Time optimal model prediction control method for tendon-driven redundant mechanical arm

Abstract

The invention provides a time optimal model prediction control method for a tendon-driven redundant mechanical arm, and belongs to the crossing field of prediction control and optimization control in the control theory. According to the method, a dynamic adjustable scale factor is introduced into a model predictive control (MPC) framework, time optimality is brought into an optimization objective, and a time optimal model predictive control (TOMPC) optimization problem including multiple physical constraints such as tendon length and speed is constructed; performing efficient iterative solution on the nonlinear optimization problem by using a sequential quadratic programming (SQP) algorithm to obtain an optimal control sequence and a scale factor in a future time domain; and an optimal control instruction at the current moment is extracted to drive the mechanical arm to move, and closed-loop rolling optimization is achieved through state feedback. According to the method, on the premise that system constraints are met, the tracking speed of the end effector is maximized, the task completion time is remarkably shortened, meanwhile, the tracking precision and operation safety are guaranteed, and the operation efficiency and economical efficiency of the redundant mechanical arm in an industrial automation scene are effectively improved.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of predictive control and optimization control in control theory, and specifically relates to a time-optimal model predictive control method for a tendon-driven redundant robotic arm. Background Technology

[0002] With the widespread application of high-degree-of-freedom underactuated robots in industrial automation, medical surgery, and complex environment operations, the position control of their end effectors has become one of the key factors affecting their practical application effectiveness. Especially under multiple physical constraints such as tendon length and drive speed, the motion control of robotic arms must balance the accuracy of path tracking, the real-time performance of operations, and the energy efficiency of system operation. In recent years, Model Predictive Control (MPC) has gradually become one of the important strategies for controlling complex robot systems due to its advantages in handling system constraints and multi-objective optimization.

[0003] Most model predictive control methods optimize predefined end effector trajector trajector trajectories. While they can effectively handle system constraints and meet performance targets, they fail to fully exploit the potential of robotic arms in terms of time optimization. Specifically, the control process typically assumes that the trajectory tracking speed is fixed or varies according to a preset pattern, failing to dynamically maximize the tracking speed of the end effector while satisfying physical constraints. This limits the efficiency and economy of the robotic arm when performing tasks. Especially in repetitive and time-sensitive production environments, trajectory tracking control lacking time optimization directly leads to longer task completion times, reduced equipment utilization, and increased production costs. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a time-optimal model predictive control method for a tendon-driven redundant robotic arm.

[0005] To achieve the above objectives, the following steps are specifically included: S1. Establish a system kinematic model: Based on the tendon length vector, after establishing the position mapping through the forward kinematic equation, the position and velocity mapping relationship of the end effector is obtained; The motion of the redundant robotic arm in this invention is controlled by forward kinematics equations. A position mapping is established using these equations, yielding the expression for the end effector position:

[0006] in, For the position of the end effector, The dimension of the end effector position. Let the tendon length vector be... Let be the dimension of the tendon length vector. This represents a nonlinear mapping from actuator space to workspace. The expression for establishing the velocity mapping relationship using differential kinematics is as follows:

[0007] in, To calculate the Jacobian matrix, To find the derivative.

[0008] S2. Constructing a discretized system model: Based on the end effector position, continuous-time model, and sampling time, a prediction model is constructed after discretization using the forward Euler method. Based on the discretized expression using the forward Euler method, the state prediction model of this invention can be obtained. The expression is:

[0009] in, For a moment hour Length, Sampling time, For sampling index number, For a moment, It predicts the time domain. To predict the number of steps, for The index.

[0010] Based on the state prediction model and the end effector position, using a first-order Taylor expansion, the expression is:

[0011] Substituting the first-order Taylor expansion expression into the state prediction model, the predicted output of the robotic arm system of this invention can be obtained. Therefore, the prediction model of this invention... The expression is: .

[0012] S3. Constructing the Time-Optimal Model Predictive Control (TOMPC) Problem: Based on the output prediction model and velocity mapping relationship, the TOMPC problem is constructed by defining the optimization objective function, integrating constraints, and performing time scaling. The time-optimal model for predicting and controlling the TOMPC problem based on a tendon-driven redundant robotic arm is expressed as follows:

[0013] in, , and These are the position tracking error weight matrix, the control input weight matrix, and the scale factor weight matrix, respectively. The conditions for TOMPC are constraints on the tendon length, tendon speed, and scaling factor of the redundant robotic arm.

[0014] S4. Transform the TOMPC problem into a standard nonlinear programming constraint problem: Based on the TOMPC problem and the prediction time domain, by defining the optimization variable vector, the TOMPC problem is transformed into a standard nonlinear programming form, resulting in a standard nonlinear programming constraint problem. The standard expression for a nonlinear programming constraint problem is as follows:

[0015] in, For state variables; and The expression is as follows: .

[0016] in, , , For constraint vectors, This is the constraint coefficient matrix.

[0017] S5. Solving using a sequential quadratic programming algorithm: Based on the standard nonlinear programming and constraint problem, a quadratic approximation function of the Lagrangian function is constructed to obtain a series of quadratic programming problems. The QP-LVI algorithm is used to solve the quadratic programming problems to obtain the search direction. Based on the search direction, search updates and convergence judgments are performed to obtain the optimal solution, thus completing the time-optimal model predictive control method for tendon-driven redundant robotic arms.

[0018] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) This invention introduces a dynamic adjustable scaling factor to achieve optimal time tracking, which significantly improves the efficiency of robotic arm operation. It solves the problem that the speed is fixed in the traditional time model predictive control (MPC) method in the predefined trajectory tracking, which cannot maximize the speed of the end effector under the premise of satisfying the constraints, resulting in long task cycle and low production efficiency. (2) This invention enhances the feasibility and safety of the system under complex constraints by constructing a time-optimal model containing multiple physical constraints to predict and control the TOMPC rolling optimization problem. It solves the problem that although the existing methods can handle some joint constraints, they are not capable of comprehensively handling multiple constraints such as tendon length, speed, and stiffness, and it is difficult to ensure the safety of the system in high-speed tracking. (3) This invention improves the real-time optimization capability and convergence speed by using the SQP algorithm to solve the nonlinear TOMPC problem. It solves the problem that the traditional time-optimal control problem is complicated to solve and difficult to complete real-time calculation within the sampling period, which limits its application in high-speed control scenarios. (4) This invention improves trajectory tracking accuracy and system robustness by explicitly combining the predicted state with the future trajectory. Attached Figure Description

[0019] Figure 1 This is a flowchart of a time-optimal model predictive control method for a tendon-driven redundant robotic arm according to the present invention. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0021] like Figure 1 As shown, a time-optimal model predictive control method for a tendon-driven redundant robotic arm specifically includes the following steps: S1. Establish a system kinematic model: Based on the tendon length vector, after establishing the position mapping through the forward kinematic equation, the position and velocity mapping relationship of the end effector is obtained; The motion of the redundant robotic arm in this invention is controlled by forward kinematics equations. A position mapping is established using these equations, yielding the expression for the end effector position:

[0022] in, For the position of the end effector, The dimension of the end effector position. Let the tendon length vector be... Let be the dimension of the tendon length vector. This represents a nonlinear mapping from actuator space to workspace. The expression for establishing the velocity mapping relationship using differential kinematics is as follows:

[0023] in, To calculate the Jacobian matrix, To find the derivative.

[0024] S2. Constructing a discretized system model: Based on the end effector position, continuous-time model, and sampling time, a prediction model is constructed after discretization using the forward Euler method. From the perspective of kinematic control, the control signal for the redundant robotic arm is the tendon velocity, so the continuous-time model expression of this invention is as follows:

[0025] in, For the control signals of the robotic arm system; The discretization expression using the forward Euler method in this invention is as follows:

[0026] In the formula, For a moment hour Length, Sampling time, For sampling index number, For a specific moment; Based on the discretized expression using the forward Euler method, the state prediction model of this invention can be obtained. The expression is:

[0027] in, It predicts the time domain. To predict the number of steps, for The index.

[0028] Based on the state prediction model and the end effector position, using a first-order Taylor expansion, the expression is:

[0029] Substituting the first-order Taylor expansion expression into the state prediction model, the predicted output of the robotic arm system of this invention can be obtained. Therefore, the prediction model of this invention... The expression is: .

[0030] S3. Constructing the Time-Optimal Model Predictive Control (TOMPC) Problem: Based on the prediction model and velocity mapping relationship, the TOMPC problem is constructed by defining the optimization objective function, integrating constraints, and performing time scaling. Expression of the prediction model This invention uses a Taylor expansion; to achieve time-optimal motion control of a tendon-driven redundant robotic arm under physical constraints, it introduces a scaling factor. To maximize the tracking speed of the end effector. It is feasible; however, it is worth noting that the time span for tracking the same expected trajectory differs at different speeds. This invention aims to achieve the same coordinate interval. , and The sampling times cannot be the same; considering Input signal Due to hardware limitations, the sampling time is set to a fixed value for control. . Sampling time Derived from the following formula:

[0031] in, To introduce a scaling factor, The desired position vector of the end effector at the current moment; The time-optimal model for predicting and controlling the TOMPC problem based on a redundant robotic arm is expressed as follows:

[0032] in, , and These are the position tracking error weight matrix, the control input weight matrix, and the scale factor weight matrix, respectively. The conditions for TOMPC are constraints on the tendon length, tendon speed, and scaling factor of the redundant robotic arm.

[0033] S4. Transform the TOMPC problem into a standard nonlinear programming constraint problem: Based on the TOMPC problem and the prediction time domain, by defining the optimization variable vector, the TOMPC problem is transformed into a standard nonlinear programming form, resulting in a standard nonlinear programming constraint problem. As can be clearly seen from the TOMPC problem, it is a rolling optimization method. To obtain its optimal solution, this invention transforms it into a nonlinear programming problem. For ease of subsequent derivation, a current state stacking vector is defined. , control input sequence vector Expected position sequence vector Scale factor sequence vector Expected velocity block diagonal matrix Time accumulation matrix Stacked vectors with initial state As shown below:

[0034] in, To control the time domain; Furthermore, the present invention provides that hour, Using the matrix designed above, the objective function TOMPC problem can be rewritten as follows:

[0035] in, The decision variables are in vector form. . and It is an identity matrix, and the subscript indicates its dimension. This is the Kronecker product. To further simplify, let the two variables... and The set is a state variable ;

[0036] in, , E is the control input selection matrix, and D is the scaling factor selection matrix; therefore, the state variables... It can be redescribed, and the state variable optimization function expression is as follows:

[0037] according to Sampling time It can be seen that, and With the expected trajectory function and variables Related; this invention concludes that when the desired trajectory is related to the variable When there is a linear relationship, the optimization function of the state variables is a quadratic programming problem; now consider the more general case, namely... and With variables There is a nonlinear relationship; therefore, it can be deduced that the TOMPC problem is a nonlinear problem, which can be further transformed into a standard form. The standard nonlinear programming constraint problem expression is as follows:

[0038] in, For state variables, For TOMPC problem functions, The vector of equality constraint functions; and The expression is as follows: .

[0039] in, For constraint vectors, This is the constraint coefficient matrix.

[0040] S5. Solving the problem using a sequential quadratic programming algorithm: Based on the standard nonlinear programming constraint problem, after defining the Lagrangian function, the quadratic programming problem is solved using the QP-LVI algorithm to obtain the search direction; based on the search direction, the search is updated and convergence is judged to obtain the optimal solution, thus completing the time-optimal model predictive control method for the tendon-driven redundant robotic arm; First, for a standard nonlinear programming constrained problem, the Lagrangian function is defined as follows:

[0041] in, and These are the Lagrange multipliers associated with equality constraints and inequality constraints, respectively. ; Using Taylor expansion, the objective function at the iteration point Simplifying to a quadratic function, the standard nonlinear programming constraint problem is reduced to a first-order linear function; therefore, the standard quadratic programming problem related to the standard nonlinear programming problem is defined as:

[0042] Among them, matrix It is a positive definite approximation of the Hessian Lagrangian function; for quadratic programming problems, the QP-LVI algorithm can find the optimal solution. ; This is considered the next search direction for the original problem; a one-dimensional search is performed on the variables of the original constrained problem in this direction to obtain an approximate solution to the original constrained problem. :

[0043] in, This represents the number of iterations in the sequential quadratic programming algorithm. This is the step size parameter for the line search process; simultaneously, This is used to form a new iteration. In the new iteration, the Hessian matrix of the positive definite quasi-Newton approximation of the Lagrange function is calculated using the BFGS method. :

[0044] in, , . and The solution is obtained directly from the solution of the quadratic programming subproblem. Finally, the iteration is repeated until the convergence condition is met, as shown below:

[0045] In summary, the sequential quadratic programming algorithm approximates the optimal solution of the original nonlinear problem with a quadratic convergence speed by sequentially solving a series of QP subproblems, thus completing the time-optimal model predictive control method for tendon-driven redundant robotic arms.

[0046] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A time-optimal model predictive control method for a tendon-driven redundant robotic arm, characterized in that, Includes the following steps: S1. Based on the tendon length vector, after establishing the position mapping through the forward kinematics equation, the position and velocity mapping relationship of the end effector is obtained; S2. Based on the end effector position, continuous-time model, and sampling time, a prediction model is constructed after discretization using the forward Euler method. S3. Based on the prediction model and velocity mapping relationship, the time-optimal model predictive control (TOMPC) problem is constructed by defining the optimization objective function, integrating constraints, and performing time scaling. S4. Based on the TOMPC problem and the prediction time domain, the TOMPC problem is transformed into a standard nonlinear programming constraint problem by defining an optimization variable vector. S5. Based on standard nonlinear programming and constraint problems, a quadratic approximation function of the Lagrange function is constructed to obtain a series of quadratic programming problems. The QP-LVI algorithm is used to solve the quadratic programming problems to obtain the search direction. Based on the search direction, search updates and convergence judgments are performed to obtain the optimal solution, thus completing the time-optimal model predictive control method for tendon-driven redundant robotic arms.

2. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, Based on the tendon length vector, after establishing the position mapping through the forward kinematics equations, the expression for the end effector position in the end effector position-velocity mapping relationship is obtained as follows: in, For the position of the end effector, For the dimension of the end effector position, Let the tendon length vector be... Let be the dimension of the tendon length vector. To drive the nonlinear mapping from the workspace to the driving space; The expression for the velocity mapping relationship is: in, To calculate the Jacobian matrix, To find the derivative.

3. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, The prediction model, constructed by discretizing the data using the forward Euler method based on the end effector position, continuous-time model, and sampling time, is expressed as follows: in, For the control signals of the robotic arm system, For sampling index number, For a moment, Sampling time, It predicts the time domain. To predict the number of steps, for index, This is the tendon length vector.

4. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, Based on the prediction model and velocity mapping relationship, by defining the optimization objective function, integrating constraints, and performing time scaling, the expression for the time-optimal model predictive control (TOMPC) problem function is constructed as follows: in, , and These are the position tracking error weight matrix, the control input weight matrix, and the scale factor weight matrix, respectively. To introduce a scaling factor, Let be the desired position vector of the end effector at the current moment. To calculate the Jacobian matrix, It predicts the time domain. To predict the number of steps, , To find the derivative, For the control signals of the robotic arm system, .

5. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, Based on the TOMPC problem and the prediction time domain, by defining an optimization variable vector, the TOMPC problem is transformed into a standard nonlinear programming form, resulting in the expression for the standard nonlinear programming constraint problem: in, For state variables, For TOMPC problem functions, Let them be the vector of equality constraint functions. For constraint vectors, This is the constraint coefficient matrix.

6. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, Based on the standard nonlinear programming and constraint problem, a quadratic approximation function of the Lagrange function is constructed, resulting in a series of quadratic programming problems. The QP-LVI algorithm is used to solve these quadratic programming problems, yielding the search direction. Define the Lagrange function as follows: in, For state variables, and These are the Lagrange multipliers associated with equality constraints and inequality constraints, respectively. , For TOMPC problem functions, Let them be the vector of equality constraint functions. For constraint vectors, This is the constraint coefficient matrix; The expression for solving the quadratic programming problem is: in, It is the Hessian positive definite approximation of the Lagrange function. This is the optimal solution. This represents the iteration point of the sequential quadratic programming algorithm. For state variables, For constraint vectors, This is the constraint coefficient matrix.

7. The time-optimal model predictive control method for a tendon-driven redundant robotic arm according to claim 1, characterized in that, In the method for time-optimal model predictive control of a tendon-driven redundant robotic arm, which involves searching, updating, and determining convergence based on the search direction to obtain the optimal solution, the convergence condition for convergence determination is as follows: In the formula, This is the optimal solution.

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