Quick and accurate control method for rope-driven tensegrity robot based on spectrum manifold

Abstract

The invention provides a rope-driven tensegrity robot rapid and accurate control method based on spectrum manifolds, and belongs to the field of tensegrity robot control. The method comprises the following steps: firstly, taking tail end position data as sampling data, performing truncation, and then removing data along a z coordinate axis direction from the truncation data to obtain training data; secondly, according to a Takens delay embedding theorem, projecting training data to a spectrum space, parameterizing a spectrum manifold and reduced-order dynamics, and finding an optimal solution; and finally, performing open-loop control to sample position data and control input of the tail end center point, projecting the data of the tail end center point to a spectrum space, converting the data into an optimal problem so as to learn a control matrix, and further constructing an optimal control problem. According to the method, dependence on high-dimensional data is low, noise sensitivity is weak, the reduced-order model has high robustness, rapid and accurate control over the tail end pose of the rope-driven tensegrity robot can be efficiently achieved, and high adaptability and expansibility are achieved.

Description

Technical Field

[0001] The present invention belongs to the field of tensegrity robot control and relates to a fast and precise control method of a rope-driven tensegrity robot based on spectral manifolds. Background Art

[0002] With the development of robotics, rope-driven tensegrity robots, with their high degrees of freedom, large deformation capabilities, and strong environmental adaptability, have demonstrated unique advantages in fields such as medical surgery, disaster relief, and industrial inspection. However, their complex rigid-flexible-soft coupling leads to strong geometric nonlinearity, high-dimensional dynamic response, and frictional hysteresis, posing significant challenges to real-time and precise control.

[0003] Currently, control methods for tensegrity robots primarily rely on full-order dynamic equations based on physical modeling, such as those described in the Chinese invention patent [CN118963116A], which numerically solve differential-algebraic equations (DAEs). However, this approach suffers from high computational complexity, slow numerical iteration speed, and the need for significant computational resources when solving high-dimensional nonlinear equations, creating a bottleneck for real-time control. A more effective approach is to use model-order reduction methods to reduce the dimensionality of the equations and thus improve efficiency. Furthermore, rope-driven tensegrity robots often use flexible materials, which exhibit creep properties and frictional hysteresis effects between joints. Theoretical modeling often ignores or only partially considers certain influencing factors. Therefore, constructing a proxy model for rope-driven tensegrity robots using data-driven methods is a preferred approach.

[0004] In recent years, data-driven model reduction has provided insights into the control of complex systems. For example, Chinese invention patent [CN119227216A] uses the POD proper orthogonal decomposition method to reduce the order of a suspension bridge model, resulting in a reduced-order finite element simulation model. However, the POD proper orthogonal decomposition method requires embedding into a full-order dynamic model and cannot directly construct a reduced-order model from data. Chinese invention patent [CN119442492A] uses DMD dynamic modal decomposition technology to derive wing flutter types from data, thereby reducing the parameterization difficulty in wing control design and improving computational efficiency. However, DMD dynamic modal decomposition technology is a linear projection-based order reduction method, making it difficult to capture the complete dynamic response of systems with strong geometric nonlinearities. Chinese patent [CN119458375A] uses neural networks to implement collaborative control of robots. While neural network models have strong nonlinear fitting capabilities, they suffer from drawbacks such as large training data requirements, high noise sensitivity, and poor physical interpretability.

[0005] Therefore, there is an urgent need to further study a data-driven, physically interpretable, strong generalization and robust low-dimensional rope-driven tensegrity robot control method to improve its practical application performance. Summary of the Invention

[0006] In response to the problems existing in the prior art, the present invention provides a fast and precise control method for a rope-driven tensegrity robot based on spectral manifolds. The present invention adopts a data-driven approach and does not rely on complex equations. A highly accurate low-dimensional control model of the rope-driven tensegrity robot is established through the spectral manifold theory. The spectral manifold is a low-dimensional invariant manifold in the spectral subspace of the rope-driven tensegrity robot. It is attractive and can effectively capture the highly nonlinear characteristics of the rope-driven tensegrity robot. By learning the position data of the end center point of the rope-driven tensegrity robot, a control-oriented reduced-order model is extracted, and the reduced-order model can retain the core physical characteristics of the system. The present invention has low dependence on high-dimensional data, weak noise sensitivity, and the reduced-order model has strong robustness, providing a new idea for solving the problems of slow response and poor control accuracy of the rope-driven tensegrity robot.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A fast and precise control method for a rope-driven tensegrity robot based on spectral manifolds, the fast and precise control method for a rope-driven tensegrity robot comprising the following steps:

[0009] Step 1: First, the rope-driven tensegrity robot is allowed to oscillate freely without any control input signal. The position data of the end is used as the sampling data. Then, the first h seconds of the sampling data are truncated to obtain the truncated data. Finally, the data along the z-axis direction is removed from the truncated data to obtain the training data. Specifically:

[0010] Step 1-1: When the rope-driven tensegrity robot is in free oscillation without control input, the position data of the end center point is used as the sampling data

[0011] Consider Figure 2 The rope-driven tensegrity robot shown in FIG is composed of a sliding rope unit and a unit cell. Figure 3 As shown, the sliding rope unit transmits power around the pulley. Figure 4 The unit cell shown in the figure is composed of rod units, spring units, and ring units. By changing the driving amount of the sliding rope unit, the rope-driven tensegrity robot can move in different ranges.

[0012] Step 1-1-1: First, apply a static force F1 to the end of the rope-driven tensegrity robot, causing the end of the rope-driven tensegrity robot to warp and the entire robot to deform significantly. Then, immediately withdraw the static force F1, and the rope-driven tensegrity robot will oscillate freely under the action of gravity. Because the rope-driven tensegrity robot is a dissipative system with damping and will stop when it reaches a stable state, during the sampling process, it is necessary to record the rope-driven tensegrity robot until it reaches a stable state as much as possible. The position data of the free oscillation of the center point of the end collected by sampling is recorded as

[0013]

[0014] Among them, x1 represents the data of the center point of the terminal along the x-axis; y1 represents the data of the center point of the terminal along the y-axis; z1 represents the data of the center point of the terminal along the z-axis; [·] T Represents the transpose of a matrix;

[0015] Step 1-1-2: Repeat the process of step 1-1-1 and apply force F in different directions to the end of the rope-driven tensegrity robot. i , where i = 1, 2, ..., l, and sample and record the position data of the end center point as the sampling data

[0016]

[0017] in, represents the data collected under the action of force F1; represents the data collected under the action of force F2; Indicates that the force F l The data collected under the action of

[0018] Step 1-2: Truncate the sample data The first h seconds of data in the truncated data

[0019] The spectrum manifold is constructed near the stable state of the rope-driven tensegrity robot. The data sampled at the initial moment will decay rapidly to the spectrum manifold in the form of exponential speed. Therefore, in order to ensure that the sampled data trajectory is closer to the spectrum manifold, the sampled data is truncated. The truncated data is obtained by using the first h seconds of data, which is recorded as

[0020]

[0021] in, Indicates that The data obtained after truncating the first h seconds of data; Indicates that The data obtained after truncating the first h seconds of data Indicates that The data obtained after truncating the first h seconds of data.

[0022] Steps 1-3: Remove truncated data The data along the z-axis direction is used to obtain the training data Y(t);

[0023] For example, the free oscillation position data of the end center point obtained in step 1-1-1 Remove z1 and only include x1 and y1. Perform the same operation and collect the obtained data as training data Y(t).

[0024] Step 2: Based on Takens' delayed embedding theory, the training data Y(t) is embedded into the spectral manifold to reveal the nonlinear dynamic characteristics implicit in the training data Y(t). First, the minimum embedding dimension p is calculated, then the number of delayed sampling times q is determined, and finally, the training data Y(t) is dimensionalized according to the delayed embedding formula to obtain the embedded data y(t). Specifically:

[0025] Step 2-1: Calculate the minimum embedding dimension p and the number of delayed embeddings q;

[0026] Since only the position data of the end center point is recorded during the sampling process, it is difficult to reflect the system characteristics of the entire rope-driven tensegrity robot. Therefore, it is necessary to introduce sufficient time-delay data to ensure that the embedding in the high-dimensional state space is sufficient. The minimum embedding dimension p is given by the following formula:

[0027] p≥2m+1; (4)

[0028] where m is the spectral manifold dimension of the rope-driven tensegrity robot.

[0029] In order to increase the redundancy and improve the embedding accuracy, an additional delay measurement is introduced. The required number of delay samples q is calculated using the following formula through the optional additional embedding coefficient E:

[0030]

[0031] in, Indicates rounding up to an integer.

[0032] The redundant information of the reconstructed training data can be enhanced by increasing E. However, it should be noted that an excessively large E may cause linearization of the data, resulting in the loss of nonlinear information in the original data, which may reduce the control accuracy of the reduced-order model. The embedding coefficient E ranges from 0 ≤ E ≤ 1000.

[0033] Step 2-2: Upscale the training data y(t) using the delayed embedding formula;

[0034] Use the delayed embedding method to transform the training data Y into a high-dimensional coordinate system. Assume that in step 1-1-1 The sampling interval is Δ seconds, and the delay embedding formula is:

[0035] y(t)=[Y(t);Y(t+Δ);…;Y(t+(q-1)Δ)]; (6)

[0036] Among them, y(t) is the data after Takens delayed embedding and dimensionality upgrading, denoted as embedded data, which represents the state information of the current moment and several subsequent moments; Y(t+Δ) represents the data after the data Y(t) is delayed by Δ seconds; Y(t+(q-1)Δ) represents the data after the data Y(t) is delayed by (q-1)Δ seconds.

[0037] Step 3: First, the embedded data y(t) is approximately projected onto the spectral subspace using principal component analysis (PCA). Then, a parameterized method is used to Taylor expand the geometric shape functions of the spectral submanifold and the dynamic equations of the reduced-order model. Second, ridge regression is used to find the optimal solution for the Taylor expansion polynomial coefficients. Finally, based on the learned spectral submanifold formed by the embedded data y(t), the relationship between the data in the z-axis direction and the spectral submanifold is learned, and the data in the z-axis direction is collectively referred to as the vector field z. Specifically:

[0038] Step 3-1: Use PCA principal component analysis on the embedded data y(t);

[0039] Using PCA principal component analysis, the embedded data y(t) is approximately projected into the spectral subspace, which is often low-dimensional. First, the embedded data y(t) is subjected to SVD singular value decomposition to solve the eigenvalues ​​and characteristic matrix.

[0040]

[0041] Among them, U and V represent the left singular matrix and the right singular matrix respectively; S represents the diagonal positive semi-definite matrix; the elements on the diagonal of S are singular values ​​and are arranged in descending order on the diagonal; u1 represents the first column vector of the left singular matrix U; u n represents the column vector of the last column of the left singular matrix U; σ1 represents the largest singular value; σ nrepresents the smallest singular value; represents the first row vector of the right singular matrix V; Row vector representing the last row of the right singular matrix V.

[0042] Left singular vector u i The basis vectors that constitute the spectral subspace and the associated singular values ​​σ i Represents the energy ratio of the basis vector. The larger the singular value, the greater the contribution to the embedded data y(t). The contribution ratio is calculated by the following formula:

[0043]

[0044] Where n represents the total number of singular values; m represents the dimension of the spectral submanifold; and a represents the set energy ratio threshold, which is generally taken as 0.99, representing the information characteristics of most embedded data y(t). This can be used to determine whether the dimension m of the spectral submanifold selected in step 2-1 meets the requirements.

[0045] Select the eigenvectors corresponding to the first m singular values ​​and record them as V m , approximately project the embedded data y(t) into the spectral subspace:

[0046]

[0047] Where x is the modal coordinate, V m It means that the spanned space is approximately the spectral subspace tangent to the spectral manifold.

[0048] Step 3-2: Using the parameterization method, Taylor expands the spectrum manifold geometry function;

[0049] First, assume that there is the following spectral manifold geometry function relationship:

[0050] y(t)=w(x); (10)

[0051] Among them, the function w(x) represents the spectral manifold geometry function, which uniquely maps the modal coordinate x to the embedded data y(t). Using the parametric method, the function w(x) is Taylor expanded into a polynomial form:

[0052]

[0053] Among them, W0 and W are coefficient matrices, W0 represents the coefficient matrix corresponding to x, and W represents The corresponding coefficient matrix; n w is the order of the Taylor expansion polynomial; Indicates from two orders to n w A series of polynomial basis functions of order. It should be noted that nw It should not be too high, as it will result in lower generalization ability of the reduced-order model. w The range is generally 3≤n w ≤15.

[0054] Finally, the ridge regression method is used to minimize the following objective function to obtain the optimal coefficient matrices W0 and W:

[0055]

[0056] in, represents the coefficient matrix corresponding to x obtained by optimization; W * Represents the optimization result and The corresponding coefficient matrix; Represents the square Frobenius norm of the matrix;

[0057] Step 3-3: Use the parameterized Taylor method to expand the dynamic equations of the uncontrolled descent model, and use the ridge regression method to find the optimal coefficient matrix;

[0058] Assume that the dynamic equations of the reduced-order model on the spectral manifold have the following relationship:

[0059]

[0060] Among them, r aut is a smooth nonlinear function; Represents x aut The time derivative, x aut is the predicted data for the uncontrolled descent-order model.

[0061] Similarly, the dynamic equations of the Taylor expansion reduced-order model are as follows:

[0062]

[0063] Among them, R0 represents the coefficient matrix corresponding to x; R represents The corresponding coefficient matrix;

[0064] Calculated using the finite difference method Finally, the optimal coefficient matrices R0 and R are obtained by minimizing the following objective function through the ridge regression method:

[0065]

[0066] in, Represents the coefficient matrix corresponding to x obtained by optimization; R * Represents the optimization result and The corresponding coefficient matrix;

[0067] Step 3-4: Parameterize the vector field z and use the ridge regression method to find the optimal coefficient matrix;

[0068] We need to find the mapping relationship between the vector field z and the modal coordinate x. Assume that the following relationship exists and expand the vector field z:

[0069]

[0070] Among them, n r is the expansion order; ζ(x) represents the mapping relationship between the modal coordinates x and z; ζ0 represents the coefficient matrix corresponding to x; ζ represents The corresponding coefficient matrix.

[0071] Find the optimal coefficient matrix through ridge regression method:

[0072]

[0073] in, represents the coefficient matrix corresponding to x obtained by optimization; ζ * Represents the optimization result and The corresponding coefficient matrix;

[0074] Step 4: First, the position data of the terminal center point and the control input signal are sampled through open-loop control. Then, the position data of the terminal center point is embedded into the spectrum manifold as control training data. Finally, the ridge regression method is used to find the optimal control matrix. Specifically:

[0075] Step 4-1: Sample the training data through open-loop control to obtain the control sampling data L' u ;

[0076] First, a set of control input signals of the rope-driven tensegrity robot is given, and the control input signals and the position data of the end center point are sampled and recorded. It should be noted that the position data of the end center point should be kept as much as possible in the truncated data in step 1-2. The position data of the end center point of the open loop control sampling is recorded as the control training data L' u :

[0077] L' u =[x' u ,y' u ,z' u ]; (18)

[0078] Among them, x' u Represents the data of the end center point along the x-axis; y' u Represents the data of the end center point along the y-axis direction; z' uRepresents the data of the end center point along the z-axis direction;

[0079] The control input signal u' is:

[0080] u'=[u'1,u'2,u'3]; (19)

[0081] Wherein, u'1 represents the displacement of the first sliding rope of the rope-driven tensegrity robot; u'2 represents the displacement of the first sliding rope of the rope-driven tensegrity robot; and u'3 represents the displacement of the first sliding rope of the rope-driven tensegrity robot.

[0082] Step 4-2: Process the control training data L' in step 4-1 through Takens delayed embedding theory u And the control input signal u', the processed control training data is recorded as L u , the control input signal is u.

[0083] Step 4-3: Use ridge regression method to learn the control matrix B r :

[0084] Assume that the reduced-order dynamic equation of the cable-driven tensegrity robot with belt control is:

[0085]

[0086] Among them, the control matrix B r Represents the influence of the control input signal on the modal coordinate x.

[0087] First, the control training data L obtained in step 4-2 u Projected into the spectrum subspace, the control mode coordinate x is obtained u ;

[0088]

[0089] Among them, V m is obtained in step 3-1;

[0090] Then, according to the dynamic equation of the reduced-order model learned in step 3-3, we have:

[0091]

[0092] Among them, among them, are the modal coordinates of the autonomous part under the influence of the control, express The derivative with respect to time.

[0093] Secondly, according to the finite difference method, we get Finally, the ridge regression method is used to obtain the control matrix B r:

[0094]

[0095] in, represents the optimal solution of the satisfied relationship;

[0096] Step 4-4: Use ridge regression method to learn the control matrix B v ;

[0097] Assume that the following mapping relationship exists:

[0098]

[0099] Among them, the control matrix B v Indicates the influence of the control input signal on the control training data; represents the integral of the control input signal from time 0 to t.

[0100] First, according to step 3-2, we have:

[0101]

[0102] in, represents the position data of the terminal center point under the influence of the control. Therefore, the ridge regression method is used to obtain the control matrix B v :

[0103]

[0104] in, Represents the optimal solution of the obtained satisfied relationship.

[0105] Combining steps 4-3, the reduced-order model of the rope-driven tensegrity robot is obtained as follows:

[0106]

[0107] where L is the position of the end center point of the cable-driven tensegrity robot.

[0108] Step 5: Based on the reduced-order model of the rope-driven tensegrity robot, construct the optimal control problem and solve it. The optimal control problem is shown in formula (28):

[0109]

[0110] Among them, δz(t f ) is the error between the estimated position of the end center point and the actual position; z(t) is the estimated position of the end center point; represents the weighted two-norm; Represents the quadratic weighted cost of the deviation at time t, which is used to weigh the cost of the deviation in the process; represents the quadratic weighted cost under the weight matrix R, which is used to weigh the cost of the control input and prevent the control signal from being too large; u(t) is the control input signal; x(0) is the initial value of the modal coordinate; L(0) is the actual initial position of the end center point; L eq Indicates the expected output value, which is a reference value.

[0111] The beneficial effects of the present invention are:

[0112] (1) The present invention establishes a reduced-order model of a rope-driven tensegrity robot in an equation-free form in a data-driven manner. The spectral manifold is a low-dimensional invariant manifold in the robot phase space, which is attractive and can capture strong nonlinear behaviors. The present invention only uses the position data of the end center point and the Takens delayed embedding theorem to reveal the nonlinear dynamic characteristics implicit in the training data, without requiring too much training data. The PCA principal component analysis is used to approximate the spectral subspace and find the dimension of the spectral manifold. Finally, a parameterization method is used to obtain the spectral manifold and the corresponding reduced-order model, which has a high physical interpretability. In addition, the method of the present invention has low sensitivity to noise, and the constructed reduced-order model has high accuracy and good robustness, and is suitable for designing the control model of the rope-driven tensegrity robot.

[0113] (2) This invention establishes a control method for a rope-driven tensegrity robot based on a spectral manifold-based reduced-order model. Using this spectral manifold-based reduced-order model, an optimal control problem is proposed. Therefore, compared to control methods using full-order models, the control method proposed in this invention exhibits superior control efficiency.

[0114] In summary, the present invention efficiently realizes the rapid and precise control of the terminal posture of the rope-driven tensegrity robot and has strong adaptability and scalability. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Figure 1 It is a specific flow chart for implementing the present invention.

[0116] Figure 2 It is a rope-driven tensegrity robot model used in the present invention.

[0117] Figure 3 is a schematic diagram of the sliding rope unit.

[0118] Figure 4 Schematic diagram of a unit cell of a rope-driven tensegrity robot.

[0119] Figure 5 It is a schematic diagram of applying static force.

[0120] Figure 6 Is the force F1, F6, F 11 ,F 16 Schematic diagram of the initial configuration of the rope-driven tensegrity robot under action.

[0121] Figure 7 It is a schematic diagram of the data along the x-axis in the sampling data.

[0122] Figure 8 It is a schematic diagram of the data along the y-axis in the sampling data.

[0123] Figure 9 It is a schematic diagram of the data along the z-axis in the sampling data.

[0124] Figure 10 It is a schematic diagram of principal component analysis of embedded data.

[0125] Figure 11 It is a comparison curve of model prediction based on spectral submanifold (SSM) and full-order model (FULL).

[0126] Figure 12 This is a schematic diagram of the results of the method of the present invention when the tracking trajectory is a circle: Figure 12 (a) in the figure is the initial state of the tensegrity robot; Figure 12 (b) in the figure is the controlled trajectory; Figure 12 (c) in the figure is the root mean square error of tracking.

[0127] Figure 13 This is a schematic diagram of the results of the method of the present invention when the tracking trajectory is a triangle: Figure 13 (a) in the figure is the initial state of the tensegrity robot; Figure 13 (b) in the figure is the controlled trajectory; Figure 13 (c) in the figure is the root mean square error of tracking.

[0128] Figure 14 This is a schematic diagram of the results of the method of the present invention when the tracking trajectory is a rectangle: Figure 14 (a) in the figure is the initial state of the tensegrity robot; Figure 14 (b) in the figure is the controlled trajectory; Figure 14 (c) in the figure is the root mean square error of tracking.

[0129] Figure 15 This is a schematic diagram of the results of the method of the present invention when the tracking trajectory is a pentagram: Figure 15 (a) in the figure is the initial state of the tensegrity robot; Figure 15 (b) in the figure is the controlled trajectory; Figure 15 (c) in the figure is the root mean square error of tracking. DETAILED DESCRIPTION

[0130] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0131] by Figure 2 The rope-driven tensegrity robot shown in the figure is the research object of the present invention, combined with Figure 1 Flowchart to verify the method involved in the present invention. Figure 2 The design concept of the rope-driven tensegrity robot shown is derived from the human spine, and its main structure consists of unit cells and sliding rope units. Figure 3 The sliding rope unit is shown, Figure 4 The unit cell is shown to be composed of rod elements, spring elements, and ring elements. Sliding rope elements are routed around the pulleys of a rope-driven tensegrity robot, enabling flexible motion across a wide range of space. The robot, as a whole, presents a complex rigid-flexible-soft coupled, high-dimensional nonlinear system. This rope-driven tensegrity robot exhibits strong geometric nonlinearities and high-dimensionality, making real-time dynamic simulation computations difficult. Due to its extensive use of flexible components, it exhibits poor robustness under open-loop control. Furthermore, the strong coupling of nonlinear friction hysteresis and redundant drives poses challenges for high-precision control. To address these issues, the present invention proposes a fast and precise control method for a rope-driven tensegrity robot based on spectral manifolds. The proposed method samples the position data of the terminal center point without control input and successfully extrapolates this data to a controlled reduced-order model based on spectral manifold theory. Based on the existing controlled reduced-order model, an optimal control problem is constructed, achieving fast and precise control of the rope-driven tensegrity robot. The proposed method is experimentally verified using the trajectory tracking control of the terminal pose of a rope-driven tensegrity robot as an example.

[0132] A fast and precise control method for a tensegrity robot based on spectral manifolds comprises the following steps:

[0133] Step 1: First, the rope-driven tensegrity robot is allowed to oscillate freely without any control input signal. The position data of the center point of the end is sampled as the sampling data. Then, the first h seconds of the sampled data are truncated to obtain the truncated data. Finally, the data along the z-axis direction is removed from the truncated data to obtain the training data. Specifically:

[0134] Step 1-1: Sample the position data of the end center point when the rope-driven tensegrity robot is in free oscillation without control input as the sampling data

[0135] Consider Figure 2 The rope-driven tensegrity robot shown in FIG is composed of a sliding rope unit and a unit cell. Figure 3 As shown, the sliding rope unit transmits power around the pulley. Figure 4 The unit cell is shown to be composed of rod elements, spring elements, and ring elements. The rope-driven tensegrity robot can be moved in different ranges by varying the driving force of the sliding rope. The differential-algebraic equations (DAEs) for the full-order model of the rope-driven tensegrity robot are derived using the second-kind Lagrange method and are of the following form:

[0136]

[0137] Where M is the mass matrix, q is the generalized coordinate, and f e are the generalized internal force vector and generalized external force vector of the system, Φ(q,t) is the constraint matrix, λ is the Lagrange multiplier associated with the constraint, ∈f e is the control input term, and when ∈ = 0, it indicates the undriven state. According to the generalized-α method, the dynamic simulation of the tensegrity robot can be performed.

[0138] Step 1-1-1: First, Figure 5 As shown, a static force F1 is applied to the end of the rope-driven tensegrity robot, causing a certain amount of warping of the end of the rope-driven tensegrity robot and a large deformation of the whole. Subsequently, the static force F1 is immediately withdrawn, and the rope-driven tensegrity robot is caused to oscillate freely under the action of gravity. Because the rope-driven tensegrity robot is usually a dissipative system with damping and will stop when it reaches a stable state, during the sampling process, it is necessary to record as much as possible until the rope-driven tensegrity robot reaches a stable state. The dynamic simulation time is selected as T = 200s, and the simulation step size is Δ = 0.02s. The position data of the free oscillation of the center point of the end collected by sampling is recorded as

[0139]

[0140] Among them, x1 represents the data of the center point of the end along the x-axis; y1 represents the data of the center point of the end along the y-axis; z1 represents the data of the center point of the end along the z-axis; Represents the transpose of a matrix;

[0141] Step 1-1-2: Repeat the process of step 1-1-1 and apply force F in different directions to the end of the rope-driven tensegrity robot. i ,i=1,2,…,20, that is, 20 sets of data are collected. Figure 6 Shown in force F1, F6, F 11 ,F 16The configuration of the rope-driven tensegrity robot is recorded under the action of the rope. The position data of the terminal center point is sampled and recorded. The 1st, 3rd, 6th, 8th, 11th, 13th, 16th and 18th data sets of the 20 sampling data sets are selected as the test data sets, and the remaining 12 groups are used as sampling data for training.

[0142]

[0143] in, Figure 7 、 Figure 8 and Figure 9 The data along the x, y and z coordinate axes in the sampled data are shown respectively; represents the data collected under the action of force F1; represents the data collected under the action of force F2; Indicates that the force F l The data collected under the action of

[0144] Step 1-2: Truncate the sample data The first 30 seconds of data in the

[0145] The spectrum manifold is constructed near the stable state of the rope-driven tensegrity robot. The data sampled at the initial moment will decay rapidly to the spectrum manifold in the form of exponential speed. Therefore, in order to ensure that the sampled data trajectory is closer to the spectrum manifold, the sampled data is truncated. The truncated data is obtained by using the first 30 seconds of data, which is recorded as

[0146]

[0147] in, Indicates that The data obtained after truncating the first 30 seconds of data; Indicates that Data obtained after truncating the first 30 seconds of data Indicates that The data is obtained by truncating the first 30 seconds of data.

[0148] Steps 1-3: Remove truncated data The data along the z-axis direction is used to obtain the training data Y(t);

[0149] Remove truncated data Y t (t) Data along the z-axis. For example, the free oscillation position data of the end center point collected in step 1-1-1 Remove z1 and only include x1 and y1. Don't consider it for now. Figure 9 The data in Perform the same operation and collect the obtained data as training data Y(t).

[0150] Step 2: Based on Takens' delayed embedding theory, the training data Y(t) is embedded into the spectral manifold to reveal the nonlinear dynamic characteristics implicit in the training data Y(t). First, the minimum embedding dimension p is calculated, then the number of delayed sampling times q is determined, and finally, the training data Y(t) is dimensionalized according to the delayed embedding formula to obtain the embedded data y(t). Specifically:

[0151] Step 2-1: Calculate the minimum embedding dimension p and the number of delayed embeddings q;

[0152] Since only the position data of the terminal center point is recorded during the sampling process, it is difficult to reflect the system characteristics of the entire rope-driven tensegrity robot. Therefore, it is necessary to introduce sufficient time-delay data to ensure sufficient embedding in the high-dimensional state space. The dimension of the spectral manifold of the rope-driven tensegrity robot is selected as: m = 4. The minimum embedding dimension p is given by the following formula:

[0153] p≥2m+1; (4)

[0154] Then in this embodiment, p≥2m+1=2×4+1=9;

[0155] In order to increase the redundancy and improve the accuracy of embedding, additional delay measurements can be introduced. Select the embedding coefficient E = 0 and use the following formula to calculate the required number of delay samples q:

[0156]

[0157] in, Indicates rounding up to an integer. In this embodiment,

[0158] Step 2-2: Upscale the training data y(t) using the delayed embedding formula;

[0159] The training data Y is transformed into a high-dimensional coordinate system using the delayed embedding method. The delayed embedding formula of this embodiment is obtained by formula (6) as follows:

[0160] y(t)=[Y(t);Y(t+0.02);…;Y(t+0.08)];

[0161] Among them, y(t) is the data after Takens delayed embedding and dimensionality upgrading, denoted as embedded data, which represents the state information of the current moment and several subsequent moments; Y(t+0.02) represents the data after the data Y(t) is delayed by 0.02 seconds; Y(t+0.08) represents the data after the data Y(t) is delayed by 0.08 seconds.

[0162] Step 3: First, the embedded data y(t) is approximately projected onto the spectral subspace using principal component analysis (PCA). Then, a parameterized method is used to Taylor expand the geometric shape functions of the spectral submanifold and the dynamic equations of the reduced-order model. Second, ridge regression is used to find the optimal solution for the Taylor expansion polynomial coefficients. Finally, based on the learned spectral submanifold formed by the embedded data y(t), the relationship between the data in the z-axis direction and the spectral submanifold is learned, and the data in the z-axis direction is collectively referred to as the vector field z. Specifically:

[0163] Step 3-1: Use PCA principal component analysis on the embedded data y(t);

[0164] Using PCA principal component analysis, the embedded data y(t) is approximately projected into the spectral subspace, which is often low-dimensional. First, the embedded data y(t) is subjected to SVD singular value decomposition to solve the eigenvalues ​​and characteristic matrix.

[0165]

[0166] Among them, U and V represent the left singular matrix and the right singular matrix respectively; S represents the diagonal positive semi-definite matrix; the elements on the diagonal of S are singular values ​​and are arranged in descending order on the diagonal; u1 represents the first column vector of the left singular matrix U; u 10 represents the column vector of the last column of the left singular matrix U; σ1 represents the largest singular value; σ 10 Represents the smallest singular value; v1 T represents the first row vector of the right singular matrix V; Represents the row vector of the last row of the right singular matrix V. σ1=4.668,σ2=4.5383,σ3=0.9252,σ4=0.8996,σ5=9.3871×10 -4 ,σ6=8.9709×10 -4 ,σ7=1.3262×10 -4 ,σ8=1.2712×10 -4 ,σ9=2.0597×10 -6 ,σ 10 =1.8036×10 -6 .

[0167] Left singular vector u iThe basis vectors that constitute the spectral subspace and the associated singular values ​​σ i Represents the energy ratio of the basis vector. The larger the singular value, the greater the contribution to the embedded data y(t). The contribution ratio is calculated by the following formula:

[0168]

[0169] In this embodiment, like Figure 10 As shown in the figure, when the dimension is 4, the variance is 99.98%. This shows that the dimension of the spectrum manifold selected in step 2-1 is 4, which meets the requirements.

[0170] Select the eigenvectors corresponding to the first four singular values ​​and record them as V m , approximately project the embedded data y(t) into the spectral subspace:

[0171] x=V m T y(t); (9)

[0172] Here, x is the modal coordinate, V m It means that the spanned space is approximately the spectral subspace tangent to the spectral manifold.

[0173] Step 3-2: Using the parameterization method, Taylor expands the spectrum manifold geometry function;

[0174] First, assume that there is the following spectral manifold geometry function relationship:

[0175] y(t)=w(x); (10)

[0176] Among them, the function w(x) represents the geometric shape function of the spectral manifold, which uniquely maps the modal coordinate x to the embedded data y(t). Using the parametric method, the function w(x) is Taylor expanded into a polynomial form, and the order of the Taylor expansion polynomial is selected as 3:

[0177] y(t)=w(x):=W0x+Wx 2:3 ; (11)

[0178] Among them, W0 and W are coefficient matrices, W0 represents the coefficient matrix corresponding to x, and W represents The corresponding coefficient matrix; 2:3 Represents x 2 ,x 3 .

[0179] Finally, the ridge regression method is used to minimize the following objective function to obtain the optimal coefficient matrices W0 and W:

[0180]

[0181] in, represents the coefficient matrix corresponding to x obtained by optimization; W * Represents the optimized result and x 2:3 The corresponding coefficient matrix; Represents the square Frobenius norm of the matrix;

[0182] Table 1 shows the average relative error of each data set of the training spectrum manifold geometry function;

[0183] Training set [%] Test set [%] Geometric shapes 2.678e-4 3.5917e-4

[0184] The average relative error of each data set of the training spectrum manifold geometry function in Table 1 can be seen intuitively that the average relative error does not exceed 0.01%.

[0185] Step 3-3: Use the parameterized Taylor method to expand the dynamic equations of the uncontrolled descent model, and use the ridge regression method to find the optimal coefficient matrix;

[0186] Assume that the dynamic equations of the reduced-order model on the spectral manifold have the following relationship:

[0187]

[0188] Among them, r aut is a smooth nonlinear function; Represents x aut The time derivative, x aut is the predicted data for the uncontrolled descent-order model.

[0189] Similarly, the dynamic equations of the Taylor expansion reduced-order model are as follows:

[0190]

[0191] Among them, R0 represents the coefficient matrix corresponding to x; R represents x 2:3 The corresponding coefficient matrix;

[0192] Calculated using the finite difference method Finally, the optimal coefficient matrices R0 and R are obtained by minimizing the following objective function through the ridge regression method:

[0193]

[0194] in, Represents the coefficient matrix corresponding to x obtained by optimization; R * Represents the optimization result and The corresponding coefficient matrix;

[0195] Table 2 shows that the reduced-order model shows high accuracy on both the training set and the test set.

[0196] Table 2 shows the average relative error of each data set for training the reduced-order model;

[0197] Training set [%] Test set [%] Reduced-order model 0.3657 0.3852

[0198] Step 3-4: Parameterize the vector field z and use the ridge regression method to find the optimal coefficient matrix;

[0199] We need to find the mapping relationship between the vector field z and the modal coordinate x. Assume that the following relationship exists and Taylor expand the vector field z, choosing the expansion order as 3:

[0200]

[0201] Among them, ζ(x) represents the mapping relationship between the modal coordinates x and z; ζ0 represents the coefficient matrix corresponding to x; ζ represents x 2:3 The corresponding coefficient matrix. In this embodiment, z=ζ(x):=ζ0x+ζx 2:3 .

[0202] Find the optimal coefficient matrix through ridge regression method:

[0203]

[0204] in, represents the coefficient matrix corresponding to x obtained by optimization; ζ * Represents the optimization result and The corresponding coefficient matrix;

[0205] Table 3 shows the average relative error of each data set for the parameterized vector field z;

[0206] Training set [%] Test set [%] Vector field z 0.0059 0.0071

[0207] Table 3 shows the average relative error of the parameterized vector field z training set and test set. It can be found that the trained z = ζ(x) has good accuracy.

[0208] Step 4: First, the position data of the terminal center point and the control input signal are sampled through open-loop control. Then, the position data of the terminal center point is embedded into the spectrum manifold as control training data. Finally, the ridge regression method is used to find the optimal control matrix. Specifically:

[0209] Step 4-1: Sample the training data through open-loop control to obtain the control sampling data L' u ;

[0210] First, a set of control input signals of a rope-driven tensegrity robot is given, and the driving amplitude is 0.01m. Then:

[0211]

[0212] The simulation time is 50s and the simulation step is 0.001s. Sample and record the control input signal and the position data of the end center point. The position data of the end center point sampled by the open loop control is recorded as the control training data L' u :

[0213] L' u =[x' u ,y' u ,z' u ]; (18)

[0214] Among them, x' u Represents the data of the end center point along the x-axis; y' u Represents the data of the end center point along the y-axis direction; z' u Represents the data of the end center point along the z-axis direction;

[0215] The control input signal u' is:

[0216] u'=[u'1,u'2,u'3]; (19)

[0217] Wherein, u'1 represents the displacement of the first sliding rope of the tensegrity robot; u'2 represents the displacement of the first sliding rope of the tensegrity robot; and u'3 represents the displacement of the first sliding rope of the tensegrity robot.

[0218] Step 4-2: Process the control training data L' in step 4-1 through Takens delayed embedding theory u And the control input signal u', the processed control training data is recorded as L u , the control input signal is u.

[0219] Step 4-3: Use ridge regression method to learn the control matrix B r :

[0220] Assume that the reduced-order dynamic equation of the cable-driven tensegrity robot with belt control is:

[0221]

[0222] Among them, the control matrix B r Represents the influence of the control input signal on the modal coordinate x.

[0223] First, the control training data L obtained in step 4-2 uProjected into the spectrum subspace, the control mode coordinate x is obtained u ;

[0224] x u =V m T L u ; (twenty one)

[0225] Among them, V m is obtained in step 3-1;

[0226] Then, according to the dynamic equation of the reduced-order model learned in step 3-3, we have:

[0227]

[0228] in, are the modal coordinates of the autonomous part under the influence of the control, express The derivative with respect to time.

[0229] Secondly, according to the finite difference method, we get Finally, the ridge regression method is used to obtain the control matrix B r :

[0230]

[0231] in, represents the optimal solution of the satisfied relationship;

[0232] Step 4-4: Use ridge regression method to learn the control matrix B v ;

[0233] Assume that the following mapping relationship exists:

[0234]

[0235] Among them, the control matrix B v Indicates the influence of the control input signal on the control training data; represents the integral of the control input signal from time 0 to t.

[0236] First, according to step 3-2, we have:

[0237]

[0238] in, represents the position data of the terminal center point under the influence of the control. Therefore, the ridge regression method is used to obtain the control matrix B v :

[0239]

[0240] in, Represents the optimal solution of the obtained satisfied relationship.

[0241] Table 4 is the matrix B r and B v Root mean square error (RMSE);

[0242] <![CDATA[B r [%]]]> <![CDATA[B v [%]]]> RMSE 6.68 1.41

[0243] The matrix B is summarized in Table 4. r and B v The root mean square error (RMSE) is around 5%, which shows that a better control model has been obtained through training.

[0244] Combining steps 4-3, the reduced-order model of the rope-driven tensegrity robot can be obtained as:

[0245]

[0246] Where L is the position of the center point of the end of the rope-driven tensegrity robot.

[0247] Here, we verify the accuracy and generalization ability of the reduced-order model obtained below, laying the foundation for the tracking control later. First, verify the accuracy of the reduced-order model. Given the control input:

[0248]

[0249] The simulation time is 10s and the step size is 0.001s. Figure 11 As shown in Figure 2, by comparing with the full-order model, it can be found that the reduced-order model has good accuracy. Furthermore, the generalization ability of the reduced-order model is verified.

[0250] Gradually increase the control amplitude ∈. As shown in Table 5, it can be found that the reduced-order model based on the spectral manifold has good generalization ability and can successfully predict the trajectory within the range of the driving amplitude of 0.03m.

[0251] Table 5 shows the root mean square error (RMSE) and average relative error (REL) of the reduced-order model with different driving amplitudes;

[0252] Drive amplitude RMSE[%] REL[%] 0.005 <![CDATA[8.7×10 -4 ]]> 0.2626 0.015 0.0022 0.7092 0.02 0.0058 0.7092 0.03 0.028 8.8058

[0253] Step 5: Based on the reduced-order model of the rope-driven tensegrity robot, construct an optimal control problem and solve it. The optimal control problem is as follows:

[0254]

[0255] Among them, δz(t f) is the error between the estimated position of the end center point and the actual position; z(t) is the estimated position of the end center point; represents the weighted two-norm; Represents the quadratic weighted cost of the deviation at time t, which is used to weigh the cost of the deviation in the process; represents the quadratic weighted cost under the weight matrix R, which is used to weigh the cost of the control input and prevent the control signal from being too large; u(t) is the control input signal; x(0) is the initial value of the modal coordinate; L(0) is the actual initial position of the end center point; L eq Indicates the expected output value, which is a reference value.

[0256] Next, consider the following four control tasks, tracking circles in space (such as Figure 12 (a), (b), (c)), triangle (such as Figure 13 (a), (b), (c)), rectangle (such as Figure 14 (a), (b), (c)) and the five-pointed star (such as Figure 15 (a), (b), and (c) in the figure). It can be seen that the proposed method can effectively track the trajectory of the rope-driven tensegrity robot for different tasks. Furthermore, the RMSE for all four control tasks is far less than 1%, demonstrating good generalization and control accuracy. Furthermore, Table 6 shows a comparison with the instantaneous optimal control method based on a full-order model. The proposed method has excellent control efficiency, laying the foundation for achieving real-time control tasks.

[0257] Table 6 shows the time comparison of the method of the present invention and the instantaneous optimal control method based on the full-order model to complete a single control task;

[0258] round triangle rectangle Pentagram The method of the present invention 2.82 2.86 3.14 2.78 Full-order model[s] 94.28 90.24 72.97 84.29

[0259] The above-described embodiments merely express the implementation methods of the present invention, but should not be understood as limiting the scope of the present invention. It should be pointed out that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention, which all fall within the scope of protection of the present invention.

Claims

1. A fast and precise control method for a rope-driven tensegrity robot based on spectral manifolds, characterized in that: The method for rapid and precise control of a rope-driven tensegrity robot comprises the following steps: Step 1: First, make the rope-driven tensegrity robot oscillate freely without any control input signal, and use the position data of the end as the sampling data. Then cut off the first h seconds of the sampled data to get the truncated data. Finally, the data along the z-axis is removed from the truncated data to obtain the training data Y(t); Step 2: Based on Takens’ delayed embedding theory, the training data Y(t) is embedded into the spectral manifold to obtain the nonlinear dynamic characteristics implicit in the training data Y(t). First, the minimum embedding dimension p is calculated, then the number of delayed sampling times q is determined, and finally, the training data Y(t) is dimensionalized according to the delayed embedding formula to obtain the embedded data y(t). Step 3: First, the embedded data y(t) obtained in step 2 is approximately projected onto the spectral subspace. Then, a parameterized method is used to Taylor expand the geometric shape function of the spectral submanifold and the dynamic equations of the reduced-order model. Third, a ridge regression method is used to find the optimal solution for the Taylor expansion polynomial coefficients. Finally, based on the spectral submanifold formed by the embedded data y(t), the relationship between the data in the z-axis direction and the spectral submanifold is learned, and the data in the z-axis direction is collectively referred to as the vector field z. Step 4: First, through open-loop control, the position data of the terminal center point and the control input signal are sampled. Then, the position data of the terminal center point is embedded into the spectrum manifold as control training data. Finally, the ridge regression method is used to find the optimal control matrix. Step 5: Based on the reduced-order model of the cable-driven tensegrity robot, construct the optimal control problem and solve it.

2. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 1, characterized in that: The step 1 is specifically as follows: Step 1-1: Get sample data Specifically: Step 1-1-1: First, a static force F1 is applied to the end of the rope-driven tensegrity robot, and then the static force F1 is withdrawn. Under the action of gravity, the rope-driven tensegrity robot is made to oscillate freely. When the robot is in a stable state, sampling is performed and the position data of the free oscillation of the center point of the end collected by sampling is recorded as Among them, x1 represents the data of the center point of the terminal along the x-axis; y1 represents the data of the center point of the terminal along the y-axis; z1 represents the data of the center point of the terminal along the z-axis; [·] T Represents the transpose of a matrix; Step 1-1-2: Repeat the process of step 1-1-1 and apply force F in different directions to the end of the rope-driven tensegrity robot. i , where i = 1, 2, ..., l, and sample and record the position data of the end center point as the sampling data in, represents the data collected under the action of force F1; represents the data collected under the action of force F2; Indicates that the force F l The data collected under the action of Step 1-2: Get the truncated data Specifically: Using truncated sampling data The truncated data is obtained by using the first h seconds of data, which is recorded as in, Indicates that The data obtained after truncating the first h seconds of data; Indicates that The data obtained after truncating the first h seconds of data Indicates that The data obtained after truncating the first h seconds of data; Step 1-3: Get training data Y(t).

3. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 1, characterized in that: The step 2 is specifically as follows: Step 2-1: Obtain the minimum embedding dimension p and the number of delayed embeddings q; The minimum embedding dimension p is given by the following formula: p≥2m+1; (4) Where m is the spectral manifold dimension of the rope-driven tensegrity robot; The delayed sampling times q is given by the following formula: in, Indicates rounding up to an integer; E is the embedding coefficient, ranging from 0≤E≤1000; Step 2-2: Upgrade the dimension of the training data y(t); Use the delayed embedding formula to transform the training data into a high-dimensional coordinate system; assuming that in step 1-1-1 The sampling interval is Δ seconds, and the delay embedding formula is: y(t)=[Y(t);Y(t+Δ);…;Y(t+(q-1)Δ)]; (6) Among them, y(t) is the data after Takens delayed embedding and dimensionality upgrading, denoted as embedded data, which represents the state information of the current moment and several subsequent moments; Y(t+Δ) represents the data after the data Y(t) is delayed by Δ seconds; Y(t+(q-1)Δ) represents the data after the data Y(t) is delayed by (q-1)Δ seconds.

4. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 3, characterized in that: The step 3 is as follows: Step 3-1: Use PCA principal component analysis on the embedded data y(t); PCA principal component analysis is used to approximately project the embedded data y(t) into the spectral subspace. First, SVD singular value decomposition is performed on the embedded data y(t) to solve the eigenvalue and characteristic matrix. Among them, U and V represent the left singular matrix and the right singular matrix respectively; S represents the diagonal positive semi-definite matrix; the elements on the diagonal of S are singular values ​​and are arranged in descending order on the diagonal; u1 represents the first column vector of the left singular matrix U; u n represents the column vector of the last column of the left singular matrix U; σ1 represents the largest singular value; σ n Represents the smallest singular value; v1 T represents the first row vector of the right singular matrix V; The row vector representing the last row of the right singular matrix V; u i represents the left singular vectors, which constitute the basis vectors of the spectral subspace, and the associated singular values ​​σ i Represents the energy ratio of the basis vector. The larger the singular value, the greater the contribution to the embedded data y(t); Select the eigenvectors corresponding to the first m singular values ​​and record them as V m , approximately project the embedded data y(t) into the spectral subspace: Where x is the modal coordinate, V m It means that the spanned space is approximately the spectral subspace tangent to the spectral manifold; Step 3-2: Using the parameterization method, Taylor expands the spectrum manifold geometry function; First, assume that there is the following spectral manifold geometry function relationship: y(t)=w(x); (10) Among them, the function w(x) represents the spectral manifold geometry function, which uniquely maps the modal coordinate x to the embedded data y(t); Secondly, using the parametric method, Taylor expands the function w(x) into a polynomial form: Among them, W0 and W are coefficient matrices, W0 represents the coefficient matrix corresponding to x, and W represents The corresponding coefficient matrix; n w is the order of the Taylor expansion polynomial; Indicates from two orders to n w A series of polynomial basis functions of order; Finally, the ridge regression method is used to minimize the following objective function to obtain the optimal coefficient matrices W0 and W: in, represents the coefficient matrix corresponding to x obtained by optimization; W * Represents the optimization result and The corresponding coefficient matrix; Represents the square Frobenius norm of the matrix; Step 3-3: Use the parameterized Taylor method to expand the dynamic equations of the uncontrolled descent model, and use the ridge regression method to find the optimal coefficient matrix; Assume that the dynamic equations of the reduced-order model on the spectral manifold have the following relationship: Among them, r aut is a smooth nonlinear function; Represents x aut The time derivative, x aut is the predicted data of the uncontrolled reduced-order model; Similarly, the dynamic equations of the Taylor expansion reduced-order model are as follows: Among them, R0 represents the coefficient matrix corresponding to x; R represents The corresponding coefficient matrix; Calculated using the finite difference method Finally, the optimal coefficient matrices R0 and R are obtained by minimizing the following objective function through the ridge regression method: in, Represents the coefficient matrix corresponding to x obtained by optimization; R * Represents the optimization result and The corresponding coefficient matrix; Step 3-4: Parameterize the vector field z and use the ridge regression method to find the optimal coefficient matrix; We need to find the mapping relationship between the vector field z and the modal coordinate x. Assume that the following relationship exists and expand the vector field z: Among them, n r is the expansion order; ζ(x) represents the mapping relationship between the modal coordinates x and z; ζ0 represents the coefficient matrix corresponding to x; ζ represents The corresponding coefficient matrix; Find the optimal coefficient matrix through ridge regression method: in, represents the coefficient matrix corresponding to x obtained by optimization; ζ * Represents the optimization result and The corresponding coefficient matrix.

5. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 4, characterized in that: In step 3-1, the contribution ratio of the singular value to the embedded data y(t) is calculated by the following formula: Where n represents the total number of singular values; m represents the dimension of the spectral submanifold; and a represents the set energy ratio threshold, which is generally taken as 0.99, representing the information characteristics of most embedded data y(t). This can be used to determine whether the dimension m of the spectral submanifold selected in step 2-1 meets the requirements.

6. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 4, characterized in that: In step 3-2, n w The range is 3≤n w ≤15.

7. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 4, characterized in that: The step 4 is specifically as follows: Step 4-1: Sample the training data through open-loop control to obtain the control sampling data L' u ; First, a set of control input signals of the rope-driven tensegrity robot is given, and the control input signals and the position data of the end center point are sampled and recorded; the position data of the end center point sampled by the open-loop control is recorded as the control training data L' u : L' u =[x' u ,y' u ,z' u ]; (18) Among them, x' u Represents the data of the end center point along the x-axis; y' u Represents the data of the end center point along the y-axis direction; z' u Represents the data of the end center point along the z-axis direction; The control input signal u' is: u'=[u'1,u'2,u'3]; (19) Wherein, u'1 represents the displacement of the first sliding rope of the rope-driven tensegrity robot; u'2 represents the displacement of the first sliding rope of the rope-driven tensegrity robot; u'3 represents the displacement of the first sliding rope of the rope-driven tensegrity robot; Step 4-2: Process the control training data L' in step 4-1 through Takens delayed embedding theory u And the control input signal u', the processed control training data is recorded as L u , the control input signal is u; Step 4-3: Use ridge regression method to learn the control matrix B r : Assume that the reduced-order dynamic equation of the cable-driven tensegrity robot with belt control is: Among them, the control matrix B r represents the influence of the control input signal on the modal coordinate x; First, the control training data L obtained in step 4-2 u Projected into the spectrum subspace, the control mode coordinate x is obtained u ; Among them, V m is obtained in step 3-1; Secondly, according to the dynamic equation of the reduced-order model in step 3, we have: Among them, among them, are the modal coordinates of the autonomous part under the influence of the control, express derivative with respect to time; Again, according to the finite difference method, we get Finally, the ridge regression method is used to obtain the control matrix B r : Among them, B r is the control matrix, which represents the influence of the control input signal on the modal coordinate x; represents the optimal solution; Step 4-4: Use ridge regression method to learn the control matrix B v ; Assume that the following mapping relationship exists: Among them, the control matrix B v Indicates the influence of the control input signal on the control training data; represents the integral of the control input signal from time 0 to t; First, according to step 3-2, we have: in, represents the position data of the terminal center point under the influence of the control; therefore, the control matrix B is obtained by the ridge regression method v : in, represents the optimal solution of the satisfied relationship; Combining steps 4-3, the reduced-order model of the rope-driven tensegrity robot is obtained as follows: where L is the position of the end center point of the cable-driven tensegrity robot.

8. The method for fast and precise control of a rope-driven tensegrity robot based on spectral manifolds according to claim 7, characterized in that: In step 5, the optimal control problem is shown in formula (28): Among them, δz(t f ) is the error between the estimated position of the end center point and the actual position; z(t) is the estimated position of the end center point; represents the weighted two-norm; Represents the quadratic weighted cost of the deviation at time t, which is used to weigh the cost of the deviation in the process; represents the quadratic weighted cost under the weight matrix R, which is used to weigh the cost of the control input and prevent the control signal from being too large; u(t) is the control input signal; x(0) is the initial value of the modal coordinate; L(0) is the actual initial position of the end center point; L eq Indicates the expected output value, which is a reference value.

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