A real-time trajectory tracking control method for a rope-driven soft robotic arm

Abstract

The application provides a real-time trajectory tracking control method for a rope-driven soft robot arm and belongs to the technical field of soft robot motion control. First, a target function of a trajectory tracking optimal control problem is established according to a target trajectory. Second, a dynamics model of the rope-driven soft robot arm is established by introducing a strain constraint based on a position dynamics method. Third, a reduced order matrix is established by using a modal derivative to realize model reduction of the rope-driven soft robot arm, and the calculation of nonlinear terms is reduced by coefficient combination. Fourth, a calculation formula of trajectory tracking control input is established by using the target function. Finally, the deformation of the soft robot arm is solved by a numerical integration method. The application establishes a simulation framework of the rope-driven soft robot arm based on the position dynamics method to solve the simulation and control problems of the soft robot arm, and aims to provide a new strategy of a complete soft robot arm model verification and real-time control to solve the problem of interaction between the soft robot arm and the environment.

Description

Technical Field

[0001] This invention belongs to the field of soft robot control technology, and relates to a method for real-time trajectory tracking control of a rope-driven soft robotic arm. Background Technology

[0002] Soft robots, typically made of flexible materials, possess high adaptability and compatibility to unknown environments, thus effectively compensating for the limited application scope and insufficient human-robot interaction of traditional rigid robots. However, soft robots face significant challenges in modeling, simulation, and control, thus limiting their practical application. A major challenge is that soft robots have infinite degrees of freedom, and the selected materials and structural deformation processes exhibit significant nonlinearities. This increases the complexity of modeling, greatly reduces simulation efficiency, and is detrimental to control.

[0003] Accurate modeling and efficient simulation are crucial for solving soft robot control problems. In recent years, increasing research has focused on soft robot modeling. Piecewise constant curvature models, beam models, and Cosserat models have been used to construct accurate soft robot dynamic models. However, the most commonly used method remains the finite element method (FEM) because it can utilize the constitutive relations of real materials for modeling. The FEM method can be used to develop high-fidelity models for a wide variety of soft robots and can be directly incorporated into the design process. While some techniques directly use the FEM model for inverse kinematics based on control or trajectory optimization, the high dimensionality (e.g., thousands to tens of thousands of degrees of freedom) and nonlinearity of material and geometric deformations of the FEM model make the design of real-time dynamic controllers challenging. Therefore, most research employs appropriate orthogonal decomposition (POD) methods to reduce the model's order, improving computational efficiency by reducing generalized coordinates. POD primarily involves recording all possible motion deformations of the structure through offline simulation or experiments, and then sampling these records to construct a representative low-dimensional space. In fact, for the POD method, the formation of the subspace is a step-by-step learning process. In order to create a subspace that accurately reflects the motion, all possible motions of the global model must be required. Thus, the process of constructing the reduced-order matrix is ​​very complex and time-consuming. Summary of the Invention

[0004] This invention proposes a real-time trajectory tracking control method for tethered soft robotic arms. This method is based on extended position dynamics and introduces strain constraints to address the problem that geometric constraints cannot reflect mechanical performance. It also facilitates coefficient merging of nonlinear terms after model order reduction, reducing computational load and improving efficiency. The combination of the reduced-order model and instantaneous optimal control aims to provide a real-time control method for tethered soft robotic arms, enabling them to achieve high-precision tracking control of target trajectories and facilitating interaction with the environment during experiments.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A real-time trajectory tracking control method for a tethered soft manipulator includes the following steps: First, an objective function for the optimal control problem of trajectory tracking is established based on the target trajectory. Second, a dynamic model of the tethered soft manipulator is established by introducing strain constraints based on position dynamics. Third, a reduced-order matrix is ​​established using modal derivatives to reduce the order of the tethered soft manipulator model, while coefficient merging reduces the calculation of nonlinear terms. Then, the calculation formula for the trajectory tracking control input is established using the objective function. Finally, the deformation of the soft manipulator is solved using numerical integration. Specifically, the method includes the following steps:

[0007] Step 1: Establish the objective function for the optimal control problem of trajectory tracking of the rope-driven soft robotic arm.

[0008] Since the target trajectory is related to time t, its trajectory tracking model is represented as:

[0009]

[0010] In the formula, J is the objective function. For the tracked trajectory at t n+1 The position at time, y n+1 For the end point of the controlled soft robotic arm at t n+1 The position of time, u n+1 For in t n+1 The control vector at time step, C is the output matrix, x n+1 For soft robotic arms in t n+1 The node position at time t, Q and R are constant coefficient matrices.

[0011] Step 2: Introduce strain constraints and construct the dynamic equations of the rope-driven soft robotic arm.

[0012] Considering that the Extended Position Dynamics (XPBD) method is simple to model, has strong robustness, and is computationally fast, this invention adopts this method to model the soft robotic arm. The internal energy of the soft robotic arm is E(x) = 1 / 2C(x). T α -1 C(x) is determined by the constraint function C = [C1(x)C2(x)...C... Q (x)] T The structure is as follows: α is the inverse of the block symmetric flexibility matrix, i.e., the stiffness matrix, and its dynamic equations are as follows:

[0013]

[0014] In the formula, x = [x1x2…x] num ] T is a column vector representing the node positions; M is the acceleration; M is the mass matrix; For the internal forces of the system, among which The gradient operator is represented, E(x) represents the internal energy of the system; F ext (x) represents the external force acting on the system.

[0015] Because the constraints of the traditional XPBD method are geometric constraints, they cannot reflect the mechanical properties of the system (e.g., stress, strain) and cannot provide a reference for the selection of practical materials. Therefore, this invention uses St. Venant-Kirchhoff materials to introduce strain constraints for modeling the soft robotic arm. The strain energy function E of the St. Venant-Kirchhoff material is... s as follows:

[0016]

[0017] In the formula, Ψ s Let V be the strain energy density, and V be the volume of the initial tetrahedral mesh. Let θ be the Lamé constant and ε be the strain matrix, then the strain constraint is C = [ε]. 11 ε 22 ε 33 ε 12 ε 13 ε 23 ] T `tr` is the operator for the tensor trace; `α` represents the block symmetric compliance matrix; `Ω` represents the integration region; and `x` represents the node position.

[0018] Considering the deformation of the soft robotic arm driven by the rope, to simplify the modeling, we assume that the rope length is not stretchable and construct a rope length constraint H. j A Lagrange multiplier is introduced to apply the load. The load is applied at each point the rope passes through the robotic arm using p. i This indicates that the point where the rope is pulled is marked as p. pull The fixed endpoint of the soft robotic arm is marked as p1. The distance u that the rope pulling point moves is... j The magnitude of the external force is determined by the rope's lack of elasticity; therefore, its length always remains constant relative to its initial length l. int Same. Rope drive constraint is:

[0019]

[0020] For a tethered soft manipulator, after the drive is introduced in the form of constraints, the dynamic equation of the tethered soft manipulator is:

[0021]

[0022] The s ropes construct the rope-driven constraint matrix H, where λ is an s×1 dimensional Lagrange multiplier vector. xTo constrain the Jacobian matrix.

[0023] Step 3: The modal derivative method is used to reduce the order of the dynamic equations of the rope-driven soft robotic arm.

[0024] Considering that flexible robotic arms mainly achieve target motion through their own deformation, which involves material and geometric nonlinearities, simulation is difficult and time-consuming, hindering real-time control. Therefore, this invention employs the modal derivative method to reduce the order of the model of the rope-driven soft robotic arm's dynamic equations, thereby reducing computational load and improving computational efficiency.

[0025] First, a reduced-order matrix is ​​created. The nodal displacement d is expanded in the initial state using a second-order Maclaurin expansion of the generalized coordinate q:

[0026]

[0027] In the formula, q = [q1q2...q g ] T It is a column vector of generalized coordinates, and its first derivative is a linear mode Ф. i The second derivative is called the modal derivative Ψ. ij . q i Represents the i-th generalized coordinate; Ф i The following generalized eigenvalue problems can be solved:

[0028]

[0029] Given that d is a function of q, w i It is the i-th order vibration frequency. Ф i (q) can also be viewed as a function of q. Modal derivative Ψ ij It can be considered as Ф i (q) is the derivative of q. Differentiating equation (7) yields:

[0030]

[0031] The inertial correlation term in the above equation has been proven to be negligible, Ψ ij It has symmetry (Ψ) ij =Ψ ji This can be obtained through the following equation:

[0032] Ψ ij =-K -1 (Ν:Φ i )Φ j (9)

[0033] In the formula, N is the Hessian matrix of stiffness, and K is the tangent stiffness matrix. Selecting the first g linear modes, they and all their corresponding modal derivatives together form the basis of the motion subspace A of the soft robotic arm. Principal component analysis (PCA) is used to prevent low-frequency linear modes and their corresponding modal derivatives from being masked by high-frequency modes. However, g+g(g+1) / 2 dimensions are still not conducive to real-time simulation; therefore, singular value decomposition (SVD) is used. Select The first r columns form a reduced-order matrix U, where r is the order of the reduction. Generally, the smaller the r is, the higher the computational efficiency, but the accuracy will decrease significantly.

[0034] Then, the dynamic equations of the rope-driven soft robotic arm are reduced in order. x n+1 =Uq n+1 Substituting +x0 into formula (5), where x0 is the initial position of the soft robotic arm node, multiplied by U on the left... T The reduced-order equations of motion are as follows:

[0035]

[0036] These are the reduced mass matrix, internal force matrix, and constraint Jacobian matrix, respectively. It's important to note that the model reduction only applies to the generalized coordinates of the global model; the tethered constraints are still based on the global model and are not reduced in order. Therefore, the tethered constraints are described in the global space.

[0037] Step 4: Implement trajectory tracking control for the cable-driven soft robotic arm based on model order reduction.

[0038] For the reduced-order dynamic equations of the cable-driven soft robotic arm in step three, the direct integration method is used to discretize them in the time domain. The nonlinear algebraic equations within each time step are iterated and then solved through step-by-step integration. This invention employs a central difference scheme, so at time t... n+1 At time t, after discretizing DAEs, the following system of nonlinear algebraic equations can be obtained:

[0039]

[0040] The above equation contains q n+1 , λ n+1 and There are a total of 2r+s unknowns, and let... The above system of nonlinear algebraic equations can then be simplified to a system of nonlinear algebraic equations in z, which can be solved using the Newton–Raphson iteration:

[0041]

[0042] Based on the optimal control objective function established in step one, the performance index depends only on the unknown control variable u.n+1 This is relevant, therefore, by minimizing the performance index, we can obtain:

[0043]

[0044] For a tethered soft robotic arm, its control variable u n+1 This represents the length of tension at the end of each rope; this term is included in the rope drive constraint H, as well as the rope drive constraint and the output variable y. n+1 Both are built based on a global model, therefore y n+1 It can be represented as:

[0045]

[0046] Taken from z n+1 The first r rows, x0, represent the position of the node in the initial state of the soft robotic arm. Therefore, the instantaneous optimal control input can be obtained as:

[0047]

[0048] in:

[0049]

[0050] The r×1 dimensional column vector is taken from z n The first r rows, Γ1 is an r×2r dimensional matrix, taken from... The first r rows and the first 2r columns. Θ is a 2r×1 dimensional column vector, taken from f(z). n The first 2r rows of Γ2. Γ2 is an r×s dimensional matrix, taken from... The first r rows and columns from 2r+1 to 2r+s. Remove the control variable u from the constraint function H n+1 The remaining terms form an s×1 dimensional vector. Then, the resulting control variable u... n+1 Substitute into formula (12) to update the current state variable z. n+1 Finally, the relative error of the state variable z between the two iterations of the Newton-Raphson method is used as the convergence criterion to complete the trajectory tracking control solution in each time step, and the reduced-order result is transformed into a global deformation.

[0051] Furthermore, in step two, the strain constraint C is determined by the Green's strain ε = 0.5 (F) in continuum mechanics. T It is constructed from FI. I is the identity matrix, F = PQ -1 It is the deformation gradient of the tetrahedral mesh, and P = [x1 - x0x2 - x0x3 - x0] is the deformation matrix. It is a reference matrix. and Here, x0, x1, x2, and x3 are the initial node configurations, and x0, x1, x2, and x3 are the corresponding positions after deformation. Then constraint C can be expressed as: C = [ε] 11 ε 22 ε 33 ε 12 ε 13 ε 23 ] T .

[0052] Furthermore, in step two, the block-symmetric compliance matrix α can be expressed as:

[0053]

[0054] Where V represents the volume of the tetrahedron; θ represents Lamé constant.

[0055] Furthermore, in step three, the motion subspace A of the soft robotic arm is represented as:

[0056]

[0057] Where w1 represents the first-order frequency; w j w represents the j-th frequency; i This represents the i-th frequency.

[0058] Furthermore, in step three, the reduced-order internal force array and overall stiffness matrix It can be written in polynomial form, represented as:

[0059]

[0060]

[0061] q i For the reduced-order generalized coordinates, Coff1, Coff2, and Coff3 are the coefficients of the first, second, and third terms in each polynomial of the reduced-order internal force, respectively.

[0062] This invention employs the XPBD method to construct a dynamic model of a soft robotic arm, while introducing strain constraints to construct internal energy. Considering that the soft robotic arm has infinite degrees of freedom and strong nonlinearity, which is not conducive to real-time simulation control and interaction with the environment, this invention reduces the order of the model through modal derivatives, and proposes a control algorithm based on this, enabling the soft robotic arm to achieve high-precision real-time trajectory tracking.

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

[0064] (1) This invention constructs a new framework for real-time trajectory tracking control of a rope-driven soft robotic arm. Compared with existing nonlinear finite element methods, this invention employs an extended position dynamics-based modeling and analysis method and introduces strain constraints. The modeling process is simple, universal, and easy to operate, and overcomes the shortcomings of traditional extended position dynamics-based methods, which do not include physical quantities and therefore cannot perform mechanical performance analysis. Furthermore, based on the introduced strain constraints, each term in the internal force matrix and stiffness matrix can be written in polynomial form. After model order reduction, coefficient merging can greatly reduce the number of nonlinear terms, thereby reducing the computational load of iterative updates and improving computational efficiency.

[0065] (2) Based on the framework for constructing a rope-driven soft robotic arm, this invention further provides formulas for model reduction and the application of instantaneous optimal control in the reduced model. The framework mainly relies on configuration and material properties for creating the reduction matrix, without simulating the entire model, which greatly simplifies the reduction process. It can also achieve high-precision tracking control of the target trajectory in real time, which helps with real-time manipulation and interaction with the environment in experiments. Attached Figure Description

[0066] Figure 1 This is a flowchart illustrating the implementation of the present invention.

[0067] Figure 2 It is a model of a rope-driven soft robotic arm.

[0068] Figure 3 This is a dynamic image showing the trajectory tracking of a rope-driven soft robotic arm. Figure 3 (a) Animated graph of a rope-driven soft robotic arm tracking a circular trajectory; Figure 3 (b) Animated graph of a rope-driven soft robotic arm tracking a square trajectory.

[0069] Figure 4 Figure (a) shows the rope length driving diagram for the rope-driven soft robotic arm tracking a circular trajectory; Figure (b) shows the rope length driving diagram for the rope-driven soft robotic arm tracking a square trajectory.

[0070] Figure 5 These are error diagrams for the trajectory tracking of a tethered soft robotic arm. Figure (a) shows the error diagram for the tethered soft robotic arm tracking a circular trajectory; Figure (b) shows the error diagram for the tethered soft robotic arm tracking a square trajectory. Detailed Implementation

[0071] In the medical field, surgical soft robotic arms are becoming increasingly sophisticated, requiring less jitter and more precise end-effector positioning. The most basic function of a soft robotic arm is to receive specified commands and move to a designated location or track a specified trajectory. Therefore, accurate trajectory tracking and positioning are prerequisites and guarantees for performing these more complex tasks. However, modeling soft robotic arms is complex; excessive degrees of freedom and strong nonlinearity affect computational efficiency and are detrimental to control implementation. Therefore, to overcome these problems, this invention utilizes the XPBD method and introduces strain constraints to construct a soft robotic arm model, achieving real-time trajectory tracking control based on model order reduction.

[0072] The invention will be further explained below in conjunction with a commonly used rope-driven elephant trunk-like soft robotic arm.

[0073] Combination Figure 1 The specific embodiments of the present invention are as follows:

[0074] The first step is to provide the target trajectory, objective function, and constraints for the trajectory tracking problem of the rope-driven soft robotic arm, based on the trajectory tracking problem to be solved.

[0075] The second step is to establish a dynamic model of the cable-driven soft robotic arm based on the extended position dynamics method. The specific steps are as follows:

[0076] (1) Give the mass matrix, strain energy constraint, generalized force vector, initial velocity and initial configuration of the soft robotic arm;

[0077] (2) Give the rope drive constraint of the soft manipulator and the Jacobian matrix of the rope drive constraint.

[0078] (3) Give the collision coefficient of the soft robotic arm's self-contact collision;

[0079] (4) Based on steps (1)-(3), establish the dynamic equations of the rope-driven soft robotic arm.

[0080] The third step is to use the modal derivative method to establish a reduced-order formula for the dynamic equations of the cable-driven soft robotic arm. The specific steps are as follows:

[0081] (1) Establish the linear modal calculation formula Ф for the general form of the rope-driven soft robot arm;

[0082] (2) Using the modal derivative method, a second-order approximate expansion is performed on the basis of linear order reduction to obtain the formula for calculating the modal reciprocal Ψ;

[0083] (3) Scaling of linear modes and modes using principal component analysis (PCA);

[0084] (4) Construct a reduced-order subspace A, and use SVD decomposition to establish a reduced-order matrix U;

[0085] (5) Establish the general form of the reduced-order equations for the dynamics of the rope-driven soft robotic arm.

[0086] The fourth step is to discretize the system of differential-algebraic equations in the time domain, obtaining equations with q... n+1 , λ n+1 A system of nonlinear algebraic equations with unknowns;

[0087] Fifth step, t n The state quantity q at time t n As the initial value for the Newton-Raphson method iteration, the instantaneous optimal control u is obtained. n+1 ;

[0088] Step 6: The instantaneous optimal control u n+1 Substitute into the nonlinear equations and update the current state variable q. n+1 ;

[0089] Step 7: Repeat steps 5 and 6, stepping over time, until the simulation of all time is completed.

[0090] Simulation Example: Using the method of this invention, a numerical simulation was conducted on a three-rope driven elephant trunk-like soft robotic arm.

[0091] Figure 2 The model is a rope-driven soft robotic arm with an overall length L of 0.43m. Each soft segment has a length l1 of 0.025m, a spacing l2 of 0.02m, an included angle α of 45°, a left-end radius of 0.035m, and a right-end radius of 0.015m. The material used is Flexible PLA with a Young's modulus of 46.575 MPa. All ropes are made of fiberglass steel wire with a cross-sectional radius of 3mm and a Young's modulus of 73 GPa. Assume the rightmost end of the model is fixed, and at point p... pull A point is applied to the rope to drive translation, allowing the soft robotic arm to bend along one side of the rope. Since the rope's mass is relatively light, it can be ignored. By controlling the different rope extension lengths, the robotic arm's end effector can move along a predetermined trajectory. Quasi-static and dynamic simulations are performed using the method of this invention, assuming zero gravity.

[0092] Figure 3The figures show dynamic diagrams of a three-rope driven soft robotic arm tracking a circular trajectory (a) and a square trajectory (b). Initially, the soft robotic arm is suspended vertically, with the ends of the three ropes at the same horizontal level. Then, the end of rope 1 rises, generating tension, while the ends of ropes 2 and 3 descend. The soft robotic arm bends towards the side where rope 1 is located until its free end reaches a certain height, allowing it to reach the radius of the tracked trajectory. By controlling the driving force of the three ropes, the soft robotic arm continuously changes its bending direction while maintaining a constant end height, thus achieving circular trajectory tracking.

[0093] Figure 4 The graph shows the change in rope drive over time when a three-rope driven soft robotic arm tracks a trajectory. (a) shows the rope drive when the soft robotic arm tracks a circular trajectory; (b) shows the rope drive when the soft robotic arm tracks a square trajectory. As can be seen from the graph, from 0 to 8s, except for the end of rope 1 which stretches upwards, the ends of the other two ropes contract downwards, providing a certain bending angle and height for the free end of the soft robotic arm. Afterwards, the ends of the three ropes alternately stretch upwards, causing the bending direction of the soft robotic arm to gradually move closer to the side where the stretched rope is located. The graph shows that the maximum length of upward stretching and downward contraction of each rope is the same, indicating that the free end of the soft robotic arm remains on the same horizontal plane during trajectory tracking. After completing one lap of trajectory tracking, the free end of the soft robotic arm returns to its position at 8s, at which point the stretch of the three rope ends is the same as at 8s.

[0094] Figure 5 This is a graph showing the tracking trajectory error of the end effector of a three-wire-driven soft robotic arm. As can be seen from the graph, the overall error is within 6e. -15 The simulation results demonstrate that the real-time trajectory tracking control method proposed in this invention has high computational accuracy. Table 1 also shows the simulation times for two trajectory tracking scenarios. Considering that a higher order of reduction leads to higher deformation accuracy but also higher computational cost, a 50-order reduction was selected after testing, which accurately describes the deformation of the soft robotic arm while ensuring computational efficiency. The soft robotic arm was divided into 22,639 units, and a 50-order reduction simulation control was performed. As can be seen from the table, the simulation time for both trajectory tracking scenarios is within 60 seconds, demonstrating that the real-time trajectory tracking control method proposed in this invention has high computational efficiency.

[0095] This invention simplifies the modeling process of soft robotic arms and greatly improves simulation efficiency. In the medical field, this invention can achieve high-precision positioning and control of surgical soft robotic arms, making it possible for doctors to remotely operate soft robotic arms and interact with the environment.

[0096] Table 1 Simulation time under different trajectory tracking control conditions

[0097]

[0098] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A real-time trajectory tracking control method for a rope-driven soft robotic arm, characterized in that, First, based on the target trajectory, an objective function for the optimal control problem of trajectory tracking is established. Second, based on the position dynamics method, strain constraints are introduced to establish a dynamic model of the tethered soft manipulator. Next, modal derivatives are used to establish a reduced-order matrix, thereby reducing the order of the tethered soft manipulator model, while coefficient merging reduces the calculation of nonlinear terms. Then, using the objective function, a calculation formula for the trajectory tracking control input is established. Finally, the deformation of the soft manipulator is solved using numerical integration. Specifically, the following steps are included: Step 1: Establish the objective function for the optimal control problem of trajectory tracking of the rope-driven soft robotic arm; Due to the target trajectory and time The trajectory tracking model is represented as follows: （1） In the formula, J Let be the objective function. For the tracked trajectory in t n+1 The position at time, y n+1 For the end point of the controlled soft robotic arm at t n+1 The position of time, u n+1 In order to be in t n+1 The control vector at time step, C is the output matrix, x n+1 For soft robotic arms in t n+1 The node position at time t, where Q and R are constant coefficient matrices; Step 2: Introduce strain constraints and construct the dynamic equations of the cable-driven soft robotic arm; The soft robotic arm is modeled using the Extended Position Dynamics (XPBD) method; the system internal energy of the soft robotic arm. E (x) = 1 / 2C(x) T α -1 C(x), by the constraint function C=[ C 1(x) C 2(x) ... C Q (x)] T The structure is as follows: α is the inverse of the block symmetric flexibility matrix, i.e., the stiffness matrix, and its dynamic equations are as follows: （2） In the formula, x = [x1x2…x] num ] T is a column vector representing the node positions; For acceleration; M is the mass matrix; F int (x)= x E (x) represents the internal forces of the system, where x This represents the gradient operator. E (x) represents the internal energy of the system; F ext (x) represents the external force acting on the system; The soft robotic arm is modeled using strain constraints introduced by St. Venant-Kirchhoff material; the strain energy function of St. Venant-Kirchhoff material... E s as follows: （3） In the formula, Ψ s For strain energy density, V The initial configuration is a tetrahedral mesh volume. and Let be the Lamé constant, and ε be the strain matrix, then the strain constraint is C = [ ε 11 ε 22 ε 33 ε 12 ε 13 ε 23 ] T ; tr Operators for tensor traces; Ω represents the block symmetric compliance matrix; Ω represents the integration region. Indicates the node position; For a tethered soft manipulator, to simplify modeling, the drive is introduced in the form of constraints. The dynamic equations of the tethered soft manipulator are then: （5） s The ropes construct the rope-driven constraint matrix H, where λ is... s A 1×1-dimensional Lagrange multiplier vector, H x To constrain the Jacobian matrix; Step 3: Use the modal derivative method to reduce the order of the dynamic equations of the cable-driven soft robotic arm. First, a reduced-order matrix is ​​created; the nodal displacements d are expanded in the initial state using a second-order Maclaurin expansion of the generalized coordinates q: （6） In the formula, q = [ q 1 q 2 ... q g ] T It is a column vector of generalized coordinates, and its first derivative is a linear mode Ф. i The second derivative is called the modal derivative Ψ. ij ; Indicates the first i A generalized coordinate; Ф i The following generalized eigenvalue problem can be solved: （7） Given that d is a function of q, w i yes i First vibration frequency; Ф i (q) can be considered as a function of q; modal derivative Ψ ij Considered to be Ф i (q) is the derivative of q; differentiating formula (7) yields: （8） The inertial correlation term in the above equation has been proven to be negligible, Ψ ij It has symmetry (Ψ) ij = Ψ ji This can be obtained through the following equation: （9） where N is the Hessian matrix of stiffness and K is the tangent stiffness matrix; select the first g order linear modes, then it and all its corresponding modal derivatives together form the basis of the motion subspace A of the soft robotic arm; use principal component analysis to prevent low-frequency linear modes and their corresponding modal derivatives from being masked by high-frequency modes; however g + g ( g +1) / 2 dimensions are still not conducive to real-time simulation, so singular value decomposition is used, that is , select the first r columns to form the reduced-order matrix U, r is the reduced-order number; Then, the dynamic equations of the rope-driven soft robotic arm are reduced in order; x n+1 = Uq n+1 Substituting +x0 into formula (5), x0 is the initial position of the soft robotic arm node, multiplied by U on the left. T The reduced-order equations of motion are as follows: （10） , These are the mass matrix, internal force matrix, and constraint Jacobian matrix after order reduction, respectively. The model order reduction only applies to the generalized coordinates of the global model. The rope-driven constraints are still based on the global model and are not reduced in order. Therefore, the rope-driven constraints are described in the global space. Step 4: Implement trajectory tracking control for the cable-driven soft robotic arm based on model order reduction. For the reduced-order dynamic equations of the cable-driven soft robotic arm in step three, the direct integration method is used to discretize them in the time domain. The nonlinear algebraic equations within each time step are iterated and then solved through step-by-step integration. Using a central difference scheme, the equations are then solved... At time t, after discretizing DAEs, the following system of nonlinear algebraic equations can be obtained: （11） The above equation contains , and 2 in total r + s An unknown variable, and let Then the above nonlinear algebraic equation system can be simplified to a nonlinear algebraic equation system in z, which can be solved by Newton–Raphson iteration: （12） Based on the optimal control objective function established in step one, the performance index depends only on the unknown control variable u. n+1 This is relevant, therefore, by minimizing the performance index, we can obtain: （13） For a tethered soft robotic arm, its control variable u n+1 This represents the length of tension at the end of each rope; this term is included in the rope drive constraint H, as well as the rope drive constraint and the output variable y. n+1 Both are built based on a global model, therefore y n+1 It can be represented as: （14） Taken from z n+1 The former r Okay, x0 is the position of the node in the initial state of the soft robotic arm; from this, the instantaneous optimal control input can be obtained as: （15） in: （16） for r The 1-dimensional column vector is taken from z. n The former r OK, for r ×2 r dimensional matrix, taken from The former r row and the first 2 r List; 2 r A 1-dimensional column vector, taken from The first 2 r OK; for r × s dimensional matrix, taken from The former r line and from the 2nd r+ 1 to 2 r+s List; Remove the control variable u from the constraint function H n+1 The remaining terms constitute s A ×1 dimensional vector; then the resulting control variable u n+1 Substitute into formula (12) to update the current state variable z. n+1 Finally, the relative error of the state variable z between the two iterations of Newton-Raphson is used as the convergence criterion to complete the trajectory tracking control solution in each time step, and the reduced-order result is transformed into a global deformation.

2. The real-time trajectory tracking control method for a rope-driven soft robotic arm according to claim 1, characterized in that, In step two, the strain constraint C is the Green strain from continuum mechanics. Constructed from; I is the identity matrix, F=PZ -1 It is the deformation gradient of the tetrahedral mesh. It is a deformed matrix. It is a reference matrix. , , and Given the initial node configuration, x0, x1, x2, and x3 represent the corresponding positions after deformation; then constraint C can be expressed as: ; Furthermore, in step two, the block-symmetric compliance matrix α can be expressed as: （17） in, V Represents the volume of a tetrahedron; and This represents Lamé's constant.

3. The real-time trajectory tracking control method for a rope-driven soft robotic arm according to claim 1, characterized in that, In step three, the motion subspace A of the soft robotic arm is represented as follows: （18） in, Indicates the first-order frequency; Indicates the first j First frequency; Indicates the first i First frequency.

4. The real-time trajectory tracking control method for a rope-driven soft robotic arm according to claim 1, characterized in that, In step three, the reduced-order internal force matrix and overall stiffness matrix It can be written in polynomial form, represented as: （19） （20） q i For the generalized coordinates after order reduction, Coff 1. Coff 2. Coff 3 represents the coefficients of the first, second, and third terms in each polynomial of the reduced-order internal force.

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