A trajectory tracking control method for a tendon-driven spatial manipulator based on actual specified time

Abstract

The present invention provides a trajectory tracking control method for a tendon-driven space manipulator based on actual specified time, comprising establishing a tendon-driven space manipulator joint model to derive the dynamic equations of the tendon-driven space manipulator, and mapping the rope tension change to the joint angle change under the rotating joint; thereby achieving high-precision motion control of the manipulator; controlling the manipulator joint angle θ by an actual specified time controller to track a desired trajectory θ within a specified time. d The present invention can ensure that the tracking error remains within the specified range from beginning to end, and achieves the specified tracking accuracy within a preset time without being affected by the initial conditions. The present invention is based on a high-order all-wheel drive system method, which avoids the repeated derivation of the virtual torque during the controller design process, greatly simplifying the controller design steps and the complexity of the calculation.

Description

Technical Field

[0001] The present invention relates to the technical field of robotic arm control, and in particular to a tendon-driven spatial robotic arm trajectory tracking control method based on actual specified time. Background Art

[0002] Space manipulators play an important role in a range of aerospace activities, such as on-orbit servicing[1], space assembly, and active space debris removal. With the development of aerospace technology, future space missions such as large-inertia rolling target capture, on-orbit assembly of large spacecraft structures, asteroid capture and mining have put forward higher requirements on the load capacity and operating distance of space manipulators.

[0003] Traditional space robotic arms consist of a rotating joint consisting of an electric motor, a reducer and a brake, and a carbon fiber composite tube. The output torque is amplified by the gear system at the rotating joint. The mass of the rotating joint accounts for 85% to 90% of the total mass of the robotic arm, and the weight cost cannot be ignored.

[0004] With the rapid development of tethered manipulator technology in recent years, tethered transmissions have enabled the separation of the drive and motion components, reducing the size of the manipulator. However, existing tethered manipulators suffer from short operating ranges and weak load-bearing capacity. To address the shortcomings of traditional tethered manipulators in long-distance, high-torque space missions, NASA has proposed a new tendon-driven space manipulator.

[0005] However, due to the large size of the robotic arm and the large number of flexible components, its drive requires the coordinated control of multiple ropes and multiple motors. At the same time, it is necessary to overcome the elastic deformation caused by the force on the ropes and the influence of the strong coupling motion between adjacent arm rods. Therefore, how to ensure the reliability of tendon driving performance and achieve high-precision robust control performance has become a difficult problem in the current research on high-speed and high-precision trajectory tracking control of new tendon-driven space robotic arms.

[0006] To address the high uncertainty of such complex systems, the high-precision requirements for multi-component collaborative driving, and the rapid convergence issues, various finite-time control strategies, including adaptive techniques, non-singular terminal sliding modes, and neural networks, have been successfully applied to trajectory tracking control of space manipulators. However, the convergence time of these control methods is extremely sensitive to the system's initial conditions, which greatly limits their application in practical engineering.

[0007] Fixed-time control provides uniformly bounded convergence time for any system initial state, ensuring that the adjustment time is determined by a single control parameter. Consequently, control schemes based on the concept of fixed-time stability are widely used in trajectory tracking control of robotic arms. While fixed-time control clearly outperforms finite-time control, the complex relationship between convergence time and control gain of these fixed-time tracking strategies makes it difficult to predetermine the required stabilization time. Furthermore, the controller design is cumbersome and complex, limiting its practical application in engineering. Summary of the Invention

[0008] In response to the shortcomings of the existing technology, the present invention provides a tendon-driven spatial robot arm trajectory tracking control method based on actual specified time. The specified time controller designed in the present invention can ensure that the system tracking error remains within the specified range from beginning to end, and achieves the specified tracking accuracy within the preset time without being affected by the initial conditions.

[0009] The technical solution of the present invention is: a tendon-driven space manipulator trajectory tracking control method based on actual specified time, comprising the following steps:

[0010] S1) Establish a tendon-driven space manipulator joint model to derive the dynamic equations of the tendon-driven space manipulator and map the rope tension change to the joint angle change under the rotation joint; thereby achieving high-precision motion control of the manipulator;

[0011] S2) Control the joint angle θ of the manipulator to track the desired trajectory θ within the specified time through the actual specified time controller d .

[0012] Preferably, in step S1), since the tendon-driven space manipulator is in an outer space environment, the gravity acting on the system is neglected, and the expression of the tendon-driven space manipulator dynamic equation is obtained as follows:

[0013]

[0014] Where M(θ) is the inertia matrix; θ is the generalized joint angle variable; represents the joint angular acceleration; represents the joint angular velocity; represents the Coriolis force and centripetal force matrix, represents external interference terms.

[0015] Preferably, in step S1), according to the dynamic equation, the relationship between the joint torque and the change in rope length is obtained as follows:

[0016] J θ f = τ s +τ θ (2);

[0017]

[0018] Where, J θ is the system Jacobian matrix, is the mapping relationship matrix of joint module i; f is the tension of each rope of the robotic arm; τ s represents the elastic loss torque of the rope; τ θ represents the joint angular torque.

[0019] Preferably, in step S1), assuming that the ropes have the same elastic modulus E and cross-sectional area R, the rope elastic loss work W s for:

[0020]

[0021] The joint angle θ is calculated by the above two sides i Differentiating, we get:

[0022]

[0023] f 11 、f 12 、f 13 、f 14 Respectively represent the tension of the four ropes of joint module 1; f n1 、f n2 、f n3 、f n4 Represents the tension of the four ropes of joint module n; l n1 、l n2 、l n3 、l n4 Represents the lengths of the four ropes of joint module n; l 11 、l 12 、l 13 、l 14 They respectively represent the lengths of the four ropes of the joint module 1.

[0024] As a preference, in step S1), the rope l i and joint angle θ i The mapping relationship between them is:

[0025]

[0026] Where, l i1 、l i2 、l i3 、l i4 They represent the lengths of the four ropes of joint module i; L i and L i+1represents the distance between the center of the drum on link i and link i+1 and the center of the sling shaft; r is the radius of the rope retracting drum; S is the distance between the end of the sling and the joint shaft; θ is the angle between the line connecting the end of the spreader to the spreader shaft and the spreader axis; i is the joint rotation angle between adjacent links.

[0027] Preferably, in step S1), by simultaneously taking the derivative of both sides of equation (3), we can obtain:

[0028] dl i =J qi dθ i ;

[0029]

[0030] Where, J qi Map the Jacobian matrix for joint module i; is the joint angle between the spreader and the connecting rod; T is the transposition operation;

[0031] Preferably, in step S1), the distance S between the end of the sling and the joint shaft is:

[0032]

[0033] The angle between the line connecting the end of the spreader to the spreader shaft and the spreader axis for:

[0034]

[0035] Where h and d are the height and width of one side of the spreader, respectively.

[0036] As a preference, in step S2), the robot arm joint angle tracking error is defined for:

[0037]

[0038] Where, θ is the joint angle of the robot arm; θ d To track the desired trajectory.

[0039] As a preference, in step S2), in order to make the tracking error of the robot arm joint angle Converge to the specified accuracy within the specified time, and introduce the specified time performance constraint function β(t):

[0040]

[0041] Where 0<T<∞ indicates the pre-specified convergence time, 0<ε<∞ indicates the tracking accuracy when the system is stable; 2p≥n+1 and p is a positive integer;

[0042] β(t) decreases monotonically from infinity to a specified tracking accuracy ε at t∈[0,T] and remains at ε when t>T. Therefore, the tracking error satisfies the following constraints:

[0043]

[0044] By adjusting the pre-specified convergence time T and the tracking accuracy ε when the system is stable, the joint angle tracking accuracy can be satisfied within the specified time T.

[0045] As a preference, in step S2), a new variation function is introduced to achieve the tracking error To constrain:

[0046]

[0047] Among them, h(γ(t)) is a composite function; is the boundary function; c>0 is the control parameter; a>cε 2 is the control parameter; ξ1(t) is the performance function.

[0048] Preferably, in step S2), the performance function ξ1(t) has the following properties:

[0049] 1) Only when When ξ1(t)=0;

[0050] 2) When When ξ1(t)→∞;

[0051] 3) When γ(t)>a, h(γ(t))=1, so

[0052] Preferably, in step S2), by taking the derivative of ξ1(t) in formula (4), we can obtain:

[0053]

[0054] in:

[0055]

[0056] Where, is the derivative of the performance function ξ1(t); Tracking error The derivative of ; h is the abbreviation of h(γ(t)); p is the control parameter; is the derivative of the performance constraint function β(t) at a specified time;

[0057] Preferably, the tendon-driven spatial manipulator dynamics equation of formula (1) is rewritten into an n-order system form as follows:

[0058]

[0059] Where, is the joint angular velocity error of the i-th joint module; is the joint angle error of the i+1th joint module; f′ represents the control input; Indicates the system state quantity; represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term; y(t) represents the system output.

[0060] As a preference, assume that the unknown time-varying interference term d i (t) is bounded, that is, there exists a positive number D i , so that |d i (t)|<D i ;

[0061] definition Combining formula (21), we can get the system equation for tracking error:

[0062]

[0063] but:

[0064]

[0065] By taking the derivative of formula (23) and combining it with formula (22), we get:

[0066]

[0067] Then the high-order system equation is obtained as:

[0068]

[0069] Where, for The derivative of ξ1 (i) for The i-1 derivative of

[0070] represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term.

[0071] As a preference, when i=n, ​​we obtain:

[0072]

[0073] Therefore, the control law of the actual specified time controller is:

[0074]

[0075]

[0076] Where, f i ′、f i ′0、f i1 ′ are control inputs respectively; A is the control parameter matrix; ξ i is the intermediate variable of the performance function; is the performance parameter; η i >0 is the control parameter.

[0077] The beneficial effects of the present invention are:

[0078] 1. The present invention can ensure that the tracking error remains within the specified range from beginning to end, and achieves the specified tracking accuracy within a preset time without being affected by the initial conditions;

[0079] 2. The present invention is based on a high-order all-wheel drive system method, which avoids the repeated derivation of virtual torque in the controller design process, greatly simplifying the controller design steps and the complexity of calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 A schematic diagram of the tendon-driven spatial robotic arm joint model constructed for the present invention;

[0081] Figure 2 Schematic diagram of a tendon-driven space robot arm according to Example 2 of the present invention;

[0082] Figure 3 2 is a response curve diagram of joint angle 1 under different initial values ​​in Example 2 of the present invention;

[0083] Figure 4 2 is a response curve diagram of joint angle 2 under different initial values ​​in Example 2 of the present invention;

[0084] Figure 5 This is a graph showing the system output response under different performance parameters in Example 2 of the present invention;

[0085] Figure 6 Graph showing the output response curve of joint angle 1 according to embodiment 1 of the present invention and without the actual specified time;

[0086] Figure 7Graph showing the output response curves of embodiment 1 of the present invention and joint angle 2 without the actual specified time;

[0087] Figure 8 This is a comparison diagram of the joint angle 1 output response between Example 1 of the present invention and the model-free time delay estimation specified time performance control method;

[0088] Figure 9 This is a comparison diagram of the joint angle 2 output response of Example 1 of the present invention and the model-free time delay estimation specified time performance control method;

[0089] Figure 10 This is a comparison chart of the specified time tracking performance between Example 1 of the present invention and model-free delay estimation;

[0090] Figure 11 This is a diagram showing the trajectory tracking of the robotic arm in Example 3 of the present invention;

[0091] Figure 12 This is a diagram showing the error during robot tracking in Example 3 of the present invention. DETAILED DESCRIPTION

[0092] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings:

[0093] Example 1

[0094] This embodiment provides a method for tracking and controlling a tendon-driven spatial manipulator trajectory based on actual specified time, comprising the following steps:

[0095] S1) Establish a tendon-driven spatial manipulator joint model, such as Figure 1 As shown in the figure, the dynamic equations of the tendon-driven spatial manipulator are derived, and the rope tension changes are mapped to the joint angle changes under the rotation joint; thus achieving high-precision motion control of the manipulator;

[0096] In this embodiment, since the tendon-driven space manipulator is in an outer space environment, the gravity acting on the system is neglected, and the expression of the tendon-driven space manipulator dynamic equation is obtained as follows:

[0097]

[0098] Where M(θ) is the inertia matrix; θ is the generalized joint angle variable; represents the joint angular acceleration; represents the joint angular velocity; represents the Coriolis force and centripetal force matrix, represents external interference terms.

[0099] According to the dynamic equation, the relationship between the joint torque and the change in rope length is:

[0100] J θ f = τ s +τ θ (2);

[0101]

[0102] Where, J θ is the system Jacobian matrix, is the mapping relationship matrix of joint module i; f is the tension of each rope of the robotic arm; τ s represents the elastic loss torque of the rope; τ θ represents the joint angular torque.

[0103] Assuming the ropes have the same elastic modulus E and cross-sectional area R, the rope elastic loss work W s for:

[0104]

[0105] The joint angle θ is calculated by the above two sides i Differentiating, we get:

[0106]

[0107] f 11 、f 12 、f 13 、f 14 Respectively represent the tension of the four ropes of joint module 1; f n1 、f n2 、f n3 、f n4 Represents the tension of the four ropes of joint module n; l n1 、l n2 、l n3 、l n4 Represents the lengths of the four ropes of joint module n; l 11 、l 12 、l 13 、l 14 They respectively represent the lengths of the four ropes of the joint module 1.

[0108] Among them, the rope l i and joint angle θ i The mapping relationship between them is:

[0109]

[0110] Where, l i1 、l i2 、l i3 、l i4 They represent the lengths of the four ropes of joint module i; Li and L i+1 represents the distance between the center of the drum on link i and link i+1 and the center of the sling shaft; r is the radius of the rope retracting drum; S is the distance between the end of the sling and the joint shaft; θ is the angle between the line connecting the end of the spreader to the spreader shaft and the spreader axis; i is the joint angle between adjacent links, h and d are the height and width of one side of the spreader, respectively.

[0111] By taking the derivative of both sides of formula (3), we can get:

[0112] dl i =J qi dθ i ;

[0113]

[0114] Where, J qi Map the Jacobian matrix for joint module i; is the joint angle between the spreader and the connecting rod; T is the transposition operation.

[0115] S2) Control the joint angle θ of the manipulator to track the desired trajectory θ within the specified time through the actual specified time controller d .

[0116] This embodiment defines the robot arm joint angle tracking error for:

[0117]

[0118] Where, θ is the joint angle of the robot arm; θ d To track the desired trajectory.

[0119] In order to make the tracking error of the robot arm joint angle Converge to the specified accuracy within the specified time, and introduce the specified time performance constraint function β(t):

[0120]

[0121] Where 0<T<∞ indicates the pre-specified convergence time, 0<ε<∞ indicates the tracking accuracy when the system is stable; 2p≥n+1 and p is a positive integer;

[0122] β(t) decreases monotonically from infinity to a specified tracking accuracy ε at t∈[0,T] and remains at ε when t>T. Therefore, the tracking error satisfies the following constraints:

[0123]

[0124] By adjusting the pre-specified convergence time T and the tracking accuracy ε when the system is stable, the joint angle tracking accuracy can be satisfied within the specified time T.

[0125] Regarding the specified time performance constraint function β(t), it has several main performance characteristics.

[0126] 1) The convergence time T and convergence accuracy ε are given in advance;

[0127] 2) The initial value of β(t) is infinite, so the initial value of the system's tracking error always remains within the bounds, thus obtaining a global result;

[0128] 3) The trajectory of the tracking error always remains within the constraints.

[0129] In addition, this embodiment also introduces a new variation function to achieve the tracking error To constrain:

[0130]

[0131] Among them, h(γ(t)) is a composite function; is the boundary function; c>0 is the control parameter; a>cε 2 is the control parameter; ξ1(t) is the performance function.

[0132] Wherein, the performance function ξ1(t) has the following properties:

[0133] 1) Only when When ξ1(t)=0;

[0134] 2) When When ξ1(t)→∞;

[0135] 3) When γ(t)>a, h(γ(t))=1, so

[0136] Preferably, in step S2), by taking the derivative of ξ1(t) in formula (4), we can obtain:

[0137]

[0138] in:

[0139]

[0140] Where, is the derivative of the performance function ξ1(t); Tracking error The derivative of ; h is the abbreviation of h(γ(t)); p is the control parameter; is the derivative of the performance constraint function β(t) at a specified time.

[0141] In this embodiment, the tendon-driven spatial manipulator dynamics equation of formula (1) is rewritten into an n-order system form as follows:

[0142]

[0143] Where, is the joint angular velocity error of the i-th joint module; is the joint angle error of the i+1th joint module; f′ represents the control input; Indicates the system state quantity; represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term; y(t) represents the system output.

[0144] Assumption 1: Assume that the unknown time-varying interference term d i (t) is bounded, that is, there exists a positive number D i , so that |d i (t)|<D i ;

[0145] definition Combining formula (21), we can get the system equation for tracking error:

[0146]

[0147] but:

[0148]

[0149] By taking the derivative of formula (23) and combining it with formula (22), we get:

[0150]

[0151] Then the high-order system equation is obtained as:

[0152]

[0153] Where, is the derivative of the performance function ξ1(t); for The derivative of represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term.

[0154] When i=n, ​​we get:

[0155]

[0156] Therefore, the control law of the actual specified time controller is:

[0157]

[0158] Where, f i ′、f i ′0、f i1 ′ are control inputs respectively; A is the control parameter matrix; ξ i It is the intermediate auxiliary variable of the control function; is the control parameter; η i >0 is the control parameter.

[0159] Example 2

[0160] like Figure 2 As shown, this embodiment is based on the Matlab simulation platform to perform trajectory tracking control on the tendon-driven spatial manipulator of the dual-joint module. The physical structure parameters of the manipulator are shown in Table 1:

[0161] Table 1 Physical structure parameters of the robotic arm

[0162]

[0163]

[0164] In the numerical simulation, the fourth-order Rung-Kuta method is used to solve the differential equation, and the control parameters are selected as: a = 5, c = 100, p = 2, η = 0.02; A is selected 0～1 =[26.59]

[0165] To prove the global stability of the proposed control scheme, the system response simulation results under different initial value conditions are given. The specified convergence time T = 1s, the steady-state tracking accuracy ε = 0.01 (rad); given different initial values ​​of the joint angle θ 01 =[-0.2 0.6] T θ 02 =[-0.4 0.8] T θ 03 =[-0.6 1.0] T θ 04 =[0.7 -0.7] T θ 05 =[0.9 -0.9] T ;The desired trajectory of the robot arm is set to:

[0166]

[0167] The simulation results are as follows Figure 3 and Figure 4 As shown, where e represents the error, from Figure 3 、 Figure 4 It can be seen that for different large ranges of initial value conditions, the system error can eventually reach the specified accuracy within the preset time, which effectively verifies the effectiveness of the control strategy.

[0168] To further illustrate that different convergence times and tracking accuracies can be specified by selecting corresponding performance parameters, this embodiment sets the tracking errors under two performance parameters: T = 1, ε = 0.01 and T = 0.5, ε = 0.005. The corresponding system response simulation results are shown in Figure 2. Figure 5 shown. Figure 5 It is verified that the control strategy can specify different convergence times and tracking accuracies simply by selecting different performance parameters without switching the controller.

[0169] At the same time, this embodiment is compared with the high-order all-wheel drive system control method without actual specified time, and the system output response simulation results Figure 6-7 As shown in the figure, Cable1-Cable4 represent rope 1-rope 4 respectively, and in the figure, "with PPTC" represents the control method of embodiment 1, while "without PPTC" represents the high-order full-drive system control method without actual specified time; Figure 6 、 Figure 7 The method of Example 1 is compared with a high-order all-wheel drive system method without actual specified time and performance. It can be seen that the system tracking error will be quickly adjusted before reaching the specified time to ensure that the error accuracy always remains within the specified range. This comes with the result that the control input will have a sudden change.

[0170] In addition, this embodiment introduces a saturation function to limit the rope tension. The saturation function is described as follows:

[0171]

[0172] Among them, u maxi ,u mini are the upper and lower limits of the rope tension of the i-th joint module;

[0173] When f i When the limit is exceeded, a sharp corner will appear. To solve this problem, this embodiment introduces a smoothing function as follows:

[0174]

[0175] Where, is a saturation function; is the saturation upper limit; umini is the lower saturation limit;

[0176] After introducing the saturation function, the method of Example 1 is compared with the model-free delay estimation specified time performance control method for simulation, and the results are shown in Table 1. Figure 8 and Figure 9 During the simulation, the parameters were adjusted appropriately to obtain the best tracking performance under the same maximum control torque amplitude. In this case, the tracking accuracy and the specified time were set to T = 1, ε = 0.005, and the initial state of the system was set to θ0 = [0.6-0.7] T , the rope tension limit is set to u maxi =1000N,u mini =-1000N.

[0177] Figure 8 、 Figure 9 After introducing the saturation function, the method of Example 1 is compared with the model-free time delay estimation specified time performance control method for simulation, where Cable1-Cable4 represent ropes 1-4 respectively, TDE represents the time delay terminal sliding mode controller, and HOFA represents the high-order all-wheel drive system controller;

[0178] The simulation results show that the tracking error converges more quickly under the high-order all-wheel drive system approach, and the control input adjustment is much smoother before the specified time. In summary, the high-order all-wheel drive system approach has significant advantages in terms of actual specified time and performance control strategy, effectively achieving high-speed and high-precision trajectory tracking control of tendon-driven spatial manipulators and making it more suitable for practical engineering applications.

[0179] This example also introduces the integrated absolute error (IAE) and time-weighted integrated absolute error (ITAE) to quantitatively evaluate the tracking performance of the two controllers. The results are shown in Figure 10 The definitions of the two evaluation indicators are as follows:

[0180]

[0181] Among them, IAE qi For ITAE qi for;e i is the error; is the integral amount.

[0182] Example 3

[0183] This embodiment builds a single-joint module platform, in which the arm length is 1955mm, the hoist length is 640mm, and the hoist width is 101mm. The single-joint module is independently driven by four motors with four ropes, and a support wheel is installed at the bottom of the robotic arm to balance the influence of gravity.

[0184] The sampling period is set to 0.05 seconds, the preset time is set to T = 10 seconds, the preset accuracy is set to ε = 4 degrees, and the expected trajectory of the joint module is set to:

[0185]

[0186] The trajectory tracking of the robot arm is as follows Figure 11 As shown; the error when the robot is tracking is as follows Figure 12 As shown, it can be seen that the system quickly converges to the specified accuracy range within the preset time.

[0187] The above embodiments and descriptions are only for explaining the principles and best embodiments of the present invention. Without departing from the spirit and scope of the present invention, the present invention may be subject to various changes and improvements, which shall fall within the scope of the invention to be protected.

Claims

1. A method for tracking and controlling the trajectory of a tendon-driven space manipulator based on actual specified time, characterized in that: It includes the following steps: S1), establish a tendon-driven space manipulator joint model to derive the dynamic equation of the tendon-driven space manipulator, and map the change of rope tension to the change of joint angle under the rotary joint, so as to complete the high-precision motion control of the manipulator; S2) Control the joint angle θ of the manipulator to track the desired trajectory θ within the specified time through the actual specified time controller d ; The details are as follows: Define the robot arm joint angle tracking error for: In the formula, θ is the manipulator joint angle; θ d To track the desired trajectory; In order to make the tracking error of the robot arm joint angle Converge to the specified accuracy within the specified time, and introduce the specified time performance constraint function β(t): Among them, 0 < T < ∞ represents the pre-specified convergence time, and 0 < ε < ∞ represents the tracking accuracy when the system is stable; 2p ≥ n + 1 and p is a positive integer; β(t) monotonically decreases from infinity to the specified tracking accuracy ε when t ∈ [0, T], and remains at ε when t > T. Therefore, the tracking error satisfies the following constraints: By adjusting the pre-specified convergence time T and the tracking accuracy ε when the system is stable, the joint angle tracking accuracy can be satisfied within the specified time T. By introducing a new change function to achieve the tracking error To constrain: Among them, h(γ(t)) is a composite function; is the boundary function; c>0 is the control parameter; a>cε 2 is the control parameter; ξ1(t) is the performance function; By taking the derivative of ξ1(t) in formula (4), we get: Where: Where, is the derivative of the performance function ξ1(t); Tracking error The derivative of ; h is the abbreviation of h(γ(t)); p is the control parameter; is the derivative of the performance constraint function β(t) at a specified time.

2. The method for tracking and controlling a tendon-driven spatial manipulator trajectory based on actual specified time according to claim 1, characterized in that: In step S1), since the tendon-driven space manipulator is in the outer space environment, the gravity acting on the system is ignored, and the expression of the dynamic equation of the tendon-driven space manipulator is: Where M(θ) is the inertia matrix; θ is the generalized joint angle variable; represents the joint angular acceleration; represents the joint angular velocity; represents the Coriolis force and centripetal force matrix, represents the external interference term; J θ is the system Jacobian matrix; τ s represents the elastic loss torque of the rope; f is the tension of each rope of the manipulator.

3. The method for tracking and controlling the trajectory of a tendon-driven space manipulator based on actual specified time according to claim 2, characterized in that: In step S1), according to the dynamic equation, the relationship between the joint torque and the change of rope length is: J θ f=τ s +t θ (2); Where, J θ is the system Jacobian matrix, is the mapping relationship matrix of joint module i; f is the tension of each rope of the robotic arm; τ s represents the elastic loss torque of the rope; τ θ represents the joint angular torque.

4. The method for tracking and controlling a tendon-driven spatial manipulator trajectory based on actual designated time according to claim 3, characterized in that: In step S1), assuming that the ropes have the same elastic modulus E and cross-sectional area R, the rope elastic loss work W s for: The joint angle θ is calculated by the above two sides i Differentiating, we get: f 11 、f 12 、f 13 、f 14 Respectively represent the tension of the four ropes of joint module 1; f n1 、f n2 、f n3 、f n4 Represents the tension of the four ropes of joint module n; l n1 、l n2 、l n3 、l n4 Represents the lengths of the four ropes of joint module n; l 11 、l 12 、l 13 、l 14 They represent the lengths of the four ropes of the joint module 1 respectively; S is the distance between the end of the sling and the joint shaft.

5. The method for tracking and controlling the trajectory of a tendon-driven space manipulator based on actual designated time according to claim 4, characterized in that: In step S1), the rope 1 i and joint angle θ i The mapping relationship between them is: Where, l i1 、l i2 、l i3 、l i4 They represent the lengths of the four ropes of joint module i; L i and L i+1 represents the distance between the center of the drum on link i and link i+1 and the center of the sling shaft; r is the radius of the rope retracting drum; S is the distance between the end of the sling and the joint shaft; θ is the angle between the line connecting the end of the spreader to the spreader shaft and the spreader axis; i is the joint angle between adjacent links; By taking the derivative of both sides of formula (3), we get: dl i =J qi dθ i ; Where, J qi Map the Jacobian matrix for joint module i; is the joint angle between the spreader and the connecting rod; T represents the transpose operation.

6. The method for tracking and controlling a tendon-driven spatial manipulator trajectory based on actual designated time according to claim 2, characterized in that: Rewrite the dynamic equation of the tendon-driven space manipulator in formula (1) into the form of an n-order system as follows: Where, is the joint angular velocity error of the i-th joint module; is the joint angle error of the i+1th joint module; f′ represents the control input; Indicates the system state quantity; represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term; y(t) represents the system output.

7. The method for tracking and controlling a tendon-driven spatial manipulator trajectory based on actual designated time according to claim 6, characterized in that: Assume that the unknown time-varying interference term d i (t) is bounded, that is, there exists a positive number D i , so that |d i (t)|<D i ; definition Combining formula (21), we can get the system equation for tracking error: Then: By taking the derivative of formula (23) and combining it with formula (22), we get: Furthermore, the high-order system equation is obtained as: Where, for The derivative of ξ1 (i) for The i-1 derivative of represents the unknown gain function; is the nonlinear uncertain continuous function of the system, d i (t) represents the unknown time-varying interference term; When i = n, we get: Therefore, the control law of the actual specified time controller is: Where, f i ′、f′ i0 、f′ i1 are the control inputs respectively; a is the control parameter matrix; ξ i is the intermediate variable of the performance function; is the performance parameter; η i >0 is the control parameter.

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