An optimal control method for robot trajectory tracking based on all-wheel drive system

Abstract

The present invention discloses an optimal control method for trajectory tracking of a robot arm based on a full-drive system method, which mainly solves the problem that the existing trajectory tracking control method cannot handle the physical constraints of the robot arm. The method comprises the following steps: S1, establishing a complex dynamic model of an uncertain robot arm; S2, designing an m-order high-order disturbance observer; S3, using the high-order disturbance observer to design a composite trajectory tracking controller based on the full-drive system method; S4, proposing an optimal control problem based on minimizing tracking errors and multiple types of physical constraints; S5, obtaining the optimal control method for trajectory tracking of the robot arm. In the controller design process of the present invention, a high-order disturbance observer is constructed to handle friction, uncertain dynamics and external disturbances; a gradient-based optimization parameter setting method is adopted to provide a reasonable criterion for the selection of control gain. In addition, under the premise of selecting the optimal control gain, the joint angle, joint angular velocity and control input torque constraints are met.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot arm control, and in particular, relates to a robot arm trajectory tracking optimal control method based on an all-wheel drive system method. Background Art

[0002] In recent years, robotic arms have been increasingly used in the manufacturing industry, especially in complex working environments, to effectively perform repetitive or dangerous tasks. However, during the operation of robotic arms, they will face challenges such as unmodeled dynamics, friction, external disturbances, and physical constraints. These factors significantly affect the effect of trajectory tracking control, so improving the trajectory tracking performance of robotic arms has received widespread attention.

[0003] Although many research results are devoted to improving tracking accuracy and safety, these methods are often accompanied by increased controller complexity and reduced system reliability. Therefore, the current research trend is to develop a simple and effective control strategy to achieve high-precision trajectory tracking of the robot arm. There are still many scientific problems to be solved in this field, which provides a broad space for exploration for future research.

[0004] Existing trajectory tracking control methods for uncertain manipulators mainly include robust control, adaptive control, sliding mode control, and anti-disturbance control. However, most research methods require the original manipulator model to be converted into a state-space model, and have the following limitations: (1) Accurate information of the manipulator's dynamic model is usually required, such as the center of mass position and inertia tensor; (2) The first-order state-space model is generally used to describe the manipulator's dynamic model, resulting in the loss of the physical meaning of the original system during the process of model reduction; (3) For complex nonlinear systems, it is very difficult to solve the feasible solution of the nonlinear differential equation, and thus it is difficult to give a global stabilization control law.

[0005] The existing trajectory tracking control method of uncertain manipulator has the following technical problems:

[0006] (1) Existing trajectory tracking control methods cannot handle the physical constraints of the robot arm;

[0007] (2) The robot is affected by friction, uncertain dynamics, and external disturbances during its motion, which significantly reduces its tracking performance. However, existing trajectory tracking methods cannot effectively handle these problems.

[0008] (3) The control gain of the robot trajectory tracking control law is usually selected based on engineering experience. Therefore, the control gain selection is difficult and requires a lot of adjustment time to achieve high accuracy, which affects the practical application. Summary of the invention

[0009] The purpose of the present invention is to provide an optimal control method for robot arm trajectory tracking based on an all-wheel drive system method, mainly to solve the problem that the existing trajectory tracking control method cannot handle the physical constraints of the robot arm.

[0010] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0011] A robot arm trajectory tracking optimal control method based on an all-wheel drive system method comprises the following steps:

[0012] S1, considering the uncertain dynamics, friction, external disturbances and multiple types of physical constraints of the manipulator in actual engineering, establish a complex dynamic model of the uncertain manipulator;

[0013] S2, design an m-th order high-order disturbance observer for the complex dynamics model of uncertain manipulator;

[0014] S3, considering the complex dynamics model of the uncertain manipulator, designs a composite trajectory tracking controller based on the full-drive system method according to the structure of the high-order disturbance observer;

[0015] S4, proposes an optimal control problem based on minimizing tracking error and multiple types of physical constraints;

[0016] S5, solving the optimal control problem based on the parameter adjustment algorithm to obtain the optimal control method for robot arm trajectory tracking.

[0017] Furthermore, in the present invention, the multiple types of physical constraints include joint angle limit constraints, joint angular velocity limit constraints and control input torque limit constraints.

[0018] Furthermore, in the present invention, the mathematical expression of the complex dynamics model of the uncertain manipulator is as follows:

[0019]

[0020] in and are the joint angle, joint angular velocity and joint angular acceleration respectively, To control the input torque, is the friction force, is an external disturbance; and They represent the nominal inertial parameter matrix, the centrifugal and Coriolis force parameter matrix, and the gravity parameter matrix respectively; and They are the uncertain inertia parameter matrix, the uncertain centrifugal force and Coriolis force parameter matrix, and the uncertain gravity parameter matrix; among them, the joint angle, joint angular velocity, and control input torque should satisfy:

[0021] q min ≤q i ≤q max ,i=1,2,...,n, (2)

[0022]

[0023] τ min ≤τ i ≤τ max ,i=1,2,...,n, (4)

[0024] where q min , τ min and q max , τ max They represent the lower and upper limits of the joint angle, joint angular velocity and control input torque respectively; then formula (1) can be rewritten as:

[0025]

[0026] in represents the total unmodeled dynamics, parameter uncertainties, and external disturbances; assuming that τ d is m-th order differentiable, that is Where ρ is a positive constant.

[0027] Furthermore, in step S2, in order to facilitate the design of the high-order disturbance observer and the composite trajectory tracking controller, it is defined as:

[0028]

[0029] in, express The estimated value of is the estimation error. In the case of i=1, is τ d An estimated value of

[0030] For the uncertain complex dynamics model of the manipulator in formula (1), the expression of the m-order high-order disturbance observer is as follows:

[0031]

[0032] in, is the selected parameter matrix, and p i ,i=1,2,...,m is an auxiliary vector.

[0033] Furthermore, in step S3, the design steps of the composite trajectory tracking controller are as follows:

[0034] S31, establish the subsystem for estimating errors:

[0035] Taking the time derivative of equation (7) and substituting it into equation (5) and equation (8), we get:

[0036]

[0037] Substituting formula (11) into formula (6), we can get:

[0038]

[0039] According to equations (9) and (10), the following subsystem of estimation error is obtained:

[0040]

[0041] S32, using the error subsystem, we get the estimated error system:

[0042] First define:

[0043]

[0044] Then, using equations (12) and (13), the estimated error system is:

[0045]

[0046] in Select the adaptive parameter matrix L i ,i=1,2,...,m, to ensure that the matrix L e The eigenvalues ​​of are all negative;

[0047] like satisfy Where μ is a positive constant, then there exists a positive matrix satisfy Defining the Lyapunov function Then the time derivative is:

[0048]

[0049] Therefore, based on the comparison theorem, we can get:

[0050]

[0051] Select L i ,i=1,2,...,m to ensure μ τ >λ max (P 1 ),but

[0052]

[0053] Therefore, the estimation error is ultimately bounded and will eventually converge to a bounded region near the origin;

[0054] S33, according to the high-order disturbance observer defined by equations (7) to (10), a composite trajectory tracking controller based on the full drive system method is designed:

[0055] Defining performance metrics but:

[0056]

[0057] Define the joint angle tracking error e = qq r , where q r is the desired trajectory of the joint angle; then the mathematical expression of the composite trajectory tracking controller is as follows:

[0058]

[0059] Among them, the control gain K is set 0 and K 1 To ensure the stability of the closed-loop system, and at the same time make the tracking error arbitrarily small by properly adjusting the control gain;

[0060] Substituting equation (18) into equation (5), we can obtain

[0061]

[0062] in:

[0063] Furthermore, in step S33, the control gain matrix K 0 and K 1 The calculation process of the parameter expression is as follows:

[0064] First, we introduce the following lemma: For any selected matrix All matrices K 0 , K 1 and a non-singular matrix satisfy

[0065] A(K 0～1 )=VFV -1 (20)

[0066] detV≠0 (21)

[0067] Thus it is expressed as:

[0068] [K 0 K 1 ]=-WF 2 V -1 ,V=[WT (WF) T ] T (twenty two)

[0069] in is a matrix that ensures that V satisfies (21). In particular, the parameter matrix is ​​chosen as follows:

[0070] W=[W 1 W 2 ],F=diag(F 1 ,F 2 )(twenty three)

[0071] W 1 =diag(w 1 ,w 2 ,...,w n )

[0072] W 2 =diag(w n+1 ,w n+2 ,...,w 2n )

[0073] F 1 =diag(σ 1 ,σ 2 ,...,σ n )

[0074] F 2 =diag(σ n+1 ,σ n+2 ,...,σ 2n )

[0075] Then the control gain K 0 and K 1 It can be written as:

[0076]

[0077] Therefore, the control gain K 0 and K 1 The closed-loop system eigenvalue σ 1 ,σ 2 ,...,σ 2n Determine, which satisfies σ 1 ,σ 2 ,...,σ 2n <0; define the decision parameter vector σ=[σ 1 ,σ 2 ,...,σ 2n ] T .

[0078] Furthermore, the specific process in step S4 is as follows:

[0079] S41, set the optimal control gain K 0 and K 1 :

[0080] First, remove the uncertain inertia parameter matrix ΔM(q), uncertain centrifugal force and Coriolis force parameter matrix in equation (1): The uncertain gravity parameter matrix ΔG(q) and the external disturbance d, equation (19) is abbreviated as:

[0081]

[0082] S42, based on minimizing the tracking error and satisfying the physical constraints, the optimal control problem is proposed as follows:

[0083]

[0084] st(18),(21),(22),(25)

[0085] q min ≤q j ≤q max ,j=1,2,...,n,(27)

[0086]

[0087] τ min ≤τ j ≤τ max ,j=1,2,...,n, (29)

[0088] By optimizing σ, the optimal performance index J(σ) is obtained while satisfying constraints (27)-(29);

[0089] Constraints (27)-(29) can be written as the following one-sided inequalities:

[0090]

[0091] where q j represents the jth joint angle, g i represents the i-th constraint of a one-sided inequality; based on the constraint approximation method, g i ≥0,i=1,2,...,6n is approximated as:

[0092]

[0093] in

[0094]

[0095] in and is a positive number, so the optimal problems (26)-(29) can be transformed into:

[0096]

[0097] st(18),(21),(22),(25)

[0098] G i ≥0,i=1,2,...,6n (33)

[0099] The optimization problem can be called problem According to nonlinear programming theory, the problem The solvability of depends on the existence of the gradients of the objective function and the constraints; therefore, the cost function J and the constraints G i ,i=1,2,...,6n The gradient of the decision parameter σ satisfies:

[0100]

[0101] in And 0 ,λ i satisfy:

[0102]

[0103] in,

[0104] Furthermore, the adjustment algorithm in step S5 is as follows:

[0105] Among them, the input is σ 0 , the output is σ * ;

[0106] initialization: σ=σ 0 ;

[0107] Step 1. If Then jump to step 3;

[0108] Step 2. Otherwise stop and output σ * =σ;

[0109] Step 3. Use the initial value σ and output Solving the problem

[0110] Step 4. If Then jump to step 6;

[0111] Step 5. Otherwise, jump to step 7;

[0112] Step 6. Setup And jump to step 1;

[0113] Step 7. Setup And jump to step 3.

[0114] Compared with the prior art, the present invention has the following beneficial effects:

[0115] (1) The present invention is based on the full-drive system method. In the modeling process, there is no need to convert the original system into a first-order state space model, thereby maintaining the physical meaning of the variables and system matrices and simplifying the stability analysis and controller design.

[0116] (2) Compared with the traditional all-wheel drive system method, the present invention adopts a high-order disturbance observer in the controller design process to deal with model uncertainty, friction and external disturbances, thereby improving the disturbance estimation accuracy and tracking accuracy.

[0117] (3) The gradient-based optimal parameter setting method proposed in the present invention provides the optimal control gain while satisfying the limitations of joint angle, joint angular velocity and control input torque, thereby effectively improving the performance of the control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0118] Figure 1 Tracking curves for the joint angles of the robot arm.

[0119] Figure 2 It is the joint angular velocity tracking curve of the robot arm.

[0120] Figure 3 It is the joint angle tracking error curve of the robot arm.

[0121] Figure 4 Input torque curve for the control of the robot arm. DETAILED DESCRIPTION

[0122] The present invention is further described below in conjunction with the accompanying drawings and embodiments. The embodiments of the present invention include but are not limited to the following embodiments.

[0123] Example

[0124] The present invention discloses an optimal control method for robot trajectory tracking based on an all-drive system method, and the model considered is a robot model with model uncertainty, friction and external disturbance. Considering that the robot arm in actual engineering is affected by uncertain dynamics, friction, external disturbance and various types of physical constraints (joint angle limit, joint angular velocity limit and control input torque limit), the complex dynamic model of the uncertain robot arm is specifically in the following form:

[0125]

[0126] in and are the joint angle, joint angular velocity and joint angular acceleration respectively, To control the input torque, is the friction force, is an external disturbance; and They represent the nominal inertial parameter matrix, the centrifugal and Coriolis force parameter matrix, and the gravity parameter matrix respectively; and They are the uncertain inertia parameter matrix, the uncertain centrifugal force and Coriolis force parameter matrix, and the uncertain gravity parameter matrix; among them, the joint angle, joint angular velocity, and control input torque should satisfy:

[0127] q min ≤q i ≤q max ,i=1,2,...,n, (38)

[0128]

[0129] τ min ≤τ i ≤τ max ,i=1,2,...,n, (40)

[0130] where q min , τ min and q max , τ max They represent the lower and upper limits of the joint angle, joint angular velocity and control input torque respectively. Further, the robot model (37) can be written as:

[0131]

[0132] in represents the total unmodeled dynamics, parameter uncertainties, and external disturbances. Assume that τ d is m-th order differentiable, that is Where ρ is a positive constant.

[0133] In order to facilitate the design of high-order disturbance observers and composite controllers, it is defined as follows:

[0134]

[0135] in, express The estimated value of is the estimation error. In the case of i=1, is τd The estimated value of .

[0136] For the robotic arm system (37), an m-order high-order disturbance observer is designed as follows:

[0137]

[0138]

[0139] in is the selected parameter matrix, and p i ,i=1,2,...,m is an auxiliary vector.

[0140] Taking the time derivative of (43) and substituting it into (41) and (44), we can obtain:

[0141]

[0142] In addition, substituting (47) into (42) yields:

[0143]

[0144] According to (45) and (46), the following subsystem of estimation error is obtained:

[0145]

[0146] definition:

[0147]

[0148] Then using (48) and (49), the estimated error system is:

[0149]

[0150] in Therefore, the parameter matrix L can be appropriately selected i ,i=1,2,...,m,to ensure L e The eigenvalues ​​of are all negative.

[0151] like satisfy Where μ is a positive constant, then there exists a positive matrix satisfy Defining the Lyapunov function Then the time derivative is:

[0152]

[0153] Therefore, based on the comparison theorem, we can get:

[0154]

[0155] If you can choose L i ,i=1,2,...,m to ensure μ τ >λ max (P 1 ),but

[0156]

[0157] Therefore, the estimation error is ultimately bounded and will eventually converge to a bounded region near the origin. Define performance metrics So:

[0158]

[0159] Define the joint angle tracking error e = qq r , where q r is the desired trajectory of the joint angle. Then the composite trajectory tracking control scheme based on the all-wheel drive system method can be designed as follows.

[0160] Considering the uncertain manipulator system (37), using the high-order disturbance observer (43)-(46), the following composite trajectory tracking controller based on the full-drive system method is designed:

[0161]

[0162] The control gain K 0 and K 1 It needs to be reasonably selected to ensure the stability of the closed-loop system. At the same time, by properly adjusting the control gain, the tracking error can be made arbitrarily small.

[0163] The tracking error is arbitrarily small. The proof is as follows: Substituting controller (54) into (41) yields:

[0164]

[0165] in:

[0166] For the tracking error system (19), define the Lyapunov function V(ψ) = ψ T P e ψ, then the time derivative is:

[0167]

[0168] in express The upper bound of 0 and K 1 To ensure According to the comparison theorem, we can conclude that:

[0169]

[0170] If μ e >λ max (P e ) can be obtained:

[0171]

[0172] because The trajectory tracking error is bounded, so it can be made arbitrarily small by properly adjusting the control gain.

[0173] Next, the control gain matrix K is given 0 and K 1 The parameter expression of , first introduce the following lemma: for any selected matrix All matrices K 0 , K 1 and a non-singular matrix satisfy

[0174] A(K 0～1 )=VFV -1 (56)

[0175] detV≠0 (57)

[0176] Thus it is expressed as:

[0177] [K 0 K 1 ]=-WF 2 V -1 ,V=[W T (WF) T ] T (58)

[0178] in is a matrix that ensures that V satisfies (57). In particular, the parameter matrix is ​​chosen as follows:

[0179] W=[W 1 W 2 ],F=diag(F 1 ,F 2 ) (59)

[0180] W 1 =diag(w 1 ,w 2 ,...,w n )

[0181] W 2 =diag(wn+1 ,w n+2 ,...,w 2n )

[0182] F 1 =diag(σ 1 ,σ 2 ,...,σ n )

[0183] F 2 =diag(σ n+1 ,σ n+2 ,...,σ 2n )

[0184] Then the control gain K 0 and K 1 It can be written as:

[0185]

[0186] Therefore, the control gain K 0 and K 1 The closed-loop system eigenvalue σ 1 ,σ 2 ,...,σ 2n Determine, which satisfies σ 1 ,σ 2 ,...,σ 2n <0. Define the decision parameter vector σ=[σ 1 ,σ 2 ,...,σ 2n ] T .

[0187] In order to select the optimal control gain K 0 and K 1 , while making the joint angles, joint angular velocities and input torques satisfy the constraints, an optimal parameter tuning method will be developed below.

[0188] Since it is difficult to find a set of optimized control gains to satisfy all uncertain scenarios, in addition, due to the existence of high-order disturbance observers, uncertainties and external disturbances can be compensated under the action of disturbance observers. Therefore, removing the uncertainty ΔM(q) in system (37), ΔG(q) and external disturbance d, the system (55) can be simplified as:

[0189]

[0190] In order to minimize the tracking error and satisfy the physical constraints, the optimal control problem is proposed as follows:

[0191]

[0192] st(18),(21),(22),(25)

[0193] q min ≤q j ≤q max ,j=1,2,...,n, (63)

[0194]

[0195] τ min ≤τ j ≤τ max ,j=1,2,...,n, (65)

[0196] Specifically, the optimal performance index J(σ) can be obtained by optimizing σ while satisfying constraints (63)-(65).

[0197] Then constraints (63)-(65) can be written as the following one-sided inequalities:

[0198]

[0199] where q j represents the jth joint angle, g i represents the i-th constraint of the one-sided inequality. Based on the constraint approximation method, g i ≥0,i=1,2,...,6n can be approximated as:

[0200]

[0201] in

[0202]

[0203] in and is a positive number. Therefore, the optimal problem (62)-(65) can be transformed into

[0204]

[0205] st(18),(21),(22),(25)

[0206] G i ≥0,i=1,2,...,6n (69)

[0207] The optimization problem described can be called the problem According to nonlinear programming theory, the problem The solvability of depends on the existence of the gradients of the objective function and the constraints. Therefore, the cost function J and the constraint G i,i=1,2,...,6n The gradient of the decision parameter σ satisfies:

[0208]

[0209] in And 0 ,λ i satisfy:

[0210]

[0211] in

[0212] To solve the problem Need to adjust the parameters appropriately and Therefore, the following algorithm can be proposed:

[0213] Algorithm 1: Solve the optimal control problem (62)-(62), with input σ 0 , the output is σ * .

[0214] initialization: σ=σ 0 .

[0215] Step 1. If Then skip to step 3.

[0216] Step 2. Otherwise stop and output σ * =σ.

[0217] Step 3. Use the initial value σ and output Solving the problem

[0218] Step 4. If Then skip to step 6.

[0219] Step 5. Otherwise, jump to step 7.

[0220] Step 6. Setup And jump to step 1.

[0221] Step 7. Setup And jump to step 3.

[0222] The following is a specific experiment to verify the above method:

[0223] In the experimental case, the Franka robot arm was selected to verify the control method. The specific values ​​of the Franka robot arm control parameters are shown in Table 1. The simulation environment is Matlab2019a, and the optimization parameter setting method is used to solve the optimization control problem.

[0224] Table 1 Franka robot arm control parameters

[0225]

[0226] The desired joint angle tracking trajectory is set as:

[0227]

[0228] The joint angle tracking results in the Franka robot arm experiment are as follows Figure 1 The joint angular velocity tracking results are shown in Figure 2 The experimental results show that the designed controller can ensure that the Franka robot moves accurately along the desired trajectory while satisfying the constraints of joint angles and joint angular velocities. Figure 3 and Figure 4 The joint angle tracking error and input torque are shown respectively. In addition, the experiment also ensures that the control input torque is within the constraint range. In summary, the proposed composite trajectory tracking control method based on the all-wheel drive system method can ensure that the robot arm achieves accurate trajectory tracking and meets physical constraints.

[0229] From the perspective of controller design, the composite trajectory tracking controller designed by the full drive system method has a simple structure and retains the physical meaning of variables and system matrices. In the controller design process, a high-order disturbance observer is constructed to deal with friction, uncertain dynamics and external disturbances; a gradient-based optimization parameter setting method is adopted to provide a reasonable criterion for the selection of control gain. In addition, under the premise of selecting the optimal control gain, the joint angle, joint angular velocity and control input torque constraints are met.

[0230] The above embodiment is only one of the preferred implementation modes of the present invention and should not be used to limit the protection scope of the present invention. Any changes or modifications that are made to the main design concept and spirit of the present invention and have no substantive significance, and the technical problems they solve are still consistent with the present invention, should be included in the protection scope of the present invention.

Claims

1. A robot arm trajectory tracking optimal control method based on an all-wheel drive system method, characterized in that: The following steps are involved: S1, considering the uncertain dynamics, friction, external disturbance and multiple types of physical constraints of the manipulator in actual engineering, a complex dynamic model of the uncertain manipulator is established; wherein the mathematical expression of the complex dynamic model of the uncertain manipulator is as follows: in and are the joint angle, joint angular velocity and joint angular acceleration respectively, To control the input torque, is the friction force, is an external disturbance; and They represent the nominal inertial parameter matrix, the centrifugal and Coriolis force parameter matrix, and the gravity parameter matrix respectively; They are the uncertain inertia parameter matrix, the uncertain centrifugal force and Coriolis force parameter matrix, and the uncertain gravity parameter matrix; among them, the joint angle, joint angular velocity, and control input torque should satisfy: q min ≤q i ≤q max ,i=1,2,...,n, (2) t min ≤τ i ≤τ max ,i=1,2,...,n, (4) where q min , and q max , τ max They represent the lower and upper limits of the joint angle, joint angular velocity and control input torque respectively; then formula (1) can be rewritten as: in represents the total unmodeled dynamics, parameter uncertainties, and external disturbances; assuming that τ d is m-th order differentiable, that is Where ρ is a positive constant; S2, design an m-order high-order disturbance observer for the complex dynamics model of the uncertain manipulator; in order to facilitate the design of the high-order disturbance observer and the composite trajectory tracking controller, define: in, express The estimated value of is the estimation error. When i=1, is τ d An estimated value of For the uncertain complex dynamics model of the manipulator in formula (1), the expression of the m-order high-order disturbance observer is as follows: in, i=1,2,...,m is the selected parameter matrix, and p i ,i=1,2,...,m is the auxiliary vector; S3, considering the complex dynamics model of the uncertain manipulator, a composite trajectory tracking controller based on the full-drive system method is designed according to the structure of the high-order disturbance observer; S4, proposes an optimal control problem based on minimizing tracking error and multiple types of physical constraints; S5, solving the optimal control problem based on the parameter adjustment algorithm to obtain the optimal control method for robot arm trajectory tracking.

2. The optimal control method for robot trajectory tracking based on the all-wheel drive system method according to claim 1, characterized in that: Multiple types of physical constraints include joint angle limit constraints, joint angular velocity limit constraints and control input torque limit constraints.

3. The optimal control method for robot trajectory tracking based on the all-wheel drive system method according to claim 2 is characterized in that: In step S3, the design steps of the composite trajectory tracking controller are as follows: S31, establish the subsystem for estimating errors: Taking the time derivative of equation (7) and substituting it into equation (5) and equation (8), we get: Substituting formula (11) into formula (6), we can get: According to equations (9) and (10), the following subsystem of estimation error is obtained: S32, using the error subsystem, we get the estimated error system: First define: Then, using equations (12) and (13), the estimated error system is: in Select the adaptive parameter matrix L i ,i=1,2,...,m, to ensure that the matrix L e The eigenvalues ​​of are all negative; like satisfy Where μ is a positive constant, then there exists a positive matrix satisfy Defining the Lyapunov function Then the time derivative is: Therefore, based on the comparison theorem, we can get: Select L i ,i=1,2,...,m to ensure μ τ >λ max (P1), then Therefore, the estimation error is ultimately bounded and will eventually converge to a bounded region near the origin; S33, using the high-order disturbance observer defined by equations (7) to (10), a composite trajectory tracking controller based on the full-drive system method is designed: Defining performance metrics but: Define the joint angle tracking error e = qq r , where q r is the desired trajectory of the joint angle; then the mathematical expression of the composite trajectory tracking controller is as follows: Among them, the control gain matrices K0 and K1 are set to ensure the stability of the closed-loop system, and the tracking error is made arbitrarily small by appropriately adjusting the control gain; Substituting formula (18) into formula (5), we can obtain: in:

4. The optimal control method for robot arm trajectory tracking based on the all-wheel drive system method according to claim 3 is characterized in that: In step S33, the calculation process of the parameter expressions of the control gain matrices K0 and K1 is as follows: First, we introduce the following lemma: For any selected matrix All matrices K0, K1 and non-singular matrices satisfy A(K 0～1 )=VFV -1 (20) detV≠0 (21) Thus it is expressed as: [K0 K1]=-WF 2 V -1 ,V=[W T (WF) T ] T (22) in is a matrix that ensures that V satisfies (21). In particular, the parameter matrix is ​​selected as follows: W=[W1 W2],F=diag(F1,F2) (23) W1=diag(w1,w2,...,w n ) W2=diag(in n+1 ,In n+2 ,...,In 2n ) F1=diag(σ1,σ2,...,σ n ) F2=diag(σ n+1 ,s n+2 ,...,s 2n ) Then the control gain matrices K0 and K1 can be written as: Therefore, the control gain matrices K0 and K1 are composed of the closed-loop system eigenvalues ​​σ1, σ2, ..., σ 2n Determine, which satisfies σ1,σ2,...,σ 2n <0; define the decision parameter vector σ=[σ1,σ2,...,σ 2n ] T .

5. The optimal control method for robot trajectory tracking based on the all-wheel drive system method according to claim 4 is characterized in that: The specific process in step S4 is as follows: S41, set the optimal control gain matrices K0 and K1: First, remove the uncertain inertia parameter matrix ΔM(q), uncertain centrifugal force and Coriolis force parameter matrix in equation (1): The uncertain gravity parameter matrix ΔG(q) and the external disturbance d, equation (19) is abbreviated as: S42, based on minimizing the tracking error and satisfying the physical constraints, the optimal control problem is proposed as follows: st(18),(21),(22),(25) q min ≤q j ≤q max ,j=1,2,...,n, (27) t min ≤τ j ≤τ max ,j=1,2,...,n, (29) By optimizing σ, the optimal performance index J(σ) is obtained while satisfying constraints (27)-(29); Constraints (27)-(29) can be written as the following one-sided inequalities: where q j represents the jth joint angle, g i represents the i-th constraint of a one-sided inequality; based on the constraint approximation method, g i ≥0,i=1,2,...,6n is approximated as: in in and is a positive number, so the optimal control problems (26)-(29) can be transformed into: st(18),(21),(22),(25) G i ≥0,i=1,2,...,6n (33) The optimal control problem can be called problem According to nonlinear programming theory, the problem The solvability of depends on the existence of the gradients of the objective function and the constraints; therefore, the cost function J and the constraints G i ,i=1,2,...,6n The gradient of the decision parameter σ satisfies: in And λ0,λ i satisfy: in, 6. The optimal control method for robot trajectory tracking based on the all-wheel drive system method according to claim 5, characterized in that: The adjustment algorithm in step S5 is as follows: Among them, the input is σ0 and the output is σ * ; initialization: σ=σ0; Step 1. If Then jump to step 3; Step 2. Otherwise stop and output σ * =σ; Step 3. Use the initial value σ and output Solving the problem Step 4. If Then jump to step 6; Step 5. Otherwise, jump to step 7; Step 6. Setup And jump to step 1; Step 7. Setup And jump to step 3.

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