Soft robot optimization modeling and robust control method based on fusion prediction equation

Abstract

A soft robot optimization modeling and robust control method based on a fusion prediction equation, comprising: first deriving measurement coordinates on the basis of a fusion prediction equation; then designing an observation function on the basis of the measurement coordinates; then identifying a Koopman model on the basis of the observation function; and finally designing a robust model prediction controller on the basis of the Koopman model. The soft robot optimization modeling and robust control method provides a fusion prediction equation and a derivation method therefor, and accurate, abundant, and non-redundant measurement coordinates can be derived on the basis of the equation, overcoming the problem of single measurement coordinates in a soft robot system, facilitating simplifying the design process of an observation function, further improving the precision of a Koopman model of the soft robot, achieving precise motion control of the soft robot, and enhancing the robustness of a control system.

Description

Optimal modeling and robust control method for soft robots based on fused predictive equations Technical Field

[0001] The present invention belongs to the technical field of soft robot modeling and control, and particularly relates to a soft robot optimization modeling and robust control method based on a fusion prediction equation. Background Art

[0002] Robotics plays a vital role in manufacturing, healthcare, and service sectors. Traditional rigid robotic arms excel in force output and precision, but their rigid materials and structures make them inadequate for complex and changing environments. Soft robots, integrated actuators and sensors built from highly elastic materials, are thriving in rescue missions, medical rehabilitation, human-robot interaction, and other fields thanks to their inherent flexibility and adaptability, presenting enormous application prospects.

[0003] However, although soft robots have many advantages in theory, their precise modeling and control still face severe challenges. The reasons are: 1. The flexible material of the soft robot body exhibits complex hysteresis nonlinearity, and it is extremely difficult to establish a mechanism model based on the mechanical properties and geometric structure of the material. 2. The driving mode of soft robots is usually fluid-driven, and dynamic modeling requires solving complex fluid dynamics problems. 3. Although modeling is achieved by considering the complex nonlinearity of materials and drives, this method is not universal and is difficult to apply to soft robot systems with customized designs of materials, structures, and drives. Moreover, the complex form of the mechanism model results in high computational costs and cannot be applied to model-based real-time control technology. The above reasons have led to the fact that most current soft robots can only achieve rough motion function verification and are difficult to be accurately controlled, which limits their practical application.

[0004] Fortunately, Koopman operator theory demonstrates that mapping nonlinear dynamics into a high-dimensional Koopman space via an observation function allows for high-fidelity global linearization. This allows the application of sophisticated linear control methods to control complex nonlinear systems. Furthermore, this data-driven approach avoids tedious mechanism modeling and is adaptable to custom-designed soft robotic systems. Currently, several published technologies have applied Koopman operator theory to the modeling and control of soft robotics.

[0005] However, these technologies still have two unresolved problems:

[0006] 1. There is no universal method for designing observation functions for Koopman modeling of soft robots. The quality of the observation function directly affects the accuracy of the Koopman model. Current observation functions are often selected based on experience, resulting in high design costs and no guaranteed effectiveness.

[0007] 2. Control design based on the Koopman model usually adopts standard model predictive control, linear quadratic regulator or other linear control methods, which have poor robustness and are difficult to cope with parameter uncertainty and external disturbances.

[0008] Summary of the Invention

[0009] To solve the above problems, the present invention discloses a soft robot optimization modeling and robust control method based on fusion prediction equations, which is used to solve the observation function design problem in the Koopman modeling of the soft robot, thereby optimizing the accuracy of the Koopman model, further realizing the precise motion control of the soft robot, and enhancing the robustness of the control system.

[0010] To achieve the above object, the technical solution of the present invention is as follows:

[0011] The soft robot optimization modeling and robust control method based on the fusion prediction equation includes the following steps:

[0012] S1, deriving measurement coordinates based on the fusion prediction equation;

[0013] S2. Design observation function based on measurement coordinates;

[0014] S3, identifying the Koopman model based on the observation function;

[0015] S4. Design a robust model predictive controller based on the Koopman model.

[0016] Furthermore, the step S1 of deriving the measurement coordinates based on the fusion prediction equation includes:

[0017] S11. Derive a fusion prediction equation. The purpose of the fusion prediction equation is to obtain a measurable physical quantity that is closely related to the predicted controlled quantity and use it as a measurement coordinate to design an observation function to fully capture the system dynamics and improve the accuracy of Koopman modeling. Based on the idea of ​​data fusion, the fusion prediction equation achieves optimized prediction based on multiple hypothetical models. Its derivation is based on the general incremental equation: θ(k+1)=θ(k)+Δθ(k+1)

[0018] Where θ is the controlled variable, usually the curvature or end position of the soft robot, and the key to predicting θ(k+1) is to accurately estimate the increment Δθ(k+1) of the controlled variable. Furthermore, Δθ(k+1) is estimated based on multiple hypothetical models, assuming that model 1 is a constant increment model: Δθ1(k+1)=Δθ(k)=θ(k)-θ(k-1)

[0019] Among them, Δθ1(k+1) is the estimated value of the controlled variable increment obtained by linear extrapolation based on the hypothetical model 1. This full-range uniform speed assumption is only applicable to extremely short sampling periods or systems with stable operation;

[0020] Relatively accurately, assuming that Model 2 is based on the uniform speed assumption within a single-step sampling period:

[0021] in, is the differential of the controlled quantity in the current sampling period, which can be estimated using the classical Kalman filter, unscented Kalman filter, and tracking differentiator. T is a sampling period. Assuming that the system moves at a constant speed during the sampling period, the estimated value of the second controlled quantity increment Δθ2(k+1) can be obtained.

[0022] The above two hypothetical models only reflect the kinematic relationship without considering the dynamics. For general soft robotic systems, the system input is usually not strictly proportional to the controlled quantity. Based on this, hypothetical model 3 is constructed: Δθ3(k+1)=εΔu(k)=ε(u(k)-u(k-1))

[0023] Where u is the input of the system and ε is the proportional coefficient. The model assumes that the input and output of the system are directly proportional. The estimates of the above three hypothetical models are not completely accurate. We will use the idea of ​​data fusion to achieve an optimized estimate: Δθ(k+1)=Δθ1(k+1)+α(Δθ2(k+1)-Δθ1(k+1))+β(Δθ3(k+1)-Δθ1(k+1))

[0024] Among them, α and β are weight parameters to be identified,

[0025] Substitute the incremental equation and the three hypothetical models into the above formula:

[0026] Introduce hysteresis coordinates and record them as θ(k-1)=θ D (k),u(k-1)=u D (k), derive the final fusion prediction equation:

[0027] S12. Deriving the measurement coordinates. There is no need to measure the noise of the three hypothetical models to determine the weight parameters in the fusion prediction equation. The equation already provides a set of physical quantities that are closely related to the predicted controlled quantity and can be designed as the measurement coordinate x in the Koopman modeling:

[0028] Note that the present invention only exemplifies the derivation process of the fusion prediction equation based on three commonly used assumption models. In actual operation, the assumption models can be added or modified and the fusion prediction equation can be derived according to the specific characteristics and empirical behavior of the soft robot system, so as to derive correct, rich and non-redundant measurement coordinates for the customized soft robot system.

[0029] Furthermore, the step S2 of designing an observation function based on the measurement coordinates includes:

[0030] S21. Design an initial observation function based on the measured coordinates; design a set of high-dimensional nonlinear real-valued functions based on the measured coordinates, namely the initial observation function, whose forms mainly include: monomials, polynomials, trigonometric functions, and radial basis functions.

[0031] S22. Filter the observation function based on the SINDy algorithm; the SINDy (Sparse Identification of Dynamically Dominant Terms) algorithm is a publicly available technology for data-driven inference of dynamically dominant terms. The algorithm is used to sparsely identify the dominant terms in the initial observation function and filter them into the final observation function, thereby fully capturing the dynamics of the nonlinear system in the lowest dimension.

[0032] The step S1 of deriving the measurement coordinates based on the fusion prediction equation and the step S2 of designing the observation function based on the measurement coordinates together constitute a general observation function design method for Koopman modeling of soft robots, that is, designing the observation function based on the measurement coordinates derived from the fusion prediction equation. This method can replace the traditional empirical design, can fully capture the nonlinear dynamics of the system with the lowest dimension and minimum cost, and can improve the accuracy of the Koopman model.

[0033] Furthermore, the step S3 of identifying the Koopman model based on the observation function includes:

[0034] S31. Data collection: Collect a large number of random measurement coordinate data pairs through experiments or simulations, such as (x[j], x[j+1]), j∈{1,2,···,p}, and organize them into two matrices with a sampling step evolution relationship: X1=[x[1] x[2] ··· x[p]] X2=[x[2] x[3] ··· x[p+1]]

[0035] S32, data dimension increase; based on the observation function Ψ designed in step S2, the dimensions X1 and X2 are increased: X 1lift =[Ψ(x[1]) Ψ(x[2]) ··· Ψ(x[p])] X 2lift =[Ψ(x[2]) Ψ(x[3]) ··· Ψ(x[p+1])]

[0036] Introducing the input term, X 1lift ,X2lift Expanded to: Y1=[X 1lift U] Τ Y2=[X 2lift U] Τ

[0037] Where U = [u[1] u[2] ··· u[p]]. Note that since the evolution of the system input is not considered, the same input terms are expanded.

[0038] S33. Identify the Koopman model; minimize the following objective function to obtain a finite-dimensional approximate representation of the Koopman operator

[0039] Then from Isolate the correlation matrix of the Koopman model:

[0040] Define the mapping matrix C d = [IO], establish the control-oriented Koopman model: d [k+1]=A d z d [k]+B d u[k] x[k]=C d z d [k]

[0041] Among them, z d is the state of the measurement coordinate x mapped to the high-dimensional Koopman space, A d ,B d ,C d are the matrix coefficients of the Koopman model respectively.

[0042] Furthermore, the step S4 of designing a robust model predictive controller based on the Koopman model includes:

[0043] S41, converting to Koopman incremental model; introducing augmented state z to convert the Koopman model identified in step 3 into a Koopman incremental model: z[k+1]=Az[k]+BΔu[k] x[k]=Cz[k]

[0044] Where z[k]=[z d [k] u[k-1]] Τ , Δu[k]=u[k]-u[k-1], the corresponding matrix coefficients are rewritten as: B=[B d I] Τ , C=[C dO]; Based on the Koopman incremental model, a model predictive controller is designed, which adds an integral action to the closed-loop system and improves the robustness of the system.

[0045] S42. Design dynamic constraints. Design a model predictive controller with dynamic constraints based on the Koopman incremental model. The optimization problem to be solved is:

[0046] Among them, N h For the prediction time domain, Q, R, and F are weight coefficients, and g is the dynamic constraint of Δu[k]. The value of g is always set to be positive and dynamically adjusted according to the tracking performance of the system: e[k]=|θ r [k]-θ[k]|

[0047] Among them, k g It can be considered as the stiffness of the controller, which is proportional to the response speed, b g It can be considered as the damping of the controller, which helps to reduce system oscillations. r It is the reference value of the controlled quantity for pre-tracking.

[0048] S43. Optimize and solve the control quantity and output the control quantity; transform the above optimization problem into a standard quadratic programming problem, and optimize and solve the optimal sequence of control quantity increments under dynamic constraints in each sampling period; finally, select the first value of the optimal sequence of control quantity increments and add it to the control quantity of the previous sampling period to obtain the optimal control quantity of the current sampling period, and repeat the solution process in the next sampling period.

[0049] The beneficial effects of the present invention are:

[0050] 1. The present invention proposes a fusion prediction equation and its derivation method, which can derive correct, rich and non-redundant measurement coordinates, overcoming the problem of single measurement coordinates in soft robot systems, helping to simplify the design process of observation functions and further improve the accuracy of the Koopman model of soft robots.

[0051] 2. In the present invention, the observation function design method based on the fusion prediction equation can fully capture the nonlinear dynamics of the system in the lowest dimension, which is conducive to improving the accuracy of the Koopman model of the soft robot and reducing the complexity of the controller solution. It can completely replace the traditional empirical design method and is suitable for soft robot systems with customized materials, structures, and drives.

[0052] 3. The present invention proposes a robust model predictive controller based on the Koopman model. This controller adds an integral action to the closed-loop system, which can effectively enhance the robustness of the control system to handle parameter uncertainty and external disturbances. In addition, the controller dynamically adjusts constraints and solves the optimal control instructions according to the system performance, which can significantly improve the dynamic and steady-state tracking performance of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] FIG1 is a flow chart of the soft robot optimization modeling and robust control method based on fusion prediction equations according to the present invention;

[0054] Figure 2 is a control block diagram of the soft robot optimization modeling and robust control method based on the fusion prediction equation described in the present invention applied to an actual soft robot system. DETAILED DESCRIPTION

[0055] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention.

[0056] Example 1

[0057] This embodiment describes the specific implementation details of a method for optimizing modeling and robust control of a soft robot based on a fusion prediction equation. The flowchart of the method is shown in FIG1 , and includes the following steps:

[0058] S1. Derive the measurement coordinates based on the fusion prediction equation. This step specifically includes:

[0059] S11. Derive the fusion prediction equation. The purpose of the fusion prediction equation is to obtain measurable physical quantities that are closely related to the predicted controlled quantity and use them as measurement coordinates to design observation functions to fully capture the system dynamics and improve the accuracy of Koopman modeling. Based on the idea of ​​data fusion, the fusion prediction equation achieves optimized prediction based on multiple hypothetical models. Its derivation is derived from the general incremental equation: θ(k+1)=θ(k)+Δθ(k+1)

[0060] Where θ is the controlled variable, usually the curvature or end position of the soft robot. As can be seen from the above formula, the key to predicting θ(k+1) is to accurately estimate the increment of the controlled variable Δθ(k+1). Furthermore, Δθ(k+1) is estimated based on multiple hypothetical models, assuming that model 1 is a constant increment model: Δθ1(k+1)=Δθ(k)=θ(k)-θ(k-1)

[0061] Here, Δθ1(k+1) is the estimated value of the controlled variable increment obtained by linear extrapolation based on the assumption model 1. This model assumes that the controlled variable increment is constant in each sampling period. This full-range uniform speed assumption is only applicable to very short sampling periods or systems with stable operation. Relatively accurately, another assumption model 2 is based on the uniform speed assumption within a single-step sampling period:

[0062] in, is the differential of the controlled variable in the current sampling period, which can be estimated using classical Kalman filtering, unscented Kalman filtering, and tracking differentiators, where T is one sampling period. Assuming that the system moves at a constant speed during this sampling period, an estimated value of the second controlled variable increment, Δθ2(k+1), can be obtained. The above two hypothetical models only reflect the kinematic relationship without considering dynamics. For general soft robotic systems, such as fluid-driven soft robots, the system input is usually not strictly proportional to the controlled variable. Based on this, hypothetical model 3 is constructed: Δθ3(k+1)=εΔu(k)=ε(u(k)-u(k-1))

[0063] Where u is the input of the system and ε is the proportional coefficient. The model assumes that there is a positive proportional relationship between the input and output of the system. Note that the estimates of the above three hypothetical models are not completely accurate and have their own application scenarios, but based on the idea of ​​data fusion, an optimized estimate can be achieved: Δθ(k+1)=Δθ1(k+1)+α(Δθ2(k+1)-Δθ1(k+1))+β(Δθ3(k+1)-Δθ1(k+1))

[0064] Among them, α and β are weight parameters to be identified,

[0065] Substitute the incremental equation and the three hypothetical models into the above formula:

[0066] Introduce hysteresis coordinates and record them as θ(k-1)=θ D (k),u(k-1)=u D (k), derive the final fusion prediction equation:

[0067] S12. Derive the measurement coordinates. To predict the controlled variable, the fused prediction equation fully utilizes the current controlled variable, the time-delayed controlled variable, the differential of the current controlled variable, the time-delayed input, and the current input. In fact, there is no need to measure the noise of the three hypothetical models to determine the weight parameters in the fused prediction equation. The equation already provides a set of physical quantities closely related to the predicted controlled variable, which can be designed as the measurement coordinate x in Koopman modeling:

[0068] Note that this example only illustrates the derivation of a fused prediction equation based on three commonly used hypothetical models. In practice, additional or modified hypothetical models can be added or modified to derive fused prediction equations based on the specific characteristics and empirical behavior of the soft robotic system, thereby deriving accurate, comprehensive, and non-redundant measurement coordinates for different soft robotic systems.

[0069] S2. Designing an observation function based on the measured coordinates. This step specifically includes:

[0070] S21. Design an initial observation function based on the measured coordinates. Based on the measured coordinates, design a set of high-dimensional nonlinear real-valued functions, i.e., the initial observation functions. These functions may take the form of monomials, polynomials, trigonometric functions, radial basis functions, etc. These initial observation functions, by virtue of their high dimensionality and rich measurement coordinates, capture nearly all the dynamics of the nonlinear system. However, their high dimensionality greatly increases the computational complexity of subsequent Koopman model identification and optimal control.

[0071] S22. Filter the observation function based on the SINDy algorithm. The SINDy (Sparsely Identified Dynamically Dominant) algorithm is a publicly available data-driven technique for inferring the dominant term of dynamics. This algorithm sparsely identifies the dominant term in the initial observation function and filters it as the final observation function, thereby fully capturing the dynamics of the nonlinear system in the lowest dimension.

[0072] The step S1 of deriving the measurement coordinates based on the fusion prediction equation and the step S2 of designing the observation function based on the measurement coordinates together constitute a general observation function design method for Koopman modeling of soft robots, that is, designing the observation function based on the measurement coordinates derived from the fusion prediction equation. This method can replace the traditional empirical design, can fully capture the nonlinear dynamics of the system with the lowest dimension and minimum cost, can improve the accuracy of the Koopman model, and reduce the complexity of the optimal control solution. At the same time, the data-driven form makes it adaptable to soft robot systems with customized materials, structures, and drives.

[0073] S3. Identify the Koopman model based on the observation function. This step specifically includes:

[0074] S31. Data acquisition. Through experiments or simulations, a large number of random measurement coordinate state pairs are collected, such as (x[j], x[j+1]), j∈{1,2,···,p}, and organized into two matrices with a sampling step evolution relationship: X1=[x[1] x[2] ··· x[p]] X2=[x[2] x[3] ··· x[p+1]]

[0075] S32, data dimension increase. Based on the observation function Ψ designed in step S2, the dimensions X1 and X2 are increased: X 1lift=[Ψ(x[1]) Ψ(x[2]) ··· Ψ(x[p])] X 2lift =[Ψ(x[2]) Ψ(x[3]) ··· Ψ(x[p+1])]

[0076] Furthermore, considering that the soft robot system is a controlled system, it is necessary to introduce input items, X 1lift ,X 2lift Further expansion: Y1=[X 1lift U] Τ Y2=[X 2lift U] Τ

[0077] Where U = [u[1] u[2] ··· u[p]], note that since there is no need to consider the evolution of the system input, the same input terms are expanded.

[0078] S33. Identify the Koopman model. By using the least squares method, particle swarm optimization algorithm, and other machine learning algorithms, minimize the following objective function to obtain a finite-dimensional approximate representation of the Koopman operator:

[0079] Further, from Isolate the correlation matrix of the Koopman model:

[0080] Define the mapping matrix C d =[IO].

[0081] Finally, a control-oriented Koopman model is established for the soft robot system: d [k+1]=A d z d [k]+B d u[k] x[k]=C d z d [k]

[0082] where z d is the state of the measurement coordinate x mapped to the high-dimensional Koopman space, A d ,B d ,C d are the matrix coefficients of the Koopman model respectively.

[0083] S4. Design a robust model predictive controller based on the Koopman model. This step specifically includes:

[0084] S41, convert to Koopman incremental model. The Koopman model identified in step 3 is converted to Koopman incremental model by introducing augmented state z: z[k+1]=Az[k]+BΔu[k] x[k]=Cz[k]

[0085] Where z[k]=[z d [k] u[k-1]] Τ , Δu[k]=u[k]-u[k-1], the corresponding matrix coefficients are rewritten as: B=[B d I] Τ , C=[C d O]. Based on the Koopman incremental model, a model predictive controller is designed, which adds an integral action to the closed-loop system and improves the robustness of the system.

[0086] S42. Design dynamic constraints. Based on the Koopman incremental model, a model predictive controller with dynamic constraints is designed. The optimization problem to be solved is:

[0087] Among them, N h For time-domain prediction, Q, R, and F are weight coefficients, and g is the dynamic constraint on Δu[k]. Setting g to a constant positive value balances the system's dynamic and steady-state performance. A larger g allows for drastic changes in the control increment, improving response speed but also causing system oscillations in steady state. A smaller g reduces steady-state oscillations but also slows response. Therefore, inspired by the principle of impedance control, g is set to a constant positive value and dynamically adjusted based on the system's tracking performance to improve both dynamic and steady-state performance. e[k]=|θ r [k]-θ[k]|

[0088] Among them, k g It can be considered as the stiffness of the controller, which is proportional to the response speed; b g It can be considered as the damping of the controller, which helps to reduce system oscillations. r It is the reference value of the controlled quantity for pre-tracking.

[0089] S43. Optimize and solve the control variable and output it. Using basic reasoning methods such as linear extrapolation, the above optimization problem is transformed into a standard quadratic programming problem. Then, within each sampling period, the optimal sequence of control variable increments under dynamic constraints is solved. Furthermore, the first value of the optimal sequence of control variable increments is selected and added to the control variable from the previous sampling period to obtain the optimal control variable for the current sampling period. This solution process is repeated for the next sampling period.

[0090] Example 2

[0091] This embodiment describes a specific implementation structure of a soft robot optimization modeling and robust control method based on a fusion prediction equation in practical applications. Referring to Figure 2, Figure 2 is a control block diagram of a soft robot optimization modeling and robust control method based on a fusion prediction equation applied to an actual soft robot system, including an optimized Koopman model, a robust model prediction controller, a power control element, a soft robot, and a sensor element. The advantages of the optimized Koopman model are: based on the fusion prediction equation, correct, rich, and non-redundant measurement coordinates are derived, and the SINDy algorithm is used to screen out the observation function that captures the system dynamics in the lowest dimension. Koopman modeling is performed based on the observation function, which can optimize the Koopman model accuracy of the soft robot. The robust model predictive controller is designed based on the optimized Koopman model, and adds integral action to the closed-loop system. It dynamically adjusts constraints according to the tracking performance of the system, solves the optimal control instructions and sends them to the power control element; common power control elements include electrical proportional valves, flow valves, motors, etc., and their types depend on the driving method of the soft robot. The power control element transmits power and drives the soft robot to achieve preset movement or behavior; the sensing element is used to measure the kinematic or mechanical signals of the soft robot, including curvature, end position, end output force, output bending moment, etc., and feeds back to the robust model predictive controller to form a closed-loop control system of the soft robot.

[0092] It should be noted that the above content merely illustrates the technical idea of ​​the present invention and cannot be used to limit the scope of protection of the present invention. For ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications all fall within the scope of protection of the claims of the present invention.

Claims

1. Optimal modeling and robust control method for soft robots based on fusion prediction equations, characterized by: The following steps are involved: S1, deriving measurement coordinates based on fusion prediction equations; S2, design observation function based on measurement coordinates; S3, identifying the Koopman model based on the observation function; S4. Design a robust model predictive controller based on the Koopman model.

2. The method for optimizing modeling and robust control of a soft robot based on a fusion prediction equation according to claim 1, characterized in that: The step S1 of deriving the measurement coordinates based on the fusion prediction equation includes: S11. Derivation of fusion prediction equation: Based on the idea of ​​data fusion, the fusion prediction equation realizes optimized prediction on the basis of multiple hypothesis models, and its derivation originates from the general incremental equation: θ(k+1)=θ(k)+Δθ(k+1) Wherein, θ is the controlled variable, usually the curvature or end position of the soft robot, and the key to predicting θ(k+1) is to accurately estimate the increment Δθ(k+1) of the controlled variable; further, Δθ(k+1) is estimated based on multiple assumed models, assuming that model 1 is a constant increment model: Δθ1(k+1)=Δθ(k)=θ(k)-θ(k-1) Among them, Δθ1(k+1) is the estimated value of the controlled quantity increment obtained by linear extrapolation based on the assumed model 1. This full-range uniform speed assumption is only applicable to systems with extremely short sampling periods or stable operation. Relatively accurately, assuming that model 2 is based on the uniform speed assumption within a single-step sampling period: in, is the differential of the controlled quantity in the current sampling period, estimated by the tracking differentiator. T is a sampling period. Assuming that the system moves at a constant speed during the sampling period, the estimated value of the second controlled quantity increment Δθ2(k+1) can be obtained. The above two hypothetical models only reflect the kinematic relationship without considering the dynamics. For general soft robot systems, the input of the system usually has a non-strict proportional relationship with the controlled quantity. Based on this, hypothetical model 3 is constructed: Δθ3(k+1)=εΔu(k)=ε(u(k)-u(k-1)) Among them, u is the input of the system, ε is the proportional coefficient, and the model assumes that there is a positive proportional relationship between the input and output of the system; the estimates of the above three hypothetical models are not completely accurate, and the optimization estimate will be realized based on the idea of ​​data fusion: Δθ(k+1)=Δθ1(k+1)+α(Δθ2(k+1)-Δθ1(k+1))+β(Δθ3(k+1)-Δθ1(k+1)); Among them, α and β are the weight parameters to be identified, Substitute the incremental equation and the three assumed models into the above formula: Introduce hysteresis coordinates and record them as θ(k-1)=θ D (k),u(k-1)=u D (k), derive the final fusion prediction equation: S12, deriving the measurement coordinates; there is no need to measure the noise of the above three hypothetical models to determine the weight parameters in the fusion prediction equation. The equation has provided a set of physical quantities closely related to the predicted controlled quantity, which can be designed as the measurement coordinates x in Koopman modeling:

3. The method for optimizing modeling and robust control of a soft robot based on a fusion prediction equation according to claim 1, characterized in that: The step S2 of designing an observation function based on the measured coordinates comprises: S21. Design an initial observation function based on the measured coordinates; design a set of high-dimensional nonlinear real-valued functions based on the measured coordinates, namely, the initial observation function, whose forms mainly include: monomial, polynomial, trigonometric function, radial basis function; S22, screening observation functions based on the SINDy algorithm; using the algorithm to sparsely identify the dominant terms in the initial observation function and screen them as the final observation function, so as to fully capture the dynamics of the nonlinear system in the lowest dimension; The step S1 of deriving the measurement coordinates based on the fused prediction equation and the step S2 of designing the observation function based on the measurement coordinates together constitute a general observation function design method for Koopman modeling of the soft robot, that is, designing the observation function based on the measurement coordinates derived from the fused prediction equation.

4. The method for optimizing modeling and robust control of a soft robot based on a fusion prediction equation according to claim 1, characterized in that: The step S3 of identifying the Koopman model based on the observation function comprises: S31, data collection: collect a large number of random measurement coordinate data pairs through experiments or simulations, and organize them into two matrices with a sampling step evolution relationship: X1=[x[1] x[2] ··· x[p]] X2=[x[2] x[3]···x[p+1]] S32, data dimension upgrade; based on the observation function Ψ designed in step S2, the dimension of X1 and X2 is upgraded: X 1lift =[Ψ(x[1]) Ψ(x[2]) ··· Ψ(x[p])] X 2lift =[Ψ(x[2]) Ψ(x[3]) ··· Ψ(x[p+1])] Introduce input term, X 1lift ,X 2lift Expands to: Y1=[X 1lift [U] Τ Y2=[X 2lift U] Τ Where U = [u[1] u[2] ··· u[p]], note that since the evolution of the system input is not considered, the same input terms are expanded; S33, identify the Koopman model; minimize the following objective function to obtain the finite-dimensional approximate representation of the Koopman operator Then from Isolate the correlation matrix of the Koopman model: Define the mapping matrix C d =[IO], establish the control-oriented Koopman model: z d [k+1]=A d z d [k]+B d u[k] x[k]=C d z d [k] Among them, z d is the state where the measurement coordinate x is mapped to the high-dimensional Koopman space, A d ,B d ,C d are the matrix coefficients of the Koopman model respectively.

5. The method for optimizing modeling and robust control of a soft robot based on fusion prediction equations according to claim 1, characterized in that: The step S4 of designing a robust model predictive controller based on the Koopman model includes: S41, converting to a Koopman incremental model; introducing an augmented state z to convert the Koopman model identified in step 3 into a Koopman incremental model: z[k+1]=Az[k]+BΔu[k] x[k]=Cz[k] Where z[k]=[z d [k] u[k-1]] Τ , Δu[k]=u[k]-u[k-1], the corresponding matrix coefficients are rewritten as: B=[B d I] Τ , C=[C d O]; Based on the Koopman incremental model, a model predictive controller is designed, which adds an integral action to the closed-loop system and improves the system robustness; S42, designing dynamic constraints; designing a model predictive controller with dynamic constraints based on the Koopman incremental model, and the optimization problem to be solved is: stz[k+1]=Az[k]+BΔu[k] -g≤Δu[k]≤g Among them, N h For the prediction time domain, Q, R, and F are weight coefficients, and g is the dynamic constraint of Δu[k]. The value of g is always set to be positive and dynamically adjusted with the tracking performance of the system: e[k]=|θ r [k]-θ[k]| Among them, k g It can be considered as the stiffness of the controller, which is proportional to the response speed, b g It can be considered as the damping of the controller, which helps to reduce system oscillations, θ r is the reference value of the controlled quantity to be tracked in advance; S43, optimize the solution and output the control quantity; convert the above optimization problem into a standard quadratic programming problem, and optimize the solution of the optimal sequence of control quantity increments under dynamic constraints in each sampling period; finally, select the first value of the optimal sequence of control quantity increments and add it to the control quantity of the previous sampling period to obtain the optimal control quantity of the current sampling period, and repeat the solution process in the next sampling period.

6. The specific implementation structure of the method for optimizing modeling and robust control of a soft robot based on fusion prediction equations in practical applications according to claim 1 is characterized by: It includes an optimized Koopman model, a robust model predictive controller, a power control element, a soft robot and a sensor element; the optimized Koopman model can derive the correct measurement coordinates based on the fusion prediction equation, and screen out the observation function that captures the system dynamics in the lowest dimension through the SINDy algorithm, and perform Koopman modeling based on the observation function, which can optimize the Koopman model accuracy of the soft robot; the robust model predictive controller is designed based on the optimized Koopman model, and adds integral action to the closed-loop system, and dynamically adjusts constraints according to the tracking performance of the system, solves the optimal control instruction and sends it to the power control element; the power control element transmits power and drives the soft robot to achieve preset motion or behavior; the sensor element is used to measure the kinematic or mechanical signals of the soft robot, including curvature, end position, end output force, output bending moment, and feed back to the robust model predictive controller to form a closed-loop control system of the soft robot.

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