Mechanical arm self-adaptive sliding mode control method based on preset time stability

Abstract

The invention discloses a mechanical arm self-adaptive sliding-mode control method based on preset time stability, and provides a novel self-adaptive preset time sliding-mode control method based on a nonlinear multi-joint mechanical arm system and definition of the preset time stability. By introducing adjustable parameters, the design of the controller is more flexible, and accurate trajectory tracking and force control can be realized. By designing the novel sliding mode surface and the controller, accurate trajectory tracking of the slave end mechanical arm to the master end mechanical arm can be achieved, and meanwhile the designed controller can achieve compliance control over an end effector of the mechanical arm.

Description

Technical Field

[0001] The present invention belongs to the field of industrial robot control, and particularly relates to a mechanical arm adaptive sliding mode control method based on pre-determined time stability. Background Art

[0002] With the improvement of industrial automation level and the growth of intelligent manufacturing requirements, as a high-precision dynamic execution system, the mechanical arm is increasingly widely used in fields such as assembly, welding, and medical surgery. The core task of its control system is to achieve fast response, high trajectory accuracy, and stable operation in a complex environment with parameter uncertainties, external disturbances, and model errors. Although traditional methods such as PID control and fuzzy control can meet basic control requirements, they have problems such as insufficient adaptability and large steady-state errors when facing strong disturbances such as nonlinear joint friction and load mutations.

[0003] In recent years, adaptive sliding mode control has become a research hotspot in the field of mechanical arm control due to its robustness to parameter changes and strong suppression ability for uncertainties. This method constructs a sliding mode surface to constrain the system state within a specific region and combines online parameter estimation technology to compensate for system uncertainties. However, existing adaptive sliding mode control strategies mainly focus on asymptotic stability and are difficult to ensure strict time performance indicators. Especially in high-speed motion scenarios, overshoot or oscillation phenomena are likely to occur.

[0004] At the same time, the pre-determined time stability (PTS) theory provides a new methodology for the control system to reach a stable state within a finite time. By introducing a time prediction mechanism, this theory allows the system to complete state convergence within a preset time, which is particularly suitable for real-time control tasks with strict deadlines. However, existing PTS control methods are mostly based on linear system models and are difficult to be directly applied to nonlinear mechanical arm systems. Traditional adaptive sliding mode control lacks optimized design under time constraints, and the extended application of PTS theory in nonlinear systems still faces major challenges. Therefore, there is an urgent need for a control method that combines the robustness of adaptive sliding mode control and the time prediction advantage of PTS theory to achieve fast and stable control of the mechanical arm system under dynamic disturbances. Summary of the Invention

[0005] Aiming at the above mechanical arm motion control and fast and stable time problems, the present invention provides a mechanical arm adaptive sliding mode control method based on pre-determined time stability. The present invention introduces the pre-determined time stability theory into traditional adaptive sliding mode control, enabling the trajectory tracking error of the mechanical arm to converge to zero within a pre-determined time, thereby improving the control accuracy, stability, and robustness of the entire mechanical arm system.

[0006] Technical Solution:

[0007] The present invention discloses a method for adaptive sliding mode control of a robotic arm based on predetermined time stability, comprising the following steps:

[0008] S1. Establish a dynamic model of a rigid robotic arm system with rotating joints having n degrees of freedom;

[0009] S2. Obtain the measurement information of the positions x i and angular velocities of each joint of the robotic arm through sensors, set the desired positions x d and desired velocities for each joint to be tracked, and calculate the trajectory tracking error ρ = x i - x d and the first derivative of the trajectory tracking error with respect to time

[0010] S3. Design a Lyapunov function to prove that while satisfying the system stability, it also satisfies the existing predetermined time stability theory;

[0011] S4. Design an adaptive force tracking impedance control;

[0012] S5. Design a sliding mode surface according to ρ and ;

[0013] S6. Design an adaptive predetermined time controller, and use the adaptive predetermined time controller to achieve the adaptive sliding mode control of the robotic arm based on predetermined time stability.

[0014] Preferably, the equation of the dynamic model of the rigid robotic arm system in step S1 is expressed as follows:

[0015]

[0016] where the subscript i = {m, s} respectively represents the master manipulator and the slave manipulator; represents the angular position, angular velocity, and angular acceleration of the generalized joint coordinates; respectively represent the velocity and acceleration in the Cartesian space; represents a positive definite inertia matrix; is the matrix effect of centrifugal and Coriolis forces; represents the vector of the gravity effect;

[0017] is the vector of the input torque; represents the vector of internal uncertainties and bounded external disturbances; when i = m, F i = F h , F h is the contact force applied at the master end; when i = s, F i = F e , F eis the end - environment contact force, \(J(q i )\) is the Jacobian matrix, is the derivative of \(J(q i ). The superscript - T represents the transpose matrix of the Jacobian matrix, and the superscript -1 represents the inverse matrix of the Jacobian matrix; represent the corresponding positive - definite inertia matrix, Coriolis force matrix, and gravity matrix in Cartesian space respectively.

[0018] Preferably, in step S3, if the Lyapunov function \(V\) satisfies the following conditions:

[0019]

[0020] then the system is stable within a predetermined time \(T c is the predetermined time and \(T c > 0\), \(n\) is the set control parameter and \(0\lt n\lt1\), represents the derivative of \(V\), and \(\pi\) takes 3.14.

[0021] Preferably, in step S4, design an adaptive force - tracking impedance control:

[0022] Controlling the relative position to obtain the desired interaction force is expressed as:

[0023]

[0024] where, represents the interaction force, \(F d is the desired external force, respectively represent the actual position and the desired position of the end - effector, and

[0025] and

[0026] the desired trajectory acceleration of the end - effector is calculated as follows:

[0027]

[0028] The adaptation rate of

[0029]

[0030] is designed as follows: ii (i＝1.2.3.4) is a diagonal matrix; where is the estimate of \(F e ), \(x\) is the actual position of the end of the robotic arm, \(x c is its reference position, ​They are the positive coefficient matrices of adaptive quality, damping, and stiffness respectively.

[0031] Preferably, in step S5, according to ρ and Design the sliding mode surface s i The specific steps are as follows:

[0032] The sliding mode surface is designed as follows:

[0033]

[0034] Among them, the subscript i = {m, s} represents the master manipulator and the slave manipulator respectively; ρ i = x i - x d is the position tracking error, x d is the expectation of x i , T c is the predetermined time.

[0035] Preferably, in step S6, the adaptive predetermined time controller is designed as follows:

[0036]

[0037] Among them, F ai is the total control input, is an adaptive parameter that changes with the sliding mode surface and satisfies s i is the sliding mode surface, 0 < μ < 1, θ i is the control parameter, is the estimated value of the contact force, and sgn represents the sign function.

[0038] The beneficial effects of the present invention

[0039] The present invention designs an adaptive impedance control method for a robotic arm based on predetermined-time stability, effectively solving the deficiencies in traditional control such as large tracking error, slow response speed, and long settling time. By introducing the theory of predetermined-time stability into traditional adaptive sliding mode control, the trajectory tracking error of the robotic arm converges to zero within the set precise time, thereby improving the control accuracy and stability of the entire robotic arm system. Brief Description of the Drawings

[0040] Figure 1 It is the position tracking simulation diagram of the slave robotic arm of link 1 tracking the master robotic arm.

[0041] Figure 2 It is the position tracking simulation diagram of the slave robotic arm of link 2 tracking the master robotic arm.

[0042] Figure 3It is a simulation diagram of the position tracking error of two connecting rods.

[0043] Figure 4 It is a simulation diagram of the control input of two connecting rods.

[0044] Figure 5 It is the adaptive control parameter simulation diagram. Specific implementation manners

[0045] The present invention will be further described below in conjunction with embodiments, but the protection scope of the present invention is not limited thereto:

[0046] The present invention discloses an adaptive sliding mode control method for a robotic arm based on predetermined-time stability, including the following steps:

[0047] (1) Establish a dynamic model of a rigid robotic arm system with n-degree-of-freedom revolute joints;

[0048] (2) Obtain the measurement information of the position x and angular velocity of each joint of the robotic arm through sensors, set the desired position x d and desired velocity for each joint to track, and calculate the trajectory tracking error e = x d -x and the first derivative of the trajectory tracking error with respect to time

[0049] (3) Satisfy the existing predetermined-time stability theory;

[0050] (4) Design an adaptive force tracking impedance control;

[0051] (5) Design a sliding mode surface according to e and ;

[0052] (6) Design an adaptive predetermined-time controller, and use the adaptive predetermined-time controller to achieve the adaptive sliding mode control of the robotic arm based on predetermined-time stability.

[0053] Preferably, the specific steps of establishing the dynamic model of the rigid robotic arm system with n-degree-of-freedom revolute joints in step (1) are as follows:

[0054]

[0055] Here,

[0056] where the subscript i = {m, s} represents the master manipulator and the slave manipulator respectively; represents the angular position, angular velocity, and angular acceleration of the generalized joint coordinate; represents a positive definite inertia matrix;

[0057] is the centrifugal and Coriolis matrix effect; vector representing the gravity effect; is the vector of the input torque;

[0058] is expressed as the vector of internal uncertainties and bounded external disturbances. F m = F h , F s = F e . F h is the contact force applied at the master end, F e is the slave end environmental contact force, J(q i ) is the Jacobian matrix.

[0059] Preferably, the specific steps of proposing a predetermined-time stability theory in the step (3) are as follows:

[0060] For a general autonomous dynamical system, if the Lyapunov function V satisfies the following conditions:

[0061]

[0062] then the system is stable within a predetermined time, T c > 0 is the predetermined time and 0 < n < 1.

[0063] Preferably, the specific steps of designing an adaptive force tracking impedance control in the step (4) are as follows:

[0064] Controlling the relative position to obtain the desired interaction force can be expressed as:

[0065]

[0066] where represents the interaction force, F d is the desired external force, respectively represent the actual position and the desired position of the end effector, and (desired inertia) (desired damping) and (desired stiffness) are diagonal positive definite matrices.

[0067] According to the dynamic relationship, the desired trajectory acceleration of the end effector can be calculated as follows:

[0068]

[0069] The adaptive rate of

[0070]

[0071] Here, \(W = diag[W ii (i＝1.2.3.4) is a diagonal matrix. Among them, is the estimated value of \(F e , \(x\) is the actual position of the end of the robotic arm, and \(x c is its reference position. are respectively the positive coefficient matrices of adaptive mass, damping, and stiffness.

[0072] Preferably, in the step (5), according to \(e\) and design the sliding mode surface \(s i as follows:

[0073] The sliding mode surface is designed as follows:

[0074]

[0075] Among them, the subscript \(i = \{m, s\}\) respectively represents the master manipulator and the slave manipulator; \(\rho i =x i -x di is the position tracking error, \(x di is the expectation of \(x i , T c is a predetermined time.

[0076] Preferably, in the step (6), the specific steps of designing the adaptive predetermined time controller are as follows:

[0077] The controller is designed as follows:

[0078]

[0079] Here, \(F a is the total control input, where: s i is the sliding mode surface, \(0 < \mu < 1\), \(\theta i is the control parameter, is the estimated value of the contact force.

[0080] To verify the effect of the solution designed in this application, the following simulation verification was carried out:

[0081] Combined with Figure 1 , a control block diagram was built based on Matlab / Simulink to simulate and verify the present invention. The red dashed line is the position trajectory of the main end link 1, and the black solid line is the position trajectory of the slave end link 1 tracking the main end. It can be seen that the slave robotic arm link 1 tracks the trajectory of the main robotic arm link 1 at 0.5 s, indicating that the designed controller has good trajectory tracking ability.

[0082] ​Combined with Figure 2 , a control block diagram is built based on Matlab / Simulink to simulate and verify the present invention. The red dashed line is the position trajectory of the main-end link 2, and the black solid line is the position trajectory of the slave-end link 2 tracking the main-end. It can be seen that the slave-end robotic arm link 2 tracks the trajectory of the main-end robotic arm link 2 at 0.7 s, indicating that the designed controller can meet the requirements of position trajectory tracking.

[0083] Combined with Figure 3 , the black solid line is the trajectory tracking error curve of link 1, and the red dashed line is the trajectory tracking error curve of link 2. It can be seen that the tracking errors of the two links stably approach zero at 0.7 s, indicating that the designed controller can finally complete error convergence and meet the requirements of the stability of the control system.

[0084] Combined with Figure 4 , the control inputs of the two links are shown as follows. The black solid line is the control input curve of link 1, and the red dashed line is the control input curve of link 2. It can be seen that the control inputs of the two links stably approach zero at 0.5 s, indicating that the designed controller meets the requirements of the stability of the control system.

[0085] Combined with Figure 5 , the adaptive parameters of the main-end and the slave-end change as shown in the figure. The black solid line is the change curve of the adaptive parameter , and the red dashed line is the change curve of the adaptive parameter . It can be seen that the adaptive parameters change within a certain range, making the control system achieve final stability.

[0086] The predetermined time T c set in this article = 0.8 s, and other parameter settings are as follows: μ = 0.3, n = 0.65, W = [5 55 5].

[0087] The specific embodiments described in this article are only examples to illustrate the spirit of the present invention. Those skilled in the technical field to which the present invention belongs can make various modifications or supplements to the described specific embodiments or use similar ways to replace them, but will not deviate from the spirit of the present invention or exceed the scope defined by the appended claims.

Claims

1. A method for adaptive sliding mode control of a robotic arm based on a predetermined time stability, characterized in that The following steps are involved: S1. Establish a dynamic model of a rigid manipulator system with n degrees of freedom rotation joints; S2. Obtain the position x of each joint of the robotic arm through sensors i and angular velocity The measurement information sets the expected position x of each joint tracking d and expected speed And calculate the robot trajectory tracking error ρ = x i -x d and the first-order derivative of the trajectory tracking error with respect to time S3. Design a Lyapunov function to prove that the system stability is satisfied while satisfying the existing predetermined time stability theory; S4, design adaptive force tracking impedance control; S5, according to ρ and Design sliding surface; S6. Design an adaptive scheduled time controller and use it to implement adaptive sliding mode control of the robotic arm based on scheduled time stability.

2. The method according to claim 1, characterized in that The equation of the rigid manipulator system dynamics model in step S1 is expressed as follows: Wherein, the subscript i = {m, s} represents the master manipulator and the slave manipulator respectively; Represents the angular position, angular velocity, and angular acceleration of generalized joint coordinates; They represent velocity and acceleration in Cartesian space respectively; represents the positive definite inertia matrix; are the centrifugal and Coriolis force matrix effects; A vector representing the effect of gravity; is the vector of input torque; It is represented by the internal uncertainty and the bounded external disturbance vector; when i = m, F i =F h , F h is the contact force applied by the master end; when i = s, F i =F e , F e is the contact force from the end environment, J(q i ) is the Jacobian matrix, is J(q i ), the derivative of -T represents the transposed matrix of the Jacobian matrix, with the superscript -1 represents the inverse matrix of the Jacobian matrix; represents the corresponding positive definite inertia matrix, Coriolis force matrix, and gravity matrix in Cartesian space respectively.

3. The method according to claim 1, characterized in that In step S3, if the Lyapunov function V satisfies the following conditions: Then the system is stable within the predetermined time, T c is the scheduled time and T c >0, n is the set control parameter and 0<n<1, It means to take the derivative with respect to V, and π is taken as 3.

14.

4. The method according to claim 1, characterized in that In step S4, the adaptive force tracking impedance control is designed: Controlling the relative position to obtain the desired interaction force is expressed as: in, represents the interaction force, F d is the desired external force, denote the actual position and the desired position of the end effector, respectively, and and Expected trajectory acceleration of the end effector The calculation is as follows: The adaptation rate is designed as follows: Here W = diag[W ii ] (i＝1.2.3.4) is a diagonal matrix; F e The estimated value of x is the actual position of the end of the robot arm, and x c For its reference position, are the adaptive mass, damping and stiffness positive coefficient matrices respectively.

5. The method according to claim 1, characterized in that In step S5, according to ρ and Design sliding surfaces i The specific steps are as follows: The sliding surface design is as follows: Wherein, the subscript i = {m, s} represents the master manipulator and the slave manipulator respectively; ρ i =x i -x d is the position tracking error, x d is x i expectations, c3>0,n∈(0,1),T c For scheduled time.

6. The method according to claim 1, characterized in that In step S6, the adaptive scheduled time controller is designed as follows: Among them, F ai is the total control input, is an adaptive parameter that changes with the sliding surface and satisfies s i is the sliding surface, 0<μ<1, θ i is the control parameter, is the estimated contact force, and sgn represents the sign function.

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