Event triggering model predictive control method of soft robot

Abstract

The invention relates to the field of soft-bodied robot control, in particular to a soft-bodied robot trajectory tracking control method based on event triggering model predictive control, which comprises the following steps: step 1, on the basis of a segmented constant curvature hypothesis of a soft-bodied robot, considering system uncertainty including unmodeled dynamics and exogenous disturbance; establishing a software robot dynamic model and converting the software robot dynamic model into a state space model; 2, designing an event triggering model prediction control strategy based on the state space model of the soft robot; 3, a dynamic event trigger threshold value and a dynamic prediction time domain method are introduced to improve the calculation efficiency; 4, the recursion feasibility and closed-loop stability of a control strategy are ensured through parameter design; and 5, performing simulation analysis on the end tracking of the soft robot by applying the designed event triggering model prediction control strategy.

Description

Technical Field

[0001] This invention relates to the field of soft robot control, and more specifically to an event-triggered model predictive control method for soft robots. Background Technology

[0002] Due to the flexible materials and special structural design used in soft robots, they have unlimited degrees of freedom, high flexibility and strong environmental adaptability. When facing special and complex tasks such as space exploration, medical treatment and rescue, they have obvious advantages over traditional rigid robots.

[0003] However, due to their characteristics of infinite degrees of freedom and nonlinearity, traditional control methods struggle to achieve high-precision control of the end effector's tracking. The system control of soft robots shares some similarities with traditional rigid robotic arms, but also exhibits significant differences. Because soft robots are nonlinear time-varying systems with infinite degrees of freedom, traditional control methods such as PID control are insufficient for achieving precise and efficient control, failing to meet practical requirements.

[0004] To achieve precise control, Model Predictive Control (MPC) is widely used to predict the future state of a system and adjust the input accordingly. However, since solving MPC requires a large amount of complex calculations, and soft robots are characterized by complex dynamic models and nonlinear time-varying systems, computational efficiency must be considered when using MPC to control soft robots to avoid affecting the real-time tracking effect of the control due to excessive computational resource consumption.

[0005] Against this backdrop, in order to achieve precise control of the end effector of a soft robot, a novel control strategy needs to be designed to meet the requirements of accurate and stable end effector tracking control, while maximizing computational efficiency. Summary of the Invention

[0006] The purpose of this invention is to provide an event-triggered model predictive control method for soft robots, which significantly reduces the computational burden of model predictive control and improves the real-time performance of the control system while ensuring trajectory tracking accuracy.

[0007] To achieve this goal, event-triggered control strategies are introduced into MPC control. Event-triggered control refers to designing an event-triggered mechanism that, when triggered by a system state, implements corresponding control strategies. Therefore, by designing an effective event-triggered mechanism and adjusting the event trigger threshold, it is possible to reduce the number of optimization calculations while ensuring control accuracy meets requirements, thereby improving control efficiency. Furthermore, by introducing dynamic event trigger thresholds and variable prediction time domain methods, computational efficiency is further improved.

[0008] The objective of this invention is achieved through the following technical solution:

[0009] An event-triggered model predictive control method for a soft robot, the method comprising the following steps:

[0010] Step 1: Based on the piecewise constant curvature assumption of the soft robot, considering the system uncertainties including unmodeled dynamics and exogenous disturbances, establish the dynamic model of the soft robot and transform it into a state-space model;

[0011] Step 2: Based on the state-space model of the soft robot, design an event-triggered predictive control strategy;

[0012] Step 3: Introduce dynamic event triggering thresholds and dynamic prediction time domain methods to improve computational efficiency.

[0013] Step 4: Ensure the recursive feasibility and closed-loop stability of the control strategy through parameter design;

[0014] Step 5: Apply the designed event-triggered model predictive control strategy to perform simulation analysis on the end-effector tracking of the soft robot;

[0015] The piecewise constant curvature assumption of the soft robot is as follows: Under the assumption of piecewise constant curvature, the soft robot is divided into a model with multiple constant curvatures, wherein the curvature of each segment is variable in time but constant in space, and each segment is connected end to end and smooth.

[0016] In step one, the system uncertainties, including unmodeled dynamics and exogenous disturbances, have The dynamic model of the soft robot segment is as follows:

[0017] (1)

[0018] in, This represents the spatial pose angle of the soft robot. Represents the real number field. and Let represent the first and second time derivatives of the pose angle, respectively. Represents the inertia matrix. It combines Coriolis force and centrifugal force Simulating the effects of gravity, and These are the stiffness matrix and the damping matrix, respectively. Input containing torque Mapping to the configuration space enables the continuum robot to be fully actuated;

[0019] Before designing an MPC strategy, the control system must first be defined and described. For a general nonlinear system, the definition is as follows: For system status, To control the input, then there is

[0020] (2)

[0021] Accordingly, the reference target trajectory is defined as Similarly,

[0022] (3)

[0023] Based on this, in order to achieve tracking of the reference trajectory, a definition is defined. Let the tracking error be the system state, then we have

[0024] (4)

[0025] (5)

[0026] Assuming the tracking error for the system state satisfy

[0027] (6)

[0028] Then there is , These represent the actual pose angle and the reference pose angle of the soft robot, respectively. This represents the tracking error of the soft robot's pose angle. Define the input torque. ,in, Indicates circling Moment in the axial direction; Indicates circling Moment in the axial direction; Indicates along Force in the axial direction. Define the input torque. and control input The relationship is , It is a symmetric positive definite matrix. Based on the above definitions, a state-space model of the tracking error and control input of the soft robot is established.

[0029] In step two, the design process of the event-triggered model predictive control is as follows:

[0030] For any nonlinear system It has an initial value Define a cost function of the following form. for:

[0031] (7)

[0032] in, For the first The moment when the optimization problem is solved; For the first The prediction time domain for solving the optimization problem; To determine the time points in the process of solving the optimization problem, . , and These represent the weight matrices for the state term, input term, and terminal state term in the cost function, respectively. and Indicates in arrive In the interval Real-time control inputs and system status. Terminal set area. The radius is set to ,Right now

[0033] (8)

[0034] To incorporate event-triggered strategies into model predictive control, the event triggering mechanism and timing are first defined. The criterion for determining the triggering timing is the actual system state. With predicted state The magnitude of the error between them defines the trigger time as:

[0035] (9)

[0036] in, For the system number The moment this event is triggered; Indicates the system at the 1st Second trigger The difference between the actual state error and the predicted state error at time step Norm; This indicates the designed event trigger threshold.

[0037] control system After the event is triggered, the future is obtained by solving an optimization problem. Time control input At the next triggering moment Previously, that is And the system status has not entered the terminal set, that is At that time, the system will continue using the previous time. The optimized control input; and when the system state enters the designed terminal set. The control input will switch to using a state feedback control law.

[0038] (10)

[0039] Therefore, the interval between two consecutive event trigger times must be less than or equal to... Therefore, the next triggering time is designed. for:

[0040] (11)

[0041] At the same time, the MPC optimization control problem must be solved at the beginning of system control, that is, it is required that... This will inevitably be triggered at the start of system control. Based on the above, the design of the predictive control strategy for the event-triggered model is complete.

[0042] In step three, the design process for the dynamic event triggering threshold and the dynamic prediction time domain is as follows:

[0043] Set event trigger threshold Defined as:

[0044] (12)

[0045] in, Indicates the first Next and first The time interval between the triggering of this event ; , , For design parameters, requirements .

[0046] By recording the trigger times of two consecutive events and The trigger time interval can be obtained. Therefore, the prediction time domain is designed as follows:

[0047] (13)

[0048] in, and The first , Predicted temporal domain at the time of event triggering; Design parameters to control the magnitude of the forward propagation in the prediction time domain, ; Design parameters to control the degree of time-domain shrinkage in prediction, Through the analysis of and The adjustment of the prediction time domain can control the magnitude of the descent rate, thereby achieving a balance between control accuracy and computational speed. The dynamic prediction time domain will be updated at each event trigger moment, i.e., as defined by formula (11). The prediction time domain is updated at each time step. And in this optimization solution, that is, the first... The suboptimal solution uses the prediction time domain. Solve the optimization problem.

[0049] In step four, to ensure the recursive feasibility and closed-loop stability of the control system, the design parameters are as follows:

[0050] For the weight matrix , and All matrices must be positive definite matrices and satisfy the following conditions:

[0051] (14)

[0052] in, For the equivalent matrix, ; Let Lyapunov be the system's Lyapunov function.

[0053] Near the system's equilibrium point, linearizing the system using the Jacobian matrix yields the following results:

[0054] in, Let be the linearized matrix of the system state at the origin. ;

[0055] To control the linearization of the input matrix at the origin, For this continuous-time system of the soft robot, the Riccati equations must be satisfied:

[0056] (15)

[0057] The weight matrix of the system's final state can be obtained from this continuous-time Riccati equation. The corresponding feedback matrix Defined as This makes the closed-loop system matrix It satisfies the Hurwitz condition.

[0058] To ensure that the control system meets input-state stability (ISS), the following design parameters need to be adjusted: terminal set radius. Event trigger threshold parameter and Dynamic prediction of time-domain parameters , and To satisfy the difference of the Lyapunov function as Satisfying the form of:

[0059] (16)

[0060] in, For class function; For class function.

[0061] In step five, the event-triggered model predictive control strategy is applied to control the end effector tracking of the soft robot.

[0062] By minimizing the cost function in formula (7), the control input sequence in the prediction time domain is obtained. After inputting the first term of the control sequence and obtaining the corresponding system state, the event trigger is determined, and the dynamic event trigger threshold and prediction time domain are adjusted, and the next optimization calculation is performed. In this way, the simulation of the end effector tracking of the soft robot is realized.

[0063] The beneficial effects of this invention are as follows:

[0064] Considering that the complex modeling of soft robots affects the system's tracking accuracy and computational efficiency, while ensuring the tracking accuracy of the soft robot's end-effector tracking control, an event-triggered strategy is introduced and the event triggering threshold and dynamic prediction time domain are designed to improve the system's computational efficiency and achieve accurate and efficient trajectory tracking.

[0065] By designing dynamic event trigger thresholds, while ensuring that the event trigger thresholds have adjustable upper and lower limits, the thresholds are made adaptable to changes in system state, reducing the number of optimization calculations in the control process. Through the design of the dynamic prediction time domain, while ensuring that the optimization problem has a solution, the prediction time domain decreases with each iteration, improving the efficiency of solving the optimization problem and thus improving the overall computational efficiency of the control system. Attached Figure Description

[0066] The present invention will now be described in further detail with reference to the accompanying drawings and specific implementation methods.

[0067] Figure 1 This is a schematic diagram of the predictive control method for event-triggered models of soft robots according to the present invention;

[0068] Figure 2 This is a graph showing the end effect tracking performance of the soft robot of the present invention.

[0069] Figure 3 This is a state change curve of the end-effector tracking system of the soft robot of the present invention;

[0070] Figure 4 This is a graph showing the change in the trigger threshold of the end-effector tracking event of the soft robot of the present invention.

[0071] Figure 5 This is a time-domain variation curve of the end-effector tracking dynamic prediction of the soft robot of the present invention;

[0072] Figure 6 This refers to the triggering time and interval of the end-effector tracking event of the soft robot of the present invention;

[0073] Figure 7 This is a three-dimensional comparison diagram of the end-effector tracking and PID control effects of the soft robot of the present invention;

[0074] Figure 8 This is a system state comparison diagram of the end-effector tracking and PID control effects of the soft robot of this invention.

[0075] Figure 9 This is a comparison diagram of the end-effector tracking and MPC control effects of the soft robot of this invention;

[0076] Figure 10 This is a system state comparison diagram of the end-effector tracking and MPC control effect of the soft robot of the present invention. Detailed Implementation

[0077] The present invention will now be described in further detail with reference to the accompanying drawings.

[0078] like Figures 1 to 10 As shown, in order to solve the technical problem that "the complex modeling of soft robots makes it difficult to achieve precise control, and although traditional MPC control can guarantee accurate tracking, its computational efficiency is too low", the steps and functions of an event-triggered model predictive control method for soft robots are explained in detail below.

[0079] Compared to rigid robots, soft robots often possess infinite degrees of freedom and nonlinear, time-varying flexible characteristics. Therefore, the kinematics and dynamics of soft robots typically exhibit complex nonlinearities, making it difficult to establish accurate and efficient models. This leads to unmodeled dynamics in the soft robot modeling process. Furthermore, in practical engineering applications, exogenous disturbances, such as noise, often exist, further increasing the uncertainty deviation between the dynamic model and the actual model. Currently, the main dynamic methods are based on the Eulerian-Lagrange method under the piecewise constant curvature assumption and other variable curvature methods. Under the piecewise constant curvature assumption, the soft robot is divided into multiple segments with constant curvature, where the curvature of each segment is variable in time but invariant in space, and each segment is connected end-to-end and smooth.

[0080] An event-triggered model predictive control method for a soft robot, the method comprising the following steps:

[0081] Step 1: Based on the piecewise constant curvature assumption of the soft robot, considering the system uncertainties including unmodeled dynamics and exogenous disturbances, establish the dynamic model of the soft robot and transform it into a state-space model;

[0082] The dynamic model of a soft robot with n segments, including unmodeled dynamics and exogenous disturbances, is as follows:

[0083] (17)

[0084] in, This represents the spatial pose angle of the soft robot. Represents the real number field. and Let represent the first and second time derivatives of the pose angle, respectively. Represents the inertia matrix. It combines Coriolis force and centrifugal force Simulating the effects of gravity, and These are the stiffness matrix and the damping matrix, respectively. Inputs containing force and torque Mapping to the configuration space enables the continuum robot to be fully actuated. This indicates unmodeled dynamics. Indicates an exogenous disturbance;

[0085] Transform the dynamic model of the soft robot into a state-space model:

[0086]

[0087] (18)

[0088] definition = , = Let represent the pose angle of the soft robot and its first-order time derivative, respectively. The total uncertainty of the state-space model formula (2) is expressed as follows: For ease of control in the design, define the actual input. and control input The relationship is , It is a positive definite matrix;

[0089] Step 2: Based on the state space model of the soft robot, design an event-triggered model predictive control strategy and combine the event-triggered strategy with model predictive control;

[0090] For any nonlinear system It has an initial value For the following form of cost function Terminal set area The radius is set to and make , and The weight matrix satisfies the Riccati equation; the corresponding feedback control matrix... satisfy This makes the closed-loop system matrix It satisfies the Hurwitz condition.

[0091] By defining the trigger time: This enables the control system to After the event is triggered, the future is obtained by solving an optimization problem. Time control input At the next triggering moment Previously, that is And the system status has not entered the terminal set, that is At that time, the system will continue using the previous time. The optimized control input; and when the system state enters the designed terminal set. The control input will switch to using a state feedback control law.

[0092] (19)

[0093] Therefore, the interval between two consecutive event trigger times must be less than or equal to... Therefore, the next triggering time is designed. for:

[0094] (20)

[0095] At the same time, the MPC optimization control problem must be solved at the beginning of system control, that is, it is required that... This will inevitably be triggered at the start of system control. Based on the above, the design of the predictive control strategy for the event-triggered model is complete.

[0096] Step 3: Based on the event-triggered model predictive control strategy, to further improve the system's computational efficiency, this invention introduces a design method for dynamic event triggering thresholds and dynamic prediction time domains. The core idea of ​​this method is to adaptively adjust the event triggering thresholds and prediction time domains according to the system's current state and historical triggering information, thereby reducing unnecessary optimization calculations while ensuring control accuracy.

[0097] First, regarding the design of event triggering thresholds, this invention proposes a dynamic event triggering threshold function:

[0098] (twenty one)

[0099] in, Indicates the first Next and first The time interval between the triggering of this event ; , , For design parameters, requirements The upper and lower limits of the trigger threshold can be obtained. , By adjusting the parameters and The parameters can be adjusted to control the upper and lower limits of the event trigger threshold. It can control the event trigger threshold. The magnitude of the rate of change. This is determined by adjusting the parameters. , and It can select appropriate parameters to calculate the event trigger threshold when facing different control situations, thereby improving the adaptability of the control system.

[0100] Secondly, in designing the dynamic prediction time domain, in order to ensure that after each update of the prediction time domain... It will decrease accordingly, that is Furthermore, it requires that the predicted end time be delayed after each update, i.e. Therefore, the design parameters are required to satisfy the inequality:

[0101] (twenty two)

[0102] because satisfy .in This is the prediction time domain during the system's first optimization, and represents the design parameters. Therefore, appropriate parameters can always be selected. and This satisfies the requirements of the inequality. Furthermore, to prevent the prediction time domain from decreasing indefinitely, a minimum value in the prediction time domain needs to be designed. The final prediction time-domain expression is then:

[0103] (twenty three)

[0104] By dynamically designing the prediction time domain, computational complexity can be effectively reduced and computational efficiency significantly improved while ensuring the feasibility and stability of the control system. Furthermore, by adjusting relevant design parameters... and The design avoids instability in solving system optimization problems due to the prediction time domain decreasing too quickly.

[0105] Step 4: To ensure the recursive feasibility and closed-loop stability of the designed event-triggered model predictive control strategy, a systematic design of the relevant parameters is required. First, for the soft robot tracking error system, a Lyapunov function is designed. ,in It is a symmetric positive definite matrix. To ensure system stability, it needs to satisfy... ,in , , and The weight matrices are for the state, input, and terminal state, respectively, and all must be positive definite matrices. Near the system equilibrium point, the weights are obtained through Jacobian linearization. ,in and These are the linearized matrices representing the system state and control input, respectively. For this continuous-time system, the algebraic Riccati equations must be satisfied, from which the terminal weight matrix can be obtained. Accordingly, a terminal feedback control law is designed. Ensure closed-loop system matrix The Hurwitz condition is satisfied, thus ensuring that the system is in the terminal set. Internal asymptotic stability.

[0106] To ensure that the optimization problem has a feasible solution at every event trigger time, the relevant parameters need to be designed appropriately to satisfy recursive feasibility and closed-loop stability. The difference between the Lyapunov functions at two adjacent event trigger times is... It needs to meet the following requirements. ,in , .for The complete expression:

[0107] (twenty four)

[0108] definition , and They are respectively:

[0109] (25)

[0110] It can be obtained by transformation and scaling.

[0111] (26)

[0112] Therefore,

[0113] (27)

[0114] definition and for:

[0115] (28)

[0116] Then there is For class function, For class The function therefore satisfies input-state stability (ISS), and the control system has closed-loop stability. This can be achieved by adjusting the weight matrix. , and Event trigger threshold and dynamic prediction time domain This ensures the closed-loop stability of the control system. The closed-loop stability of the event-triggered model predictive control system was then analyzed. Through the above design, the proposed control strategy can be theoretically guaranteed for tracking control of soft robots.

[0117] Step 5: After completing the modeling and control strategy design of the soft robot system, the system is simulated and trajectory tracking is performed using code to verify the feasibility and stability of the proposed control strategy on the soft robot model. The control effect of the event-triggered model predictive control strategy is compared with that of traditional MPC and PID control to verify the advantages of the proposed control strategy in terms of control accuracy and computational efficiency. Python is used for programming, primarily employing libraries such as SciPy and NumPy for matrix calculations and optimization problems, and Matplotlib for plotting. Simulation results verify the feasibility and stability of the event-triggered model predictive control, successfully controlling the soft robot end effector to follow various preset trajectories.

[0118] The trajectory tracking curve of the soft robot is as follows Figures 2 to 3 As shown, tracking of three trajectories—circular, heart-shaped, and figure-eight—demonstrates the effectiveness of the soft robot modeling and event-triggered predictive control strategy design presented in this paper. Analysis of the tracking errors reveals that: for the circular trajectory, the tracking error is less than 0.008m in 0.35 seconds; for the heart-shaped trajectory, the tracking error is 0.007m in 0.40 seconds; and for the figure-eight trajectory, the tracking error is 0.008m in 0.50 seconds. Furthermore, the tracking error of the control system gradually decreases after these times, thus the trajectory tracking meets the control accuracy requirements.

[0119] according to Figure 4 By analyzing the design of dynamic event trigger thresholds, it can be observed that when the system changes relatively smoothly, increasing the event trigger threshold reduces the number of triggers, thereby improving control efficiency; when the system changes significantly, decreasing the event trigger threshold increases the number of triggers, achieving more precise control.

[0120] according to Figure 5 Analysis of the dynamic prediction time-domain design reveals that the prediction time domain will gradually decrease as control progresses, eventually reaching the set minimum value. However, since the prediction time domain is only updated upon each event trigger, it remains at its current value when no event occurs. This is reflected in the graph as a plateau during the decline of the prediction time domain. The advantages of this design are: in the early stages when tracking errors are large, a higher prediction time domain allows for more precise system control; after the tracking error decreases, if the prediction time domain remains large, the control mechanism can utilize the control input obtained from the previous optimization as much as possible, thereby improving control efficiency; if the prediction time domain has decreased to a smaller value, faster optimization calculations can be performed, and due to the smaller tracking error, the system's tracking stability can still be maintained.

[0121] according to Figure 6 By analyzing the event triggering time, it can be found that at the event triggering time, the system updates the event triggering threshold and the prediction time domain. (In summary...) Figures 4 to 6 This demonstrates the feasibility of the event-triggered model predictive control based on dynamic prediction time domain and dynamic event trigger threshold.

[0122] pass Figure 7 and Figure 8 Taking circular trajectory tracking as an example, a comparison of the effects of the ETMPC strategy and PID control reveals that the ETMPC strategy achieves better tracking control of the soft robot's end effector. Due to the complexity of the soft robot's dynamic model, precise trajectory tracking is difficult to achieve using PID control. Furthermore, compared to the ETMPC strategy, PID control requires significant time for adjusting controller parameters when following different trajectories, lacking versatility; while the ETMPC strategy only requires fine-tuning of some parameters to be applicable to most trajectory tracking situations. Therefore, the ETMPC strategy designed in this paper has significant advantages.

[0123] pass Figure 9 and Figure 10 The controller parameter settings used in traditional MPC are the same as those used in circular trajectory tracking, and the soft robot dynamics model and starting position are also the same. By comparing traditional MPC with the ETMPC strategy designed in this paper, it can be found that their control effects are almost identical, both achieving control of the soft robot's end effector. However, comparing their control efficiency reveals that, on the one hand, due to the existence of the event triggering mechanism, according to... Figure 6 The designed ETMPC controller was triggered 83 times, meaning it performed 83 MPC optimization problems; while the traditional MPC solves the problem at every sampling time, performing 102 MPC optimization problems, significantly reducing the number of problems. Furthermore, due to the dynamic prediction time-domain mechanism, the speed of solving each MPC optimization problem is also greatly improved. Therefore, the ETMPC strategy designed in this paper can effectively improve control efficiency while ensuring control accuracy, thus achieving efficient control.

Claims

1. An event-triggered model predictive control method for a soft robot, characterized in that: The method includes the following steps: Step 1: Based on the piecewise constant curvature assumption of the soft robot, considering the system uncertainties including unmodeled dynamics and exogenous disturbances, establish the dynamic model of the soft robot and transform it into a state-space model; Step 2: Based on the state-space model of the soft robot, design an event-triggered predictive control strategy; Step 3: Introduce dynamic event triggering thresholds and dynamic prediction time domain methods to improve computational efficiency. Step 4: Ensure the recursive feasibility and closed-loop stability of the control strategy through parameter design; Step 5: Apply the designed event-triggered model predictive control strategy to perform simulation analysis on the end-effector tracking of the soft robot.

2. The event-triggered model predictive control method for a soft robot according to claim 1, characterized in that: The segmented constant curvature assumption of the soft robot is as follows: Under the assumption of segmented constant curvature, the soft robot is divided into a model of multiple segments with constant curvature, wherein the curvature of each segment is variable in time but constant in space, and each segment is connected end to end and smooth.

3. The event-triggered model predictive control method for a soft robot according to claim 2, characterized in that: In step one, the system uncertainties, including unmodeled dynamics and exogenous disturbances, have The dynamic model of the soft robot segment is as follows: （1） in, This represents the spatial pose angle of the soft robot. Represents the real number field. and Let represent the first and second time derivatives of the pose angle, respectively. Represents the inertia matrix. It combines Coriolis force and centrifugal force Simulating the effects of gravity, and These are the stiffness matrix and the damping matrix, respectively. Input containing torque Mapping to the configuration space enables the continuum robot to be fully actuated.

4. The event-triggered model predictive control method for a soft robot according to claim 3, characterized in that: Before designing an MPC strategy, the control system must first be defined and described; for general nonlinear systems, the definition... For system status, To control the input, then there is （2） Accordingly, the reference target trajectory is defined as Similarly, （3） Based on this, in order to achieve tracking of the reference trajectory, a definition is defined. Let the tracking error be the system state, then we have （4） （5） Assuming the tracking error for the system state satisfy （6） Then there is , These represent the actual pose angle and the reference pose angle of the soft robot, respectively. This represents the tracking error of the pose angle of the soft robot; Define input torque ,in, Indicates circling Moment in the axial direction; Indicates circling Moment in the axial direction; Indicates along Force in the axial direction; define input torque and control input The relationship is , It is a symmetric positive definite matrix; based on the above definitions, a state-space model between the tracking error and control input of the soft robot is established.

5. The event-triggered model predictive control method for a soft robot according to claim 4, characterized in that: In step two, the design process for this event-triggered model predictive control is as follows: For any nonlinear system It has an initial value ; Define the cost function in the following form for: （7） in, For the first The moment when the optimization problem is solved; For the first The prediction time domain for solving the optimization problem; To determine the time points in the process of solving the optimization problem, ; , and These represent the weight matrices for the state term, input term, and terminal state term in the cost function, respectively. and Indicates in arrive In the interval Real-time control inputs and system status; terminal set area The radius is set to ,Right now （8） To introduce event-triggered strategies into model predictive control, the event triggering mechanism and event triggering time are first defined; the criterion for determining the triggering time is the actual system state. With predicted state The magnitude of the error between them defines the trigger time as: （9） in, For the system number The moment this event is triggered; Indicates the system at the 1st Second trigger The difference between the actual state error and the predicted state error at time step Norm; This indicates the designed event trigger threshold.

6. The event-triggered model predictive control method for a soft robot according to claim 5, characterized in that: control system After the event is triggered, the future is obtained by solving an optimization problem. Time control input At the next triggering time Previously, that is And the system status has not entered the terminal set, that is At that time, the system will continue using the previous time. The optimized control input; and when the system state enters the designed terminal set. The control input will switch to using a state feedback control law; that is... （10） Therefore, the interval between two consecutive event trigger times must be less than or equal to... Therefore, the next triggering time is designed. for: （11） At the same time, the MPC optimization control problem must be solved at the beginning of system control, that is, it is required that... It will definitely be triggered at the start of system control; based on the above, the design of the predictive control strategy for the event-triggered model is completed.

7. The event-triggered model predictive control method for a soft robot according to claim 6, characterized in that: In step three, the design process for the dynamic event triggering threshold and the dynamic prediction time domain is as follows: Set event trigger threshold Defined as: （12） in, Indicates the first Next and first The time interval between the triggering of this event ; , , For design parameters, requirements ; By recording the trigger times of two consecutive events and The trigger time interval can be obtained. Therefore, the prediction time domain is designed as follows: （13） in, and The first , Predicted temporal domain at the time of event triggering; Design parameters to control the magnitude of the forward propagation in the prediction time domain, ; Design parameters to control the degree of time-domain shrinkage in prediction, ; through the and The adjustment of the prediction time domain can control the magnitude of the descent rate, thereby achieving a balance between control accuracy and computation speed; the dynamic prediction time domain will be updated at each event trigger moment, i.e., as defined by formula (11). The prediction time domain is updated at each time step. And in this optimization solution, that is, the first... The suboptimal solution uses the prediction time domain. Solve the optimization problem.

8. The event-triggered model predictive control method for a soft robot according to claim 7, characterized in that: In step four, to ensure the recursive feasibility and closed-loop stability of the control system, the design parameters are as follows: For the weight matrix , and All matrices must be positive definite matrices and satisfy the following conditions: （14） in, For the equivalent matrix, ; For the Lyapunov function of the system; Near the system's equilibrium point, linearizing the system using the Jacobian matrix yields the following results: in, Let be the linearized matrix of the system state at the origin. ; To control the linearization of the input matrix at the origin, For this continuous-time system of the soft robot, the Riccati equations must be satisfied: （15） The weight matrix of the system's final state can be obtained from this continuous-time Riccati equation. The corresponding feedback matrix Defined as This makes the closed-loop system matrix It satisfies the Hurwitz condition.

9. The event-triggered model predictive control method for a soft robot according to claim 8, characterized in that: To ensure that the control system meets input-state stability (ISS), the following design parameters need to be adjusted: terminal set radius. ; Event trigger threshold parameter and Dynamic prediction of time-domain parameters , and To satisfy the difference of the Lyapunov function as Satisfying the form of: （16） in, For class function; For class function.

10. The event-triggered model predictive control method for a soft robot according to claim 9, characterized in that: In step five, the event-triggered model predictive control strategy is applied to control the end effector tracking of the soft robot. By minimizing the cost function in formula (7), the control input sequence in the prediction time domain is obtained. After inputting the first term of the control sequence and obtaining the corresponding system state, the event trigger is judged, and the dynamic event trigger threshold and prediction time domain are adjusted, and the next optimization calculation is performed. In this way, the simulation of the end effector tracking of the soft robot is realized.

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