[0045] In order to better understand the content of the patent of the present invention, the technical solutions of the present invention are further described below with reference to the accompanying drawings and specific embodiments.
[0046] like Figure 1-6 As shown, the multi-mobile robot formation method based on distributed preset time observer includes the following steps:
[0047] Step 1: Kinematics modeling: According to the operating characteristics of the multi-mobile robot, establish the kinematic equation of the multi-mobile robot. The kinematic model of the multi-mobile robot is as follows: figure 2 shown. The mobile robot is selected as a wheeled robot under the two-dimensional plane of the global coordinate system, and its kinematic equations can be described as follows:
[0048]
[0049] where i is the number of the mobile robot; N is a normal number, indicating the number of multi-mobile robots; Δ={1,2,...,N} is the number set of multi-mobile robots; x i ,y i is the position information of the ith mobile robot; θ i represents the heading angle of the ith mobile robot; v i ,ω i For the control input, it represents the speed and rotation angular velocity of the i-th mobile robot, respectively.
[0050] Step 2: Communication topology description: establish a communication network architecture between multiple mobile robots, and build a master-slave architecture of multiple mobile robots. Establish a master-slave architecture of N+1 multi-mobile robots, set one mobile robot as the master with the number 0, and the remaining N mobile robots as the slaves with the number i, i∈Δ for collaborative formation; To have a graph to express the communication topology relationship. The expression steps of the undirected weighted graph are as follows: The topological relationship between the slaves is G N ={υ,ε}, the set of N mobile robot nodes is defined as υ={υ 1 ,υ 2 ,...,υ N}, the weighted edge between mobile robots numbered i and j, i, j∈Δ is defined as ε={(υ i ,υ j )|υ i ,υ j ∈υ}. The topological relationship of the entire multi-mobile robot system is G N+1 , the description of the direct communication connection between the i-th slave mobile robot and the host mobile robot is constant b i , the communication topology relationship matrix between the host mobile robot and the slave mobile robot can be established as B=diag{b 1 ,b 2 ,...,b N}, and combined with the Laplacian matrix L in the slave communication topology, the communication topology of the entire multi-mobile robot system can be expressed as matrix M=L+B. In this embodiment, three mobile robots are selected as slaves, and the numbers are respectively slave 1, slave 2, and slave 3; the master number is master 0. Its communication topology is described as image 3 As shown, at least one slave has direct information communication with master 0.
[0051] Step 3: State Observer Design: Design a distributed preset time state observer, so that the slave can obtain the observed value of the master state within the preset time. The state observer is designed as:
[0052]
[0053] in, are the state observations of the i-th slave mobile robot to the host mobile robot; k x ,k y ,k θ 0, c x ,c y ,c θ1/λ min (M) and λ x ,λ y ,λ θ is the parameter of the observer; ρ x ,ρ y ,ρ θ is the preset time function; a ij is the element in the adjacency matrix in the undirected weighted graph.
[0054] Wherein, the preset time function is defined as follows:
[0055]
[0056] where, t k ,T k ,k∈{x,y,θ} are the initial time and the preset time interval, respectively, and h is a positive integer parameter.
[0057] The control objectives of the formation controller are:
[0058]
[0059] where δ ki ,k∈{x,y,θ} is the error system between the master and the i-th slave mobile robot, and the specific relationship is as follows
[0060]
[0061] where x i (t),y i (t), θ i (t) represent the horizontal and vertical positions and heading angles of the i-th slave mobile robot in the global coordinate system, respectively, x 0 (t),y 0 (t), θ 0 (t) represent the horizontal and vertical positions and heading angles of the host mobile robot in the global coordinate system, respectively, The target distance that needs to be maintained between the i-th slave and the master in the horizontal and vertical directions.
[0062] Step 4: Formation controller design: Input the observations to the multi-mobile robot formation controller to realize the pre-set time cooperative formation of the multi-mobile robots. The formation control method of multiple mobile robots is as follows Figure 4 As shown, the design of the formation controller is:
[0063]
[0064] where u xi ,u yi is a virtual controller, defined as follows
[0065]
[0066] Among them, a x ,a y ,a θ0 and r x ,r y ,r θ 0 is a constant coefficient; are the first derivative of the state observation value of the i-th slave mobile robot to the host mobile robot; e xi ,e yi ,e θi is the error system between adjacent mobile robots based on the described communication topology, and the specific relationship is as follows
[0067]
[0068] in, It is the target distance that needs to be maintained between the i-th slave and the j-th slave in the horizontal and vertical directions.
[0069] In this embodiment, each mobile robot is installed with a GPS positioning system, a laser radar and an inertial measurement unit including an accelerometer, a gyroscope and a magnetometer to obtain the relative position and angle between the mobile robots. When the information obtained by the GPS positioning system is not accurate enough, lidar can make up for this error.
[0070] In this embodiment, the parameters of the distributed preset time state observer are selected as h=2, k x =k y =k θ =1, c x =c y =c θ = 2. Since it is the control structure of the inner and outer loops, the preset convergence time of the inner loop attitude needs to be faster than the preset convergence time of the outer loop position, so we choose T x =T y =2s, T θ =1s. The kinematic state of the host of the entire multi-mobile robot system is
[0071]
[0072] The initial states of the master and slave are respectively x 0 (0)=0, y 0 (0)=1, θ 0 (0) = arctan1; x 1 (0)=-2, y 1 (0)=2, θ 1 (0)=2; x 1 (0)=5, y 1 (0)=1, θ 1 (0) = 0; x 1 (0)=2, y 1 (0)=-3, θ 1 (0)=1.
[0073] The proof process of the convergence of the distributed preset time state observer designed by the present invention is as follows:
[0074] definition First, we prove the observation error of slave i in the x direction at a preset time T x converges to zero. Will and T x Bringing it into the observer equation 2, we can get
[0075]
[0076] make The Lyapunov function is chosen as and get the derivation
[0077]
[0078] Among them, κ x for u 0x the upper bound. Since M is a positive definite matrix, there is an invertible matrix Ω such that M=Ω T Ω. Therefore we can get
[0079]
[0080] Integrate the above formula and bring in c x1/λ min (M), we can get
[0081]
[0082] in t∈[t 0 ,t 0 +T x ) stage, multiply the left and right sides of Equation 12 by ρ x 2
[0083]
[0084] It is further derived
[0085]
[0086] Solving the differential equation in Equation 14, we get
[0087]
[0088]
[0089] By presetting the properties of the scaling function, we can get in t∈[t 0 +T x ,t 0 +T x +T x ) stage, due to 2k x λ min (M)>0 and Then V(t)≤0, further 0≤V(t)≤V(t 0 +T x )=0. So in t∈[t 0 +T x ,t 0 +T x +T x ) stage, V(t)≡0. In the same way, we can get in t∈[t 0 +T x ,+∞) stage, V(t)≡0.
[0090] From the above, we can get at preset time T x converges to zero. Similar to the above proof method, we can prove that and respectively at the preset time T x with T θ converges to zero.
