AU2021103765A4

AU2021103765A4 - Robotic arm motion programming method based on fixed-parameter neural network

Info

Publication number: AU2021103765A4
Application number: AU2021103765A
Authority: AU
Inventors: Jilong Xie; Song Yang; Zhijun Zhang
Original assignee: South China University of Technology SCUT
Current assignee: South China University of Technology SCUT
Priority date: 2020-07-08
Filing date: 2021-06-30
Publication date: 2021-08-19
Anticipated expiration: 2029-06-30
Also published as: CN111975768B; CN111975768A

Abstract

The present invention discloses a robotic arm motion programming method based on a fixed-parameter neural network, including: Si, establishing an inverse kinematics equation of a robotic arm according to a Jacobian matrix and a preset target trajectory of an end of the 5 robotic arm; S2, establishing a physical limit double-ended inequality constraint of the robotic arm according to actual physical limit constraint parameters of joints of the robotic arm; S3, formulating the inverse kinematics equation and the physical limit double-ended inequality constraint as a time-varying quadratic programming problem; S4, designing the fixed-parameter neural network with a penalty function to solve the time-varying quadratic 0 programming problem; and S5, transferring angle information of the robotic arm as solved to a lower computer controller of the robotic arm to drive the robotic arm to move to complete a target trajectory tracking task. The present invention uses a new type of fixed-parameter neural network with a penalty function to solve the time-varying quadratic programming problem, which has faster convergence velocity and higher calculation accuracy, and can 5 effectively solve the inequality constraint in the quadratic programming scheme. Establishinga model of a robotic arm according to structural parameters ofthe robotic arm to obtain a Jacobian matrix of an end effector. and establishing an inverse kinematics equation of the robotic arm according to the Jacobian matrix and a preset target trajectory of an end of the robotic arm Establishing a physical limit double-ended inequality constraint S2 of the robotic arm on a velocity layer according to actual physical limit constraint parameters of joints of the robotic arm Formulating the inverse kinematics equation and the physical S3 limit double-ended inequality constraint as a time-varying quadratic programming problem Designing the fixed-parameter neural network with a penalty S4 function to solve the time-varying quadratic programming problem Transferring angle information of the robotic arm as solved to a S5 lower computer controller of the robotic arm to drive the robotic arm to move to complete a target trajectory tracking task FIG. 1 1/2

Description

Establishinga model of a robotic arm according to structural parameters ofthe robotic arm to obtain a Jacobian matrix of an end effector. and establishing an inverse kinematics equation of the robotic arm according to the Jacobian matrix and a preset target trajectory of an end of the robotic arm

Establishing a physical limit double-ended inequality constraint S2 of the robotic arm on a velocity layer according to actual physical limit constraint parameters of joints of the robotic arm

Formulating the inverse kinematics equation and the physical S3 limit double-ended inequality constraint as a time-varying quadratic programming problem

Designing the fixed-parameter neural network with a penalty S4 function to solve the time-varying quadratic programming problem

Transferring angle information of the robotic arm as solved to a S5 lower computer controller of the robotic arm to drive the robotic arm to move to complete a target trajectory tracking task

FIG. 1

1/2

ROBOTIC ARM MOTION PROGRAMMING METHOD BASED ON FIXED-PARAMETER NEURAL NETWORK TECHNICAL FIELD

[0001] The present invention relates to the technical field of robotic arm motions under various constraints, and in particular to a robotic arm motion programming method based on a

fixed-parameter neural network.

BACKGROUND

[0002] Repetitive motion of a robotic arm is a very common motion in industrial

production. Repetitive motion requires joints of the robotic arm to return to their initial states

after completing a periodic motion of closed end trajectory. In this case, it can be ensured that

the initial state of each periodic motion of the robotic arm is consistent. If all the joints of the

robotic arm do not return to their initial states after completing a periodic motion, it is deemed

that a joint offset phenomenon occurs. If the joint offset phenomenon occurs during the

periodic motion, the motion control accuracy of the robotic arm will be reduced, or an

additional reset adjustment process will be required, resulting in a serious decrease in

production efficiency. Therefore, in the actual control of the robotic arm, in order to realize

the periodic motion of the robotic arm, it is necessary to ensure that the robotic arm will not

undergo joint drift during a task cycle. At the same time, the actual motion of the robotic arm

may be subject to various constraints to ensure the safety of the robotic arm in practical

applications, such as physical limit constraints of the joints. Almost all robotic arms have

physical limits of joints. If the algorithm does not take into account avoidance of the physical

limits of the joints, the control amount generated by the algorithm will easily exceed the

actual joint limit of the robotic arm. The idea of a general physical avoidance algorithm of

joints is to rewrite the physical limits of the joints of the robotic arm into an inequality double-ended constraint, and then take it into consideration in a quadratic programming scheme.

[0003] The traditional kinematics solution method of the robotic arm is based on a pseudo-inverse method. However, because the pseudo-inverse method needs to calculate an inverse of a matrix, and tasks faced by the robotic arm in practical applications are becoming more and more diverse, it is difficult to process more complex tasks in real time in practical applications. Compared with the pseudo-inverse method, a robotic arm programming method based on the solving of an optimization problem is more widely used. Methods for solving the optimization problem include a numerical method solver and a neural network solver. Due to the characteristics of numerical iteration, the numerical method solver has the advantages of convenient control on a digital computer and easy generation of pulse signals for motor driving. For example, a 94LVI numerical method is used to solve a quadratic programming problem based on an equality constraint and is applied to the motion programming of the robotic arm. However, this numerical method requires a lot of computer resources to perform iterative calculations, which increases the calculation time and is difficult to apply to real-time motion programming of robotic arms. Compared with the numerical method solver, the recurrent neural network solver based on neural dynamics has a parallel computer system and is considered to be a more powerful solver for real-time optimization problems. This neural network solver is widely used in many practical applications due to the high efficiency of -0 calculation and the convenience of hardware implementation. Zhang et al. proposed a recursive neural network solver, referred to as annihilation neural network, which is used to solve time-varying quadratic programming problems ]and is used to solve redundant robotic arm motion programming problems. Due to the use of derivative information of parameters, the annihilation neural network can successfully obtain an optimal solution of the time-varying quadratic programming problem. However, this traditional annihilation neural network can only approach the theoretical optimal solution in infinite time. In order to overcome the physical constraints of robot joints, a primal-dual neural network based on linear variational inequality was proposed to solve a corresponding quadratic programming problem, and it was applied to a redundant robot end trajectory tracking task. However, LVI-PDNN cannot reach an exponential convergence velocity. In order to make full use of the characteristics of the time-varying system of the robot's motion programming problem, Zhang et al. designed a variable-parameter convergence differential neural network with super exponential convergence effect[2 ), and applied the network to the repetitive motion programming scheme of redundant robots. However, the variable-parameter convergence differential neural network cannot effectively solve the inequality constraint problems in the robotic arm motion programming scheme, such as the physical limit constraints of joints.

[0004] In summary, there is an urgent need in the industry to develop a neural network solver or solution method that has an exponential convergence velocity and effectively solves

the robotic arm motion programming problem constrained by physical limitss.

References:

[1] Zhang Y, Ge S.. A General Recurrent Neural Network Model for Time-Varying Matrix

Inversion[A]. In: 42nd IEEE International Conference on Decision and Control[C]. Maui, HI,

USA: IEEE, 2003. 6:6169-6174.

[2] Z. Zhang, Y Lu, L. Zheng, S. Li, Z. Yu, and Y Li, "A new varying-parameter

convergent-differential neural-network for solving time-varying convex QP problem

constrained by linear-equality," IEEE Transactions on Automatic Control, vol. 63, no. 12, pp.

4110-4125,2018.

SUMMARY

[0005] An objective of the present invention is to overcome the above shortcomings of the

prior art, and provide a robotic arm motion programming method based on a fixed-parameter

neural network that has an exponential convergence velocity and can effectively solve the

physical limit constraints.

[0006] The objective of the present invention is achieved through the following technical

solutions.

[0007] A robotic arm motion programming method based on a fixed-parameter neural

network, including:

[0008] S1, establishing a model of a robotic arm according to structural parameters of the

robotic arm to obtain a Jacobian matrix of an end effector, and establishing an inverse kinematics equation of the robotic arm according to the Jacobian matrix and a preset target trajectory of an end of the robotic arm;

[00091 S2, establishing a physical limit double-ended inequality constraint of the robotic arm on a velocity layer according to actual physical limit constraint parameters of joints of the robotic arm;

[00101 S3, formulating the inverse kinematics equation and the physical limit double-ended inequality constraint as a time-varying quadratic programming problem, and solving a motion programming problem of the robotic arm constrained by physical limits by using a quadratic programming scheme, wherein a repetitive motion index is used as a performance index for the time-varying quadratic programming problem;

[00111 S4, designing the fixed-parameter neural network with a penalty function to solve the time-varying quadratic programming problem; and

[0012] S5, transferring angle information of the robotic arm as solved to a lower computer controller of the robotic arm to drive the robotic arm to move to complete a target trajectory tracking task.

[0013] Preferably, the physical limit constraint parameters of the joints in step S2 comprise: a joint angle constraint and an angular velocity constraint; and the physical limit double-ended inequality constraint of the robotic arm established on the velocity layer is:

[0014] where (- E R" and + Rn are defined as:

_-, if 6 [i,6+] r710-, if 6 E [6-, - 1 ] +- + if 0E -, ZU2 ] 207if C ZU 2 ]

[00151 where 6 E Rn represents angles of the joints of the robotic arm, 6- and

6+represent upper and lower physical limits of the angles of the joints the robotic arm

respectively; 6 E Rn represents angular velocities of the joints of the robotic arm, and O and 0+ represent upper and lower physical limits of the angular velocities of the joints of

A the robotic arm respectively; r/1 = 1 - (sin (fl(sin (fl(6- 1)/(6- - 71)

12 = 1- (sin (f#(sin (fl(6 - o 2 )/(0+ - 0 2 )))2))2

zu=a, 06

c2 2

[0016] where f is a positive number in an interval of [0, 1].

[00171 Preferably, step S3 includes:

[00181 converting the physical limit double-ended inequality constraint<(-6 <( into a physical limit one-sided inequality constraint; and

[0019] formulating the inverse kinematics equation and the physical limit one-sided inequality constraint as the time-varying quadratic programming problem.

[00201 Preferably, the physical limit one-sided inequality constraint is:

K6 < e ,

[0021] where K = [E, -E]T E R2nxn , _ - r E R2n , where E

represents an identity matrix of n X n; and

[00221 a control scheme of formulating the inverse kinematics equation and the physical

limit one-sided inequality constraint as the time-varying quadratic programming problem is:

min. b 2 S.t. JE =PE

Kb <e

[0023] where b = (6(t) - 6(0)), where the parameter OJ is a response coefficient

of joint offset.

[00241 Preferably, the penalty function is an exponential penalty function P(t), and an

expression of the exponential penalty function P(t) is:

2n

P(t) = p e-a "i-Kjb) i=1

[00251 where (' > 0, p is a positive number close to 0, ei and Ki are i-dimension

row vectors of vector e and matrix K respectively.

[0026] Preferably, step S4 includes:

[00271 S41, converting an inequality constraint term in the control scheme of the time-varying quadratic programming problem into a penalty term in the performance index by

using the exponential penalty function P(t), namely:

-T. 66T min. 2 +b T+P(t)

S.t. JE6 =E

[0028] S42, converting the quadratic programming problem into a matrix equation using a Lagrangian multiplier method:

B(t)y(t)=G(t)

[00291 where specific content of each symbol is as follows:

B(t) = E JE E R (n+m)x(n+m), y(t)= E Rn+n, jE - mXm 2n-16 2n

G(t) = b - pa e-(ei-Ki) Ki) E Rn+m TE

[0030] where A is a Lagrangian operator, and an error function is defined as

Eft)= B(t)y(t) - G(t) C R"-""

[0031] in order to make an error converge to zero, the following neural dynamics criterion is adopted: dE(t)/dt = -y-i(E(t) ),

[0032] where (): R -+ R" is an activation function, and V(-) must be a

monotonically increasing odd function; and the neural dynamics criterion, and similar terms are combined to obtain the fixed-parameter neural network with a penalty strategy:

R (t)f (t) = -(t)y(t) - yP(B (t)y(t) - G (t)) + V(t),

[0033] where

R(t) E + po 2 1e-(i-Ki-) - KiK) JE

T(t): 2n -FG-iO IE Omxm

11(t) [b- Puz,=i fe-(~i) (ki - i&

[0034] then a first n element of a solution of the fixed-parameter neural network is an

optimal solution *,which is integrated to obtain 0*, and 0* is an optimal joint angle of

the robotic arm.

[00351 Preferably, the structural parameters include a length of a connecting rod and a

number of degrees of freedom.

[0036] Compared with the existing robotic arm motion programming scheme, the present

invention has the following advantages:

[00371 1. Compared with the classic recursive neural network solver, the present

invention uses a new type of fixed-parameter neural network with a penalty function to solve

the time-varying quadratic programming problem, which has faster convergence velocity and

higher calculation accuracy, and can effectively solve the inequality constraint in the quadratic

'7 programming scheme.

[0038] 2. Compared with the traditional pseudo-inverse-based method, the present

invention solves the motion programming problem of the robotic arm through a quadratic

programming scheme, which has the characteristics of good real-time performance and can

take into consideration a variety of constraints.

[0039] 3. Compared with a numerical method solver, the present invention belongs to a neural network solver, which has fast calculation velocity and higher efficiency.

BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The accompanying drawings constituting a part of the present application are used to provide a further understanding of the present invention. The exemplary embodiments and

descriptions of the present invention are used to explain the present invention, and do not

constitute an improper limitation of the present invention. In the attached picture:

[0041] FIG. 1 is a flowchart of a robotic arm motion programming method based on a

fixed-parameter neural network according to the present invention.

[0042] FIG. 2 (a) is a diagram of a motion trajectory of a robotic arm according to the

present invention.

[0043] FIG. 2(b) is a diagram of an initial state and an end state of the robotic arm

according to the present invention.

[0044] FIG. 2(c) is a diagram of a motion trajectory of an end effector of the robotic arm

according to the present invention.

[0045] FIG. 2(d) is a diagram of a position error of the end effector of the robotic arm

according to the present invention.

[0046] FIG. 2(e) is a state diagram of joints of the robotic arm according to the present

invention.

[0047] FIG. 2(f) is a state diagram of an angular velocity of a third joint of the robotic arm

according to the present invention.

DETAILED DESCRIPTION

[00481 The present invention will be further described below in conjunction with the drawings and embodiments.

[0049] As shown in FIG. 1, the present invention provides a robotic arm motion programming method based on a fixed-parameter neural network, which is mainly composed

of three parts: problem raising, problem transformation, and problem solving. First, an inverse

kinematics equation is established on a velocity layer according to a preset end trajectory of a

robotic arm and a Jacobian matrix, and a motion programming problem of the robotic arm is

designed as a time-varying convex quadratic programming problem, in which repetitive

motion of the robotic arm is designed as an optimization index, an inverse kinematics

equation is designed as an equality constraint, and a physical limit constraint is designed as a

double-ended inequality constraint. The quadratic programming problem is transformed into a

time-varying matrix equation using a Lagrangian equation. Finally the matrix equation is

solved by the designed fixed-parameter neural network with a penalty function. The details

are as follows:

[0050] First, referring to FIG. 2(a), consider the inverse kinematics equation of the robotic

arm as:

f (0)= , (1)

[00511 where # E R" represents angular velocities of joints of the robotic arm,

r E R 7is the preset end trajectory of the robotic arm, and f() is a non-linear mapping

equation. m represents a dimension of a working space of the robotic arm. If it is a

three-dimensional space, then m = 3. n is the number of joints of the robotic arm.

Equation (1) indicates to solve angle information of each joint of the robotic arm at a

a corresponding time from a known end position. Due to the nonlinearity of the function f(0 in equation (1), it is difficult to obtain an analytical solution of equation (1). Therefore, in order to linearize equation (1), the problem is considered on the velocity layer. Derivatives of both sides of equation (1) with respect to time are solved as,

JE E , (2)

[00521 where JE E Rx" is the Jacobian matrix of the robotic arm, and 0 E R' and

fE E R' are derivatives of the joint angle and the end trajectory of the robotic arm with

respect to time, respectively.

[0053] Almost all robotic arms have physical limits of joints. If the algorithm does not take into account avoidance of the physical limits of joints, the control amount generated by

the algorithm will easily exceed the actual joint limit of the robotic arm. The idea of a general

physical avoidance algorithm of joints is to rewrite the physical limits of joints of the robotic

arm into an inequality double-ended constraint:

[0054] where 0 E R" represents angles of the joints of the robotic arm, and 0- and

0+ represents upper and lower physical limits of the angles of the joints the robotic arm

respectively. 0- and 0+ represent upper and lower physical limits of the joint angular

velocity of the robotic arm respectively. Considering that the inverse kinematics problem of

the robot herein is solved at the velocity layer, the above equation is transformed into a

double-ended constraint on the velocity layer:

1in

[0055] where ( and (+are defined as:

_ _ b-, if 0 C[U1,0*] 7716-, if 0 E [0-,ZDF]

+ *, if 0 C [0-z 1 ]

[00561 where:

71 = 1 - (sin(f7r(sin(C(0 - wu1 )/(0 - -u,))) 2 ))2

2= 1 - (sin(fin(sin(fl(0 - w 2 )/(0+ - u 2 )))2)) , zi= a 1 0

a2 0+ %u=

[0057] # is a positive number in an interval [0, 1]. In order to simplify the calculation,

the two-sided inequality constraint is transformed into a one-sided inequality constraint, namely:

Ki9<e

[00581 where K = [E, -E] R 2 nxne= + - r-]T R 2 , and the quadratic

programming problem of the robotic arm constrained by the physical limit in this case is:

mi. r . T 2 (3)

s.t. E E(4)

K e(5)

[0059] where b is designed as b = o((t) - 0(0)), and the parameter & is a

response coefficient of joint offset. The repetitive motion of the robotic arm is shown in FIG. 2(b). When the end of the robotic arm completes a periodic motion (represented by a butterfly

curve here), and when the robotic arm performs repetitive motion (i.e., the parameter o is

not zero, here let o = 6), after the robotic arm completes a periodic motion, an end position

of each joint is consistent with an initial position, as shown in FIG. 2(b). The actual joint

angle of the robotic arm is constrained as shown in FIG. 2(e), where a joint 03 is constrained

the most. An angular velocity O of a third joint is shown in FIG. 2(f).

[00601 In order to solve the above time-varying quadratic programming problems (3)-(5), an exponential penalty function is designed. Its function is to transform the time-varying quadratic programming problem with an inequality constraint and an equality constraint into a time-varying quadratic programming problem only with an equality constraint, and the inequality constraint is equivalent to a penalty term in an objective function (performance

index, a function following min.). The expression of the exponential penalty function P(t)

is: 2n

P(t) = p e-9 -K

[0061] where U> 0, p is a positive number close to 0, and ej and Ki are the i

-dimension row vector of vector e and matrix K respectively. After using the penalty

function P(t), the time-varying quadratic programming problems (3)-(5) may be

transformed into:

. 2 + b T + P(t) m2 (6)

S. t. JE# = (7)

[0062] The above quadratic programming problem may be transformed into a matrix equation by a Lagrangian multiplier method. A Lagrangian function is constructed as follows,

L(,A,t)= -+b# +P(t)+A(E (E)

[00631 where A is a Lagrange multiplier. A partial derivative of equation (8) is solved as,

fL(OAt) = + b+ TA +p1(e-iNWt - K1) = 0

-B - = JEO TE'.

[00641 The equation system (9) may be expressed as the following time-varying matrix equation,

B(t)y(t)=G(t) ( 10)

[00651 where

B(t) = E JE E R (n+m) X(n+) , y(t)= ERn+m, JE OkXk IA

G(t) = -b -pa e-"(K") - K(t)) E Rn+m

rE

[0066] In order to solve the matrix equation (10), an error function is defined as:

E it) = B (t)-yft) - G (t) , (11)

[0067] Through a neural dynamics method, errors are designed to converge to zero in the

following way,

du__ = -Y(Ef(t)), (12)

1'

[0068] where y is a parameter that adjusts a convergence rate, and 4(-) is an

activation function. Equation (11) is substituted into equation (12), and similar terms are

merged to obtain a fixed-parameter neural network solver with a penalty function, that is,

R(t)f(t)= -T(t)y(t) - y(B(t)y(t) - G(t)) + V(t) , (13)

[00691 where

R(t) [1 -KTKK) 0i] IE ?nxml 2n 2 -a(ev -K 0) - KT T (t). - a eK K)

! V(t) = b ia=1ie(ii) Ik' E T' ~

[0070] From equation (13), an optimal solution y* of the matrix equation (10) may be

solved. A first n term thereof is an optimal solution #* of quadratic programming (3)-(5).

$* may be integrated to obtain an optimal solution O*of the redundant joint angle of the

robotic arm. FIG. 2(c) shows a motion trajectory diagram of an end effector of the robotic arm

when performing a butterfly trajectory tracking task. It can be seen that an actual path of the

end effector can well track a expected path (the preset target end trajectory of the robotic arm).

FIG. 2(d) shows a position error of the robotic arm in tracking the target trajectory. From FIG.

2(d), it can be seen that the position error always remains at a level around 10-7m, which

basically meets the requirements of most practical applications.

[00711 The above specific implementations are preferred embodiments of the present

invention and do not limit the present invention. Any other changes or other equivalent replacement methods that do not deviate from the technical solutions of the present invention are included in the scope of protection of the present invention.

Claims

What is claimed is: 1. A robotic arm motion programming method based on a fixed-parameter neural network, comprising: Si, establishing a model of a robotic arm according to structural parameters of the robotic arm to obtain a Jacobian matrix of an end effector, and establishing an inverse kinematics equation of the robotic arm according to the Jacobian matrix and a preset target trajectory of an end of the robotic arm; S2, establishing a physical limit double-ended inequality constraint of the robotic arm on a velocity layer according to actual physical limit constraint parameters of joints of the robotic arm; S3, formulating the inverse kinematics equation and the physical limit double-ended inequality constraint as a time-varying quadratic programming problem, and solving a motion programming problem of the robotic arm constrained by physical limits by using a quadratic programming scheme, wherein a repetitive motion index is used as a performance index for the time-varying quadratic programming problem; S4, designing the fixed-parameter neural network with a penalty function to solve the time-varying quadratic programming problem; and S5, transferring angle information of the robotic arm as solved to a lower computer controller of the robotic arm to drive the robotic arm to move to complete a target trajectory tracking task.
2. The robotic arm motion programming method based on a fixed-parameter neural network according to claim 1, wherein the physical limit constraint parameters of the joints in step S2 comprise: a joint angle constraint and an angular velocity constraint; and the physical limit double-ended inequality constraint of the robotic arm established on the velocity layer is:

where E R nand (+ E Rn are defined as:

_ 0-, if 6 E [ui, 0+] 1116-, if E [0 -, Z1 ] = + if 0 [0 -, 2 2] 1726+, if6 E [Z2, 0+1

where 0 E Rn represents angles of the joints of the robotic arm, 6- and 6+

represent upper and lower physical limits of the angles of the joints the robotic arm

respectively; E R represents angular velocities of the joints of the robotic arm, and O and $+ represent upper and lower physical limits of the angular velocities of the joints of

the robotic arm respectively;

71 = 1 - (sin (fl(sin (#3(6- xi)/(6- - i)))2))2

q2 = 1 - (sin (fl(sin (fl(6 - -o2 )/(0+ _ 2)))2)2

where f is a positive number in an interval of [0, 1].
3. The robotic arm motion programming method based on a fixed-parameter neural network according to claim 2, wherein step S3 comprises:

converting the physical limit double-ended inequality constraint (- ( into a

physical limit one-sided inequality constraint; and formulating the inverse kinematics equation and the physical limit one-sided inequality constraint as the time-varying quadratic programming problem.
4. The robotic arm motion programming method based on a fixed-parameter neural network according to claim 3, wherein the physical limit one-sided inequality constraint is:

K <e ,

where K = [E, -E]T E R 2 nne + _ - r R 2 n, where E represents an

identity matrix of n X n; and

a control scheme of formulating the inverse kinematics equation and the physical limit

1'7 one-sided inequality constraint as the time-varying quadratic programming problem is: *T m . 2 +b S. t. JE$ = E

Kb ; e

where b = o)(6(t) - 6(0)), where the parameter to is a response coefficient of

joint offset.
5. The robotic arm motion programming method based on a fixed-parameter neural

network according to claim 4, wherein the penalty function is an exponential penalty function

P(t), and an expression of the exponential penalty function P(t) is:

2n

P(t) =p e-"(ei-Kib) i=1

where o > 0, p is a positive number close to 0, ei and Ki are i-dimension row

vectors of vector e and matrix K respectively.
6. The robotic arm motion programming method based on a fixed-parameter neural

network according to claim 5, wherein step S4 comprises:

S41, converting an inequality constraint term in the control scheme of the time-varying

quadratic programming problem into a penalty term in the performance index by using the

exponential penalty function P(t), namely:

.T TT 66 0 $ T min. 2 +b + P(t) S.t. ]E = E

S42, converting the quadratic programming problem into a matrix equation using a

Lagrangian multiplier method:

B (t)y (t)= G(t)

1Q where specific content of each symbol is as follows:

B(t) = [E E IE R(n+m)x(n+m), y(t)= e Rn+n,

- 2n

G(t) [b - Pa e-u(ei-Kib) • Kbj) E Rn+m

rE

where A is a Lagrangian operator, and an error function is defined as

Eft)= B(t)yt)- G(t) E R+*"

in order to make an error converge to zero, the following neural dynamics criterion is

adopted:

dE(t)/dt = -Ylf(E(t)

where (): R` - g+m is an activation function, and V(-) must be a

monotonically increasing odd function; and the neural dynamics criterion, and similar terms

are combined to obtain the fixed-parameter neural network with a penalty strategy:

R(t)p(t) = -T(t)y(t) - y(B(t)y(t) - G(t)) +V(t),

where

R (t) [E±Pr i ~ i~ KTKE O~n] ~E + 2 _"e-("l) - KK )

IE O MXM R(t) Pa (kT 2n -Te-j)IV

15 VT(t). - O2 1f- r I 7j

then a first n element of a solution of the fixed-parameter neural network is an optimal solution #*which is integrated to obtain 0*, and * is an optimal joint angle of the robotic arm.
7. The robotic arm motion programming method based on a fixed-parameter neural

network according to claim 1, wherein the structural parameters comprise a length of a

connecting rod and a number of degrees of freedom.

In