Markov jump-based mechanical arm preset time trajectory tracking control method

Abstract

The invention provides a Markov jump-based mechanical arm preset time trajectory tracking control method, which comprises the following steps of: calculating a trajectory tracking error by establishing a two-degree-of-freedom mechanical arm dynamic model with a Markov jump characteristic; a trajectory tracking error is subjected to nonlinear transformation, a nonsingular terminal sliding mode surface is constructed in combination with a preset convergence time upper limit function, and then a control torque is synthesized through an equivalent control item and a robust switching item containing a sliding mode surface sign function and a switching gain to drive a mechanical arm. And the trajectory tracking error is difficult to converge in the preset time.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of intelligent control of mechanical arms, and particularly relates to a preset time trajectory tracking control method for a mechanical arm based on Markov jumps. BACKGROUND

[0002] As a core actuator in the fields of modern industrial automation, intelligent manufacturing, space operation and medical surgery, the trajectory tracking control performance of a mechanical arm directly determines the task execution accuracy and intelligent level of the entire system. The core goal of trajectory tracking control is to design an efficient control law to enable the end effector of the mechanical arm to quickly and accurately track the preset time-varying desired trajectory, thereby meeting the operation requirements in complex scenarios. In the actual operation of the mechanical arm, the system parameters are prone to random mutations due to factors such as load changes, component wear and work mode switching. Such systems with random mode switching characteristics can be accurately described by Markov jump system theory. Markov jump system theory provides a reliable mathematical framework for describing dynamic systems with multiple mode switching, and has been widely applied in modeling complex systems such as mechanical arms with parameter mutations.

[0003] To cope with the inherent strong coupling, nonlinearity and external disturbance of the mechanical arm system, the existing technology uses sliding mode control to drive the system state to converge to a sliding mode surface by constructing a discontinuous control law, thereby achieving trajectory tracking.

[0004] However, the sliding mode control has singularity near the equilibrium point and chattering problem. When the system parameters of the mechanical arm jump according to Markov, the sliding mode control cannot balance the robustness and preset time convergence characteristics of the system, and cannot strictly guarantee that the trajectory tracking error converges to zero within the preset time, resulting in poor predictability of the system response and difficulty in meeting the strict requirements of trajectory tracking of the mechanical arm in high-risk and high-precision fields such as intelligent manufacturing and on-orbit servicing. SUMMARY

[0005] The present application provides a preset time trajectory tracking control method for a mechanical arm based on Markov jumps, which solves the problem of trajectory tracking error not converging within the preset time when the system parameters jump according to Markov.

[0006] In one aspect, the present application provides a preset time trajectory tracking control method for a mechanical arm based on Markov jumps, comprising: establishing a two-degree-of-freedom mechanical arm dynamics model with Markov jump characteristics, obtaining joint state data of the mechanical arm, calculating a trajectory tracking error in combination with a predefined desired trajectory; performing nonlinear transformation on the trajectory tracking error, combining the transformed result with a function of the upper limit of the preset convergence time, constructing a non-singular terminal sliding mode surface, and obtaining a sliding mode surface function; substitute the sliding mode surface function into a preset equivalent control law formula to calculate an equivalent control term; introduce a robust term including a sign function of the sliding mode surface function and a switching gain to generate a robust switching term; add the equivalent control term and the robust switching term to synthesize a control torque, and input the control torque into a dynamics system of the robot arm to drive the robot arm joint to move, so that a joint movement trajectory of the robot arm tracks the desired trajectory.

[0007] Optionally, a dynamics model of the robot arm with Markov jump characteristics is established, and joint state data of the robot arm is obtained, including: a two-degree-of-freedom robot arm dynamics model with Markov jump characteristics is established; wherein the two-degree-of-freedom robot arm dynamics model is: wherein, is a joint angle position vector, and are an angular velocity vector and an angular acceleration vector, respectively, is a positive definite inertia matrix, is a centrifugal force and Coriolis force matrix, is a gravity matrix, is a control torque, is a continuous-time Markov chain taking values in a finite set .

[0008] Optionally, the desired trajectory is combined to calculate a trajectory tracking error, including: determining a desired joint angle position and a desired angular velocity corresponding to the predefined desired trajectory; obtaining the joint angle position vector and the angular velocity vector during operation of the robot arm; differentially calculating the joint angle position vector and the desired joint angle position to obtain an angle position tracking error; differentially calculating the angular velocity vector and the desired angular velocity to obtain an angular velocity tracking error; generating a trajectory tracking error based on the angle position tracking error and the angular velocity tracking error.

[0009] Optionally, the trajectory tracking error is nonlinearly transformed, and the transformed result is combined with a function of a preset upper limit of convergence time to construct a non-singular terminal sliding mode surface to obtain a sliding mode surface function, including: setting a segmented critical value , an upper limit of convergence time , and a transformation parameter ​; wherein the piecewise critical value is an arbitrarily small positive number, the upper bound of convergence time is a positive number, the transformation parameter has a value range of positive numbers between 0 and 1; extracting an angular position tracking error from the trajectory tracking error , calculating the absolute value of the angular position tracking error ; comparing the absolute value of the angular position tracking error with the piecewise critical value ; if , a fractional power and sign function containing the transformation parameter is used to nonlinearly transform ; wherein the nonlinear change is: ; if , a linear term and a quadratic term combination is used to nonlinearly transform ; wherein the nonlinear change form is: ; wherein , ; a positive adjustment parameter is set, combined with the upper bound of convergence time , the transformation parameter , a function of the upper bound of convergence time is constructed; extracting the angular velocity tracking error in the trajectory tracking error , superimposing the nonlinearly transformed angular position tracking error result, the corresponding coefficient term, the corresponding coefficient term, multiplying the function of the upper bound of convergence time, and finally adding the angular velocity tracking error to obtain the sliding mode surface function.

[0010] wherein the sliding mode surface function is: ; wherein , , ， ， ， ， ， ， It is an arbitrarily small positive integer. This is the preset upper limit for convergence time.

[0011] Optionally, the sliding surface function is substituted into a preset equivalent control law formula to calculate the equivalent control term, including: Obtain the desired angular acceleration data corresponding to the predefined desired trajectory. and expected angular velocity data ; Extracting the upper limit of convergence time from the sliding surface function Transformation parameters Adjusting parameters Determine the nonlinear transformation coefficients ; Calling the positive definite inertia matrix in the dynamic model of a two-degree-of-freedom manipulator with Markov jump properties Centrifugal force and Coriolis force matrix Gravity matrix ; Desired angular acceleration data Expected angular velocity data Upper limit of convergence time Transformation parameters Adjusting parameters Nonlinear transformation coefficients Positive definite inertial matrix Centrifugal force and Coriolis force matrix Gravity matrix Substituting into the equivalent control term formula, we obtain the equivalent control term; The formula for the equivalent control term is: .

[0012] Optionally, a robust switching term is generated by introducing a sign function including the sliding surface function and a robust term for the switching gain, including: Set two positive gain switching parameters , And satisfy ; Set preset time parameters ; Obtain the state data of the sliding surface function and calculate the norm of the sliding surface function. ; Based on preset calculation rules, the norm of the sliding surface function is... With two switching gain parameters respectively , Perform adaptation operations to obtain intermediate variables related to the convergence law; The positive number switching gain parameter, the preset time parameter and the norm are substituted into a robust switching term formula to obtain a robust switching term; The robust switching term formula is: .

[0013] Optionally, the equivalent control term and the robust switching term are added to synthesize a control torque, and the control torque is input into a mechanical arm dynamics system to drive the mechanical arm joint to move, so that the joint movement trajectory tracks the expected trajectory, including: The equivalent control term and the robust switching term are extracted, and the equivalent control term and the robust switching term are directly superimposed through algebraic addition operation to obtain a control torque; The control torque is sent to a two-degree-of-freedom mechanical arm dynamics system with Markov jump characteristics as an input signal, replaces the corresponding control torque parameter in the two-degree-of-freedom mechanical arm dynamics model, and continuously adjusts the joint movement state until the joint movement trajectory and the predefined expected trajectory are consistent.

[0014] Optionally, it further includes: A first Lyapunov function is constructed based on a sliding surface function and a positive definite inertia matrix; The time derivative of the first Lyapunov function is calculated and simplified by using the dynamics characteristics of the mechanical arm, the minimum and maximum eigenvalues of the positive definite inertia matrix in all modes are extracted, and the convergence characteristics of the first Lyapunov function are verified; A second Lyapunov function is constructed based on an angle position tracking error, and an error correlation is obtained in combination with the convergence characteristics of the sliding surface; The error correlation is substituted into the sliding surface function to verify the convergence characteristics of the second Lyapunov function; If the convergence characteristics of the first Lyapunov function and the second Lyapunov function are both convergent, a simulation model is constructed.

[0015] Optionally, the simulation model includes: Physical parameters of the two-degree-of-freedom mechanical arm are set, including the mass of each link, the length of each link, the moment of inertia of each link around the center of mass, and the acceleration of gravity; System parameters of the Markov jump system are defined, and a value set of the Markov chain is determined; the value set includes three elements corresponding to three operating modes of the Markov jump system; The transition probability matrix between the three operating modes is set, and the corresponding parameters of the positive definite inertia matrix, the centrifugal force and Coriolis force matrix, and the gravity matrix are configured for each operating mode; The integration of non-singular terminal sliding mode surface construction logic, equivalent control item calculation logic and robust switching term generation logic forms the execution flow of the controller; A signal interaction path between the controller and the robot model is established, so that the control torque output by the controller is input to the two-degree-of-freedom robot dynamics model, and the robot model can feed back joint angle position data and angular velocity data to the controller; The desired trajectory of the robot end effector is set in the simulation platform; The physical parameters, Markov jump system parameters, execution flow of the controller, robot dynamics model and desired trajectory are imported into a simulation tool to form a simulation model.

[0016] Optionally, the method further comprises: A simulation result of the simulation model is obtained; It is determined whether the simulation result meets a preset condition; If the simulation result meets the preset condition, a control parameter corresponding to the simulation result is obtained; The control parameter is configured into a robot control system, a control flow is started, and the robot is driven to perform a trajectory tracking task.

[0017] In another aspect, the present application also provides an electronic device comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the Markov jump-based robot preset time trajectory tracking control method according to any one of the above.

[0018] In another aspect, the present application also provides a non-transitory computer readable storage medium having a computer program stored thereon, wherein the computer program is executable by a processor to implement the Markov jump-based robot preset time trajectory tracking control method according to any one of the above.

[0019] In another aspect, the present application also provides a computer program product comprising a computer program, wherein the computer program is executable by a processor to implement the Markov jump-based robot preset time trajectory tracking control method according to any one of the above.

[0020] The application provides a preset time trajectory tracking control method for a mechanical arm based on Markov jump. BRIEF DESCRIPTION OF DRAWINGS

[0021] In order to more clearly illustrate the technical solutions in the application or the prior art, the drawings needed in the embodiments or the prior art description will be briefly introduced. Obviously, the drawings in the following description are some embodiments of the application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0022] Figure 1 Fig. 1 is a flowchart of a preset time trajectory tracking control method for a mechanical arm based on Markov jump provided by an embodiment of the application; Figure 2 Fig. 2 is a schematic diagram of a two-link rigid mechanical arm model provided by an embodiment of the application; Figure 3 Fig. 3 is a schematic diagram of a mechanical arm joint angle tracking provided by an embodiment of the application; Figure 4 Fig. 4 is a schematic diagram of a mechanical arm joint angle tracking error provided by an embodiment of the application; Figure 5 Fig. 5 is a schematic diagram of a mechanical arm end position provided by an embodiment of the application; Figure 6 Fig. 6 is a schematic diagram of a mechanical arm joint torque provided by an embodiment of the application; Figure 7 Fig. 7 is a schematic diagram of a mechanical arm mode switching provided by an embodiment of the application; Figure 8 Fig. 8 is a schematic diagram of the structure of an electronic device provided by an embodiment of the application. DETAILED DESCRIPTION

[0023] In order to make the objects, technical solutions and advantages of the present application clearer, the technical solutions in the present application will be described clearly and completely below in conjunction with the drawings in the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor fall within the scope of protection of the present application.

[0024] Figure 1 is a flowchart of a preset time trajectory tracking control method for a Markov jump-based mechanical arm provided by an embodiment of the present application.

[0025] As shown in Figure 1 , the preset time trajectory tracking control method for a Markov jump-based mechanical arm provided by an embodiment of the present application mainly includes the following steps: 101. A two-degree-of-freedom mechanical arm dynamics model with Markov jump characteristics is established, joint state data of the mechanical arm is obtained, a trajectory tracking error is calculated in combination with a predefined desired trajectory.

[0026] The two-degree-of-freedom mechanical arm dynamics model with Markov jump characteristics can describe the dynamic characteristics of the mechanical arm under different modes. Different modes can be caused by external interference, load change or system internal fault and other factors. In the model establishment process, the physical parameters of the mechanical arm, such as link mass, length, moment of inertia, are defined in detail, and the value set of the Markov chain and its transition probability matrix are set to accurately reflect the random switching process between system modes. Through the two-degree-of-freedom mechanical arm dynamics model, the joint angle, angular velocity and other state data of the mechanical arm can be obtained in real time, providing a basis for the calculation of the subsequent trajectory tracking error.

[0027] A two-link mechanical arm is taken as a simulation analysis embodiment, as shown in Figure 2 . Wherein and respectively represent the mass of the two sections of the mechanical arm, and respectively represent the length of the two sections of the mechanical arm, and represent the moment of inertia of the mechanical arm around the center of mass, is the acceleration of gravity, and are the joint angles of the two sections of the mechanical arm, and the two sections of the mechanical arm can independently and freely move.

[0028] For a two-link rigid mechanical arm, the two-link rigid mechanical arm dynamics equation can be obtained from the Lagrange equation. Considering the system mode jump caused by load change, the two-degree-of-freedom mechanical arm dynamics model with Markov jump characteristics is established as: ; wherein, is a joint angle position vector, and are an angular velocity vector and an angular acceleration vector, respectively, is a positive definite inertia matrix, is a centrifugal and Coriolis force matrix, is a gravity matrix, is a control torque, is a continuous-time Markov chain taking values in a finite set to describe the random jumps of the system among three different modes.

[0029] Further, in combination with a predefined desired trajectory, a trajectory tracking error is calculated, including: determining an expected joint angle position and an expected angular velocity corresponding to the predefined desired trajectory; obtaining a joint angle position vector and an angular velocity vector during operation of the robot arm; calculating a difference between the joint angle position vector and the expected joint angle position to obtain an angle position tracking error; calculating a difference between the angular velocity vector and the expected angular velocity to obtain an angular velocity tracking error; generating a trajectory tracking error based on the position tracking error and the angular velocity tracking error.

[0030] Specifically, a state variable is selected, wherein, is a joint angle position vector, is an angular velocity vector. The dynamics model of the two-degree-of-freedom robot arm can be converted into the following expression: ; Further define a tracking error, wherein, is calculated by a difference between an actual joint angle position vector and an expected joint angle position , reflecting the deviation degree in the position dimension, is calculated by a difference between an actual joint angular velocity vector and an expected angular velocity , reflecting the deviation degree in the velocity dimension.

[0031] Substitute the angle position tracking error and the angular velocity tracking error into the converted state space expression, and introduce the second derivative of the expected joint angle, to finally derive the trajectory tracking error of the robot arm. The trajectory tracking error expression is: .

[0032] 102. Nonlinearly transforming the trajectory tracking error and combining the transformed result with a function of the preset upper limit of the convergence time to construct a nonsingular terminal sliding mode surface, to obtain a sliding mode surface function.

[0033] The construction of the nonsingular terminal sliding mode surface provides an important reference for subsequent trajectory tracking control, and helps to improve the accuracy and stability of the trajectory tracking of the robot arm.

[0034] Specifically, the trajectory tracking error is nonlinearly transformed, and the transformed result is combined with a function of the preset upper limit of the convergence time to construct a nonsingular terminal sliding mode surface, to obtain a sliding mode surface function, including: setting a segmented critical value , an upper limit of the convergence time , and a transformation parameter ; wherein the segmented critical value is an arbitrarily small normal number, the upper limit of the convergence time is a positive number, and the transformation parameter has a value range of a positive number between 0 and 1; extracting an angular position tracking error from the trajectory tracking error, and calculating the absolute value of the angular position tracking error ; comparing the absolute value of the angular position tracking error with the segmented critical value ; if , nonlinearly transforming using a fractional power containing the transformation parameter and a sign function; wherein the nonlinear transformation is: ; if , nonlinearly transforming using a combination of a linear term and a quadratic term; wherein the nonlinear transformation form is: ; wherein , ; setting a positive adjustment parameter , , constructing a function of the upper limit of the convergence time in combination with the upper limit of the convergence time , the transformation parameter ; extracting an angular velocity tracking error from the trajectory tracking error, and combining the nonlinearly transformed angular position tracking error result , a corresponding coefficient term ,The corresponding coefficient terms are superimposed, multiplied by a function of the upper limit of the convergence time, and finally multiplied by the angular velocity tracking error The sliding mode surface function is obtained by adding the above two terms. The sliding mode surface function is: ； ， ， ， ， ， ， is an arbitrarily small positive number, is the preset upper limit of the convergence time.

[0035] It can be understood that, in order to understand the formula, in the sliding mode surface function, and in the trajectory tracking error have the same meaning, except that the content in the parentheses is omitted for ease of writing and understanding.

[0036] 103. Substitute the sliding mode surface function into the preset equivalent control law formula to calculate the equivalent control term.

[0037] Specifically, substitute the sliding mode surface function into the preset equivalent control law formula to calculate the equivalent control term, including: Obtain the expected angular acceleration data corresponding to the predefined expected trajectory and the expected angular velocity data ; Extract the upper limit of the convergence time , the transformation parameter , the adjustment parameter , and determine the nonlinear transformation coefficient ; Call the positive definite inertia matrix in the two-degree-of-freedom robotic arm dynamics model with Markov jump characteristics , the centrifugal force and Coriolis force matrix , the gravity matrix ; Substitute the expected angular acceleration data , the expected angular velocity data , the upper limit of the convergence time , the transformation parameter , the adjustment parameter , and the nonlinear transformation coefficient ​​​, a positive definite inertia matrix , a centrifugal and Coriolis force matrix , a gravity matrix Substitute the equivalent control term formula to obtain the equivalent control term; The equivalent control term formula is: .

[0038] 104, introduce a robust term including a sign function of the sliding surface function and a switching gain to generate a robust switching term.

[0039] Specifically, the robust term including the sign function of the sliding surface function and the switching gain is introduced to generate the robust switching term, including: Two positive switching gain parameters , are set, and satisfy ; A preset time parameter is set; Obtain the state data of the sliding surface function, and calculate the norm of the sliding surface function ; Based on the preset operation rule, the norm of the sliding surface function is respectively adapted and operated with the two switching gain parameters , to obtain the approaching law related intermediate variable; Substitute the positive switching gain parameter, the preset time parameter and the norm into the robust switching term formula to obtain the robust switching term; The robust switching term formula is: .

[0040] 105, add the equivalent control term and the robust switching term to synthesize the control torque, and input the control torque to the mechanical arm dynamics system to drive the mechanical arm joint to move, so that the mechanical arm joint movement trajectory tracks the expected trajectory.

[0041] The control torque can represent the specific force size and direction required by the mechanical arm in the trajectory tracking process, and is a key control quantity to ensure that the mechanical arm moves accurately according to the preset trajectory.

[0042] Specifically, the equivalent control term and the robust switching term are added to synthesize the control torque, and the control torque is input to the mechanical arm dynamics system to drive the mechanical arm joint to move, so that the mechanical arm joint movement trajectory tracks the expected trajectory, including: Extract the equivalent control term and the robust switching term, and directly superimpose the equivalent control term and the robust switching term by algebraic addition operation to obtain the control torque; The control torque is sent to the two-degree-of-freedom robot arm dynamics system with Markov jump characteristics as an input signal, replaces the corresponding control torque parameter in the two-degree-of-freedom robot arm dynamics model, continuously adjusts the joint motion state of the robot arm, and until the joint motion trajectory is consistent with the predefined expected trajectory.

[0043] The equivalent control term calculated by the preset equivalent control law formula and the robust switching term generated after introducing the sign function and switching gain of the sliding mode surface function are extracted, and the calculation basis of the two control data is consistent with the Markov jump mode in which the robot arm is currently located.

[0044] Then, the equivalent control term and the robust switching term are directly superimposed by using algebraic addition operation to synthesize the control torque capable of driving the joint motion of the robot arm, and the control torque has the dual effects of offsetting the inherent dynamics of the system and suppressing external disturbances.

[0045] Then, the synthesized control torque is sent to the two-degree-of-freedom robot arm dynamics system with Markov jump characteristics as an input signal, replaces the original control torque parameter in the system; under the action of the control torque, the robot arm dynamics system combines the positive definite inertia matrix, the centrifugal force and Coriolis force matrix, the gravity matrix and other dynamics parameters corresponding to the current mode, solves the joint angular acceleration in real time through the internal dynamics equation, and then drives the continuous optimization of the joint angular velocity and the joint angular position, forms a closed loop regulation, and until the joint motion trajectory of the robot arm is accurately matched with the predefined expected trajectory and consistent.

[0046] In some embodiments, in order to verify the performance of the preset time trajectory tracking control method of the robot arm based on Markov jump, the following steps are further included: A first Lyapunov function is constructed based on the sliding mode surface function and the positive definite inertia matrix; The time derivative of the first Lyapunov function is solved and simplified by using the dynamics characteristics of the robot arm, the minimum and maximum eigenvalues of the positive definite inertia matrix in all modes are extracted, and the convergence characteristics of the first Lyapunov function; A second Lyapunov function is constructed based on the angle position tracking error, and the error correlation is obtained in combination with the convergence characteristics of the sliding mode surface; The error correlation is substituted into the sliding mode surface function to verify the convergence characteristics of the second Lyapunov function; If the convergence characteristics of the first Lyapunov function and the second Lyapunov function are both convergent, a simulation model is constructed.

[0047] Specifically, the first Lyapunov function is: ; Taking the time derivative of the first Lyapunov function, we have ; From =0, we have .

[0048] According to , since there is a Markov jump, will change with the mode, in order to get a unified stability conclusion and a unified preset upper limit of time, we need to select ; further, from , we can get ; where , ; Further arrangement gives ; where , ; Finally, we get ； where , at different mode switching moments, strictly decreasing, the convergence characteristics of the first Lyapunov function is convergence.

[0049] Construct the second Lyapunov function, where the second Lyapunov function is ; According to the sliding mode surface function, when s=0, we have ; Take the reciprocal of the time of the second Lyapunov function, and we have

[0050] Therefore, the convergence characteristics of the second Lyapunov function is convergence.

[0051] Since the convergence characteristics of the first Lyapunov function and the second Lyapunov function are both convergence, the preset time trajectory tracking control method for the mechanical arm based on Markov jump provided by the embodiment of the application not only realizes accurate tracking of the trajectory of the mechanical arm system, but also enables the trajectory tracking error to converge to a balanced state within a preset time, and the rapid convergence characteristics and high-precision tracking capability fully verify the practicability and effectiveness of the preset time trajectory tracking control method for the mechanical arm based on Markov jump.

[0052] In some embodiments, the simulation model is constructed by: Setting physical parameters of the two-degree-of-freedom mechanical arm, the physical parameters including mass of each link, length of each link, moment of inertia of each link around the center of mass, and gravitational acceleration; Defining system parameters of the Markov jump system, and determining a value set of the Markov chain; the value set including three elements corresponding to three operating modes of the Markov jump system; Setting a transition probability matrix between the three operating modes, and configuring corresponding parameters of a positive definite inertia matrix, a centrifugal force and Coriolis force matrix, and a gravity matrix for each operating mode; Integrating the non-singular terminal sliding mode surface construction logic, the equivalent control term calculation logic, and the robust switching term generation logic to form an execution process of the controller; Establishing a signal interaction path between the controller and the mechanical arm model, so that the control torque output by the controller is input to the two-degree-of-freedom mechanical arm dynamics model, and the mechanical arm model can feed back joint angle position data and angular velocity data to the controller; Setting a desired trajectory of the end effector of the mechanical arm in the simulation platform; Importing the physical parameters, the Markov jump system parameters, the execution process of the controller, the mechanical arm dynamics model, and the desired trajectory into a simulation tool to form a simulation model.

[0053] For example, the application is simulated and verified by running MATLAB / Simulink in a Windows 10 operating system environment. The simulation platform is equipped with an Intel(R) Core(TM) i5-9400F processor to ensure the calculation accuracy and real-time performance during the simulation research.

[0054] Figure 2 The physical parameters of the two-link mechanical arm are as follows: , , , , , The desired trajectory of the end effector is: , , and the initial joint angle position is: , the initial joint angular velocity is: . The parameters of the controller are: , , , , , , , . Markov jump system mode definition: mode 1 (normal load): ; mode 2 (increase 20% load): ; mode 3 (decrease 20% load): . Markov transition probability matrix: ; steady state probability distribution: .

[0055] Under the above simulation conditions, the mechanical arm control system is simulated to verify the trajectory tracking ability of the mechanical arm control system, and the simulation results are shown in Figures 3 to 7 .

[0056] Figure 3 is a comparison diagram of the joint angle tracking effect of the mechanical arm. The dashed line in the figure represents the desired joint angle target trajectory, and the solid line represents the actual tracking trajectory. By comparison, it can be observed that the actual trajectory and the target trajectory are highly coincident, indicating that the control algorithm can quickly respond to the angle command and maintain stable tracking without obvious overshoot or oscillation phenomenon, verifying the accuracy of the angle control. Figure 4 Further shows the curve of the joint angle tracking error changing with time. The error value tends to zero and the fluctuation range is small, indicating that the control system has high steady state accuracy and strong anti-interference ability. The convergence characteristics of the error curve reflect the robustness of the control algorithm. Figure 5 is a position tracking diagram of the mechanical arm end effector in the operational space. The dashed line in the figure is the desired end trajectory, and the solid line is the actual trajectory. The close tracking of the end trajectory proves that the effectiveness of the joint angle control has been transmitted to the task space, meeting the demand of complex path planning and being suitable for high-precision operation scenarios. Figure 6 shows the change curve of the driving torque required by each joint during control. The torque output is smooth and has no sudden change, indicating that the control algorithm can reasonably allocate the kinetic load and avoid torque saturation or severe fluctuation, which is beneficial to prolong the service life of the effector and ensure the safety of the system. Figure 7 reflects the switching process of the mechanical arm system between different operating modes. The time points of mode switching and the corresponding state changes are clearly indicated in the figure, indicating that the control strategy designed by the present application can adapt to various working conditions and has good self-adaptation and fault tolerance capability.

[0057] In some embodiments, the preset time trajectory tracking control method for a mechanical arm based on Markov jump provided by the embodiments of the present application further comprises: obtain a simulation result of the simulation model; determine whether the simulation result meets preset conditions; if the simulation result meets the preset conditions, obtain a control parameter corresponding to the simulation result; configure the control parameter into a mechanical arm control system, start a control process, and drive the mechanical arm to perform a trajectory tracking task.

[0058] Specifically, a simulation result of a simulation model of a combination of a preset time sliding mode controller under a Markov jump system and a two-degree-of-freedom mechanical arm is obtained. The simulation result includes a mechanical arm joint angle tracking curve, trajectory tracking error change data, joint driving torque fluctuation, system modal switching time sequence, and stability performance.

[0059] Then, it is determined whether the simulation result meets preset conditions according to a preset control performance evaluation standard. For example, the preset conditions include that a trajectory tracking error converges to an allowable error range within a preset upper limit of time, a joint driving torque output is smooth and has no sudden change, there is no obvious chattering phenomenon in the system running process, and there is no significant fluctuation in tracking performance when the Markov modal switches.

[0060] If the simulation result meets all the preset conditions, a control parameter corresponding to the simulation result is extracted. The control parameter includes a segmented critical value required for sliding mode surface construction, a preset convergence time upper limit, a transformation parameter, a positive number adjustment parameter, and two switching gain parameters and a preset time parameter of a robust switching term, and all verified effective parameters.

[0061] Finally, the extracted control parameter is accurately configured into a parameter setting module of an actual mechanical arm control system to ensure that the parameter values are completely consistent with those in the simulation verification stage. Then, a mechanical arm control process is started, so that the mechanical arm adjusts the joint motion state in real time under the driving of a control torque integrated with an equivalent control term and a robust switching term, and finally accurately performs a trajectory tracking task, thereby realizing stable matching of the joint motion trajectory and a predefined expected trajectory.

[0062] Figure 8 is a structural schematic diagram of an electronic device provided by an embodiment of the present application.

[0063] As Figure 8As shown, the electronic device can include a processor 810, a communications interface 820, a memory 830, and a communications bus 840, wherein the processor 810, the communications interface 820, and the memory 830 complete mutual communication through the communications bus 840. The processor 810 can invoke a logic instruction in the memory 830 to execute the preset time trajectory tracking control method of the Markov jump-based robot arm.

[0064] In addition, the logic instruction in the memory 830 described above can be implemented in the form of a software function unit and sold or used as an independent product, and can be stored in a computer-readable storage medium. Based on such understanding, the technical solutions of the present application essentially or the part that contributes to the prior art or part of the technical solutions can be embodied in the form of a software product. The computer software product is stored in a storage medium, and includes several instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present application. The aforementioned storage medium includes: a U disk, a mobile hard disk, a read-only memory (ROM, Read-Only Memory), a random access memory (RAM, Random Access Memory), a magnetic disk or an optical disk, and various media that can store program codes.

[0065] On the other hand, the present application also provides a computer program product, which includes a computer program, the computer program can be stored on a non-transitory computer readable storage medium, and the computer program is executed by a processor, so that the computer can execute the Markov jump-based robot arm preset time trajectory tracking control method provided by the above-mentioned methods.

[0066] In yet another aspect, the present application also provides a non-transitory computer readable storage medium having a computer program stored thereon, the computer program being executed by a processor to implement the Markov jump-based robot arm preset time trajectory tracking control method provided by the above-mentioned methods.

[0067] The device embodiments described above are only schematic, wherein the units shown as separate components can or can not be physically separated, and the components shown as units can or can not be physical units, i.e., they can be located in one place, or distributed on multiple network units. Part or all of the modules can be selected to achieve the purpose of the present embodiment scheme according to actual needs. Those skilled in the art can understand and implement it without creative labor.

[0068] Those skilled in the art can clearly understand the technical solutions of the various embodiments from the above description of the embodiments, and the various embodiments can be implemented by means of software with the necessary general hardware platforms, and of course, can also be implemented by hardware. Based on such understanding, the above technical solutions, essentially or in other words, the part of the prior art that makes a contribution, can be embodied in the form of a software product, which can be stored in a computer readable storage medium, such as a ROM / RAM, a magnetic disk, an optical disk, and the like, and includes a number of instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0069] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, rather than limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement for some technical features therein; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present application.

Claims

1. A method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps, characterized in that, include: A dynamic model of a two-degree-of-freedom robotic arm with Markov jump characteristics is established, the joint state data of the robotic arm is obtained, and the trajectory tracking error is calculated by combining the predefined expected trajectory. The trajectory tracking error is nonlinearly transformed, and the transformed result is combined with a function of a preset upper limit of convergence time to construct a non-singular terminal sliding surface and obtain the sliding surface function. Substitute the sliding surface function into the preset equivalent control law formula to calculate the equivalent control term; A robust switching term is generated by introducing a sign function that includes the sliding surface function and a robust term for the switching gain. The equivalent control term is added to the robust switching term to synthesize a control torque, which is then input into the robotic arm dynamics system to drive the robotic arm joints to move, so that the robotic arm joint movement trajectory tracks the desired trajectory.

2. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, Establish a dynamic model of the robotic arm with Markov jumping characteristics, and obtain the joint state data of the robotic arm, including: Establish a dynamic model of a two-degree-of-freedom manipulator with Markov jump characteristics; The dynamic model of the two-degree-of-freedom robotic arm is as follows: ； in, This is the joint angle position vector. and These are the angular velocity vector and the angular acceleration vector, respectively. It is a positive definite inertial matrix. The matrix represents the centrifugal force and the Coriolis force. For the gravity matrix, To control the torque, For a finite set A continuous-time Markov chain that takes values ​​from the given values.

3. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, The calculation of trajectory tracking error, based on a predefined expected trajectory, includes: Determine the desired joint angle position and desired angular velocity corresponding to the predefined desired trajectory; Obtain the joint angle position vector and the angular velocity vector during the operation of the robotic arm; The difference between the joint angle position vector and the desired joint angle position is calculated to obtain the angle position tracking error; The difference between the angular velocity vector and the desired angular velocity is calculated to obtain the angular velocity tracking error. Based on the angular position tracking error and the angular velocity tracking error, a trajectory tracking error is generated.

4. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, The trajectory tracking error is nonlinearly transformed, and the transformed result is combined with a function of a preset upper limit for convergence time to construct a non-singular terminal sliding surface, resulting in a sliding surface function, including: Set segmented critical values Upper limit of convergence time and transformation parameters ; where the segmented critical value For arbitrarily small positive constants, the upper limit of convergence time is... For positive numbers, transform parameters The value of is a positive number between 0 and 1; Extract the angle position tracking error from the trajectory tracking error. Calculate the angle and position tracking error The absolute value; Angular position tracking error absolute value and piecewise critical value Compare; like Then, the transformation parameter is used. Fractional powers and sign functions Perform a nonlinear transformation; where the nonlinear transformation is: ； like Then, the form of a combination of linear and quadratic terms is used to... Perform a nonlinear transformation; the nonlinear transformation takes the form of: ； in, , ; Set positive adjustment parameters , Combined with the upper limit of convergence time Transformation parameters Construct a function that sets the upper limit of the convergence time; Extracting angular velocity tracking error from trajectory tracking error The nonlinearly transformed angle position tracking error results, Corresponding coefficient terms The corresponding coefficients are summed, multiplied by a function of the upper limit of convergence time, and finally multiplied by the angular velocity tracking error. Adding them together yields the sliding surface function. The sliding surface function is: ； in, , , ， ， ， ， ， ， It is an arbitrarily small positive integer. This is the preset upper limit for convergence time.

5. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 2, characterized in that, Substituting the sliding surface function into the preset equivalent control law formula, the equivalent control terms are calculated, including: Obtain the desired angular acceleration data corresponding to the predefined desired trajectory. and expected angular velocity data ; Extracting the upper limit of convergence time from the sliding surface function Transformation parameters Adjusting parameters Determine the nonlinear transformation coefficients ; Calling the positive definite inertia matrix in the dynamic model of a two-degree-of-freedom manipulator with Markov jump properties Centrifugal force and Coriolis force matrix Gravity matrix ; Desired angular acceleration data Expected angular velocity data Upper limit of convergence time Transformation parameters Adjusting parameters Nonlinear transformation coefficients Positive definite inertial matrix Centrifugal force and Coriolis force matrix Gravity matrix Substituting into the equivalent control term formula, we obtain the equivalent control term; The formula for the equivalent control term is: 。 6. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, Introducing a robust term that includes the sign function of the sliding surface function and the switching gain, a robust switching term is generated, including: Set two positive gain switching parameters , And satisfy ; Set preset time parameters ; Obtain the state data of the sliding surface function and calculate the norm of the sliding surface function. ; Based on preset calculation rules, the norm of the sliding surface function is... With two switching gain parameters respectively , Perform adaptation operations to obtain intermediate variables related to the convergence law; Substituting the positive switching gain parameter, the preset time parameter, and the norm into the robust switching term formula, we obtain the robust switching term; The robust switching term formula is as follows: 。 7. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, The equivalent control term and the robust switching term are added to synthesize a control torque, which is then input to the robotic arm dynamics system to drive the robotic arm joints to move, so that the joint motion trajectory of the robotic arm tracks the desired trajectory, including: Extract the equivalent control term and the robust switching term, and directly superimpose the equivalent control term and the robust switching term through algebraic addition to obtain the control torque; The control torque is sent as an input signal to the two-degree-of-freedom manipulator dynamics system with Markov jump characteristics, replacing the corresponding control torque parameters in the two-degree-of-freedom manipulator dynamics model, and continuously adjusting the joint motion state of the manipulator until the joint motion trajectory is consistent with the predefined desired trajectory.

8. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 1, characterized in that, Also includes: Based on the sliding surface function and the positive definite inertia matrix, the first Lyapunov function is constructed; The time derivative of the first Lyapunov function is calculated and simplified using the dynamic characteristics of the robotic arm. The minimum and maximum eigenvalues ​​of the positive definite inertia matrix under all modes are extracted to verify the convergence characteristics of the first Lyapunov function. Based on the angular position tracking error, a second Lyapunov function is constructed, and the error correlation is obtained by combining the convergence characteristics of the sliding surface. Substitute the error correlation into the sliding surface function to verify the convergence characteristics of the second Lyapunov function; If both the first and second Lyapunov functions exhibit convergence, then a simulation model is constructed.

9. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 8, characterized in that, The simulation model construction includes: Set the physical parameters of the two-degree-of-freedom robotic arm, including the mass of each link, the length of each link, the moment of inertia of each link about its center of mass, and the acceleration due to gravity. Define the system parameters of the Markov jump system and determine the set of values ​​for the Markov chain; the set of values ​​includes three elements corresponding to the three operating modes of the Markov jump system. Define the transition probability matrix between the three operating modes, and configure the corresponding parameters of the positive definite inertia matrix, centrifugal force and Coriolis force matrix, and gravity matrix for each operating mode; The non-singular terminal sliding surface construction logic, equivalent control term calculation logic, and robust switching term generation logic are integrated to form the controller's execution flow; Establish a signal interaction path between the controller and the robotic arm model, so that the control torque output by the controller is input to the two-degree-of-freedom robotic arm dynamics model, and the robotic arm model can feed back joint angle position data and angular velocity data to the controller. Set the desired trajectory of the robotic arm's end effector in the simulation platform; The physical parameters, Markov jump system parameters, controller execution flow, robotic arm dynamics model, and desired trajectory are imported into a simulation tool to form a simulation model.

10. The method for tracking and controlling a robotic arm's preset time trajectory based on Markov jumps according to claim 9, characterized in that, Also includes: Obtain the simulation results of the simulation model; Determine whether the simulation results meet the preset conditions; If the simulation results meet the preset conditions, then the control parameters corresponding to the simulation results are obtained; Configure the control parameters into the robotic arm control system, start the control process, and drive the robotic arm to perform trajectory tracking tasks.

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