the situation of degenerative conditions；Below, This degenerate case is carried out to detailed analysis；

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>c&theta;</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> <msub> <mi>s&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>c&theta;</mi> <mn>23</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>s&theta;</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>s&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>c&theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>n</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>x</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>y</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>o</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mi>z</mi> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mi>z</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mmultiscripts> <mi>T</mi> <mn>6</mn> <mn>3</mn> </mmultiscripts> </mrow>

Based on joint shaft 4 for coordinate, by s θ_s=O, c θ₅=± 1 tries to achieveTransition matrix：

Make (1,2), (3,2) are equal obtains：

-o_xsθ₁+o_ycθ₁=c θ₄sθ₆±sθ₄cθ₆

θ₄±θ₆=Atan2 (o_xsθ₁-o_ycθ₁,-o_xcθ₁cθ₂₃-o_ysθ_1cθ₂₃+o_zsθ₂₃)

By previously mentioned o_x5、o_y5、o_z5Parameter, θ can be obtained₆：

θ₆=Atan2 (o_x, o_x5)

It can to sum up obtain：

Wherein：The selection of the positive negativity of above formula is by c θ₅Positive and negative judge；

2) analysis that special circumstances are under degenerate state

Above the Singularity of the six axles six degree of freedom serial manipulator is generally analyzed；In fact, When the robot motion is to a certain particular point, unusual situation also occurs；Due to such case only exist with it is a certain The state of class location point；Therefore it is referred to as Singularity in particular cases；

Situation 1：Work as d₂=d₃, and the axis of joint shaft 1 and 4 (or can not solve θ when coincidence, under this state₁And θ₄； Such case there is more than among the robot model after change, in original robot model and Singularity be present Problem；

Such case can occur in both cases：First, such location point is the transit point of movement locus；Second, such position Point is movement locus starting point；

If such location point is the transit point of movement locus, can solve this problem using " inheritance act "；Also It is to say, when there is such case, θ₁Value acquiescence inherit θ under neighbouring position dotted state₁Value, and then to θ₄Enter Row solves；

If such location point is the starting point of movement locus, can solve this problem using " Reconstruction Method "；Also It is to say, when there is such case, θ₁Value be defaulted as 0, so as to afterwards to θ₄Solved；

Situation 2：Work as d₂=d₃, and when the axis coincidence of joint shaft 1 and 6, θ can not be solved under this state₁And θ₆；This feelings Condition is existed only among the robot model after conversion, is that Singularity problem is not present in original robot model； But in above-mentioned Analysis of Inverse Kinematics, the state of this kind of Singularity still occurs；

In this case, can solve this problem using " sciagraphy "；That is, when there is such case Wait, θ can be obtained by the projection that robot is formed in xoy planes₁Numerical value.