A multi-robot real-time control and collaborative containment method based on distributed optimization

Abstract

The present invention discloses a multi-robot real-time control and collaborative containment method based on distributed optimization. First, a distributed optimization system based on a time-varying objective function and subject to equality constraints is constructed, wherein the objective function is the sum of the local objective functions of each robot, the local objective function contains the local information of the robot, and the communication between the robots is carried out through a connected undirected graph; then, a distributed control method with a specified time convergence is used to solve the distributed optimization system to obtain the optimal position information of the containment robot; finally, after solving the optimal position information of the robot obtained by the distributed optimization system, a real-time control algorithm related to position and posture is used to make the containment robot move to the optimal position. When the present invention uses a distributed control method to solve the containment problem, the solution obtained is the target position coordinates of the robot, and the present invention enables the robot to track the target position by real-time control.

Description

Technical Field

[0001] The present invention belongs to the field of distributed optimization and robot collaborative control, and relates to a multi-robot real-time control and collaborative containment method based on distributed optimization. Background Art

[0002] Distributed optimization based on multi-agent systems has received widespread attention in academia due to its application potential, such as transportation systems, multi-robot systems, and smart grids. In recent years, many distributed optimization algorithms have been proposed. From a methodological point of view, existing distributed optimization algorithms include alternating direction multiplier method, projected subgradient method, primal-dual method, neurodynamic method, etc. With the development of control theory, distributed optimization algorithms continue to develop based on control and optimization theory, and time-independent objective functions are proposed. For example, a distributed control method proposed by researchers can ensure that robots can reach the target position collaboratively and only rely on the local communication network. A finite-time convergence solution has also been proposed to solve the distributed rendezvous problem, which enables multiple robots to rendezvous by minimizing the distance performance function.

[0003] Previous research has focused on distributed optimization problems with non-time-varying objective functions, but optimization problems with time-varying objective functions and constraints also have a wide range of applications, such as real-time resource scheduling problems, traffic network control, and formation problems for multi-robot systems. Time-varying distributed optimization problems are more complex than parameter optimization problems because their optimal solutions vary over time. However, time-varying distributed optimization problems have attracted the attention of more and more scholars. For example, some researchers have proposed a discrete-time distributed prediction and correction method to solve unconstrained time-varying optimization problems. By adding a Newton correction mechanism, this method has been extended to continuous-time multi-agent systems. In particular, some have proposed a gradient-based distributed optimization scheme to track the optimal solution of a time-varying quadratic optimization problem, whose communication is in a non-directed graph. Others have proposed a projection-based method that can converge consistently to an approximate optimal solution. In order to asymptotically track the optimal solution, sliding models and matrix terms have been proposed to solve restricted distributed optimization problems.

[0004] Although many time-varying distributed optimization schemes have been proposed, their finite-time convergence can often only be guaranteed in theory. Therefore, in the practical application of distributed optimization, finite-time convergence is still an important research topic. For example, while optimizing the objective function, it is required to obtain the target formation and reach the tracking speed within a finite or fixed time. Not only that, the convergence time is also related to the initial position of the multi-agent system, so the convergence time cannot be accurately evaluated if the initial position cannot be accurately measured. Some studies have given a definition of specified time convergence, and the agent can converge to a consistent state within a specified time. However, the above-mentioned specified time convergence research is not based on distributed optimization. In addition, in the process of applying theoretical results to real objects, more limiting factors need to be considered, such as: robot speed limit. Summary of the invention

[0005] Purpose of the invention: In response to the current problem of multi-robot collaborative containment, the present invention proposes a multi-robot real-time control and collaborative containment method based on distributed optimization, which controls multiple robots in real time and completes the containment and tracking of the target.

[0006] Technical solution: The present invention provides a multi-robot real-time control and collaborative containment method based on distributed optimization, which specifically includes the following steps:

[0007] (1) Construct a distributed optimization system based on a time-varying objective function and subject to equality constraints. The system uses a time-varying function as the objective function and the constraint condition as an equality. The objective function is the sum of the local objective functions of each robot. The local objective function contains the local information of the robot. The communication between the robots is carried out through a connected undirected graph.

[0008] (2) using a distributed control method that converges within a specified time to solve the distributed optimization system and obtain the optimal position information of the encircling robot, wherein the distributed control method can enable the encircling robot to encircle the target robot within a specified time and continuously follow the target robot in the formation;

[0009] (3) Solve the distributed optimization system to obtain the optimal position information of the robot, and use a real-time control algorithm related to position and posture to make the containment robot move to the optimal position.

[0010] Furthermore, the distributed optimization system described in step (1) is described as follows:

[0011] x′ i (t) = u i (t),(i=1,2,...,n)

[0012]

[0013] stx i-x j -q ij =0,(i=1,2,...,n,j∈N i )

[0014] in, q ij ∈R k , x i is the trajectory of robot i, u i (t) is the controller to be designed, f i (x i ,t):R k ×R + →R is the local objective function, which is related to the specific optimization problem. is the overall objective function.

[0015] Furthermore, the optimal position information in step (2) is the target position that the containment robot will reach at the next moment during the moving process.

[0016] Furthermore, the implementation process of step (2) is as follows:

[0017] Design the following distributed controller:

[0018]

[0019] in:

[0020]

[0021]

[0022]

[0023] Among them, η, α1, α2 are the weight coefficients of functions φ1, φ2, φ3, and their values ​​can be set according to actual conditions; t f is the convergence time, this value can be set manually; I k is the k-order identity matrix, 1 k is a k-dimensional column vector of all 1s, sgn(·) is a vector-valued sign function, which is defined as: for any have diag(·) is the mapping from vector function to matrix function, which is defined as: Vector Functions and matrix functions is the objective function f i (x i (t),t) first and second order partial derivatives;

[0024] The fourth-order Runge-Kutta method is used to solve the discrete form of the distributed control method:

[0025]

[0026] Among them, h is the step size of the algorithm.

[0027] Furthermore, the implementation process of step (3) is as follows:

[0028]

[0029] Among them, v i 、w i is the linear velocity and angular velocity of robot i, k v , k e is the system design parameter, (x i ,y i ) is the current position of robot i, is the target position of robot i, distance Let θ i is the rotation angle of robot i, then △θ i =α i -θ i .

[0030] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are as follows: Compared with other multi-agent systems, the present invention is dedicated to the optimization problem of time-varying objective functions, and its scope of application is wider, which can make up for the limitations of non-time-varying objective functions; Compared with other multi-agent distributed optimization systems, the initial position of the multi-agent system in the present invention is randomly set, and the selection of its initial position does not affect the convergence and convergence time of the system, which further expands the application scope and practicality of the present invention; Compared with other distributed optimization algorithms, the distributed control method proposed in the present invention is a distributed collaborative method with specified time convergence. Compared with other fixed time convergence and finite time convergence algorithms, the algorithm proposed in the present invention has more precise and practical control over the convergence time; The real-time control algorithm of the present invention controls the robot to converge to the optimal position. This process overcomes the speed limitation of the robot when tracking the target position, so that the theoretical algorithm is guaranteed in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is a flow chart of the present invention;

[0032] Figure 2 is a schematic diagram of the Turtlebot3 device used in the present invention;

[0033] Figure 3 At 0s, gazebo simulation diagram of the blocking robot and the target robot;

[0034] Figure 4 Gazebo simulation diagram of the blocking robot and the target robot at the 10s moment;

[0035] Figure 5 Gazebo simulation diagram of the blocking robot and the target robot at the 20s moment;

[0036] Figure 6 Gazebo simulation diagram of the blocking robot and the target robot at the 30s moment;

[0037] Figure 7 Gazebo simulation diagram of the blocking robot and the target robot at the 40s moment;

[0038] Figure 8 Gazebo simulation diagram of the blocking robot and the target robot at the 50s moment;

[0039] Fig. 9 At time 0s, the physical picture of the blocking robot and the target robot;

[0040] Fig.10 At the 10s moment, the actual image of the blocking robot and the target robot;

[0041] Fig.11 At the 20s moment, the actual image of the blocking robot and the target robot;

[0042] Fig.12 At the 30s moment, the actual image of the blocking robot and the target robot;

[0043] Fig.13 At the 40s moment, the actual image of the blocking robot and the target robot;

[0044] Fig.14 Actual picture of the encircling robot and the target robot at the 50s moment. DETAILED DESCRIPTION

[0045] The present invention is further described in detail below in conjunction with the accompanying drawings:

[0046] like Figure 1 As shown, the present invention proposes a multi-robot real-time control and collaborative containment method based on distributed optimization. Compared with other distributed optimization algorithms, it is dedicated to a multi-agent system with a time-varying objective function. Compared with other distributed optimization algorithms, it has the advantage of converging within a specified time, and has a better scope of application and practicality. It specifically includes the following steps:

[0047] Step 1: Construct a distributed optimization system based on a time-varying objective function and subject to equality constraints. The system uses a time-varying function as the objective function and an equality as the constraint condition. The objective function is the sum of the local objective functions of each robot. The local objective function contains the local information of the robot. The communication between the robots is carried out through a connected undirected graph. Establish a distributed optimization system:

[0048] x′ i (t) = u i (t),(i=1,2,...,n)

[0049]

[0050] stx i -x j -q ij =0,(i=1,2,...,n,j∈N i )

[0051] in, q ij ∈R k , x i is the trajectory of robot i, u i (t) is the controller to be designed, f i (x i ,t):R k ×R + →R is the local objective function, which is related to the specific optimization problem. is the overall objective function.

[0052] like Figure 2 As shown in the figure, the TurtleBot3Burger non-holonomic constrained wheeled robot is used as the experimental object. The number of target robots and blocking robots is 1 and 3 respectively. The trajectory information of the target robot is c(t) = (sin(t), t) T , then the time-varying objective function is: The adjacency matrix of the communication graph G between the containment robots is: 0 means no information transfer between robots, 1 means information transfer between robots, thus the constraint condition can be obtained:

[0053] q 12 =(-1.2,-2.08) T ,q 13 =(-2.4,0) T ,q 23 =(-1.2,2.08) T ,q 21 =-q 12 ,q 31 =-q 13,q 32 =-q 23 , which can be shortened to:

[0054] x1-x2-(-1.2,-2.08) T =0,x1-x3-(-2.4,0) T =0,x2-x3-(-1.2,2.08) T =0, that is:

[0055] x′ i =u i (t),(i=1,2,3)

[0056]

[0057]

[0058] Step 2: Use a distributed control method with specified time convergence to solve the distributed optimization system and obtain the optimal position information of the containment robot, which is the target position that the containment robot will reach at the next moment in the moving process. The distributed control method can make the containment robot form a containment around the target robot within a specified time and continue to follow the target robot in this formation.

[0059] Designing Distributed Controllers:

[0060]

[0061]

[0062]

[0063]

[0064] Among them, η, α1, α2 are the weight coefficients of functions φ1, φ2, φ3, and their values ​​can be set according to actual conditions; t f is the convergence time, this value can be set manually; I k is the k-order identity matrix, 1 k is a k-dimensional column vector of all 1s, sgn(·) is a vector-valued sign function, which is defined as: for any have diag(·) is the mapping from vector function to matrix function, which is defined as: Vector Functions and matrix functions is the objective function f i (x iHere we select the parameters η=1.3,a1=a2=1.3,β=0.5 and set the convergence time t f =10s.

[0065] The distributed control method is used to solve the distributed optimization system. That is, the distributed control method proposed in this paper is the solution of the distributed optimization system. It is given in the form of ordinary differential equations. Since the ordinary differential equations are more complicated, their continuous solution expression cannot be given, so the discretized differential equations are: i =u i , (i=1,2,...,n) and solve it using the fourth-order Runge-Kutta method:

[0066]

[0067] Set the iteration step length h = 0.5s, and the Runge-Kutta algorithm input x0 is the current position (x, y) of the containment robot. The optimal position after the time interval h is calculated based on the current position

[0068] Step 3: Solve the distributed optimization system to obtain the optimal position information of the robot, and use a real-time control algorithm related to position and posture to make the containment robot move to the optimal position.

[0069] Tracking the optimal position through real-time control algorithms

[0070]

[0071] Among them, v i 、w i is the linear velocity and angular velocity of robot i, k v , k e is the system design parameter, (x i ,y i ) is the current position of robot i, is the target position of robot i, distance Let θ i is the rotation angle of robot i, then △θ i =α i -θ i ; Select parameter k v =0.6, k e =0.0006.

[0072] Figures 3 to 8The simulation results of the algorithm in ROS are shown. The initial positions of the robot are (-0.8, -0.8), (-1.2, 0.2), (1.2, -0.8), and the initial position of the target is (0, 0). The robot positions at t = 0s, 10s, 20s, 30s, 40s, and 50s are shown. It can be seen that at t = 40s, the robot forms a containment effect on the target and tracks the target robot in the specified formation.

[0073] Figures 9 to 14 This is the physical simulation result of the algorithm. The initial positions are (-0.8, -0.8), (-1.2, 0.2), and (1.2, -0.8). The initial position of the target is (0, 0). The moments corresponding to the pictures are 0s, 10s, 20s, 30s, 40s, and 50s. It can be seen that the target is blocked at 30s, and the robot tracks the target thereafter.

[0074] It can be seen from the above theories, ROS simulations, and actual results that the multi-robot real-time control and collaborative containment method based on distributed optimization proposed in the present invention is precise and widely applicable. It can effectively contain and track the target robot in a specific formation, and the effect in actual application scenarios is also satisfactory.

Claims

1. A multi-robot real-time control and collaborative containment method based on distributed optimization, characterized in that: The following steps are involved: (1) Construct a distributed optimization system based on a time-varying objective function and subject to equality constraints, where the objective function is the sum of the local objective functions of each robot. The local objective function contains the local information of the robot, and the communication between the robots is carried out through a connected undirected graph; (2) using a distributed control method that converges within a specified time to solve the distributed optimization system and obtain the optimal position information of the blocking robot, wherein the distributed control method can enable the blocking robot to block the target robot within a specified time and continuously follow the target robot in a blocking formation; (3) Solving the distributed optimization system to obtain the optimal position information of the robot, and using a real-time control algorithm related to position and posture to make the containment robot move to the optimal position; The distributed optimization system described in step (1) is described as follows: x′ i (t)=u i (t),(i=1,2,..,n) s.t.x i -x j -q ij =0,(i=1,2,...,n,j∈N i ) in, x i is the trajectory of robot i, u i (t) is the controller to be designed, f i (x i , t):R k ×R + →R is the local objective function, is the overall objective function; The implementation process of step (2) is as follows: Design the following distributed controller: in: Among them, η, α1, α2 are the weight coefficients of functions φ1, φ2, φ3 respectively; t f is the convergence time, this value can be set manually; I k is the k-order identity matrix, 1 k is a k-dimensional column vector of all 1s, sgn(·) is a vector-valued sign function, which is defined as: for any have diag(·) is the mapping from vector function to matrix function, which is defined as: Vector Functions and matrix functions is the objective function f i (x i (t),t) first and second order partial derivatives; Distributed control method is used to solve the distributed optimization system, discretizing the differential equation: x′ i =u i ,, i = 1,2,...,n), and the fourth-order Runge-Kutta method is used to solve the discrete form of the distributed control method: Among them, h is the step size of the algorithm, that is, the time interval. The input x0 of the Runge-Kutta algorithm is the current position (x, y) of the containment robot. The optimal position after the time interval h is calculated by the current position. The implementation process of step (3) is as follows: Tracking the optimal position through real-time control algorithms Among them, v i 、w i is the linear velocity and angular velocity of robot i, k v , k e is the system design parameter, (x i ,y i ) is the current position of robot i, is the target position of robot i, distance Let θ i is the rotation angle of robot i, then Δθ i =α i -θ i .

2. According to the method of multi-robot real-time control and coordinated containment based on distributed optimization in claim 1, it is characterized in that: The optimal position information in step (2) is the target position that the containment robot will reach at the next moment during the moving process.

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