Koopman-based iterative learning model predictive control method for pneumatic soft robots
By combining iterative learning control and model predictive control, and introducing the Koopman operator to construct a linear predictive model, the problem of high model accuracy dependence in pneumatic soft actuator control methods is solved, achieving rapid improvement in control performance and enhanced robustness.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NANJING INST OF TECH
- Filing Date
- 2026-01-28
- Publication Date
- 2026-06-05
AI Technical Summary
Existing control methods for pneumatic soft actuators suffer from high dependence on model accuracy, high computational complexity, and unstable control performance. In particular, when the initial control effect is poor or the disturbance is large, it is difficult to balance the requirements of control response speed and real-time performance.
By combining iterative learning control and model predictive control, the Koopman operator is introduced to construct a linear predictive model. The control input is then corrected using an iterative learning mechanism, reducing the dependence on an accurate model and improving control performance.
This has enabled a rapid improvement in the control performance of pneumatic soft actuators, reduced the difficulty of modeling nonlinear systems, and enhanced the synergy and robustness of control effects.
Smart Images
Figure CN122151498A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of soft robot control technology, and in particular to a predictive control method for pneumatic soft robots based on an iterative learning model using Koopman. Background Technology
[0002] Pneumatic soft actuators, due to their high compliance and safety in human-machine interaction, have broad application prospects in rehabilitation assistance, wearable robots, and flexible manipulation. However, pneumatic soft actuators are usually made of flexible materials, and their dynamic processes exhibit strong nonlinearity, time-varying characteristics, and significant modeling uncertainties, making accurate modeling and high-performance control challenging. To address the complex nonlinear dynamic characteristics of pneumatic soft actuators, researchers have proposed various system modeling methods, including continuum models based on physical mechanisms, pseudo-rigid body models, and data-driven modeling methods. However, these methods still have limitations in terms of model accuracy, computational complexity, and compatibility with model-based predictive control methods. In recent years, Koopman operator theory has provided a new modeling approach for constructing predictive models with linear forms by mapping the dynamics of nonlinear systems to a high-dimensional linear space.
[0003] In terms of control methods, research on control for pneumatic soft actuators mainly falls into two categories: model-based control methods and model-independent control methods. Model predictive control (MPC), due to its ability to explicitly handle system constraints and its good predictive performance, is widely used in soft robot control research. However, the control performance of MPC is highly dependent on the accuracy of the predictive model. When there are deviations in the model or changes in system parameters, it can easily lead to a decrease in trajectory tracking performance or even a reduction in system stability. To reduce the dependence on accurate models, iterative learning control (ILC), as a typical model-independent control method, is used to improve the control performance of systems with repetitive operation characteristics. ILC iteratively corrects the control input using historical operational error information, gradually improving the system's tracking accuracy. However, traditional ILC lacks explicit characterization of the system's dynamic characteristics, and its control performance is highly sensitive to initial inputs. When the initial control effect is poor or the disturbance is large, it often requires many iterations to converge, making it difficult to balance control response speed and real-time requirements. Summary of the Invention
[0004] Purpose of the invention: The purpose of this invention is to provide a predictive control method for pneumatic soft robots based on an iterative learning model using Koopman, which can rapidly improve the control performance of pneumatic soft actuators while reducing dependence on high-precision system models.
[0005] Technical solution: The predictive control method for pneumatic soft robots based on the iterative learning model of Koopman, as described in this invention, includes the following steps:
[0006] (1) System initialization and initial model construction;
[0007] (2) Model predictive control calculation based on predictive models;
[0008] (3) System response acquisition and tracking error calculation;
[0009] (4) Input and model correction based on iterative learning control;
[0010] (5) Model update and iterative prediction control execution.
[0011] Further, in step (1), the system initialization and initial model construction include:
[0012] (21) The Koopman operator of the system; the system state of the bending actuator is represented as follows , It refers to the angle of the finger joint; the system control input is denoted as... and satisfy ,in This indicates the input air pressure to the air chamber;
[0013] A nonlinear system with external input is represented in the following form:
[0014]
[0015] in Indicates that the system is in The state at any given moment; Indicates the system's control input; Represents the dynamics of a controlled system;
[0016] Koopman operator for controlled systems An observation function that acts on both the system state and the input; the observation function is composed of a vector function. Representation; Koopman operator Acting on the observation function It has the following linear expression form:
[0017]
[0018] in This represents the approximation error of the finite-dimensional Koopman operator;
[0019] Assume the system state and the observation function of the input. It has the following forms:
[0020]
[0021] Where vector It consists of the observation function of the state, and the vector It consists of the system's control inputs;
[0022] Based on the representation of the observation function, the Koopman operator of the system can be expressed in the following form:
[0023]
[0024] Among them, matrix and The observation function describes the evolution of the system state; represents the system input. Submatrices of the evolution process;
[0025] Based on the above formula, the linear state-space equation of the controlled system is as follows:
[0026]
[0027] Where the matrix and These represent the state matrix and input matrix of the model, respectively.
[0028] (22) Numerical computation of the Koopman operator; given a dataset satisfying the dynamic equations. The state matrix in the Koopman operator model is estimated using the least squares method. and input matrix This method estimates the matrix by minimizing the sum of squared residuals of the Koopman operator model. and The expression is as follows:
[0029]
[0030] Among the symbols Represents the 2-norm; matrix , and According to the dataset The calculation yields the following expression:
[0031]
[0032]
[0033]
[0034] Where the matrix and Observation functions containing system state, matrices Includes system input;
[0035] The matrix is obtained by solving the least squares problem using normal equations. and The expression is as follows:
[0036]
[0037] in , , Represents the Moore–Penrose generalized inverse of the matrix.
[0038] Further, in step (2), the model predictive control calculation based on the prediction model includes solving for the control input within the current prediction period based on the Koopman prediction model.
[0039] Further, in step (3), the system response acquisition and tracking error calculation includes:
[0040] The actual output state of the pneumatic soft actuator is obtained under the control input calculated by model predictive control.
[0041] Obtain the model's predicted output or the target reference trajectory, and calculate the corresponding tracking error based on the deviation between the actual output state and the target reference trajectory.
[0042] Furthermore, in step (4), the input and model correction based on iterative learning control includes introducing an iterative learning mechanism to correct the control input and prediction model based on the tracking error.
[0043] Furthermore, the iterative learning control is based on the following update law:
[0044]
[0045]
[0046] like ,
[0047] in It is a pseudo-partial derivative; , It is a sequence of constant step size and , , It is a positive weighting factor; It is a small positive number; , The first Input pressure and output angle at each iteration; It is the reference angle that the actuator needs to track; yes and The error between them.
[0048] Furthermore, in step (5), the model update and iterative predictive control execution includes feeding back the model parameters and control inputs corrected by iterative learning to the model predictive control module, and repeatedly executing the model predictive control calculation in subsequent control cycles.
[0049] Further, in step (5), the iterative learning model prediction control includes:
[0050] (81) Model predictive control, the extended model of the system is as follows:
[0051]
[0052]
[0053] in and These represent the state and process noise of the extended model, respectively. This represents the modeling error of the system. The process noise driving the modeling error follows a Gaussian distribution with a mean of zero. and These represent the system's process noise and measurement noise, respectively. Represents the control vector; , , The state matrix, control matrix, and output matrix of the extended model are represented by the following formulas:
[0054]
[0055]
[0056]
[0057] in, , and These represent the system's state matrix, input matrix, and output matrix, respectively. Represents the dimension of the state space. Represents the dimension of the subspace. It controls the dimension of the input. This indicates the number of times the current time t is passed within the prediction space. One discrete control step;
[0058] Based on the established system extension model, online estimation of the system extension state is performed; the entire estimation process is divided into two stages: prior estimation and posterior estimation.
[0059] (82) Simulated predictive controller: Based on the iterative learning law of model parameters, the updated model is substituted into the internal MPC predictive controller, and its cost function is consistent with the internal model:
[0060]
[0061] in and These are the weighting factors for the output and input, respectively; For prediction in the time domain; It is the first The reference angle that the stepper needs to track. It is the first The actual angle at the next iteration;
[0062] (83) Input iterative learning law; use typical iterative learning methods for linearized models:
[0063]
[0064]
[0065] in It is a constant; It is the reference angle that the actuator needs to track. It is the actual angle at the k-th iteration; It is the error between the reference angle and the actual angle; It is the learning law for the k-th iteration.
[0066] Further, in step (81), the prior estimate and the posterior estimate are:
[0067] Prior estimation: Calculate the prior estimates of the system's extended state and covariance matrix based on the extended model. and The calculation process is as follows:
[0068]
[0069]
[0070] in , These represent the state matrix and control matrix of the extended model, respectively. Represents the control vector; matrix Indicates process noise covariance;
[0071] Posterior estimation: using system measurements Compute the posterior estimates of the extended state and covariance matrix of the system. and To improve the accuracy of prior estimates, the calculation process is as follows:
[0072]
[0073]
[0074]
[0075]
[0076] in Indicates the output matrix. This represents the output matrix of the extended model; Represents Kalman gain; matrix Indicates measurement noise covariance; This indicates the error between the actual angle and the model angle.
[0077] Furthermore, in step (82), the constraints of the cost function are as follows:
[0078]
[0079]
[0080]
[0081]
[0082] in , and These represent the system's state matrix, input matrix, and output matrix, respectively. Indicates the state of the system; Indicates the actual output angle; This is the control input for the system.
[0083] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: 1. By combining iterative learning control with model predictive control, the control performance of pneumatic soft actuators is synergistically improved; 2. By introducing the Koopman operator and integrating it with iterative learning model predictive control, the difficulty of modeling nonlinear systems is reduced and the overall control effect is improved. Attached Figure Description
[0084] Figure 1This is a block diagram of the control system for the pneumatic soft robot of the present invention.
[0085] Figure 2 This is a flowchart of the ILMPC model of the present invention. Detailed Implementation
[0086] The technical solution of the present invention will be further described below with reference to the accompanying drawings.
[0087] Combination Figure 1 and Figure 2 The Koopman-based iterative learning model predictive control method for pneumatic soft robots includes the following steps:
[0088] Step 1: System initialization and initial model construction;
[0089] (1) The Koopman operator of the system; the system state of the bending actuator can be expressed as:
[0090]
[0091] System control input is denoted as , It refers to the angle of the finger joints, and it satisfies... ,in This indicates the input air pressure of the air chamber.
[0092] To define the Koopman operator for the system, the nonlinear system with external input is first represented in the following form:
[0093]
[0094] in Indicates that the system is in The state at any given moment. This represents the system's control input. This represents the dynamics of a controlled system.
[0095] Koopman operator for controlled systems An observation function that acts on both the system state and the input, the observation function being a vector function Represented by the Koopman operator. Acting on the observation function This describes the evolution of the observation function and has the following linear expression:
[0096]
[0097] in This represents the approximation error of the finite-dimensional Koopman operator. Theoretically, when the dimension of the subspace is... As it approaches infinity, the approximation error It tends towards zero.
[0098] To obtain the linear state-space equations of a controlled system, we assume the system state and the observation functions of the input. It has the following forms:
[0099]
[0100] Where vector It consists of the observation function of the state, and the vector It consists of the system's control inputs.
[0101] Based on the representation of the observation function, the Koopman operator of the system can be expressed in the following form:
[0102]
[0103] Among them, matrix and The evolution of the observation function that describes the system state. Indicates the description of system input The submatrix of the evolution process is usually determined by the system's controller.
[0104] Based on the above formula, the linear state-space equation of the controlled system can be obtained using the Koopman operator as follows:
[0105]
[0106] Where the matrix and These represent the state matrix and input matrix of the model, respectively, and can be identified using numerical calculation methods such as the least squares method.
[0107] (2) Numerical computation of the Koopman operator;
[0108] Given a dataset that satisfies the dynamic equations The state matrix in the Koopman operator model is estimated using the least squares method. and input matrix This method estimates the matrix by minimizing the sum of squared residuals of the Koopman operator model. and The expression is as follows:
[0109]
[0110] Among the symbols Represents the 2-norm. Matrix , and According to the dataset The calculation yields the following expression:
[0111]
[0112]
[0113]
[0114] Where the matrix and Observation functions containing system state, matrices Includes system input;
[0115] Solving the least squares problem using normal equations yields the matrix. and The expression is as follows:
[0116]
[0117] in , , Represents the Moore–Penrose generalized inverse of the matrix.
[0118] Step 2: Model predictive control calculation based on the predictive model;
[0119] Based on the Koopman operator This is the reference angle that the actuator needs to track. The cost function of MPC is:
[0120]
[0121] Constraints:
[0122]
[0123]
[0124]
[0125]
[0126] in, and These are the weighting coefficients for the output and input, respectively. For prediction in the time domain.
[0127] Step 3: System response acquisition and tracking error calculation;
[0128] Under the control input calculated by the model predictive control, the pneumatic soft actuator is driven to run, and the actual output state of the actuator in the current operating cycle is collected; at the same time, the model predictive output or target reference trajectory is acquired, and the corresponding tracking error is calculated based on the deviation between the actual output state and the model predictive output or target reference trajectory.
[0129] Step four: Input and model correction based on iterative learning control;
[0130] The Koopman operator method can establish an approximate linear model for pneumatic soft actuators, but the resulting mathematical model will still differ somewhat from the actual system. Such model errors can easily lead to suboptimal performance of model-based control strategies during tracking. Iterative learning control (ILC) is well-suited for systems with repetitive operation characteristics, gradually improving control tracking performance through the repetitiveness of system operation. ILC is a control method that does not rely on an accurate model and requires only a small number of system parameters to implement control.
[0131] This method employs the model-free ILC method proposed in the two-stage optimal iterative learning control of nonlinear non-affine discrete-time systems. The ILC controller is based on the following update law:
[0132]
[0133]
[0134] like ,
[0135] in It is a pseudo-partial derivative; , It is a sequence of constant step size and , , It is a positive weighting factor; It is a small positive number; , The first Input pressure and output angle at each iteration; It is the reference angle that the actuator needs to track; yes and The error between them.
[0136] Step 5: Model update and iterative prediction control execution.
[0137] Based on MPC and ILC, the proposed Iterative Learning Model Predictive Control (ILMPC) method is designed to fully leverage the advantages of MPC and ILC while overcoming their shortcomings.
[0138] (1) MPC predictive controller; for simultaneously estimating the system's observation function and modeling error The extended model of the system is as follows:
[0139]
[0140]
[0141] in and These represent the state and process noise of the extended model, respectively. This represents the modeling error of the system. The process noise driving the modeling error follows a Gaussian distribution with a mean of zero. and These represent the system's process noise and measurement noise, respectively. Represents the control vector; , , The state matrix, control matrix, and output matrix of the extended model are represented by the following formulas:
[0142]
[0143]
[0144]
[0145] in, , and These represent the system's state matrix, input matrix, and output matrix, respectively. Represents the dimension of the state space. Represents the dimension of the subspace. It controls the dimension of the input. This indicates the number of times the current time t is passed within the prediction space. One discrete control step;
[0146] Based on the established system extended model, the Kalman filter, a method widely used in linear state estimation, is selected to perform online estimation of the system extended state. The entire estimation process is divided into two stages: prior estimation and posterior estimation. The calculation flow is shown below:
[0147] (1.1) Prior estimation: Calculate the prior estimates of the extended state and covariance matrix of the system based on the extended model. and The calculation process is as follows:
[0148]
[0149]
[0150] in , These represent the state matrix and control matrix of the extended model, respectively. Represents the control vector; matrix Indicates process noise The covariance.
[0151] (1.2) Posterior estimation: using the system's measurements Compute the posterior estimates of the extended state and covariance matrix of the system. and To improve the accuracy of prior estimates, the calculation process is as follows:
[0152]
[0153]
[0154]
[0155]
[0156] in Indicates the output matrix. This represents the output matrix of the extended model; Represents Kalman gain; matrix Indicates measurement noise covariance; This indicates the error between the actual angle and the model angle.
[0157] Using the estimation method described above, the extended state of the system can be estimated within each sampling period. . It includes estimates of the actual system state and modeling errors. Estimates based on the extended state. The designed active model predictive controller can reduce the impact of system modeling errors and external disturbances, and improve the accuracy and robustness of traditional model predictive controllers.
[0158] (2) Simulated predictive controller; based on the iterative learning law of model parameters, the updated model is substituted into the internal MPC predictive controller. Its cost function is consistent with the internal model:
[0159]
[0160] in and These are the weighting factors for the output and input, respectively; For prediction in the time domain; It is the first The reference angle that the stepper needs to track. It is the first The actual angle at the next iteration;
[0161] Constraints:
[0162]
[0163]
[0164]
[0165]
[0166] in , and These represent the system's state matrix, input matrix, and output matrix, respectively. Indicates the state of the system; Indicates the actual output angle; This is the control input for the system.
[0167] (3) Input iterative learning law; for linearized models, a typical iterative learning method is adopted:
[0168]
[0169]
[0170] in It is a constant; It is the reference angle that the actuator needs to track. It is the actual angle at the k-th iteration; It is the error between the reference angle and the actual angle; It is the learning law for the k-th iteration.
[0171] Using a pneumatic soft hand rehabilitation robot as an example, this paper describes the predictive control method based on the iterative learning model of the Koopman operator. The pneumatic soft rehabilitation hand adopts a wearable flexible glove structure, and its driving unit is a double-layer lattice structure pneumatic soft actuator. This actuator is made of flexible material and arranged along the back of the fingers, consistent with the natural movement direction of the human finger, thus enabling assistance and guidance of finger joint movement while ensuring flexibility and wearing comfort.
[0172] The dual-layer lattice structure pneumatic soft actuator includes a flexion airbag layer and an extension airbag layer, with the two airbags stacked along the thickness direction of the actuator. When inflated, the flexion airbag drives the finger to flex in the flexion direction, and the extension airbag drives the finger to extend in the extension direction. By coordinating the input air pressure of the two types of airbags, bidirectional continuous movement of the finger joints within a certain angle range can be achieved. The bending angle output by the actuator exhibits a significant nonlinear relationship with the input air pressure and is affected by factors such as material properties, manufacturing errors, and external loads.
[0173] In this rehabilitation hand system, a pneumatic soft actuator is connected to a power control element via an air circuit. The power control element adjusts the input air pressure into each air chamber according to control commands. Simultaneously, the system is equipped with angle or displacement sensors to collect real-time data on the actual movement of the fingers and generate feedback. Combined with... Figure 1 The control system block diagram shown includes a predictive model module, an iterative learning model predictive control module, a power control element, and a sensor feedback unit. The predictive model module constructs a Koopman predictive model based on the input and output data of the pneumatic soft actuator. The iterative learning model predictive control module generates control commands based on the predictive model and combines feedback information to correct the model and control input, thus forming a closed-loop control system.
[0174] Based on the framework of the rehabilitation hand structure and control system, this method combines model predictive control with iterative learning mechanism and introduces the Koopman operator to model the actuator dynamics, thereby realizing closed-loop control of the pneumatic soft rehabilitation hand.
Claims
1. A predictive control method for pneumatic soft robots based on an iterative learning model using Koopman, characterized in that, Includes the following steps: (1) System initialization and initial model construction; (2) Model predictive control calculation based on predictive models; (3) System response acquisition and tracking error calculation; (4) Input and model correction based on iterative learning control; (5) Model update and iterative learning model prediction control execution.
2. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (1), the system initialization and initial model construction include: (21) The Koopman operator of the system; the system state of the bending actuator is represented as follows , It refers to the angle of the finger joint; the system control input is denoted as... and satisfy ,in This indicates the input air pressure to the air chamber; A nonlinear system with external input is represented in the following form: in Indicates that the system is in The state at any given moment; Indicates the system's control input; Represents the dynamics of a controlled system; Koopman operator for controlled systems An observation function that acts on both the system state and the input; the observation function is composed of a vector function. Representation; Koopman operator Acting on the observation function It has the following linear expression form: in This represents the approximation error of the finite-dimensional Koopman operator; Assume the system state and the observation function of the input. It has the following forms: Where vector It consists of the observation function of the state, and the vector It consists of the system's control inputs; Based on the representation of the observation function, the Koopman operator of the system can be expressed in the following form: Among them, matrix and The observation function describes the evolution of the system state; represents the system input. Submatrices of the evolution process; Based on the above formula, the linear state-space equation of the controlled system is as follows: Where the matrix and These represent the state matrix and input matrix of the model, respectively. (22) Numerical computation of the Koopman operator; given a dataset satisfying the dynamic equations. The state matrix in the Koopman operator model is estimated using the least squares method. and input matrix This method estimates the matrix by minimizing the sum of squared residuals of the Koopman operator model. and The expression is as follows: Among the symbols Represents the 2-norm; matrix , and According to the dataset The calculation yields the following expression: Where the matrix and Observation functions containing system state, matrices Includes system input; The matrix is obtained by solving the least squares problem using normal equations. and The expression is as follows: in , , Represents the Moore–Penrose generalized inverse of the matrix.
3. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (2), the model predictive control calculation based on the prediction model includes solving for the control input within the current prediction period based on the Koopman prediction model.
4. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (3), the system response acquisition and tracking error calculation includes: The actual output state of the pneumatic soft actuator is obtained under the control input calculated by model predictive control. Obtain the model's predicted output or the target reference trajectory, and calculate the corresponding tracking error based on the deviation between the actual output state and the target reference trajectory.
5. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (4), the input and model correction based on iterative learning control includes introducing an iterative learning mechanism to correct the control input and prediction model based on the tracking error.
6. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 5, characterized in that, The iterative learning control is based on the following update law: like , in It is a pseudo-partial derivative; , It is a sequence of constant step size and , , It is a positive weighting factor; It is a small positive number; , The first Input pressure and output angle at each iteration; It is the reference angle that the actuator needs to track; yes and The error between them.
7. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (5), the model update and iterative predictive control execution includes feeding back the model parameters and control inputs corrected by iterative learning to the model predictive control module, and repeating the model predictive control calculation in subsequent control cycles.
8. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 1, characterized in that, In step (5), the iterative learning model prediction control includes: (81) Model predictive control, the extended model of the system is as follows: in and These represent the state and process noise of the extended model, respectively. This represents the modeling error of the system. The process noise driving the modeling error follows a Gaussian distribution with a mean of zero. and These represent the system's process noise and measurement noise, respectively. Represents the control vector; , , The state matrix, control matrix, and output matrix of the extended model are represented by the following formulas: in, , and These represent the system's state matrix, input matrix, and output matrix, respectively. Represents the dimension of the state space. Represents the dimension of the subspace. It controls the dimension of the input. This indicates the number of times the current time t is passed within the prediction space. One discrete control step; Based on the established system extension model, online estimation of the system extension state is performed; the entire estimation process is divided into two stages: prior estimation and posterior estimation. (82) Simulated predictive controller: Based on the iterative learning law of model parameters, the updated model is substituted into the internal MPC predictive controller, and its cost function is consistent with the internal model: in and These are the weighting factors for the output and input, respectively; For prediction in the time domain; It is the first The reference angle that the stepper needs to track. It is the first The actual angle at the next iteration; (83) Input iterative learning law; use typical iterative learning methods for linearized models: in It is a constant; It is the reference angle that the actuator needs to track. It is the actual angle at the k-th iteration; It is the error between the reference angle and the actual angle; It is the learning law for the k-th iteration.
9. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 8, characterized in that, In step (81), the prior estimate and the posterior estimate are: Prior estimation: Calculate the prior estimates of the system's extended state and covariance matrix based on the extended model. and The calculation process is as follows: in , These represent the state matrix and control matrix of the extended model, respectively. Represents the control vector; matrix Indicates process noise covariance; Posterior estimation: using system measurements Compute the posterior estimates of the extended state and covariance matrix of the system. and To improve the accuracy of prior estimates, the calculation process is as follows: in Indicates the output matrix. This represents the output matrix of the extended model; Represents Kalman gain; matrix Indicates measurement noise covariance; This indicates the error between the actual angle and the model angle.
10. The predictive control method for iterative learning model of a pneumatically driven soft robot according to claim 8, characterized in that, In step (82), the constraints of the cost function are as follows: in , and These represent the system's state matrix, input matrix, and output matrix, respectively. Indicates the state of the system; Indicates the actual output angle; This is the control input for the system.