Optimization Modeling and Robust Control Method of Soft Robot Based on Fusion Prediction Equation

Abstract

The present invention discloses an optimization modeling and robust control method for soft robots based on a fusion prediction equation. First, the measured coordinates are derived based on the fusion prediction equation; then an observation function is designed based on the measured coordinates; next, the Koopman model is identified based on the observation function; finally, a robust model predictive controller is designed based on the Koopman model. The present invention proposes a fusion prediction equation and its derivation method. This equation can derive correct, rich, and non-redundant measured coordinates, overcoming the problem of single measured coordinates in soft robot systems, helping to simplify the design process of the observation function, further improving the accuracy of the Koopman model of soft robots, realizing the precise motion control of soft robots, and enhancing the robustness of the control system.

Description

Technical Field

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

[0002] Robot technology plays an important role in the manufacturing, medical, and service fields. Traditional rigid robotic arms excel in force output intensity and motion accuracy. However, due to their high-stiffness materials and structures, they are difficult to perform well in complex and variable environments. As a driving and sensing integrated device with high-elasticity materials as its main body, soft robots are booming in fields such as rescue missions, medical rehabilitation, and human-computer interaction due to their inherent flexibility and adaptability, showing great application prospects.

[0003] However, despite the many theoretical advantages of soft robots, their accurate modeling and control still face severe challenges. The reasons are as follows: 1. The flexible materials of the soft robot body exhibit complex hysteretic nonlinearities, making it extremely difficult to establish a mechanism model based on the mechanical properties and geometric structures of the materials. 2. The driving method 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 nonlinearities 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 technologies. The above reasons have led to the situation that most current soft robots can only achieve rough motion function verification and are difficult to be precisely controlled, limiting their practical applications.

[0004] Fortunately, the Koopman operator theory proves that by mapping the nonlinear dynamics to a high-dimensional Koopman space through an observation function, high-fidelity global linearization can be achieved, and then mature linear control methods can be applied to control complex nonlinear systems. In addition, this method is data-driven, avoiding cumbersome mechanism modeling and being adaptable to soft robot systems with customized designs. Currently, there are already some publicly available technologies applying the Koopman operator theory to the modeling and control of soft robots.

[0005] However, there are still two unsolved problems in these technologies:

[0006] 1. For the Koopman modeling of soft robots, there is no general method for designing observation functions. The quality of the observation function directly affects the accuracy of the Koopman model, and current observation functions are usually selected based on experience, with high design costs and no guarantee of effectiveness.

[0007] 2. The 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 uncertainties and external disturbances. Summary of the Invention

[0008] To solve the above problems, the present invention discloses an optimized modeling and robust control method for soft robots based on a fusion prediction equation, which is used to solve the problem of the observation function design in the Koopman modeling of soft robots, thereby optimizing the accuracy of the Koopman model, further realizing the precise motion control of soft robots, and enhancing the robustness of the control system.

[0009] To achieve the above object, the technical solution of the present invention is as follows:

[0010] An optimized modeling and robust control method for soft robots based on a fusion prediction equation, comprising the following steps:

[0011] S1. Derive the measurement coordinates based on the fusion prediction equation;

[0012] S2. Design an observation function based on the measurement coordinates;

[0013] S3. Identify the Koopman model based on the observation function;

[0014] S4. Design a robust model predictive controller based on the Koopman model.

[0015] Further, the step S1 of deriving the measurement coordinates based on the fusion prediction equation includes:

[0016] S11. Derive the fusion prediction equation; the purpose of the fusion prediction equation is to obtain measurable physical quantities closely related to the predicted controlled quantity, and use them as the measurement coordinates to design the observation function, so as to fully capture the system dynamics and improve the accuracy of Koopman modeling; based on the idea of data fusion, the fusion prediction equation realizes optimized prediction on the basis of multiple assumed models, and its derivation originates from the general incremental equation:

[0017] θ(k + 1) = θ(k) + △θ(k + 1)

[0018] where θ is the controlled quantity, usually the curvature or end position of the soft robot, and the key to predicting θ(k + 1) is to accurately estimate the increment △θ(k + 1) of the controlled quantity; further, estimate △θ(k + 1) based on multiple assumed models, and the assumed model 1 is a constant increment model:

[0019] △θ1(k + 1) = △θ(k) = θ(k) - θ(k - 1)

[0020] Among them, △θ1(k + 1) is the estimated value of the increment of the controlled variable obtained by linear extrapolation based on the said Hypothetical Model 1. This full-range uniform motion assumption only applies to extremely short sampling periods or systems operating in a steady state;

[0021] Relatively precisely, Hypothetical Model 2 is based on the uniform motion assumption within a single sampling period:

[0022]

[0023] Among them, is the differential of the controlled variable in the current sampling period, which can be estimated by classical Kalman filter, unscented Kalman filter, or tracking differentiator. T is a sampling period. Assuming that the system moves at a uniform speed during this sampling period, the estimated value of the second increment of the controlled variable △θ2(k + 1) can be obtained;

[0024] 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 variable. Based on this, Hypothetical Model 3 is constructed:

[0025] △θ3(k + 1) = ε△u(k) = ε(u(k) - u(k - 1))

[0026] Among them, u is the input of the system, and ε is the proportionality coefficient. This model assumes that there is a direct proportional relationship between the input and output of the system; The estimations of the above three hypothetical models are not completely accurate. An optimized estimation will be achieved based on the idea of data fusion:

[0027] △θ(k + 1) = △θ1(k + 1) + α(△θ2(k + 1) - △θ1(k + 1)) + β(△θ3(k + 1) - △θ1(k + 1))

[0028] Among them, α and β are weight parameters to be identified,

[0029] Substitute the incremental equation and the three hypothetical models into the above formula:

[0030]

[0031] Introduce the hysteresis coordinate and denote θ(k - 1) = θ D (k), u(k - 1) = u D (k), and derive the final fusion prediction equation:

[0032]

[0033] S12. Export 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. This equation has provided a set of physical quantities closely related to the predicted controlled quantity, which can be designed as the measurement coordinate x in Koopman modeling:

[0034]

[0035] Note that the present invention only exemplifies the derivation process of the fusion prediction equation based on the three common hypothetical models. In actual operation, the hypothetical models can be added or modified according to the specific characteristics and empirical behaviors of the soft robot system, and the fusion prediction equation can be derived, so as to export correct, rich and non-redundant measurement coordinates for the customized soft robot system.

[0036] Further, the step S2 of designing the observation function based on the measurement coordinates includes:

[0037] S21. Design the initial observation function based on the measurement coordinates; design a set of high-dimensional non-linear real-valued functions based on the measurement coordinates, that is, the initial observation function, and its forms mainly include: monomials, polynomials, trigonometric functions, and radial basis functions.

[0038] S22. Screen the observation function based on the SINDy algorithm; the SINDy (Sparse Identification of Nonlinear Dynamics) algorithm is a publicly known technology for data-driven speculation of the dominant terms of dynamics. Use this 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 with the lowest dimension.

[0039] 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 the smallest cost, and can improve the accuracy of the Koopman model.

[0040] Further, the step S3 of identifying the Koopman model based on the observation function includes:

[0041] S31. Data acquisition; collect a large number of random measurement coordinate data pairs through experiments or simulations, in the form of (x[j], x[j + 1]), j ∈ {1, 2, …, p} and organize them into two matrices with an evolution relationship of one sampling step:

[0042] X1 = [x[1] x[2] … x[p]]

[0043] X2 = [x[2] x[3] … x[p + 1]]

[0044] S32. Data dimensionality increase; increase the dimensions of X1 and X2 based on the observation function Ψ designed in the said step S2:

[0045] X 1lift = [Ψ(x[1]) Ψ(x[2]) … Ψ(x[p])]

[0046] X 2lift = [Ψ(x[2]) Ψ(x[3]) … Ψ(x[p+1])]

[0047] Introduce the input term, X 1lift , X 2lift Expand to:

[0048] Y1 = [X 1lift U] Τ

[0049] Y2 = [X 2lift U] Τ

[0050] where U = [u[1] u[2] … u[p]]. Note that since the evolution of the system input is not considered, the same input term is expanded.

[0051] S33. Identify the Koopman model; obtain a finite-dimensional approximate representation of the Koopman operator by minimizing the following objective function

[0052]

[0053] Then isolate and partition the relevant matrix of the Koopman model from :

[0054]

[0055] Define the mapping matrix C d = [I O], and establish a control-oriented Koopman model:

[0056] z d [k+1] = A d z d [k] + B d u[k]

[0057] x[k] = C d z d [k]

[0058] where z d is the state obtained by mapping the measurement coordinate x to the high-dimensional Koopman space, A d , Bd , C d are the matrix coefficients of the Koopman model respectively.

[0059] Furthermore, the step S4 of designing a robust model predictive controller based on the Koopman model includes:

[0060] S41. Transforming into a Koopman incremental model; introducing an augmented state z to transform the Koopman model identified in step 3 into a Koopman incremental model:

[0061] z[k + 1] = Az[k] + B△u[k]

[0062] x[k] = Cz[k]

[0063] where z[k] = [z d [k] u[k - 1]] Τ , △u[k] = u[k] - u[k - 1], and the corresponding matrix coefficients are rewritten as: B = [B d I] Τ , C = [C d O]; Designing a model predictive controller based on this Koopman incremental model adds an integral action to the closed-loop system, which can improve the system robustness.

[0064] S42. Designing dynamic constraint conditions; designing a model predictive controller with dynamic constraint conditions based on the Koopman incremental model, and the optimization problem to be solved is:

[0065]

[0066] s.t. z[k + 1] = Az[k] + B△u[k]

[0067] -g ≤ △u[k] ≤ g

[0068] where N h is the prediction horizon, Q, R, F are weight coefficients, g is the dynamic constraint of △u[k], and the value of g is always set to be positive and dynamically adjusted according to the tracking performance of the system:

[0069]

[0070] e[k] = |θ r [k] - θ[k]|

[0071]

[0072] where k g can be regarded as the stiffness of the controller, which is proportional to the response speed, and b gIt can be regarded as the damping of the controller, which helps to reduce system oscillation, θ r is the reference value of the controlled quantity to be pre-tracked.

[0073] S43. Optimize and solve to output the control quantity; transform the above optimization problem into a standard quadratic programming problem. In each sampling period, optimize and solve the optimal sequence of the control quantity increment under dynamic constraints; finally, select the first value of the optimal sequence of the control quantity increment 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 this solution process in the next sampling period.

[0074] The beneficial effects of the present invention are as follows:

[0075] 1. The present invention proposes a fusion prediction equation and its derivation method. This equation can derive correct, rich and non-redundant measurement coordinates, overcoming the problem of single measurement coordinates in the soft robot system, helping to simplify the design process of the observation function, and further improving the Koopman model accuracy of the soft robot.

[0076] 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 with the lowest dimension, which is beneficial to improving the Koopman model accuracy of the soft robot and reducing the solution complexity of the controller. It can completely replace the traditional empirical design method and is suitable for soft robot systems with customized materials, structures and drives.

[0077] 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 uncertainties and external disturbances. In addition, this controller dynamically adjusts the constraints according to the system performance and solves the optimal control instruction, which can significantly improve the dynamic and steady-state tracking performance of the system. Description of the Drawings

[0078] Attached Figure 1 is the flow chart of the optimization modeling and robust control method for the soft robot based on the fusion prediction equation of the present invention;

[0079] Attached Figure 2 is the control block diagram of the optimization modeling and robust control method for the soft robot based on the fusion prediction equation of the present invention applied to an actual soft robot system. Detailed Embodiments

[0080] The following further clarifies the present invention in conjunction with the drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and not to limit the scope of the present invention.

[0081] Embodiment 1

[0082] This embodiment describes the specific implementation details of an optimization modeling and robust control method for soft robots based on a fusion prediction equation. The flowchart of this method is as shown in Figure 1 and includes the following steps:

[0083] S1. Derive the measurement coordinates based on the fusion prediction equation. This step specifically includes:

[0084] S11. Derive the fusion prediction equation. The purpose of the fusion prediction equation is to obtain measurable physical quantities closely related to the predicted controlled quantity, and use them as the measurement coordinates to design an observation function, so as to fully capture the system dynamics and improve the Koopman modeling accuracy. Based on the idea of data fusion, the fusion prediction equation realizes optimized prediction on the basis of multiple hypothetical models, and its derivation originates from the general incremental equation:

[0085] θ(k + 1) = θ(k) + △θ(k + 1)

[0086] where θ is the controlled quantity, usually the curvature or the end position of the soft robot. It can be seen from the above formula that the key to predicting θ(k + 1) is to accurately estimate the increment △θ(k + 1) of the controlled quantity. Further, based on multiple hypothetical models, △θ(k + 1) is estimated. The hypothetical model 1 is a constant increment model:

[0087] △θ1(k + 1) = △θ(k) = θ(k) - θ(k - 1)

[0088] where △θ1(k + 1) is the estimated value of the increment of the controlled quantity obtained by linear extrapolation based on the hypothetical model 1. This model assumes that the increment of the controlled quantity in each sampling period is constant. This full-range uniform motion assumption only applies to extremely short sampling periods or systems operating stably. Relatively accurately, another hypothetical model 2 is based on the uniform motion assumption within a single sampling period:

[0089]

[0090] where is the differential of the controlled quantity in the current sampling period, which can be estimated by classical Kalman filter, unscented Kalman filter, or tracking differentiator. T is a sampling period. Assuming that the system moves uniformly in this sampling period, the estimated value △θ2(k + 1) of the second increment of the controlled quantity can be obtained. The above two hypothetical models only reflect the kinematic relationship and do not consider the dynamics. For general soft robot systems, such as fluid-driven soft robots, the input of the system usually has a non-strict proportional relationship with the controlled quantity. Based on this, the hypothetical model 3 is constructed:

[0091] △θ3(k + 1) = ε△u(k) = ε(u(k) - u(k - 1))

[0092] Among them, \(u\) is the input of the system, and \(\varepsilon\) is the proportionality coefficient. This model assumes a direct proportional relationship between the input and output of the system. Note that the estimations of the above three assumed models are not completely accurate and have their respective application scenarios. However, based on the idea of data fusion, an optimized estimation can be achieved:

[0093] \(\Delta\theta(k + 1)=\Delta\theta_1(k + 1)+\alpha(\Delta\theta_2(k + 1)-\Delta\theta_1(k + 1))+\beta(\Delta\theta_3(k + 1)-\Delta\theta_1(k + 1))\)

[0094] Among them, \(\alpha\) and \(\beta\) are weight parameters to be identified.

[0095] Substitute the incremental equation and the three assumed models into the above formula:

[0096]

[0097] Introduce the hysteresis coordinate and denote \(\theta(k - 1)=\theta\) D (k), \(u(k - 1)=u\) D (k), and derive the final fusion prediction equation:

[0098]

[0099] S12. Derive the measurement coordinates. To achieve the prediction of the controlled quantity, the fusion prediction equation makes full use of the current controlled quantity, the time-delayed controlled quantity, the differential of the current controlled quantity, the time-delayed input, and the current input. In fact, it is not necessary to measure the noises of the above three assumed models to determine the weight parameters in the fusion prediction equation. This equation already provides a set of physical quantities closely related to the predicted controlled quantity, which can be designed as the measurement coordinate \(x\) in the Koopman modeling:

[0100]

[0101] Note that this embodiment only gives an example of the derivation process of the fusion prediction equation based on the commonly used three assumed models. In actual operation, the assumed models can be added or modified according to the specific characteristics and empirical behaviors of the soft robot system, and the fusion prediction equation can be derived to obtain correct, rich, and non-redundant measurement coordinates for different soft robot systems.

[0102] S2. Design an observation function based on the measurement coordinates. This step specifically includes:

[0103] S21. Design the initial observation function based on the measurement coordinates. Design a set of high-dimensional non-linear real-valued functions based on the measurement coordinates, which is the initial observation function. The form can include monomials, polynomials, trigonometric functions, radial basis functions, etc. The initial observation function captures almost all the dynamics of the non-linear system by virtue of its high-dimensional characteristics and rich measurement coordinates. However, the high-dimensional characteristics will greatly increase the computational complexity of subsequent Koopman model identification and optimal control.

[0104] S22. Screen the observation function based on the SINDy algorithm. The SINDy (Sparse Identification of Nonlinear Dynamics) algorithm is a publicly available technique for data-driven speculation of the dominant terms of dynamics. Use this 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 non-linear system with the lowest dimension.

[0105] 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 non-linear dynamics of the system with the lowest dimension and the least cost, can improve the accuracy of the Koopman model, and reduce the complexity of solving the optimal control. At the same time, the data-driven form enables it to adapt to soft robot systems with customized materials, structures, and drives.

[0106] S3. Identify the Koopman model based on the observation function. This step specifically includes:

[0107] S31. Data acquisition. Collect a large number of random measurement coordinate state pairs, in the form of (x[j], x[j + 1]), j ∈ {1, 2, …, p} through experiments or simulations, and organize them into two matrices with an evolution relationship of one sampling step:

[0108] X1 = [x[1] x[2] … x[p]]

[0109] X2 = [x[2] x[3] … x[p + 1]]

[0110] S32. Data dimensionality increase. Increase the dimensions of X1 and X2 based on the observation function Ψ designed in the step S2:

[0111] X 1lift = [Ψ(x[1]) Ψ(x[2]) … Ψ(x[p])]

[0112] X 2lift = [Ψ(x[2]) Ψ(x[3]) … Ψ(x[p + 1])]

[0113] Furthermore, considering that the soft robot system is a controlled system, an input term, \(X\), needs to be introduced. 1lift , \(X\) 2lift is further expanded to:

[0114] \(Y1 = [X\) 1lift \(U][\) Τ

[0115] \(Y2 = [X\) 2lift \(U][\) Τ

[0116] where \(U = [u[1]\ u[2]\ \cdots\ u[p]]\). Note that since the evolution of the system input does not need to be considered, the same input term is expanded.

[0117] S33. Identify the Koopman model. Through other machine learning algorithms such as the least squares method and the particle swarm optimization algorithm, minimize the following objective function to obtain a finite-dimensional approximate representation of the Koopman operator.

[0118]

[0119] Furthermore, from , isolate and partition the relevant matrices of the Koopman model:

[0120]

[0121] Define the mapping matrix \(C\) d \(= [I\ O]\).

[0122] Finally, for the soft robot system, establish a control-oriented Koopman model:

[0123] \(z\) d [k + 1] = A d \(z\) d [k] + B d u[k]

[0124] x[k] = C d \(z\) d [k]

[0125] where \(z\) d is the state obtained by mapping the measurement coordinate \(x\) into the high-dimensional Koopman space, and \(A\) d , \(B\) d , \(C\) d are the matrix coefficients of the Koopman model, respectively.

[0126] S4. Design a robust model predictive controller based on the Koopman model. This step specifically includes:

[0127] S41. Transform into a Koopman incremental model. The Koopman model identified in Step 3 is transformed into a Koopman incremental model by introducing the augmented state z:

[0128] z[k + 1]= Az[k]+ BΔu[k]

[0129] x[k]= Cz[k]

[0130] where z[k]=[z d [k] u[k - 1]] Τ , Δu[k]= u[k]- u[k - 1], and the corresponding matrix coefficients are rewritten as: B =[B d I] Τ , C =[C d O]. Design a model predictive controller based on this Koopman incremental model, which adds an integral action to the closed-loop system and can improve the system robustness.

[0131] S42. Design dynamic constraint conditions. Design a model predictive controller with dynamic constraint conditions based on the Koopman incremental model. The optimization problem to be solved is:

[0132]

[0133] s.t. z[k + 1]= Az[k]+ BΔu[k]

[0134] -g ≤ Δu[k]≤ g

[0135] where N h is the prediction horizon, Q, R, F are weight coefficients, g is the dynamic constraint of Δu[k]. Set g to be always positive, and its value balances the dynamic and steady-state performance of the system. A larger g allows the control increment to change drastically, which helps to improve the response speed, but it is also easy to cause system oscillation at steady state; a smaller g helps to reduce the steady-state oscillation but has a slow response. Therefore, inspired by the impedance control idea, set the value of g to be always positive and dynamically adjusted according to the tracking performance of the system to improve both the dynamic and steady-state performance of the system.

[0136]

[0137] e[k]=|θ r [k]- θ[k]|

[0138]

[0139] where k g can be considered as the stiffness of the controller, which is proportional to the response speed; b gIt can be regarded as the damping of the controller, which helps to reduce system oscillation, θ r is the reference value of the controlled quantity to be pre-tracked.

[0140] S43. Optimize and solve to output the control quantity. By 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 the control quantity increment under dynamic constraints is solved. Further, the first value of the optimal sequence of the control quantity increment is selected and added to the control quantity of the previous sampling period, that is, the optimal control quantity of the current sampling period is obtained, and this solving process is repeated in the next sampling period.

[0141] Embodiment 2

[0142] This embodiment describes the specific implementation structure of an optimized modeling and robust control method for a soft robot based on a fusion prediction equation in practical applications. Refer to Figure 2 , Figure 2 is a control block diagram of an optimized modeling and robust control method for a soft robot based on a fusion prediction equation applied to an actual soft robot system, including an optimized Koopman model, a robust model predictive controller, a power control element, a soft robot, and a sensing element. The superiority of the optimized Koopman model lies in: deriving correct, rich, and non-redundant measurement coordinates based on the fusion prediction equation, and screening out the observation function that captures the system dynamics with the lowest dimension through the SINDy algorithm. Based on this observation function, Koopman modeling is performed, which can optimize the accuracy of the Koopman model of the soft robot. The robust model predictive controller is designed based on the optimized Koopman model, adds an integral action to the closed-loop system, and dynamically adjusts the constraints according to the tracking performance of the system, solves the optimal control instruction and sends it to the power control element; common power control elements include electro-hydraulic proportional valves, flow valves, motors, etc., and their types depend on the driving mode of the soft robot. The power control element transmits power and drives the soft robot to achieve preset motions or behaviors; 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.

[0143] It should be noted that the above content only illustrates the technical idea of the present invention and cannot be used to limit the protection scope of the present invention. For those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements all fall within the protection scope of the claims of the present invention.

Claims

1. An optimization modeling and robust control method for soft robots based on a fusion prediction equation, characterized in that: It includes the following steps: S1. Derive the measurement coordinates based on the fusion prediction equation; including: S11. Derive the 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 stems from the incremental equation: θ(k + 1) = θ(k) + Δθ(k + 1) where θ is the controlled variable, which is the curvature or the 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, estimate Δθ(k + 1) based on multiple hypothesis models. Hypothesis model 1 is a constant increment model: Δθ1(k + 1) = Δθ(k) = θ(k) - θ(k - 1) where Δθ1(k + 1) is the estimated value of the increment of the controlled variable obtained by linear extrapolation based on the hypothesis model 1. This full-range uniform speed assumption is only applicable to extremely short sampling periods or systems operating stably; Hypothesis model 2 is based on the uniform speed assumption within a single sampling period: Among them, is the differential of the controlled variable in the current sampling period, which is estimated by a tracking differentiator. T is a sampling period. It is assumed that the system moves at a constant speed in this sampling period to obtain the estimated value Δθ2(k + 1) of the second increment of the controlled variable. Construct hypothesis model 3: Δθ3(k + 1) = εΔu(k) = ε(u(k) - u(k - 1)) where u is the input of the system and ε is the proportionality coefficient. This model assumes a proportional relationship between the input and output of the system; The estimations of the above three hypothesis models are not completely accurate, and optimized estimation 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)); where α and β are weight parameters to be identified, Substitute the incremental equation and the three hypothesis models into the above formula: Introduce the hysteresis coordinates and denote them as θ(k - 1) = θ D (k), u(k - 1) = u D (k), and derive the final fusion prediction equation: S12. Derive the measurement coordinates; It is not necessary to measure the noises of the above three hypothesis models to determine the weight parameters in the fusion prediction equation. This equation has provided a set of physical quantities closely related to the predicted controlled variable, which is designed as the measurement coordinate x in Koopman modeling: S2. Design the observation function based on the measurement coordinates; including: S21. Design the initial observation function based on the measurement coordinates; Design a set of high-dimensional non-linear real-valued functions based on the measurement coordinates, that is, the initial observation function, and its forms include: monomials, polynomials, trigonometric functions, radial basis functions; S22. Screen the observation function based on the SINDy algorithm; Use this 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 non-linear system with the lowest dimension; 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; S3. Identify the Koopman model based on the observation function; including: S31. Data acquisition; Collect a large number of random measurement coordinate data pairs through experiments or simulations, and organize them into two matrices with an evolution relationship of one sampling step: X1 = [x[1] x[2] ··· x[p]] X2 = [x[2] x[3] ··· x[p + 1]] S32. Data dimensionality increase; increase the dimensions of X1 and X2 based on the observation function Ψ designed in step S2: X 1lift = [Ψ(x[1])Ψ(x[2])···Ψ(x[p])] X 2lift = [Ψ(x[2])Ψ(x[3])···Ψ(x[p + 1])] Introduce the input item, X 1lift , X 2lift Extended to: Y1 = [X 1lift U] Τ Y2 = [X 2lift U] Τ where U = [u[1] u[2] ··· u[p]]. Since the evolution of the system input is not considered, the same input terms are extended; S33. Identify the Koopman model; obtain a finite-dimensional approximate representation of the Koopman operator by minimizing the following objective function Then isolate and partition the correlation matrix of the Koopman model from : Define the mapping matrix C d = [IO], and establish a 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 obtained by mapping the measurement coordinate x into the high-dimensional Koopman space, and A d , B d , C d are the matrix coefficients of the Koopman model, respectively; S4. Design a robust model predictive controller based on the Koopman model; including: S41. Transform into a Koopman incremental model; introduce the augmented state z to transform the Koopman model identified in step 3 into a Koopman incremental model: z[k + 1] = Az[k] + BΔu[k] x[k] = Cz[k] where \(z[k]=[z d [k]u[k - 1]] Τ \), \(\Delta u[k]=u[k]-u[k - 1]\), and the corresponding matrix coefficients are rewritten as: \(B = [B d I] Τ \), \(C = [C d O]\); A model predictive controller is designed based on this Koopman increment model, adding integral action to the closed-loop system and improving the system robustness; S42. Design dynamic constraint conditions; design a model predictive controller with dynamic constraint conditions based on the Koopman incremental model. The optimization problem to be solved is: s.t. z[k + 1] = Az[k] + BΔu[k] -g ≤ Δu[k] ≤ g Among them, N h is the prediction horizon, Q, R, and F are weight coefficients, g is the dynamic constraint of Δu[k], and the value of g is set to be always positive and dynamically adjusted according to the tracking performance of the system: e[k] = |θ r [k] - θ[k]| where k g is the stiffness of the controller, which is proportional to the response speed, and b g is the damping of the controller, which helps to reduce system oscillation, and θ r is the reference value of the controlled variable for pre-tracking; S43. Optimize and solve to output the control quantity; transform the above optimization problem into a standard quadratic programming problem. In each sampling period, optimize and solve the optimal sequence of the control quantity increment under dynamic constraint conditions; finally, select the first value of the optimal sequence of the control quantity increment 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 this solution process in the next sampling period.

2. The specific implementation structure of a method for optimizing the modeling and robust control of a soft robot based on a fusion prediction equation in practical applications according to claim 1, characterized in that: It includes an optimized Koopman model, a robust model predictive controller, a dynamic control element, a soft robot, and a sensing 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 with the lowest dimension through the SINDy algorithm. Based on this observation function, Koopman modeling can be carried out to optimize the accuracy of the Koopman model of the soft robot; the robust model predictive controller is designed based on the optimized Koopman model, adds integral action to the closed-loop system, and dynamically adjusts the constraints according to the tracking performance of the system, solves the optimal control instruction and sends it to the dynamic control element; the dynamic control element transmits power and drives the soft robot to achieve preset motions or behaviors; 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, and feedbacks to the robust model predictive controller to form a closed-loop control system of the soft robot.

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