Cooperative control method and system of spatial flexible multi-arm robot using stress stiffening effect

Abstract

The invention relates to a collaborative control method and system for a space flexible multi-arm robot using the stress stiffening effect, and relates to the field of collaborative control of space flexible multi-arm robots. The invention solves the problem that the research on robots considering flexibility in the prior art mainly focuses on a single robotic arm, and there is still little research on multi-arm space robots, especially those that can switch configurations between open chain and closed chain. The method comprises: based on the two topological configurations of the space flexible multi-arm robot, namely the open chain and the self-closed chain, respectively establishing the dynamic model of the space multi-arm robot considering joint flexibility in the two states of open chain and self-closed chain; introducing the internal force stiffening effect, and determining whether there is internal stress to complete the stiffness analysis of the space multi-arm robot considering flexible joints in the self-closed chain configuration; designing a composite control law for the space multi-arm robot considering flexible joints based on the singular perturbation method, and is also applicable to the field of improving control accuracy in the process of orbital service and orbital capture.

Description

Technical Field

[0001] The present invention relates to the technical field of collaborative control of space flexible multi-arm robots, and in particular to a collaborative control method and system for space flexible multi-arm robots utilizing stress stiffening effect. Background Art

[0002] Currently, there are hundreds of high-value spacecraft in geosynchronous orbit. However, these spacecraft often fail due to structural failures, control system malfunctions, fuel depletion, and other reasons. Due to the lack of effective on-orbit repair methods, they can only be replaced by launching new satellites, which significantly increases the cost of development and launch. Therefore, how to stably capture failed spacecraft and perform space operations such as disassembly, repair, and life extension has become a key focus of future space technology research in various countries. As a new type of on-orbit capture tool, space multi-arm robots offer greater stability, flexibility, and reliability than traditional single-arm robots. Current research focuses on the collaborative motion planning, collaborative force control, and collaborative perception of space multi-arm robots, and has achieved rich research results.

[0003] In the study of the dynamics of space multi-arm robots, space robots are classified as floating base systems, in which the motions and forces of the manipulator, actuator, and base interact, a phenomenon known as "dynamic coupling." A generalized Jacobian matrix suitable for floating space manipulators is derived for solving satellite attitude control problems. Prior art proposes a generalized relative Jacobian method for relative motion planning of two arms in inertial space. Furthermore, the closed-loop system formed by the space multi-arm robot during target capture is considered, and the dynamic coupling effects between open-loop and closed-loop systems are analyzed. Based on the kinematic and dynamic equations of a free-floating multi-arm space robot with closed-loop constraints, the mapping relationship from joint space to task space is described. The impact dynamics, post-capture stability, and maneuver transfer of the space multi-arm robot system during capture are unified into a single dynamic framework. The flexibility of space manipulators, especially joint flexibility, cannot be ignored in practical tasks. This is especially true when capturing large payloads with unknown dynamic properties, where joint flexibility significantly impacts capture accuracy. To address this issue, prior art has gradually refined the dynamic models of flexible joints and proposed modeling methods that are more consistent with practical systems.

[0004] In summary, existing research on flexible robots mainly focuses on single robotic arms. There is still little research on multi-arm space robots, especially those that can switch configurations between open and closed chains.

[0005] In the study of control methods for space multi-arm robots, many researchers have applied nonlinear control methods such as adaptive control, robust control, and sliding mode control, achieving promising results. Furthermore, some researchers have considered the closed-loop kinematic constraints of the multi-arm system and proposed a series of coordinated control methods to improve the control performance of the multi-arm robot. Prior art proposes a dual-arm coordinated motion planning and compliant control method that uniquely defines the relative Jacobian matrix of the null space projection and the relative operating forces between the two arms. A dual-arm cooperative gripping control method based on model predictive control has also been proposed, as well as a dual-arm coordinated "region-directed capture" method for capturing non-cooperative tumbling targets. However, most of these studies consider the space multi-arm robot as a rigid body and fail to consider the control errors caused by flexibility. In research on control methods for space robots that consider flexibility, the dynamic model of the multi-arm system becomes a more complex, high-order model due to the modeling of flexible joints, posing significant challenges and requirements for the control system. PD control with flexibility compensation treats the error caused by flexibility as a disturbance term, which can suppress the control error of flexible joints to a certain extent. However, it does not consider the dynamic model of the flexible joints, resulting in poor tracking dynamic performance. Singular perturbation control decouples the flexible joint model into a system with two time scales, fast and slow, offering the advantage of model order reduction. However, these methods primarily focus on single flexible joints or single manipulators with multiple flexible joints, with limited research on spatial multi-arm robots.

[0006] In summary, existing research focuses on enabling multiple robotic arms to simultaneously operate on a target satellite, such as in multi-arm gripping and hugging scenarios. However, in certain missions, such as when the target has only one nozzle for docking or is relatively small, the robotic arms have only a single point of operation. Summary of the Invention

[0007] The present invention aims to solve the problem that the existing research on flexible robots mainly focuses on a single robotic arm, and there is still little research on multi-arm space robots, especially those that can switch configurations between open and closed chains.

[0008] To solve the above technical problems, the present invention is achieved through the following technical solutions:

[0009] Solution 1: The present invention proposes a collaborative control method for a space flexible multi-arm robot using a stress stiffening effect, the collaborative control method comprising the following steps:

[0010] S1. Based on the open chain and self-closed chain topological configurations of the spatial flexible multi-arm robot, the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering joint flexibility are established respectively;

[0011] S2. Introduce the internal force stiffening effect into the dynamic model of the open chain and closed chain states described in S1, and determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration;

[0012] S3. Based on the singular perturbation method, a composite control law is designed for the space multi-arm robot considering flexible joints described in S2. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, collaborative control of the space flexible multi-arm robot that completes the internal force stiffening effect.

[0013] Furthermore, a preferred embodiment is provided, in which the stiffness analysis method of the spatial multi-arm robot considering flexible joints in a closed chain configuration when there is no internal stress in S2 is implemented by determining the stiffness matrix of the kth robotic arm in a state without pre-internal force.

[0014] Furthermore, a preferred embodiment is provided, wherein the method for determining the stiffness matrix of the k-th robotic arm in a state without pre-internal force specifically includes:

[0015] Considering the elastic deformation of the flexible joint, the pose x based on the Combine Body CB Expressed as the generalized motor angle of the k-th robotic arm and the deformation of the k-th generalized joint spring of the robotic arm Function:

[0016] x CB =f(Θ k ,ε k ) (twenty one)

[0017] Taking the differential on both sides we get:

[0018]

[0019] Where, Used to describe the posture changes of Combine Body. is the vector formed by the differential of the angle variable of the generalized joint motor of the k-th robot arm, is the vector formed by the differential deformation of the joint hinge on the kth robotic arm, δε0=0 6×1 , the kinematic Jacobian matrix of the k-th robotic arm is Respectively expressed as the generalized joint motor speed and the deformation speed of the flexible joint spring Combine Body speed the impact of;

[0020]

[0021] Where: is the torque column vector of the flexible hinge on the k-th robotic arm, where is the torque column vector on the i-th flexible hinge on the k-th robotic arm, δε k is the deformation of the joint spring;

[0022] The control torque applied by the joint motor is

[0023]

[0024] Where, To control the equivalent stiffness;

[0025] When Combine Body is subjected to external force f e When it works, the Combine Body produces δx at the end CB The corresponding joint change is (δΘ, δε), and the virtual work done by the external force is equal to the sum of the virtual work done by the internal force of the mechanism:

[0026]

[0027] Substituting (22) into (25) we obtain

[0028]

[0029] Equation (26) holds true for any virtual displacement, thus the static equilibrium equation can be obtained:

[0030]

[0031] Substituting (23), (24) and (27) into (22) yields

[0032]

[0033] Further we get:

[0034] f e =K k δx CB (29)

[0035] Where, is the stiffness matrix of the kth robotic arm in the state without pre-internal force.

[0036] Furthermore, a preferred embodiment is provided, wherein the Combine Body dynamics equation is obtained using the Newton-Euler formula.

[0037] Furthermore, a preferred embodiment is provided, wherein a stiffness analysis method of a spatial multi-arm robot with flexible joints in a closed chain configuration is performed by determining the stiffness of the space multi-arm robot with pre-loaded internal force f I The system stiffness matrix under action accomplish.

[0038] Furthermore, a preferred embodiment is provided, by determining the pre-internal force f I The system stiffness matrix under action The specific implementation method is:

[0039] Define the potential energy function And taking the differential of each term in (27) we can obtain:

[0040]

[0041] Where, is the Hessian matrix of the potential energy function ζ, where

[0042] Solving equation (31) yields

[0043]

[0044] Where,

[0045] Substitute (32) into (22) and we get

[0046]

[0047] Where,

[0048]

[0049] Where, For the pre-loaded internal force f I The stiffness matrix of the k-th robotic arm in the case of is the stiffness coefficient.

[0050] Further, a preferred embodiment is provided, wherein There is a pre-loaded internal force f I The stiffness matrix of the k-th manipulator in this case is given by f I The direction and size of the system are determined by the system configuration.

[0051] Solution 2: A collaborative control system for a space flexible multi-arm robot utilizing stress stiffening effect, the system comprising:

[0052] The model building module is used to establish the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering the flexibility of the joints based on the two topological configurations of the spatial multi-arm robot;

[0053] The stiffness analysis module is used to introduce the internal force stiffening effect into the dynamic models of the open chain and closed chain states described in the model building module, and to determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration;

[0054] A collaborative control module is used to design a composite control law for the space multi-arm robot considering flexible joints described in the stiffness analysis module based on the singular perturbation method. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, to complete the collaborative control of the space flexible multi-arm robot with the internal force stiffening effect.

[0055] The present invention is beneficial in that:

[0056] The collaborative control method of a spatial flexible multi-arm robot utilizing the stress stiffening effect described in the present invention improves the system stiffness from both the dynamics and control perspectives by combining the internal force interaction mechanism and coordinated control algorithm under a closed chain configuration, in order to reduce the control error caused by inaccurate modeling of joint flexibility and dynamics during docking and capture.

[0057] The present invention proposes a design scheme for a fast locking and releasing device for a space multi-arm robot. The device can switch between open chain and self-closing chain structures to improve the robot's adaptability in different tasks.

[0058] This paper introduces the "stress stiffening" effect from material mechanics to establish dynamic models of a spatial multi-arm robot in both open-chain and closed-chain states, taking into account joint flexibility. This effect describes the relationship between internal stress and lateral stiffness. It demonstrates that by changing the internal stress distribution within the closed-chain system of the spatial multi-arm robot, the system stiffness can be improved from a dynamic perspective.

[0059] Based on singular perturbation theory, this paper designs a composite control law for a spatial multi-arm robot that takes joint flexibility into account. This control law operates on both slow and fast time scales, taking into account both capture compliance and suppressing vibrations in the flexible joints, thereby improving the system's equivalent stiffness at the control level.

[0060] The present invention is also applicable to the field of improving control accuracy during track service and track capture. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1Schematic diagrams of two topological configurations of the spatial multi-arm robot described in embodiment one.

[0062] Figure 1 In the figure, (a) is a schematic diagram of the topological configuration of the open chain state, and (b) is a schematic diagram of the topological configuration of the closed chain state;

[0063] Figure 1 In the figure, 1 is the base, 2 is the No. 1 robot arm, 3 is the cone-rod docking mechanism, 4 is the target, 5 is the quick-install and quick-change device, and 6 is the No. 2 robot arm.

[0064] Figure 2 This is a schematic diagram of the open-chain coordinate system definition described in Implementation 11.

[0065] Figure 3 This is a schematic diagram of the open-chain coordinate system definition described in Implementation 11.

[0066] Figure 4 This is a schematic diagram of the flexible joint model described in embodiment 11.

[0067] Figure 5 Schematic diagram of the interaction force in the closed chain state according to the eleventh embodiment.

[0068] Figure 6 Schematic diagram of the terminal deformation under stress stiffening described in the eleventh embodiment.

[0069] Figure 7 This is the original block diagram of the spatial flexible multi-arm robot collaborative control system utilizing the internal force stiffening effect as described in Implementation Example 8.

[0070] Figure 8 This is a schematic diagram of the simulation process described in the eleventh embodiment.

[0071] Figure 8 In the figure, (a) is a simulation diagram of the open chain control stage at 0-5s, (b) is a simulation diagram of the closed chain formation stage at 5-10s, (c) is a simulation diagram of the closed chain docking stage at 10-15s, and (d) is a simulation diagram of the assembly control stage at 15-30s.

[0072] Figure 9 Schematic diagram of the expected trajectory of the end of the robotic arm in the open chain stage described in embodiment eleven.

[0073] Figure 9 In the figure, (a) is a schematic diagram of the desired position trajectory of the end of the robotic arm, and (b) is a schematic diagram of the desired posture trajectory of the end of the robotic arm.

[0074] Figure 10 This is a schematic diagram of the expected trajectory of the CombinedBody end in the closed-loop chain stage described in the eleventh embodiment.

[0075] Figure 10 In the figure, (a) is a schematic diagram of the expected position trajectory of CombinedBody, and (b) is a schematic diagram of the expected posture trajectory of CombinedBody.

[0076] Figure 11 Schematic diagram comparing the terminal error results of Arm_1 and Arm_2 under two controllers in the open chain stage described in embodiment eleven.

[0077] Figure 11 In the figure, (a) is a schematic diagram of the position error in the x direction of the end of Arm_1, (b) is a schematic diagram of the position error in the x direction of the end of Arm_2, (c) is a schematic diagram of the position error in the y direction of the end of Arm_1, (d) is a schematic diagram of the position error in the y direction of the end of Arm_2, (e) is a schematic diagram of the position error in the z direction of the end of Arm_1, and (f) is a schematic diagram of the position error in the z direction of the end of Arm_2.

[0078] Figure 12 Schematic diagram of the control error of CombinedBody in the closed chain state according to the eleventh embodiment.

[0079] Figure 12 In the figure, (a) is a schematic diagram of the Combined Body x-direction position error, (b) is a schematic diagram of the Combined Body quaternion error q1, (c) is a schematic diagram of the Combined Body quaternion error q2, (e) is a schematic diagram of the Combined Body z-direction position error, and (f) is a schematic diagram of the Combined Body quaternion error q3.

[0080] Figure 13 This is a schematic diagram of the joint angle of the robotic arm when the compressive stress is 300N as described in the eleventh embodiment.

[0081] Figure 13 In the figure, (a) is the connecting rod rotation angle q of Arm_1 1 Schematic diagram, (b) is the rotation angle θ of the Arm_1 motor 1 Schematic diagram, (c) is the Arm_2 connecting rod angle q 2 Schematic diagram, (d) is the rotation angle θ of the Arm_2 motor 2 Schematic diagram.

[0082] Figure 14 This is a schematic diagram of the output torque τ of the robotic arm when the compressive stress is 300N as described in the eleventh embodiment.

[0083] Figure 14 In the figure, (a) is the output torque τ of the Arm_1 motor1 Schematic diagram, (b) is the output torque τ of the Arm_2 motor 2 Schematic diagram.

[0084] Figure 15 This is a schematic diagram of the Base pose error described in the eleventh embodiment.

[0085] Figure 15 In the figure, (a) is a schematic diagram of the Base x-direction position error, (b) is a schematic diagram of the Base attitude quaternion error q1, (c) is a schematic diagram of the Base y-direction position error, (d) is a schematic diagram of the Base attitude quaternion error q2, (e) is a schematic diagram of the Base z-direction position error, and (f) is a schematic diagram of the Base attitude quaternion error q3.

[0086] Figure 16 This is a physical diagram of the ground experiment setup of the space multi-arm robot described in Implementation Method 11.

[0087] Figure 17 This is a schematic diagram of a physical simulation verification based on a ground experiment of a space multi-arm robot for the verification of the eleventh embodiment.

[0088] Figure 17 In the figure, (a) is a simulation diagram of the movement from the open-chain configuration to the docking state at 0s, (b) is a simulation diagram of the movement from the open-chain configuration to the docking state at 200s, (c) is a simulation diagram of the movement from the open-chain configuration to the docking state at 255s, and (d) is a simulation diagram of the movement from the open-chain configuration to the docking state at 375s.

[0089] Figure 18 Schematic comparison of the spatial distance error curves of the end of the robotic arm moving under load as described in the eleventh embodiment.

[0090] Figure 19 This is a schematic diagram of the change of internal force and torque over time in the self-closed chain configuration formed by the space robot through topology change as described in the eleventh embodiment.

[0091] Figure 19 In the figure, (a) is a schematic diagram of the change of internal force in the closed chain configuration with time, and (b) is a schematic diagram of the change of torque in the closed chain configuration with time. DETAILED DESCRIPTION

[0092] In order to make the purpose, technical solutions and advantages of the implementation methods of this application clearer, the technical solutions in the implementation methods of this application will be clearly and completely described below in combination with the drawings in the implementation methods of this application. Obviously, the described implementation methods are only part of the implementation methods of this application, not all of the implementation methods.

[0093] Embodiment 1: This embodiment provides a method for collaborative control of a space flexible multi-arm robot using a stress stiffening effect, the method comprising the following steps:

[0094] S1. Based on the open chain and self-closed chain topological configurations of the spatial flexible multi-arm robot, the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering joint flexibility are established respectively;

[0095] S2. Introduce the internal force stiffening effect into the dynamic model of the open chain and closed chain states described in S1, and determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration;

[0096] S3. Based on the singular perturbation method, a composite control law is designed for the space multi-arm robot considering flexible joints described in S2. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, collaborative control of the space flexible multi-arm robot that completes the internal force stiffening effect.

[0097] Implementation method 2. This implementation method is a further limitation of the collaborative control method of the space flexible multi-arm robot utilizing the stress stiffening effect described in implementation method 1. In S2, the stiffness analysis method of the space multi-arm robot considering the flexible joints in the closed chain configuration when there is no internal stress is realized by determining the stiffness matrix of the kth robotic arm in the state without pre-internal force.

[0098] Implementation 3: This implementation further limits the collaborative control method of the spatial flexible multi-arm robot using the stress stiffening effect described in Implementation 2. The method for determining the stiffness matrix of the k-th robotic arm in the state without pre-internal force specifically includes:

[0099] Considering the elastic deformation of the flexible joint, the pose x based on the Combine Body CB Expressed as the generalized motor angle of the k-th robotic arm and the deformation of the k-th generalized joint spring of the robotic arm Function:

[0100] x CB =f(Θ k ,ε k ) (twenty one)

[0101] Taking the differential on both sides we get:

[0102]

[0103] Where, Used to describe the posture changes of Combine Body. is the vector formed by the differential of the angle variable of the generalized joint motor of the k-th robot arm, is the vector formed by the differential deformation of the joint hinge on the kth robotic arm, δε0=0 6×1 , the kinematic Jacobian matrix of the k-th robotic arm is Respectively expressed as the generalized joint motor speed and the deformation speed of the flexible joint spring Combine Body speed the impact of;

[0104]

[0105] Where: is the torque column vector of the flexible hinge on the k-th robotic arm, where is the torque column vector on the i-th flexible hinge on the k-th robotic arm, δε k is the deformation of the joint spring;

[0106] The control torque applied by the joint motor is

[0107]

[0108] Where, To control the equivalent stiffness;

[0109] When Combine Body is subjected to external force f e When it works, the Combine Body produces δx at the end CB The corresponding joint change is (δΘ, δε), and the virtual work done by the external force is equal to the sum of the virtual work done by the internal force of the mechanism:

[0110]

[0111] Substituting (22) into (25) we obtain

[0112]

[0113] Equation (26) holds true for any virtual displacement, thus the static equilibrium equation can be obtained:

[0114]

[0115] Substituting (23), (24) and (27) into (22) yields

[0116]

[0117] Further we get:

[0118] f e =K k δx CB (29)

[0119] Where, is the stiffness matrix of the kth robotic arm in the state without pre-internal force.

[0120] Implementation method 4: This implementation method further limits the collaborative control method of the spatial flexible multi-arm robot using the stress stiffening effect described in implementation method 3. The Combine Body dynamics equation is obtained using the Newton-Euler formula.

[0121] Implementation 5. This implementation is a further limitation of the collaborative control method of the space flexible multi-arm robot using the stress stiffening effect described in Implementation 1. When there is an internal force in S2, the stiffness analysis method of the space multi-arm robot considering the flexible joints in the closed chain configuration is determined by determining the pre-internal force f I The system stiffness matrix under action accomplish.

[0122] Implementation 6: This implementation is a further limitation of the collaborative control method of the space flexible multi-arm robot using the stress stiffening effect described in Implementation 5. By determining the pre-internal force f I The system stiffness matrix under action The specific implementation method is:

[0123] Define the potential energy function And taking the differential of each term in (27) we can obtain:

[0124]

[0125] Where, is the Hessian matrix of the potential energy function ζ, where

[0126] Solving equation (31) yields

[0127]

[0128] Where,

[0129] Substitute (32) into (22) and we get

[0130]

[0131] Where,

[0132]

[0133] Where, For the pre-loaded internal force f I The stiffness matrix of the k-th robotic arm in the case of is the stiffness coefficient.

[0134] Implementation 7: This implementation is a further limitation of the collaborative control method of the space flexible multi-arm robot using the stress stiffening effect described in Implementation 5. There is a pre-loaded internal force f I The stiffness matrix of the k-th manipulator in this case is given by f I The direction and size of the system are determined by the system configuration.

[0135] Embodiment 8: This embodiment proposes a spatial flexible multi-arm robot collaborative control system utilizing stress stiffening effect, the system comprising:

[0136] The model building module is used to establish the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering the flexibility of the joints based on the two topological configurations of the spatial multi-arm robot;

[0137] The stiffness analysis module is used to introduce the internal force stiffening effect into the dynamic models of the open chain and closed chain states described in the model building module, and to determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration;

[0138] A collaborative control module is used to design a composite control law for the space multi-arm robot considering flexible joints described in the stiffness analysis module based on the singular perturbation method. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, to complete the collaborative control of the space flexible multi-arm robot with the internal force stiffening effect.

[0139] Implementation method 9. This implementation method proposes a computer device including a memory and a processor, wherein the memory stores a computer program. When the processor runs the computer program stored in the memory, the processor executes the method described in any one of implementation methods 1 to 7.

[0140] Implementation 10: This implementation proposes a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, the steps of the method described in any one of Implementation 1 to Implementation 7 are implemented.

[0141] Implementation 11: This implementation provides an example, which is used to explain the above implementations 1 to 8. Specifically, the example is as follows:

[0142] See also Figures 1 to 19 In this embodiment, the space multi-arm robot capture scene consists of two parts: the space multi-arm robot and the target, which is a satellite. The target is used to simulate a communication satellite in geosynchronous orbit. The space dual-arm robot uses two seven-degree-of-freedom manipulator arms to approach the target spacecraft in an open chain configuration. Figure 19 (a) A self-closing chain configuration is formed above the target, and the nozzle of the target spacecraft is docked and fixed using the docking cone rod, as shown in Figure 19 (b) shown.

[0143] The space multi-arm robot consists of four parts: a base, a quick locking and releasing device, two 7-DOF robotic arms, and a cone-rod docking device. Figure 2 The base is a cube satellite with a sailboard, which has the ability to change attitude and orbit, and is controlled by the attitude and orbit control system. It is a controlled space flight robot.

[0144] like Figure 3 and Figure 4 As shown, the center-of-mass coordinate system ∑0 of the spatial multi-arm robot is flying freely in the inertial coordinate system ∑I. The generalized coordinates of the center-of-mass coordinate system ∑0 can be expressed as Where r0∈R 3 represents the position vector of the matrix in the inertial system, Indicates the posture of the base coordinate system relative to the inertial system, expressed as the Euler angle of the zyx rotation order. In general, assuming that the space robot system has N manipulators, where the kth (k = 1, ..., N) manipulator has n degrees of freedom k , that is, with n k The base is represented as the 0th rigid body, and the path direction from the base to the end is defined as the positive direction, and the i-th axis Connected to the i-th link body, represents the joint node connecting link i-1 and link i in arm k, where Represents the root joint node of arm k. According to graph theory, node Is the parent node of the i rigid body, node It is the child node of rigid body i.

[0145] like Figure 3 and Figure 4 As shown, the definition is the coordinate system of the center of mass of rod i located at arm k; represents the center of mass of the i-th rod of arm k; Indicates connection arrive vector of Indicates connection arrive vector of Indicates connection arrive A vector of Indicates the connection base centroid to vector.

[0146] 1. Dynamic modeling of a spatial dual-arm robot with flexible joints in an open-chain state

[0147] use Represents the position vector of the end of the robot arm k in the inertial system, using r0∈R 3 It represents the position vector of the center of mass of the base in the inertial system. use express The antisymmetric matrix of .

[0148] Derived to obtain the projection of the kth robotic arm end velocity in the inertial system The conversion relationship between the k-joint velocity of the robot arm is:

[0149]

[0150] Where, Represents the projection of the position and attitude vector of the base body's center of mass in the inertial system. represents the base Jacobian matrix for the end of the k-th robotic arm. represents the Jacobian matrix of the k-th robotic arm end, where represents the Jacobian matrix of the linear velocity at the end of the k-th robotic arm, Represents the Jacobian matrix of the angular velocity of the k-th robotic arm. represents the joint velocity column vector of the k-th robotic arm, represents the velocity of the i-th joint of the robot arm k.

[0151] The velocity of the floating base in the inertial system The end velocity of all manipulators is combined Expressed in one equation, we can get

[0152]

[0153] Where, the velocity set in the joint space is expressed as The Jacobian matrix of the floating base space robot represents the mapping between the base velocity and joint velocity in the joint space and the base velocity and the end-arm velocity in the Cartesian space.

[0154] The dynamic model of the space robot is derived using the Lagrange method, and L is the Lagrange function. is the generalized force vector corresponding to the non-conservative active force of the system

[53] , T is kinetic energy, V is potential energy, and the generalized coordinates of the system are So the Lagrange function is

[0155] L=TV (3)

[0156] Taking the derivative of formula (3) with respect to time, we can get

[0157]

[0158] The total kinetic energy of the space multi-arm robot system is the sum of the kinetic energy of the base and the kinetic energy of each robotic arm.

[37] :

[0159]

[0160] Where, is the generalized inertia matrix, H0∈R 6 × 6 is the floating matrix inertia matrix, is the inertia matrix of the robot arm link, is the coupling term between the matrix and connecting rod moment of inertia.

[0161] The generalized force Q is mainly composed of the external forces / torques on multiple end effectors according to the external forces on the space multi-arm robot system. composition, When the base body is performing attitude and orbit control, the projection of the control force / torque in the inertial system is: represents the projection of the external force on the end of the k-th robotic arm in the inertial system. Q can be expressed as follows:

[0162]

[0163] This implementation is the flexibility of the robot joints, which is mainly caused by the harmonic reducer and torque sensor.

[27] Assuming that , the flexible joint can be simplified into a linear spring with neglected mass, such as Figure 5 shown.

[0164] Flexible joint dynamics model:

[0165]

[0166] Where, Represents the inertia matrix of the motor rotor. represents the main inertia of the i-th motor rotor of the k-th robotic arm, represents the motor angle column vector, represents the generalized torsional stiffness coefficient matrix of the joint, where represents the torsional stiffness coefficient of the i-th joint of the k-th robotic arm, Control torque of the robot joint.

[0167] Furthermore, since the matrix does not contain elastic elements and moves freely in 6-dimensional space, consider adding a 6-dimensional virtual flexible joint with a stiffness of 0 to the matrix. The potential energy V is written in matrix form:

[0168]

[0169] make represents the generalized motor angle of the system, represents the generalized torsional stiffness coefficient of the system. Then Equation (8) can be expressed by generalized coordinates express

[0170]

[0171] Substituting equations (5), (9) and (6) into (4), we can obtain the dynamic model of the spatial multi-arm robot with flexible joints:

[0172]

[0173] Where, is the inertia force term, is a nonlinear term, is the Coriolis force, For centrifugal force.

[0174] In order to unify the dimensions of equations (7) and (10), we can replace Represents the main inertia of the generalized motor rotor shaft of the system, represents the system generalized motor control torque, then (7) can be rewritten as

[0175]

[0176] Combining equations (10) and (11), we can obtain the dynamic equations of the spatial multi-arm robot with flexible joints in the open chain state:

[0177]

[0178] In the capture task, we are more concerned about the dynamic characteristics in the operation space, so (12) is rewritten as the dynamic equation in the operation space,

[0179]

[0180] Where, represents the system mass matrix in the operating space, represents the Coriolis force and centrifugal force matrix in the operating space, Represents the generalized coordinate column vector composed of the base body pose and each manipulator end pose in the operation space.

[0181] 2. Dynamic modeling of closed chain state of spatial multi-arm robot considering flexible joints

[0182] After forming a self-closed chain structure, such as Figure 1 As shown in (b), it can be considered that the i-th robot arm and the j-th robot arm are constrained together on a rigid body, which is called CombinedBody. The motion state of CombinedBody is determined by the ends of the two robot arms. According to Assumption 1, there is no relative velocity between the two constrained objects. Represents the projection of the combined body's center of mass velocity in the inertial system, and There are the following relationships

[0183]

[0184] Where, Represents the set of end velocities of the robot arm in the closed chain state. It represents the conversion matrix of the end velocity of the contact point between each manipulator and CombinedBody and the velocity of the center of mass of CombinedBody in the closed chain state. Usually, T Ce It is called the contact matrix. is a vector The antisymmetric matrix of .

[0185] For each end of the robot connected to the CombinedBody, there is a velocity constraint, that is, the relative velocity is 0. Taking the first and second robot arms connected to the CombinedBody as an example, the following kinematic constraints are satisfied between them:

[0186]

[0187] in is the constraint Jacobian matrix between manipulators i and j, where yes The antisymmetric matrix of .

[0188] Introducing the constraints of the base to construct a more complete system constraint equation, Equation (15) can be written as

[0189]

[0190] in, represents the generalized constraint matrix of the system, represents the Jacobian matrix J G The pseudo-inverse of . Equation (16) represents the velocity constraint relationship of the space robot terminal when the system is in a closed chain state.

[0191] The end of the k-th robotic arm interacts with the Combined Body at the constraint point It is equivalent to the end contact force in the open chain state, which is the The two are action and reaction forces. The combined body dynamics equation can be obtained using the Newton-Euler formula:

[0192]

[0193] Among them, H CB ∈R 6×6 Represents the mass matrix of CombinedBody, C CB ∈R 6 represents the combined body centrifugal force and Coriolis term, f CB ∈R 6 Represents the resultant force applied to CombinedBody, F CBext ∈R 6 Indicates the resultant force of the external forces acting on the CombinedBody.

[0194] f CB It is the multiple contact forces applied by multiple arms to the CombinedBody when the two arms form a closed chain structure. The control force formed together satisfies

[0195]

[0196] in, express If only one robotic arm is operating, a closed chain state cannot be formed. In the case of one-to-one mapping of joint torque and force projections, the inverse dynamics has a unique solution.

[0197] F ce The other part generates the contact internal force f of the object I ∈R 6N , f I They exist in pairs, with a net force of 0, and are used to generate stable internal forces, such as clamping forces. Introducing the null space matrix

[39] , F ce Can be obtained by f cb and f I express:

[0198]

[0199] in, T ce The null space matrix is ​​represented as follows:

[0200]

[0201] Since the null space matrix N(T ce ) exists, we can I Design and distribute, and f I It will not affect the control effect of the Combined Body in the inertial system.

[0202] Using the QLRD device, a spatial multi-arm robot can form a self-closed chain configuration, generating internal forces within the chain while maintaining control forces on the end points. This introduces the concept of "stress stiffening," which describes the relationship between internal forces and lateral stiffness in a structure. Analogous to a guitar string, the tighter the string, the greater the axial tension, the greater the lateral stiffness. This section focuses on the relationship between internal forces and stiffness within the chain of a spatial multi-arm robot with flexible joints, as it forms a self-closed chain for capture.

[0203] 3. Stiffness Analysis of a Spatial Multi-Arm Robot Considering Flexible Joints in a Self-Closed Chain Configuration Without Internal Forces

[0204] Considering the elastic deformation of the joints, the pose x of the Combine Body CB It can be expressed as the generalized motor angle of the k-th robotic arm and the deformation of the k-th generalized joint spring of the robotic arm Function:

[0205] x CB =f(Θ k ,ε k ) (twenty one) Taking the differential on both sides we get:

[0206]

[0207] Where: Used to describe the posture changes of Combine Body. is the vector formed by the differential of the angle variable of the generalized joint motor of the k-th robot arm, is the vector formed by the differential deformation of the joint hinge on the kth robot arm. Since the spring on the base is a virtual hinge, δε0=0 6×1 The kinematic Jacobian matrix of the k-th robotic arm is Describes the generalized joint motor speed and the deformation speed of the flexible joint spring CombineBody Speed impact.

[0208] Since the deformation of the flexible joint is usually very small (the initial deformation is 0), according to the spring model mentioned above, the relationship between deformation and torque can be expressed as

[0209]

[0210] Where: is the torque column vector of the flexible hinge on the k-th robotic arm, where is the torque column vector on the i-th flexible hinge on the k-th robotic arm, δε k is the deformation of the joint spring.

[0211] The control torque applied by the joint motor (active hinge) is

[0212]

[0213] Where, To control the equivalent stiffness.

[0214] When Combine Body is subjected to external force f e When the action is performed, due to the existence of flexible factors, the end of the Combine Body generates δx CB The corresponding joint change is (δΘ, δε). According to the principle of virtual work, the virtual work done by the external force is equal to the sum of the virtual work done by the internal force of the mechanism:

[0215]

[0216] Substituting (22) into (25) we obtain

[0217]

[0218] Equation (26) holds true for any virtual displacement, thus the static equilibrium equation can be obtained:

[0219]

[0220] Substituting (23), (24) and (27) into (22) yields

[0221]

[0222] Further we get:

[0223] f e =K k δx CB (29) Where, is the stiffness matrix of the k-th manipulator in the state without pre-internal force, which reflects the influence of the elasticity of the k-th manipulator on the end Combinebody posture under the action of external force.

[0224] Consider that all the manipulator branches are connected to the same Combinebody and thus have the same pose change δx CB The stiffness matrix of the entire system without pre-internal force only needs to be K k Add up to get

[0225]

[0226] 4. Stiffness Analysis of a Spatial Multi-Arm Robot Considering Flexible Joints in a Self-Closed Chain Configuration with Internal Forces

[0227] After applying internal force f to the Combine body I Under the premise of , assuming that an external force δf is applied to the Combine body at this time, the Combine body produces a transformation of δq, satisfying For the pre-loaded internal force f I The system stiffness matrix under the action.

[0228] According to the principle of reciprocity between strain energy and work

[40] , define the potential energy function And taking the differential of each term in (27) we can obtain:

[0229]

[0230] Where, is the Hessian matrix of the potential energy function ζ, where

[0231] Solving equation (31) we can get Where,

[0232] Substitute (32) into (22) and we get

[0233]

[0234] Where,

[0235]

[0236] Where, For the pre-loaded internal force f I The stiffness matrix of the k-th robotic arm in the case of is the stiffness coefficient. It can be found that the stiffness matrix under pre-internal force is affected These parameters further depend on f I The direction and magnitude of the system configuration. This proves that the pre-internal force f I Changing the stiffness of a spatial multi-arm robot system considering joint flexibility.

[0237] Consider that all robot branches are connected to the same Combine Body and thus have the same pose change δx CB , the stiffness matrix of the entire system with pre-internal force only needs to be Add up to get

[0238]

[0239] Although the stiffness matrix expression is very complex, by analyzing (34), it is found that the stiffness With coefficient and Negatively correlated, and and Positive correlation. At the same time, through (31) it can be found that when f I When it is orthogonal to the impact displacement δf, the modulus of ζ has a maximum or minimum value. This shows that f I The impact on the Combine Body's stiffness in the orthogonal direction is the greatest, presenting two effects: "stress stiffening" and "stress softening," similar to the tension and relaxation of a guitar string. When designing a controller, the Combine Body can be subjected to the impact force f e The direction of f is designed and controlled according to (19) I The size and direction of the impact direction are adjusted to improve the stiffness in the impact direction, thereby improving the control accuracy in the capture process from a dynamic level under the condition of model uncertainty (including uncertainty of the spatial dual-arm robot itself and missing target inertia parameters).

[0240] Because the dynamic model of a spatial multi-arm robot with flexible joints is a high-order system, direct control is difficult. Using singular perturbation methods, the flexible joint model is reduced to a low-order system, decomposing the complex high-order system into a slow subsystem and a fast subsystem, describing the large-scale rigid body motion and the elastic motion of the joints, respectively. The reduced-order slow subsystem is equivalent to a rigid-body model, while the fast subsystem is a linear system. Therefore, control algorithms can be designed for each of these two subsystems to ensure tracking accuracy and quickly suppress the chattering of the flexible arm, providing an effective control solution.

[0241] Let the flexible joint torque Then (12) can be rewritten as:

[0242]

[0243] According to the singular perturbation theory, the state variable z1 = τ is selected. Based on (36), the state space equation is obtained

[0244]

[0245] in, is called the perturbation parameter, which is a standard form of a singular perturbation system. It can be interpreted as a two-time-scale system with state variables x = [x1, x2] T is changing slowly, and the state variable z=[z1,z2] T It changes rapidly, thus dividing the system into two subsystems: fast and slow.

[0246] The kinetic equation of state at slow time scales:

[0247]

[0248] (38) is the form of the dynamic state equation of a space robot under rigid conditions.

[0249] Fast time scale t f Next f =(tt s ) / μ, the “boundary layer” system of the slow manifold about z can be defined as In this boundary layer system, by letting μ approach zero, a reduced-order model of the fast system can be obtained.

[0250]

[0251] in,

[0252] According to the derivation of the reduced-order model, the control input can be divided into a fast part and a slow part. A dual-time-scale composite control law is used to apply to the rigid dynamics part and the flexible joint dynamics part of the spatial multi-arm robot respectively. The control input τ can be expressed as follows:

[0253]

[0254] Use Slow Control To control a slow system, use fast control To control fast systems, the original fourth-order control problem is decomposed into two second-order problems, which are processed at different time scales.

[0255] 0.3 Slow part control law

[0256] 5. Impedance control in open-link state

[0257] definition Then the dynamic equation (13) in the operation space is written as:

[0258]

[0259] Among them, F τ Represents the end-operation space force. The joint torque τ generated by the external force ext and external force F e satisfy

[0260] When a spatial dual-arm robot with flexible joints performs a capture operation on a target, it is hoped that it will be flexible in an unknown contact environment. That is, the robot should have a certain degree of compliance with the working environment to prevent damage to the robot body and the operated object.

[0261] The slow-speed reduced-order model of the spatial dual-arm robot considering flexible joints has the same form as the rigid dynamics model, so the controller developed for rigid manipulator control can be used for slow-speed control.

[0262] In order to realize the impedance characteristics of the end of the robot arm, the actual end position x∈R 6+6N and the virtual equilibrium point position x d ∈R 6+6N The error between The purpose of impedance control is to achieve position at the end With F e The dynamic relationship between:

[0263]

[0264] where Λ d ∈R (6+6N)×(6+6N) , D d∈R (6+6N)×(6+6N) and K d ∈R (6+6N)×(6+6N) are the desired inertia, damping, and stiffness matrices of the end of the manipulator and are symmetric and positive definite.

[0265] Substituting (42) into (41) yields:

[0266]

[0267] In order to avoid the external force feedback term in the control input of the closed-loop system, let Λ d =Λ(x), multiply both sides by The joint input torque based on the Cartesian spatial impedance control law is obtained when the slow part of the system is in an open chain state

[0268]

[0269] 4.1.2 Impedance control in closed-loop state

[0270] After forming the self-closing chain system, the cone rod device at the end of the Combined Body is used for docking. Therefore, it is hoped that the Combined Body will exhibit impedance characteristics. (42) is slightly modified.

[0271]

[0272] in represents the position error of the Combined Body, Λ CBd ∈R 6×6 , D CBd ∈R 6×6 and K CBd ∈R 6×6 They are the expected inertia, damping and stiffness matrices of CombinedBody, all of which are symmetric positive definite matrices.

[0273] Substitute (45) into the rigid body dynamics equation (17) of CombinedBody and let Λ CBd =H CB , and the control input force f is obtained CB ∈R 6 Expressed as

[0274]

[0275] f CB The contact force F of the end of the manipulator in contact with the CombinedBody in the space dual-arm robot ce Provide. At the same time, according to the previous research, a certain internal force f is applied I, can improve the stiffness of the end, reduce the displacement caused by the capture impact, and improve the control accuracy. According to formula (19), the expected contact force F of the end of the manipulator on the CombinedBody is calculated as ced , based on the action and reaction forces, the Combined Body's reaction forces on the ends of each robotic arm The desired end force F is quickly tracked by the robot controller ed , to achieve impedance control of the Combined Body end.

[0276] Based on the dynamic equation of the manipulator, the PI controller of the force is designed, and the control force / torque F of the spatial dual-arm robot is c :

[0277]

[0278] Among them F error =F ed -F e , K p and K I are the positive definite gain matrices of the proportional and integral links of the robot arm force in the closed chain state.

[0279] Multiply both sides of (47) by The joint input torque of the slow part of the system under the end Cartesian impedance control in the closed chain state and the PI control law of the manipulator spatial operation force is obtained.

[0280]

[0281] 0.4 Fast Partial Control Law

[0282] In the previous article, the closed-loop fast dynamics is approximated as a second-order system, and according to the literature

[42] A control law for the fast system is proposed, which controls the bandwidth while maintaining the stability of the boundary layer system and is designed based on the measurable flexible joint data. Therefore, the control input of the fast part of the system is

[0283]

[0284] Where, They are the state variables related to the fast subsystem proposed in the previous article. and are the gain coefficients of the fast control law. Since the dynamic equations of the flexible joints of the spatial multi-arm robot are consistent in both the open chain state and the closed chain state, the input of the fast control part is the same. Based on the idea of ​​singular perturbation, the control strategy of the spatial multi-arm robot with flexible joints in the open chain and closed chain states is considered, as shown in Figure 7 shown.

[0285] 0.5 simulation experiment

[0286] In order to verify the correctness of the "stress stiffening" effect and composite control algorithm of the spatial multi-arm robot considering flexible joints proposed in this paper, a simulation framework based on Matlab / Simulink was established. The planning and control algorithms of the spatial dual-arm robot were implemented in Matlab / Simulink, and the dynamic modeling of the robot was implemented in Smiscpae. A simulation experiment of the spatial dual-arm robot considering flexible joints to capture the unknown dynamic parameter space target was carried out, and the simulation results of four control strategies were compared, namely: (1) operating space impedance control law, represented by "IC" (Impedance Control); (2) composite control law based on singular perturbation, represented by "CC" for Compoundcontrol, in this state there is no internal force; (3) considering the existence of 300N compressive stress in the x direction of Combined Body in the closed chain state, i.e. f I =[0,0,0,300,0,0] T The composite control law is represented by “CC(CS)” Compressive stress; (4) Considering that there is a 100N tensile stress in the x direction of CombinedBody in the closed chain state, that is, f I =[0,0,0,-100,0,0] T The composite control law is represented by “CC(TS)” Tensile stress.

[0287] The spatial dual-arm robot described in this embodiment utilizes two identical seven-DOF manipulator arms, with improved DH parameters shown in Table 1. Arm_1's end effector is a tapered rod docking mechanism, which docks and locks with the target object's nozzle. Arm_2's end effector, the active end of the QLRD, docks and locks with the passive end of the QLRD located on link1_6 on Arm_1, forming a self-closing chain loop. At this point, link1_6, link1_7, the tapered rod docking mechanism, and the passive end of the QLRD on Arm_1 together form the previously mentioned Combined Body, which is controlled by the joint action of Arm_1 and Arm_2.

[0288] Table 1 Improved DH parameters of the space dual-arm robot

[0289]

[0290] (α i is the connecting rod angle, a i is the connecting rod length, θ i is the joint angle offset, d i is the connecting rod offset)

[0291] The mass inertia parameters of each body in the space dual-arm robot are shown in Table 2. In the dynamic parameters of the flexible joint, the system generalized motor rotor shaft inertia is set Generalized torsional stiffness of the system

[0292] Table 2 Mass and inertia parameters of the space dual-arm robot

[0293]

[0294] Set the initial pose parameters of each body. The default position unit is m, the pose is expressed in Euler angles, the default order is "ZYX", and the unit is radians. The initial pose of the center of mass of the base body is In the initial configuration, the end of Arm_1, that is, the end of the cone-rod docking device, is positioned as follows: The end of Arm_2, that is, the pose of the end of the QLRD active end is The initial center of mass pose of the target is The initial stage of the base body mass center velocity v0, joint angular velocity ω0, target body mass center velocity v target All are 0.

[0295] Table 3 Control system parameter settings

[0296]

[0297] The entire simulation process is as follows: 1. Open-chain control phase (0-5s): The dual-arm space robot moves based on the open-chain control strategy. During this process, the control base remains stable near the initial position, and the end of the cone rod of Arm_1 is moved above the nozzle of the target in preparation for docking. Simultaneously, the active end of the QLRD in Arm_2 is moved above the passive end of the QLRD in Arm_1 in preparation for docking. The expected trajectory is: the end of Arm_1 moves from the waypoint [-1.158, 0.108, -1.000, -0.503, 0, pi]]. T Move to the waypoint [-1.158,0.108,-1.000,-pi / 2,0,pi] T , the end of Arm_2 is at the waypoint [1.155,-0.108,-0.800,2.639,0,pi] TMove to the waypoint [-1.158,0.108,-1.000,pi / 2,0,pi] T 2. Self-closing chain formation stage (5-10s): Arm_2 and Arm_1 form a self-closing chain system and stabilize through the closing and locking of the QLRD jaws; 3. Self-closing chain docking stage (10-15s): Operation force f is performed based on impedance control and zero space. CB Calculation and internal force f I The end of the Combinedbody is positioned at [-1.158,0.108,-1.000,-pi / 2,0,pi] T Move to pose [-1.158,0.108,-1.000,-pi / 2,0,pi] T , docking with the target's nozzle and stabilizing; 4. Assembly control phase 15-30s: Under the joint control of Arm_1 and Arm_2, the target is pulled into the vicinity of the base based on the closed-loop state control strategy, and the posture changes from [-1.158, 0.108, -1.000, -pi / 2, 0, pi] T Return to the pose [-1.158,0.108,-1.000,-pi / 2,0,pi] T Simulate the on-orbit operation process. The desired position of the base body always remains at the initial position. The trajectory sequence x is obtained by interpolating the fifth-order polynomial between each position point. e , The expected pose trajectory of Arm_1 and Arm_2 in the open chain phase after interpolation and the expected pose trajectory of Combinedbody in the closed chain phase are as follows: Figure 9 and Figure 10 The simulation process is shown in Figure 8 shown.

[0298] Figure 11 The results show a comparison of the control errors of the two controllers for the end-points Arm_1 and Arm_2 during the open-chain control phase (0-10 seconds). The figure shows that during the 0-5 second period, the two arms are in independent motion, but at the 5th second, the error briefly increases due to interference caused by QLRD contact and locking. Compared to a controller using only impedance control, the controller using the composite control law achieves smaller control errors and faster convergence at the end-points of the manipulator. This is because the fast control component of the singular perturbation theory considers the motion state of the flexible joint and provides a suppression method, which can effectively reduce the end-point jitter amplitude and vibration frequency caused by joint flexibility, thereby improving control accuracy. At the same time, the slow control component of the composite control law retains the compliant characteristics of impedance control, resulting in a damped attenuation trend in the error curves of the two manipulators.

[0299] Depend on Figure 12It can be seen that after the closed chain is formed, CombinedBody is affected by Arm_1 and Arm_2 and moves along the -z axis along the desired trajectory. The attitude error is expressed using the error quaternion

[43] Description, its imaginary part is the posture error component of each axis. At the 15th second, the Combine Body completes docking with the target and moves in the positive direction of the z axis, pulling the target closer. It can be observed from the figure that after docking, the position error of CombinedBody in the z direction is larger than that in the x and y directions, reaching 0.03m. This is because the impact interference comes from the z direction. It can be observed that compared with the case of using the "IC" controller, when the "CC" control law is adopted, the end jitter of the Combine body caused by the docking impact and joint flexibility is smaller, thereby obtaining higher control accuracy. When considering the internal force of the closed chain structure formed by the two arms, under the same control parameters, when a pressure of 300N "CC(CS)" is applied, the end posture error caused by the docking impact is smaller than that without internal force, especially in the z direction of the impact, such as Figure 12 As shown in (e), the error was reduced by 0.004m and the z-direction accuracy was improved by 13.33%. This result verifies the "stress stiffening" effect mentioned above, that is, when sufficient pressure is provided in the axial direction, that is, the x-direction, the stiffness in the lateral z-direction, that is, the impact direction, can be increased. Similarly, under the "CC(TS)" condition, the "stress relaxation" effect was also observed. When a tensile force of 100N was applied, the lateral stiffness was significantly reduced, the control error increased by 0.002m compared to the case without internal force, the stiffness decreased by 6%, and vibration was obvious. If the value of internal stress is further increased, the "stress stiffening" or "stress relaxation" effect will appear more obviously, thus proving that it is feasible to use a closed chain structure formed by two arms to generate internal force and to use the direction and magnitude of the internal force to change the stiffness in a certain direction.

[0300] Figure 14 The figure shows the motor torque output of the robot arm when a 300N pressure is applied in the self-closing chain system. It can be observed that in the open chain state, the motor output torque is small from 0 to 10 seconds. After the self-closing chain is formed, the motor output torque increases after 10 seconds due to the need to provide an internal force of 300N. In particular, the Jiont3, Jiont4, and Jiont5 joints bear a greater load, reaching 200Nm. This is also to reflect the concept that the stiffness of the system can be increased when the internal stress is sufficient. Therefore, a larger internal force value is set in the simulation. However, when considering the pressure bearing capacity of the material, the range of the internal force needs to be reasonably designed.

[0301] Figure 15The figure shows the changes in the position and posture errors of the base during the entire simulation process. As the support of the two robotic arms, the base will be affected by the reaction force of the robotic arms, and its equivalent stiffness will also change due to the internal force. It can be observed from the figure that in the "CC (CS)" case with a pressure of 300N, compared with other cases, the position and posture errors of the base are smaller when affected by the dynamic coupling interference of the two robotic arms and the target object. In particular, the position error in the z direction is reduced by 0.004m compared to the case without internal force. This shows that the presence of sufficiently large internal forces in the closed chain system can make the structure have higher stiffness, thereby reducing the interference of external disturbances.

[0302] The ground experiment of the space multi-arm robot was built as follows Figure 16 As shown in the figure. We built a space robot with two 7-DOF manipulator arms on the experimental platform. Since the research focus of this paper is on the variable topology dynamics and control of the robot, an air-floating platform was not used to simulate the microgravity environment. The base of the space robot was fixed to the experimental platform. The end of the manipulator arm contains a QLRD device, which allows the robot to switch between open-chain and closed-chain configurations. The manipulator arm and the QLRD device are both driven by servo motors. The industrial controller uses the Ethercat protocol to transmit control instructions to the motor. An OptoForce Sensing System is also installed at the end of Arm_2 to measure the internal force / torque in the closed-chain configuration to verify the "stress stiffening" effect. Four Optitrack vision sensors were built around the experimental platform to measure the position of the end of the manipulator arm.

[0303] To verify the effectiveness of the proposed theory and algorithm, the following experimental conditions were designed: Similar to the simulation conditions, from 0 to 255 seconds, the two manipulators first moved to the docking state in an open-chain configuration. From 255 to 375 seconds, the manipulators were subjected to vertical motion with a 2.5 kg load at the end using (1) single-manipulator impedance control (IC(Single)) without docking, (2) closed-chain without internal force coordination (CC(Null)), and (3) closed-chain with internal force coordination (CC(CS)) to simulate docking and transfer motion in the Z direction.

[0304] Figure 19The figure shows the spatial distance error curves for the end-of-arm motion under load under three experimental conditions. There is a fluctuating error between the actual end-of-arm position and the desired motion position. The rise in the curve is due to the time-accumulated error caused by the deviation of the load under the influence of gravity. The fluctuation in the curve is due to the flexible vibration caused by the joints and assembly clearances. A comparison shows that the error accumulation is more significant and the vibration is faster in the single-arm configuration compared to the closed-chain configuration. The closed-chain configuration exhibits less error accumulation, and this phenomenon is even more pronounced in the presence of internal forces, reducing the error by approximately 0.002 m. Experimental results demonstrate that this method can be used for target capture by a spatial multi-arm robot. The proposed method achieves high accuracy. A composite control law based on singular perturbations splits the robot into slow and fast subsystems for control, while taking into account the capture compliance and the equivalent stiffness of the flexible joints. Simulations and experiments verify the effectiveness and correctness of the proposed method. Results show that, compared to a simple impedance control law, the composite control law based on singular perturbations can effectively reduce the vibration caused by joint flexibility at the end-of-arm, thereby reducing control error. In the closed chain state, by reasonably applying internal stress, the stiffness of the system can be increased and the control error caused by the impact during capture can be reduced.

[0305] Those skilled in the art will understand that the above description is only a preferred embodiment of the present invention, and the features described in the various embodiments and / or claims of the present disclosure may be combined or coupled in various ways, even if such a combination or coupling is not explicitly described in the present disclosure. It is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art may still modify the technical solutions described in the aforementioned embodiments or make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the scope of protection of the present invention.

[0306] Although preferred embodiments of the present invention have been described, those skilled in the art may make additional changes and modifications to these embodiments once they are aware of the basic inventive concepts. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention. Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, the present invention is intended to include such changes and modifications as fall within the scope of the claims and their equivalents.

Claims

1. A collaborative control method for a space flexible multi-arm robot utilizing stress stiffening effect, characterized in that: The collaborative control method of the space flexible multi-arm robot comprises the following steps: S1. Based on the open chain and self-closed chain topological configurations of the spatial flexible multi-arm robot, the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering joint flexibility are established respectively; S2. Introduce the internal force stiffening effect into the dynamic model of the open chain and closed chain states described in S1, and determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration; S3. Design a composite control law for the space multi-arm robot considering flexible joints described in S2 based on the singular perturbation method. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, collaborative control of the space flexible multi-arm robot with internal force stiffening effect. The stiffness analysis method of the spatial multi-arm robot considering flexible joints in the closed chain configuration when there is no internal stress in S2 is determined by determining the first The stiffness matrix of the manipulator is realized by this method; Determine the first The method of calculating the stiffness matrix of a robotic arm specifically includes: Considering the elastic deformation of flexible joints, the pose based on Combine Body Expressed as Generalized motor angle of the robot arm Hedi Deformation of the generalized joint spring of the robotic arm Function: （21） Taking the differential on both sides we get: （22） Where, , used to describe the posture changes of Combine Body, For the The vector formed by the differential of the generalized joint motor angle variables of the robot arm, For the The vector formed by the differential deformation of the joint hinge on the robot arm, , No. The kinematic Jacobian matrix of the robot arm is , Respectively expressed as the generalized joint motor speed and the deformation speed of the flexible joint spring Combine Body Speed the impact of; （23） Where: For the The torque column vector of the flexible joint on the manipulator, where For the The first robot arm The column vector of torque on the flexible hinge is, is the deformation of the joint spring; The control torque applied by the joint motor is （24） Where, To control the equivalent stiffness; When Combine Body is subjected to external force When activated, Combine Body ends The corresponding joint change is , the virtual work done by the external forces is equal to the sum of the virtual work done by the internal forces of the mechanism: （25） Substituting (22) into (25) yields （26） Equation (26) holds true for any virtual displacement, thus the static equilibrium equation can be obtained: （27） Substituting (23), (24) and (27) into (22) yields （28） Further we get: （29） Where, In the state without internal stress The stiffness matrix of the robot arm.

2. The collaborative control method of a space flexible multi-arm robot using stress stiffening effect according to claim 1 is characterized in that: The Combine Body dynamics equation is obtained using the Newton-Euler formula.

3. The collaborative control method of a space flexible multi-arm robot using stress stiffening effect according to claim 1, characterized in that: The stiffness analysis method of the spatial multi-arm robot considering flexible joints in the closed chain configuration when there is internal force in S2 is obtained by Determine if there is internal stress The system stiffness matrix under action accomplish.

4. The method for collaborative control of a space flexible multi-arm robot using stress stiffening effect according to claim 3, characterized in that: By determining the internal stress The system stiffness matrix under action The specific implementation method is: Define the potential energy function , and taking the differentials of each term in (27) we get: （30） Where, , , , is the potential energy function The Hessian matrix of ; Solving equation (30) yields （31） Where, , ; Substitute (31) into (22) and we get （32） Where, （33） Where, For internal stress The situation The stiffness matrix of the robot arm, , is the stiffness coefficient.

5. The collaborative control method of a space flexible multi-arm robot utilizing stress stiffening effect according to claim 4 is characterized in that: described There is internal stress The situation The stiffness matrix of the manipulator is given by The direction and size of the system are determined by the system configuration.

6. A collaborative control system for a space flexible multi-arm robot utilizing stress stiffening effect, characterized in that: The system is implemented based on the collaborative control method according to any one of claims 1 to 5, and the system includes: The model building module is used to establish the dynamic models of the spatial multi-arm robot in the open chain and self-closed chain states considering the flexibility of the joints based on the two topological configurations of the spatial multi-arm robot; The stiffness analysis module is used to introduce the internal force stiffening effect into the dynamic models of the open chain and closed chain states described in the model building module, and to determine whether there is internal stress to complete the stiffness analysis of the spatial multi-arm robot considering flexible joints in the closed chain configuration; A collaborative control module is used to design a composite control law for the space multi-arm robot considering flexible joints described in the stiffness analysis module based on the singular perturbation method. The composite control law includes collaborative control of the stiffness of the space flexible robot system at both slow and fast time scales, that is, to complete the collaborative control of the space flexible multi-arm robot with the internal force stiffening effect.

7. A computer device comprising a memory and a processor, characterized in that A computer program is stored in the memory. When the processor runs the computer program stored in the memory, the processor executes the method according to any one of claims 1 to 5.

8. A computer-readable storage medium, characterized in that The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 5 are implemented.

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