Vibration suppression method for cooperative operation of clamping flexible beam by multiple mechanical arms

Abstract

The invention discloses a vibration suppression method for cooperative operation of multiple mechanical arms clamping a flexible beam, belongs to the technical field of mechanical arm on-orbit service control, and aims to solve the problems of consistency and vibration suppression of the multiple mechanical arms clamping the flexible beam in the space station on-orbit service task process. A distributed passive controller based on speed feedback of the tail end of the flexible beam is designed, real-time vibration information of the flexible beam is measured through a visual sensor, the deflection change rate of the tail end of the flexible beam is fed back to the distributed passive controller, and therefore vibration of the tail end of the flexible beam is restrained in the process of achieving consistency of multiple mechanical arms through the controller.

Description

Technical Field

[0001] The invention belongs to the technical field of on-orbit service control of robotic arms, and in particular relates to a vibration suppression method for collaborative operation of multiple robotic arms clamping flexible beams. Background Art

[0002] When a space manipulator performs operations such as grabbing and transporting flexible parts, it will excite the elastic vibration of the flexible load, and its vibration will in turn affect the operation accuracy of the manipulator, thereby producing a coupled dynamic effect. The highly nonlinear characteristics of this coupling behavior place extremely high demands on controller design and have attracted close attention from a large number of scientific researchers. For example, Yan et al. (Yan Z, Lai XZ, Meng QX, Wu M, She JH, Iwasaki M. Modeling, analysis, and adaptive neural modified-backstepping control of an uncertain horizontal pendubot with double flexible joints [J]. Control Engineering Practice, 2023, 139: 105647.) proposed an adaptive controller based on the backstepping method, which successfully solved the control problem of a planar underactuated manipulator with two flexible joints. Yang et al. (Yang TW, Xu WL, Han JD. Dynamic compensation control of flexible macro-micro manipulator systems [J]. IEEE Transactions on Control System Technology, 2010, 18 (1): 143-151.) used a laser diode emitter and PSD to detect the vibration of the end of the flexible arm for vibration suppression at the end of the flexible manipulator system. Stieber et al. (Stieber ME, McKay M, Vukovich G, Petriu E. Vision-based sensing and control for space robotics applications [J]. IEEE Transactions on Instrumentation and Measurement, 1999, 48 (4): 807-812.) used a reference target to measure the end vibration of the flexible arm in real time through a visual sensor and image processing algorithm.Cong et al. (Cong YZ, Du HB, Li XL, Jin X Z. Distributed bounded finite-time cooperative control algorithm for multiple nonlinear manipulators [J]. International Journal of Robust and Nonlinear Control, 2024, 34 (12): 8127-8143.) designed a bounded distributed cooperative controller by introducing a virtual leader to solve the synchronization control problem of multiple nonlinear operating systems. Wang et al. (Wang EM, Wu SN, Liu YF, Wu ZG, Liu XD. Distributed vibration control of a large solar power satellite [J]. Astrodynamics, 2019, 3: 189-203.) proposed a distributed cooperative controller for the vibration problem of multiple underactuated rigid-flexible coupled systems. The controller uses proportional and differential feedback and interactive feedback between adjacent control units to suppress vibration and achieve state consistency between multiple systems. Ortega et al. (Ortega R, Loria A, Nicklasson P J, Sira-Ramirez H. Euler-Lagrange systems [M], London: Springer, 1998: 15-37.) proposed a passive control theory based on the passive characteristics of the system. On this basis, Chen et al. (Chen T, Shan JJ, Wen H. Distributed passivity-based control for multiple flexible spacecraft with attitude-only measurements [J]. Aerospace Science and Technology, 2019, 94: 105408.) designed a distributed passive controller that only relies on attitude angle feedback.

[0003] Currently, few studies have deeply investigated the vibration suppression of flexible loads during collaborative manipulation of space manipulators, especially integrating the feedback of beam free end deflection velocity in the form of visual measurement into a passivity-based distributed controller. Summary of the invention

[0004] The present invention provides a vibration suppression method for the collaborative operation of multiple robotic arms clamping a flexible beam, which uses a visual sensor to measure real-time vibration information of the flexible beam, achieves collaborative consistency of the multiple robotic arms, and suppresses the end vibration of the flexible beam in the process of achieving consistency.

[0005] To achieve the above objectives, the present invention adopts the following technical solutions:

[0006] A method for suppressing vibration when multiple mechanical arms clamp a flexible beam in coordinated operation, comprising the following steps:

[0007] Construct a dynamic model of a robotic arm that holds multiple flexible beams;

[0008] By introducing the generalized coordinates of the virtual controller system corresponding to the generalized coordinates of the robot, the dynamic equation of the auxiliary controller with the same structure as the robot system is constructed, and the total energy equation of the controller dynamics is obtained;

[0009] The controller dynamics equation is interconnected with the robot arm dynamics equation to form a closed-loop system, and the control law based on the feedback of the deflection change rate of the end of the flexible beam is designed:

[0010]

[0011] Among them, q ai,1 …q ai,4 represents the first to fourth joint angles of the i-th robot arm, and θ i,1 …θ i,4 is θ i The four elements of Represents the rate of change of the deflection at the end of the flexible beam;

[0012] The controller is distributed passive, and the rate of change of the deflection of the end of the flexible beam is fed back to the distributed passive controller, so that the vibration of the end of the flexible beam is suppressed by the controller in the process of achieving consistency of multiple mechanical arms.

[0013] Beneficial effect: The present invention provides a vibration suppression method for the collaborative operation of multiple robotic arms clamping a flexible beam. Aiming at the consistency problem of multiple robotic arms clamping the flexible beam, a distributed passive controller based on the velocity feedback of the end of the flexible beam is designed. The real-time vibration information of the flexible beam is measured by a visual sensor, and the deflection change rate of the end of the flexible beam is fed back to the distributed passive controller, so that the vibration of the end of the flexible beam is suppressed by the controller in the process of achieving consistency of multiple robotic arms. Verification results prove that the distributed passive control method based on VDFEB feedback proposed in the present invention is feasible and can achieve consistency control and vibration suppression for multiple robotic arms clamping flexible beams. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1This is a schematic diagram of a single mechanical arm clamping a flexible beam in an embodiment of the present invention;

[0015] Figure 2 A schematic diagram of a collaborative control task scenario of a space manipulator holding multiple flexible loads in an embodiment of the present invention;

[0016] Figure 3 is an undirected communication graph between the dual robotic arms and the virtual leader in an embodiment of the present invention;

[0017] Figure 4 The experimental results under PBC control without VDFEB feedback are shown in Figure 1, where a is the joint angle of robot arm 1 and b is the joint angle of robot arm 2;

[0018] Figure 5 Graph showing the experimental results under PBC control based on VDFEB feedback in an embodiment of the present invention, where a is the joint angle of robot arm 1, and b is the joint angle of robot arm 2;

[0019] Figure 6 3 is a comparison diagram of the deflection test results of the dual robotic arms in an embodiment of the present invention, wherein a is the deflection of the flexible beam 1 , and b is the deflection of the flexible beam 2 . DETAILED DESCRIPTION

[0020] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments:

[0021] Figure 1 The dynamic equation of the single robot arm holding the flexible beam system can be expressed as

[0022]

[0023] Where M(q) is the mass inertia matrix of the system, It includes the centrifugal force and Coriolis force matrix of the system. K is the stiffness matrix. Considering that only the joints of the manipulator are drivable in the rigid-flexible coupling system of the space manipulator operating the flexible beam, the generalized force vector of the system is defined as u = [u a T ,0] T , the robot arm has four drivable joints, and considering the first-order bending mode of the flexible beam, so q = [q a ,η] T =[q 1 ,q 2 ,q 3 ,q 4 ,η] T are the generalized coordinates of the system.

[0024] Formula (1) is networked and expanded into a dynamic model of a manipulator holding N flexible beams, which can be expressed as:

[0025]

[0026] in u s =[u 1 T … u N T ] T , Here, q 1 ,q 2 ,q 3 and q 4 Rewrite q ai,1 ,q ai,2 ,q ai,3 and q ai,4 ,therefore, The subscript i indicates that these variables correspond to the i-th robot arm, where i=1,…,N.

[0027] In a multi-agent system consisting of N robotic arms holding flexible beams, the information exchange between agents can be described by a communication topology. A communication topology usually consists of a set of nodes and a set of edges. Nodes represent agents, and edges represent the paths of information flow between adjacent agents. In the graph G = (V, E), V = {1, 2…N} represents the set of all nodes. represents the set of all edges connecting nodes. Edge (i, j) represents the information flow from node i to node j, where node i is called the neighbor of node j. If all edges have both (i, j) and (j, i), then the graph G is undirected; otherwise, the graph G is directed. The neighbor set of node i is denoted by N i ={j∈V|(j,i)∈E}; from node i 1 To node i k A path is a sequence of ordered edges (i l ,i l+1 ), where l = 1,…,k-1; if there is at least one path between any two nodes, then the graph G is connected; define the matrix is the adjacency matrix of graph G. If (j,i)∈E, then a ij >0; otherwise a ij = 0. Let the in-degree of the i-th node be d i , defined as the sum of the elements in the i-th row of the adjacency matrix A, that is Define the in-degree matrix D as a diagonal matrix D = diag{d 1 ,…,d N}. Then L=DA is the Laplace matrix of graph G.

[0028] The desired position of the robot arm considered in this embodiment is determined by the virtual leader, which is represented by node 0; correspondingly, the other nodes are called followers. As long as there is a path from the leader to node i, it is said that the leader is reachable from node i. If the leader's information can be transmitted to every node, the leader is said to be globally reachable. Define B = diag{b 1 ,,b N}, where if node i can receive information from the leader without going through other nodes, then b i =1; otherwise, b i = 0. Let matrix H = L + B. In a graph G, if at least one node can receive information from the leader and the communication between followers is an undirected connected graph, then the leader is globally reachable and the matrix H is positive definite.

[0029] By introducing the generalized coordinate q of the robot as The corresponding generalized coordinates θ of the virtual controller system are used to construct the dynamic equation of the auxiliary controller with the same structure as the robotic arm system:

[0030]

[0031] in, and is a constant positive definite matrix. Therefore, the controller dynamics corresponding to the i-th robot is expressed as

[0032]

[0033] in, and is a constant positive definite matrix, i = 1,…,N; is the generalized coordinate vector of the controller dynamics equation. It can be clearly seen from equation (4) that the controller dynamics corresponding to the i-th robot arm depends not only on its own information, but also on the information of its neighbors, so the controller system is distributed.

[0034] The total energy of the controller dynamics is

[0035]

[0036] in

[0037]

[0038] and

[0039]

[0040] The control law based on the feedback of the change rate of the deflection at the end of the flexible beam is designed as follows:

[0041]

[0042] Among them, q ai,1 …q ai,4 represents the first to fourth joint angles of the i-th robot arm, and θ i,1 …θ i,4 is θ i The four elements. represents the rate of change of the end deflection of the flexible beam, A is a constant, and u as =[u a1 T ,…u aN T ] T , i=1…N; at this time, the controller dynamics equation and the robot arm dynamics equation are interconnected to form a closed-loop system, which can be expressed as

[0043]

[0044] Energy function V of the closed-loop system c Expressed as

[0045]

[0046] V c The derivative with respect to time can be expressed as

[0047]

[0048] in and u as are defined as the output and input of this system respectively, so this closed-loop system is passive.

[0049] The stability of the controller u designed above is verified by using the Lyapunov function:

[0050] First, define the following auxiliary variables

[0051]

[0052] and

[0053]

[0054] Consider the following candidate Lyapunov function

[0055]

[0056] where ψ s =[ψ 1 T ,…,ψ N T ]T ,and

[0057]

[0058] V 2 The time derivative of can be written as

[0059]

[0060] where δ s =[δ 1 T ,…,δ N T ] T , from which we can get

[0061]

[0062] and

[0063]

[0064] According to equations (17) and (18), we can get

[0065]

[0066] According to the LaSalle invariance principle, the system will asymptotically converge to the maximum invariant set Among them C c It is positive. mean Assumptions θ remains constant, so the controller dynamics described in Eq. (3) can be rewritten as

[0067]

[0068] Due to K c and are all constant positive definite matrices, so q as Keeping it unchanged, we can get

[0069] The dynamic equations of the manipulator holding multiple flexible beams can be rewritten as

[0070]

[0071] According to the dynamic equation of the robot arm, [m 12 c 12 ]There is at least one row with [m 22 c 22 ] is linearly independent. When θ i =q ai =q ad When established, η is satisfiedi ≡0 and Therefore, the only solution of equation (22) is η i ≡ 0. So when When η i ≡0 and θ i =q ai =q ad According to the LaSalle invariance principle, the system is asymptotically stable.

[0072] The following experimental results verify the proposed distributed passive control and vibration suppression strategy. VDFEB can be measured by the hand-eye camera fixed on the robot arm, and the parameters of equations (3) and (8) are set as M ci =I 4 , C ci =50I 4 and K ci =100I 4 ,in is the unit matrix, and the value of k is 1×10 4 In order to highlight the effectiveness of distributed passive control (PBC) based on the velocity of deflection at the free end of the beam (VDFEB) feedback for vibration suppression, a comparison of distributed PBC without VDFEB feedback is given. Let g(t)≡0, then the control law without VDFEB feedback can be expressed as

[0073] u as =K c (θ-q as ) (twenty three)

[0074] Now using Figure 2 The two manipulators holding the flexible beam shown in the figure verify the effectiveness of the controller proposed in equation (8) and give the control effect of PBC without VDFEB feedback. The communication topology between the two manipulators is shown in Figure 3 As shown, the initial joint angles of the two space manipulators are and the joint angle q of the virtual leader ad Set as

[0075]

[0076] And the flexible beam has no deformation at the initial moment. Figure 4 and 5The response curves of the joint angles of the manipulators under the control laws of (23) and (8) are shown respectively, which means that all joint angles of the manipulators converge to the desired positions within about 50 seconds. Obviously, both controllers can effectively drive the two manipulators to track the virtual leader to achieve the consistency of the manipulator posture.

[0077] In order to illustrate the advantage of vibration suppression based on VDFEB feedback in the proposed controller, the deflection of the flexible beam end under the control laws of Eq. (8) and Eq. (23) is compared. Figure 6 It can be seen that the control method based on VDFEB feedback has an obvious suppression effect on the vibration of the flexible beam; it can be intuitively seen that under the control law of formula (8), the vibration amplitude of the flexible beam is smaller than the vibration amplitude under the control law of formula (23), and the amplitude decays faster. In summary, the control law proposed in formula (8) can realize distributed consistency control and vibration suppression.

[0078] The above description is only a preferred embodiment of the present invention. It should be pointed out that those skilled in the art can make corresponding changes and adjustments to the technology of the present invention without departing from the basic principles of the present invention. These changes and adjustments all fall within the protection scope of the present invention.

Claims

1. A vibration suppression method for multiple mechanical arms clamping a flexible beam in collaborative operation, characterized in that: The following steps are involved: Construct a dynamic model of a robotic arm that holds multiple flexible beams; By introducing the generalized coordinates of the virtual controller system corresponding to the generalized coordinates of the robot, the dynamic equations of the auxiliary controller with the same structure as the robot system are constructed, and the total energy of the controller dynamics is obtained; The controller dynamic equations and the robot arm dynamic equations are interconnected to form a closed-loop system. A control law based on the feedback of the deflection change rate of the end of the flexible beam is designed. The deflection change rate of the end of the flexible beam is fed back to the controller. The controller is used to suppress the vibration of the end of the flexible beam in the process of achieving consistency of multiple robot arms.

2. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 1, characterized in that: The dynamic model of the manipulator holding multiple flexible beams is: in u s =[u1 T …u N T ] T , The subscript i indicates that these variables correspond to the i-th robot arm, where i=1,…,N.

3. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 2, characterized in that: The process of building a dynamic model of a robotic arm holding multiple flexible beams is as follows: Construct the dynamic equation of a single robotic arm holding a flexible beam system: Where M(q) is the mass inertia matrix of the system, Includes the centrifugal force and Coriolis force matrix of the system, K η is the stiffness matrix; Considering that only the joints of the manipulator are drivable in the rigid-flexible coupling system of the space manipulator operating the flexible beam, the generalized force vector of the system is defined as u = [u a T ,0] T , the robot arm has four drivable joints, and considering the first-order bending mode of the flexible beam, so q = [q a ,η] T =[q1,q2,q3,q4,η] T is the generalized coordinate of the system; The dynamic equations of a single robotic arm clamping a flexible beam are networked to obtain the dynamic model of N robotic arms clamping flexible beams.

4. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 1 or 3, characterized in that: The information exchange between multiple robotic arms holding flexible beams is described by a communication topology graph, and the desired positions of the robotic arms are determined by a virtual leader.

5. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 3, characterized in that: The dynamic equation of the controller is: in, and is a constant positive definite matrix.

6. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 5, characterized in that: The total energy of the controller dynamics is: in, 7. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 6, characterized in that: The energy function V of the closed-loop system c for: V c The derivative with respect to time can be expressed as in, and u as are defined as the output and input of this system respectively.

8. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 1 or 7, characterized in that: The controller system is distributed passive.

9. The vibration suppression method for cooperative operation of multiple mechanical arms clamping flexible beams according to claim 1, characterized in that: The control law based on the feedback of the deflection change rate of the flexible beam end is: Among them, q ai,1 …q ai,4 represents the first to fourth joint angles of the i-th robot arm, and θ i,1 …θ i,4 is θ i The four elements of represents the rate of change of the end deflection of the flexible beam, A is a constant, and u as =[u a1 T ,…u aN T ] T ,i=1…N.

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