A sliding mode control method for tracking a predefined time trajectory of a robotic arm

Abstract

This invention discloses a sliding mode control method for predefined time trajectory tracking of a robotic arm, belonging to the field of robot industrial assembly. First, the desired trajectory information of the robotic arm is obtained; an n-DOF robotic arm dynamic model is established and its form is transformed to obtain the transformed n-DOF robotic arm dynamic model; a predefined time extended state observer is designed to obtain estimated disturbance and actual velocity information; a non-singular predefined time sliding surface is designed; and a predefined time sliding controller is designed based on the sliding surface and the obtained estimated disturbance and velocity information, resulting in a predefined time sliding control law that achieves trajectory tracking control of the robotic arm within a predefined time. This invention can effectively achieve accurate trajectory tracking of the robotic arm and meets the requirement of predefined time convergence of the robotic arm system.

Description

Technical Field

[0001] This invention belongs to the field of robot industrial assembly technology, specifically relating to a sliding mode control method for tracking a predefined time trajectory of a robotic arm. Background Technology

[0002] Robotic arms are complex systems characterized by high precision, multiple inputs and outputs, strong coupling, and the presence of parameter perturbations, external disturbances, and modeling uncertainties. Practical engineering problems such as industrial machinery assembly, spacecraft maintenance, space exploration, military disaster relief, and domestic services all require robotic arm trajectory tracking control. Therefore, research on robotic arm trajectory tracking control with external disturbances and model uncertainties has significant practical value.

[0003] For trajectory tracking control of robotic arms with external disturbances and model uncertainties, it is necessary to compensate for the unmodeled dynamics of the robotic arm system and external environmental disturbances. Currently, most sliding mode control schemes use disturbance observers to estimate and compensate for the combined disturbances formed by external environmental disturbances and unmodeled system dynamics. However, the design of disturbance observers requires measured velocity information. The higher the accuracy of the required velocity information measurement, the higher the cost of the corresponding velocity sensor. Moreover, velocity sensors are inevitably affected by measurement noise in industrial operating environments, which leads to a decrease in system control accuracy and system stability. If the velocity sensor is damaged during actual operation, it may even cause system divergence. In addition, the existing fixed-time sliding mode control methods have the problem that the system convergence time function expression is not directly related to the system parameters, and the convergence time does not appear as an independently adjustable parameter in the system structure. For some systems that require fast stability and have clear time constraints, this control method is not widely applicable. Therefore, designing a predefined-time sliding mode controller based on a predefined-time extended state observer to improve system stability and trajectory tracking performance is a very effective approach.

[0004] Currently, there are no reports in domestic and international literature and patent documents on applying the predefined time trajectory tracking sliding mode control method to the trajectory tracking of uncertain robotic arms. Summary of the Invention

[0005] To address the aforementioned problems in existing technologies, this invention proposes a sliding mode control method for tracking a predefined time trajectory of a robotic arm. This method is rationally designed, overcomes the shortcomings of existing technologies, and achieves excellent results.

[0006] A sliding mode control method for tracking a predefined time trajectory of a robotic arm includes the following steps:

[0007] S1. Obtain the desired trajectory information of the robotic arm, including the desired angular position and desired angular velocity information of each joint of the robotic arm;

[0008] S2. Establish an n-degree-of-freedom manipulator dynamic model. After giving the lumped disturbance d formed by the nominal part and the uncertain part of the manipulator dynamic model, perform a formal transformation on the manipulator dynamic model to obtain the transformed n-degree-of-freedom manipulator dynamic model.

[0009] S3. Set a predefined time-extended state observer, estimate the lumped disturbance caused by the uncertainty of the compensation system and external interference, obtain the lumped disturbance estimate z3, estimate the actual velocity information, and obtain the velocity estimate z2.

[0010] S4. Based on the joint expected angular position and expected angular velocity information obtained in S1, establish position tracking error and velocity tracking error, design a non-singular predefined time sliding surface s, and design a predefined time sliding controller based on the velocity estimation information z2 and lumped disturbance estimation value z3 obtained from the sliding surface s and S3, and obtain the predefined time sliding control law to realize the robot arm trajectory tracking.

[0011] S5. Based on S3 to S4, the predefined time sliding mode controller law based on the predefined time extended state observer is obtained.

[0012] Further, step S2 includes the following sub-steps:

[0013] S2.1 Establish the dynamic model of the n-degree-of-freedom robotic arm, as shown below:

[0014]

[0015] In the formula, q represents the joint angle position. Indicates joint angular velocity, M(q) represents the joint angular acceleration, and M(q) represents the system inertia matrix. G(q) represents centrifugal force and Coriolis force, G(q) represents gravity, and τ represents joint torque. d This indicates interference from the external environment;

[0016] S2.2 The parameters of the robotic arm's dynamic model are rewritten as nominal and uncertain parts, as follows:

[0017]

[0018] Among them, M0(q), And G0(q) is the nominal part, ΔM(q) and ΔG(q) are the uncertain parts;

[0019] The system lumped disturbance d is:

[0020]

[0021] The dynamic model of an n-degree-of-freedom robotic arm can be rewritten in the following form:

[0022]

[0023] For system (4), the corresponding state-space expression is given as follows:

[0024] Let x1 = q, but:

[0025]

[0026] make and h|≤L h h is the derivative value of the extended state x3, L h Since h is a constant, i.e., the upper bound of h, the extended state-space expression of system (4) is as follows:

[0027]

[0028] Further, step S3 includes the following sub-steps:

[0029] S3.1 First, define the following estimation error:

[0030] e i =z i -x i (7)

[0031] Among them, z i For x i The estimated value, e i To estimate the error, i = 1, 2, 3;

[0032] Design a sliding surface s of the following form. p :

[0033] s p =c1e1+e2; (8)

[0034] Where c1 is a positive constant;

[0035] S3.2, combined with sliding surface s p Design the following predefined time-extended state observer, as shown below:

[0036]

[0037] Where ε is a constant, ε=2^((2+r) / 2), T ESO Let sig(s) be the system convergence time constant. p )=|s p |sign(s pk3 and k4 are adjustable positive constants, and r is a constant, r∈(0,1);

[0038] Based on the system state-space expression (6) and the predefined time-extended state observer (9), the error observation system expression of the predefined time-extended state observer is derived as follows:

[0039]

[0040] Among them, u a To design an auxiliary control law;

[0041] Designing the auxiliary control law and differentiating equation (8) yields:

[0042]

[0043] Auxiliary control law u a The expression is:

[0044]

[0045] Where k1 and k2 are positive constants;

[0046] S3.3. Using the predefined time-extended state observer designed in S3.2, observe the synthetic disturbance and actual speed information formed by system uncertainties and external disturbances during the robotic arm trajectory tracking control process, and construct the Lyapunov function V1 as shown below:

[0047]

[0048] Based on predefined time theory and Lyapunov's theorem, it is proved that the estimation error can be controlled by the auxiliary control law u. a Under control, it converges to the sliding surface s p Above; the estimation error on the sliding surface within a predefined time T ESO The internal energy can converge to zero.

[0049] Further, step S4 includes the following sub-steps:

[0050] S4.1 Define a nonlinear function in the following form:

[0051]

[0052] Where x is the independent variable, α is an adjustable parameter, α = δ, δ is an adjustable parameter, and δ ∈ (0, e) -1 ), ε 1p It is an adjustable parameter. ε 2p It is an adjustable parameter.

[0053] Based on the desired angular position information q obtained from S1 d and expected angular velocity information The actual position estimation information obtained by S2 is defined as follows: trajectory tracking error w and velocity tracking error. As shown below:

[0054]

[0055] S4.2. Based on the nonlinear function (14), design the following predefined time sliding surface:

[0056]

[0057] Where F(w)=[f(w1),f(w2),…,f(w) j ),…,f(w n )] T w is a matrix formed by nonlinear functions. j For the tracking error of the j-th joint, γ is a constant, γ>0; T c2 This refers to the convergence time parameter during the system's approach phase.

[0058] S4.3 Design a predefined time sliding mode controller:

[0059]

[0060] Where, τ ssw For a predefined time-stable reaching law, τ eq This is the equivalent control law, where η is a constant. K is the convergence time parameter for the sliding phase. 1p For positive constants, γ > 0.

[0061] Furthermore, S5 specifically involves constructing a Lyapunov function V2 based on the predefined time-extended state observer of S3 and the predefined time sliding mode control of S4, as shown below:

[0062]

[0063] By stabilizing the Lyapunov function V2, we obtain a predefined time sliding mode control law based on a predefined time extended state observer, namely the joint torque τ, as shown below:

[0064]

[0065] The beneficial technical effects of this invention are as follows:

[0066] This invention proposes a predefined time sliding mode control method based on a predefined time extended state observer. Based on non-singular predefined time sliding mode control, it can effectively achieve precise trajectory tracking of the robotic arm and meet the predefined time convergence requirement of the robotic arm system.

[0067] The predefined time-extended state observer designed by the method of this invention treats the lumped disturbance formed by the model uncertainty of the rigid manipulator and the unknown bounded disturbances from the outside world as a new observable state of the system for accurate observation. This is achieved by designing a predefined time double power e1 for the position observation error. 1-r +e1 1+r The formal implementation of the extended state observer enables precise observation of the actual trajectory within a pre-set time. While improving the accuracy of trajectory observation, it also further improves the observation speed of velocity information and lumped disturbance state, reduces the number of adjustable parameters of the observer, and reduces the difficulty of manual parameter tuning. Attached Figure Description

[0068] Figure 1 This is a schematic diagram of a sliding mode control method for tracking a predefined time trajectory of a robotic arm in this invention.

[0069] Figure 2 This is a model diagram of a rigid robotic arm with input saturation in an embodiment of the present invention;

[0070] Figure 3 This is a trajectory tracking curve of joint 1 of a robotic arm with predefined time convergence in an embodiment of the present invention;

[0071] Figure 4 This is a trajectory tracking curve of joint 2 of a robotic arm with predefined time convergence in an embodiment of the present invention;

[0072] Figure 5 This is a graph showing the trajectory tracking error of joint 1 of a robotic arm with predefined time convergence in an embodiment of the present invention.

[0073] Figure 6 This is a graph showing the trajectory tracking error of joint 2 of a robotic arm with predefined time convergence in an embodiment of the present invention. Detailed Implementation

[0074] The specific embodiments of the present invention will be further described below with reference to specific examples:

[0075] A sliding mode control method for tracking a predefined time trajectory of a robotic arm includes the following steps:

[0076] S1. Obtain the desired trajectory information of the robotic arm, including the desired angular position and desired angular velocity information of each joint of the robotic arm;

[0077] S2. Establish an n-degree-of-freedom manipulator dynamic model. After giving the lumped disturbance d formed by the nominal part and the uncertain part of the manipulator dynamic model, perform a formal transformation on the manipulator dynamic model to obtain the transformed n-degree-of-freedom manipulator dynamic model.

[0078] S2 includes the following sub-steps:

[0079] S2.1 Establish the dynamic model of the n-degree-of-freedom robotic arm, as shown below:

[0080]

[0081] In the formula, q represents the joint angle position. Indicates joint angular velocity, M(q) represents the joint angular acceleration, and M(q) represents the system inertia matrix. G(q) represents centrifugal force and Coriolis force, G(q) represents gravity, and τ represents joint torque. d This indicates interference from the external environment;

[0082] S2.2 The parameters of the robotic arm's dynamic model are rewritten as nominal and uncertain parts, as follows:

[0083]

[0084] Among them, M0(q), And G0(q) is the nominal part, ΔM(q) and ΔG(q) are the uncertain parts;

[0085] The system composite disturbance d is defined in the following form:

[0086]

[0087] The dynamic model of an n-degree-of-freedom robotic arm can be rewritten in the following form:

[0088]

[0089] For system (4), the corresponding state-space expression is given as follows:

[0090] Let x1 = q, but:

[0091]

[0092] make and h|≤L h h is the derivative value of the extended state x3, L h Since h is a constant, i.e., the upper bound of h, the extended state-space expression of system (4) is as follows:

[0093]

[0094] S3. Set a predefined time-extended state observer, estimate the lumped disturbance caused by the uncertainty of the compensation system and external interference, obtain the lumped disturbance estimate z3, estimate the actual velocity information, and obtain the velocity estimate z2.

[0095] S3 includes the following sub-steps:

[0096] S3.1 First, define the following estimation error:

[0097] e i =z i -x i (7)

[0098] Among them, z i For x i The estimated value, e i To estimate the error, i = 1, 2, 3;

[0099] Design a sliding surface s of the following form. p :

[0100] s p =c1e1+e2; (8)

[0101] Where c1 is a positive constant;

[0102] S3.2, combined with sliding surface s p Design the following predefined time-extended state observer, as shown below:

[0103]

[0104] Where ε is a constant, ε=2^((2+r) / 2), T ESO Let sig(s) be the system convergence time constant. p )=|s p |sign(s p k3 and k4 are adjustable positive constants, and r is a constant, r∈(0,1);

[0105] Based on the system state-space expression (6) and the predefined time-extended state observer (9), the error observation system expression of the predefined time-extended state observer is derived as follows:

[0106]

[0107] Among them, u a To design an auxiliary control law;

[0108] Designing the auxiliary control law and differentiating equation (8) yields:

[0109]

[0110] Auxiliary control law u a The expression is:

[0111]

[0112] Where k1 and k2 are positive constants;

[0113] S3.3. Using the predefined time-extended state observer designed in S3.2, observe the synthetic disturbance and actual speed information formed by system uncertainties and external disturbances during the robotic arm trajectory tracking control process, and construct the Lyapunov function V1 as shown below:

[0114]

[0115] Based on predefined time theory and Lyapunov's theorem, it is proved that the estimation error can be controlled by the auxiliary control law u. a Under control, it converges to the sliding surface s p Above; the estimation error on the sliding surface within a predefined time T ESO The internal energy can converge to zero.

[0116] S4. Based on the joint expected angular position and expected angular velocity information obtained in S1, establish position tracking error and velocity tracking error, design a non-singular predefined time sliding surface s, and design a predefined time sliding controller based on the velocity estimation information z2 and lumped disturbance estimation value z3 obtained from the sliding surface s and S3, and obtain the predefined time sliding control law to realize the robot arm trajectory tracking.

[0117] S4 includes the following sub-steps:

[0118] S4.1 Define a nonlinear function in the following form:

[0119]

[0120] Where x is the independent variable, α is an adjustable parameter, α = δ, δ is an adjustable parameter, and δ ∈ (0, e) -1 ), ε 1p It is an adjustable parameter. ε 2p It is an adjustable parameter.

[0121] Based on the desired angular position information q obtained from S1 d and expected angular velocity information The actual position estimation information obtained by S2 is defined as follows: trajectory tracking error w and velocity tracking error. As shown below:

[0122]

[0123] S4.2. Based on the nonlinear function (14), design the following predefined time sliding surface:

[0124]

[0125] Where F(w)=[f(w1),f(w2),…,f(w) j ),…,f(w n )] T w is a matrix formed by nonlinear functions. j For the tracking error of the j-th joint, γ is a constant, γ>0; T c2 This refers to the convergence time parameter during the system's approach phase.

[0126] S4.3 Design a predefined time sliding mode controller:

[0127]

[0128] Where, τ ssw For a predefined time-stable reaching law, τ eq This is the equivalent control law, where η is a constant and T c1 K is the convergence time parameter for the sliding phase. 1p For positive constants, γ > 0.

[0129] S5. Based on S3 to S4, the predefined time sliding mode controller law based on the predefined time extended state observer is obtained.

[0130] Based on the predefined time-extended state observer of S3 and the predefined time-sliding mode control of S4, the Lyapunov function V2 is constructed as follows:

[0131]

[0132] By stabilizing the Lyapunov function V2, we obtain a predefined time sliding mode control law based on a predefined time extended state observer, namely the joint torque τ, as shown below:

[0133]

[0134] like Figure 1 As shown, firstly, the joint angular position q of the robotic arm is calculated, along with the desired angular position information q. d The difference between the two is used to design a predefined time sliding surface s. Combined with the observed state of the extended state observer, a predefined time sliding controller control law τ is given to realize the trajectory tracking control of the robotic arm.

[0135] by Figure 2 Taking the rigid robotic arm model with input saturation as an example, the trajectory tracking control method proposed in this invention is verified. In the figure, m1 and m2 are the masses of the two joints, r1 and r2 are the lengths of the link, and q1 and q2 are the angular displacements of the joints. Figure 3 and Figure 4 The figure shown is a trajectory tracking curve of the two joints of the rigid robotic arm in this embodiment. Figure 3 This is a trajectory tracking curve for joint 1. Figure 4 For the joint 2 trajectory tracking curve, Figure 5 and Figure 6 For a predefined time trajectory tracking error curve, Figures 3 to 6 The invention verifies a sliding mode control strategy for predefined time trajectory tracking of a robotic arm, which has faster convergence speed, more accurate tracking performance, and stronger anti-interference capability. It ensures that the robotic arm system can achieve trajectory tracking within a pre-set time, taking into account convergence speed, tracking performance, and anti-interference capability, and realizes predefined time trajectory tracking control of a rigid robotic arm.

[0136] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A sliding mode control method for tracking a predefined time trajectory of a robotic arm, characterized in that, Includes the following steps: S1. Obtain the desired trajectory information of the robotic arm, including the desired angular position and desired angular velocity information of each joint of the robotic arm; S2. Establish a dynamic model of an n-degree-of-freedom robotic arm, and derive the lumped disturbances formed by the nominal and uncertain parts of the robotic arm's dynamic model. Next, the dynamic model of the robotic arm is transformed to obtain the transformed dynamic model of the n-degree-of-freedom robotic arm. S3. Set a predefined time-extended state observer to estimate the lumped disturbance caused by the uncertainty of the compensation system and external disturbances, and obtain the estimated value of the lumped disturbance. Estimate actual speed information and obtain speed estimates. ; S4. Based on the joint desired angular position and desired angular velocity information obtained in S1, establish position tracking error and velocity tracking error, and design a non-singular predefined time sliding surface. According to the sliding surface Speed ​​estimation information obtained with S3 and lumped disturbance estimate Design a predefined time sliding mode controller and obtain a predefined time sliding mode control law to achieve robot arm trajectory tracking; S5. Based on S3~S4, obtain the predefined time sliding mode controller law based on the predefined time extended state observer; S3 includes the following sub-steps: S3.1 First, define the following estimation error: ；(7) in, for The estimated value, To estimate the error, ; Design a sliding surface in the following form. : ；(8) in, It is a positive number; S3.2, Combining sliding surfaces Design the following predefined time-extended state observer, as shown below: ；(9) in, It is a constant. , Let be the system convergence time constant. , and It is an adjustable positive number. It is a constant. ; Based on the system state-space expression (6) and the predefined time-extended state observer (9), the error observation system expression of the predefined time-extended state observer is derived as follows: ；(10) in, To design an auxiliary control law; Designing the auxiliary control law and differentiating equation (8) yields: ；(11) Auxiliary control law The expression is: ；(12) in, , It is a positive number; S3.

3. Using the predefined time-extended state observer designed in S3.2, observe the synthetic disturbance and actual velocity information formed by system uncertainties and external disturbances during the robotic arm trajectory tracking control process, and construct the Lyapunov function. As shown below: ；(13) Based on the predefined time theory and Lyapunov's theorem, it is proved that the estimation error can be mitigated by the auxiliary control law. Converging to the sliding surface under control Above; the estimation error on the sliding surface within a predefined time. The internal energy can converge to zero.

2. The sliding mode control method for tracking a predefined time trajectory of a robotic arm according to claim 1, characterized in that, S2 includes the following sub-steps: S2.1 Establish the dynamic model of the n-degree-of-freedom robotic arm, as shown below: ；（1） In the formula, Indicates the position of the joint angle. Indicates joint angular velocity, Represents joint angular acceleration. Represents the system's inertia matrix. Representing centrifugal force and Coriolis force, Represents gravity. Indicates joint torque. This indicates interference from the external environment; S2.2 The parameters of the robotic arm's dynamic model are rewritten as nominal and uncertain parts, as follows: ；（2） in, , and For the nominal portion, , and The part that is uncertain; System lumped disturbance for: ；（3） The dynamic model of an n-degree-of-freedom robotic arm can be rewritten in the following form: ；（4） For system (4), the corresponding state-space expression is given as follows: make , ,but: ；(5) make and , , For extended state The derivative value, It is a constant. The upper bound value of the system (4) is given by the following extended state-space expression: (6)。 3. The sliding mode control method for tracking a predefined time trajectory of a robotic arm according to claim 2, characterized in that, S4 includes the following sub-steps: S4.1 Define a nonlinear function in the following form: ；(14) in, As the independent variable, It is an adjustable parameter. , It is an adjustable parameter. , It is an adjustable parameter. , It is an adjustable parameter. ; Based on the expected angular position information obtained from S1 and expected angular velocity information The actual position estimation information obtained by S2 is defined as follows: trajectory tracking error and speed tracking error As shown below: ；(15) S4.

2. Based on the nonlinear function (14), design the following predefined time sliding surface: ；(16) in, A matrix formed by nonlinear functions. In order to target the Tracking error of each joint , It is a constant. ; This refers to the convergence time parameter during the system's approach phase. S4.3 Design a predefined time sliding mode controller: ；(17) ；(18) in, For a predefined time-stable reaching law, This is an equivalent control law. It is a constant. This refers to the convergence time parameter during the sliding phase. For positive integers, , .

4. The sliding mode control method for tracking a predefined time trajectory of a robotic arm according to claim 3, characterized in that, Specifically, S5 involves constructing a Lyapunov function based on the predefined time-extended state observer in S3 and the predefined time-sliding mode control in S4. As shown below: ； (19) Make Lyapunov function As the system stabilizes, a predefined time sliding mode control law based on a predefined time-extended state observer is obtained, i.e., the joint torque. As shown below: (20)。

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