CN110134018B - Multi-foot cooperative control method of underwater multi-foot robot system - Google Patents

Abstract

A multi-legged collaborative control method for an underwater multi-legged robot system belongs to the technical field of underwater multi-legged robot collaborative control. The present invention is to solve the problem of communication delay between different mechanical legs of the multi-legged robot, which is affected by the processor's operation speed and the signal transmission path. The present invention introduces an obstacle Lyapunov function in logarithmic form, so that the trajectory tracking error of the system always meets the set error limit requirement; only the communication topology between different mechanical feet is required to be a directed graph, and only some followers can obtain it. The information of the navigator is enough, avoiding the communication burden caused by the global knowledge of the information; the selection of the input signal source as the virtual navigator makes the change of the navigator more flexible and meets the requirements of the robot for movement flexibility. The invention is suitable for the coordinated motion control of the underwater multi-legged robot.

Description

A multi-legged cooperative control method for an underwater multi-legged robot system

technical field

The invention belongs to the technical field of collaborative control of an underwater multi-legged robot.

Background technique

As a major marine country, in recent years, my country has vigorously developed the marine economy, and the development and utilization of offshore areas has become more and more important. Among them, the offshore drilling platform is the carrier for the development and utilization of marine resources, and the research on the safety of the offshore drilling platform is gradually deepening. The daily inspection and maintenance of the offshore drilling platform is very important, but due to the harsh working environment, manual maintenance is very inconvenient. With the advancement of science and technology, more attention has been paid to the research and development of underwater robots, and the advantages of multi-legged robots over traditional roller or crawler robots have gradually emerged. The design of underwater multi-legged robots has become a hot topic. The research on the control of coordinated motion of multi-legged bionic robots has attracted more and more attention of experts and scholars.

The theory currently used by the underwater multi-legged robot is the assumption of minimum energy, that is, the energy consumed by the motion of the multi-legged robot can be minimized by optimizing the shape structure and control algorithm. Multi-body system dynamics is an important theoretical basis for the research and development of multi-legged robots. Therefore, the research results of coordinated control of multi-body systems can be used for the control of underwater multi-legged robots. At present, the research on the collaborative tracking of multi-body systems mostly adopts the follower to track the leader. It only needs to accurately plan the trajectory of the leader, and other followers use their own under the action of effective control algorithms. Or the information of neighbors can track the leader. This control method is often used in multi-legged robot systems to reduce energy consumption and save costs. Among them, the signal source is virtualized as the leader, and each mechanical foot of the underwater multipedal robot is regarded as a follower with multiple degrees of freedom.

However, in actual engineering projects, there are often high requirements for control accuracy, especially in multi-legged robot systems. If the error is too large, it may cause the multi-legged robot to roll over or even become paralyzed, resulting in irreversible consequences.

Therefore, it is necessary to limit the output error, and a good method to limit the control error is the BLF (Barrier Lyapunov Function) technique. At present, an asymmetric time-varying BLF (Barrier Lyapunov function) is used to construct an adaptive controller, and the Lyapunov stability theory is applied to ensure that the output error is within the set range. However, multi-legged robots are mostly controlled by a microcomputer network, which is affected by the computing speed of the processor and the signal transmission path, which often causes communication delays between different mechanical legs.

SUMMARY OF THE INVENTION

The purpose of the present invention is to solve the problem of communication delay between different mechanical legs of the multi-legged robot, which is affected by the processor operation speed and signal transmission path, and proposes a multi-legged cooperative control method for an underwater multi-legged robot system.

The multi-legged collaborative control method of an underwater multi-legged robot system according to the present invention, the specific steps of the method include:

Step 1. Establish the dynamic model of all the mechanical feet of the underwater multi-legged robot; obtain the control system of the underwater multi-legged robot;

Step 2, establishing a directed topology structure diagram of the communication relationship between the mechanical feet in the underwater multi-legged robot system described in step 1; the root node of the directed topology structure diagram is the navigator, and each other node is a Mechanical feet, the leader is the control signal source, and a mechanical foot is a follower;

Step 3, using the distributed observer and the directed topology diagram described in step 2 to estimate the state information of the leader obtained by all nodes, and obtain the state information of the leader;

Step 4: Use the obstacle Lyapunov function to constrain the error of the pilot's state information obtained in step 3; limit the error variable within a specified range;

Step 5, using neural network technology to deal with the nonlinear uncertainty in the underwater multi-legged robot system; realize the estimation of unknown parameters;

Step 6: According to the error compensation described in step 4 and the processing of nonlinear uncertainty described in step 5, the adaptive control law of the underwater multi-legged robot control system is obtained, and the multi-legged control of the underwater multi-legged robot system is realized. Collaborative control.

The invention comprehensively considers the constant communication delay between different mechanical feet of the multi-legged robot system and the generalized interference of the dynamic model of the multi-legged robot, and at the same time introduces a logarithmic obstacle Lyapunov function to make the trajectory of the system The tracking error always meets the set error limit requirements; only the communication topology between different mechanical feet is required to be a directed graph, and only some followers can obtain the information of the leader, avoiding the communication burden caused by the global knowledge of information; Selecting the input signal source as the virtual navigator makes the change of the navigator more flexible and meets the requirements of the robot for motion flexibility.

Description of drawings

Fig. 1 is the flow chart of the multi-legged cooperative control method of the underwater multi-legged robot system according to the present invention;

Fig. 2 is a schematic diagram of a mechanical foot of an underwater multi-legged robot;

Fig. 3 is the communication topology diagram of the mechanical foot of the underwater multi-legged robot;

Fig. 4 is the structural representation of mechanical foot;

Fig. 5 is the motion trajectory curve diagram of each mechanical foot joint 1 tracking the leader joint 1;

Fig. 6 is the motion trajectory curve diagram of each mechanical foot joint 2 tracking leader joint 2;

Fig. 7 is the trajectory tracking error curve diagram of each mechanical foot joint 1 and the pilot joint 1;

Fig. 8 is the trajectory tracking error curve diagram of each mechanical foot joint 2 and the pilot joint 2;

Fig. 9 is the input control torque curve diagram at each mechanical foot joint 1;

Fig. 10 is the input control torque curve diagram at each mechanical foot joint 2;

Figure 11 is a graph showing the relationship between the auxiliary variable Z ₁₁ and the time-varying error limit;

Figure 12 is a graph of the relationship between the auxiliary variable Z ₁₂ and the time-varying error limit;

Figure 13 is a graph of the relationship between the auxiliary variable Z ₂₁ and the time-varying error limit;

Figure 14 is a graph of auxiliary variable Z ₂₂ versus time-varying error limit;

Figure 15 is a graph of the relationship between the auxiliary variable Z ₃₁ and the time-varying error limit;

Figure 16 is a graph of auxiliary variable Z ₃₂ versus time-varying error limit;

Figure 17 is a graph of the relationship between the auxiliary variable Z ₄₁ and the time-varying error limit;

FIG. 18 is a graph of auxiliary variable Z ₄₂ versus time-varying error limit.

Detailed ways

The embodiments of the present invention will be described in detail below with reference to the accompanying drawings and examples, so as to fully understand and implement the implementation process of how the present invention applies technical means to solve technical problems and achieve corresponding technical effects. The embodiments of the present application and the various features in the embodiments can be combined with each other under the premise of no conflict, and the formed technical solutions all fall within the protection scope of the present invention.

Embodiment 1: The present embodiment will be described below with reference to FIG. 1 , and a multi-legged collaborative control method of an underwater multi-legged robot system described in this embodiment. The specific steps of the method include:

Step 1. Establish the dynamic model of all the mechanical feet of the underwater multi-legged robot; obtain the control system of the underwater multi-legged robot;

Step 2, establishing a directed topology structure diagram of the communication relationship between the mechanical feet in the underwater multi-legged robot system described in step 1; the root node of the directed topology structure diagram is the navigator, and each other node is a Mechanical feet, the leader is the control signal source, and a mechanical foot is a follower;

Step 3, using the distributed observer and the directed topology diagram described in step 2 to estimate the state information of the leader obtained by all nodes, and obtain the state information of the leader;

Step 4: Use the obstacle Lyapunov function to constrain the error of the pilot's state information obtained in step 3; limit the error variable within a specified range;

Step 5, using neural network technology to deal with the nonlinear uncertainty in the underwater multi-legged robot system; realize the estimation of unknown parameters;

Step 6: According to the error compensation processing described in Step 4 and the nonlinear uncertainty processing described in Step 5, the adaptive control law of the underwater multi-legged robot control system is obtained, and the multi-functionality of the underwater multi-legged robot system is realized. Foot synergistic control.

This embodiment proposes an observer that considers the communication delay between different mechanical feet. In terms of error limitation, a logarithmic barrier Lyapunov function is introduced, and a distributed automatic Lyapunov function is designed by using the backstep method. The adaptive control algorithm is used to ensure that the trajectory tracking error of the multi-legged robot's mechanical feet to the leader converges to around 0.

The present invention uses a directed graph to describe the topology between the mechanical feet of the underwater multi-legged robot, and mainly studies the motion of the n mechanical feet of the underwater multi-legged robot; The mechanical feet can receive control signals independently and have independent movement, so they can be regarded as separate individuals. The underwater multi-legged robot system can be regarded as a multi-body system. Considering the wide application of Euler-Lagrangian equations in nonlinear multi-body systems, the Euler-Lagrange equations are used for underwater multi-legged systems. The motion of the robotic foot of the robot is described. Each mechanical foot of the underwater multipedal robot is regarded as a follower with p degrees of freedom, and the follower is denoted by 1,...,n. Because the underwater multi-legged robot has high requirements for motion flexibility, it is usually necessary to add and arrange navigators to achieve additional flexible motion, so the signal source is set as a virtual navigator, and the navigator uses n+1 express.

表示所有边的集合， A＝{a_ij}∈R^(n+1)×(n+1)表示的邻接矩阵；在点集υ中，υ_i表示水下多足机器人的第i条机械足，边(υ_i,υ_j)∈ε表示水下多足机器人的第j条机械足可以获得第i条机械足的信息。υ_i称为父节点，υ_j称为子节点，并且υ_i和υ_j是相邻的两个节点。有向图的路径为一个有限的顶点序列υ_i1,...,υ_in，满足(υ_i,υ_j)∈ε。如果有向图中除了一个节点(称为根节点(root))外，其余每个节点均有且仅有一个父节点，且存在根节点到其余任何节点的路径，则称该有向图为有向树(directed tree)。有向图的有向生成树(directed spanning tree)为包含该有向图所有节点的有向树。如果有向图存在一个为有向生成树的子图，则称该有向图具有有向生成树。定义邻接矩阵A是一个非负矩阵。当且仅当(υ_i,υ_j)∈ε时，元素a_ij＝1，否则a_ij＝0。对于所有的j＝1,...,n+1，都有a_(n+1)j＝0，定义矩阵

其中

i∈{1,...,n}， j＝1,...,n+1。当有向图ζ包含一个有向生成树成立的时候，

的分母不为0。A directed graph ζ=(υ,ε,A) is used to represent the communication relationship between a virtual pilot and the n mechanical feet of the underwater multipedal robot. In the directed graph, υ={1,2,. ...,n+1} is the set of all vertices,

represents the set of all edges, and A={a _ij }∈R ^(n+1)×(n+1) represents the adjacency matrix; in the point set υ, υ _i represents the ith mechanical foot of the underwater multipedal robot , the edge (υ _i ,υ _j )∈ε indicates that the jth mechanical foot of the underwater multipod robot can obtain the information of the ith mechanical foot. υ _i is called the parent node, υ _j is called the child node, and υ _i and υ _j are two adjacent nodes. The path of a directed graph is a finite sequence of vertices υ _i1 ,...,υ _in , satisfying (υ _i ,υ _j )∈ε. If each node in a directed graph has one and only one parent node except one node (called the root node), and there is a path from the root node to any other node, then the directed graph is called Directed tree. A directed spanning tree of a directed graph is a directed tree that contains all the nodes of the directed graph. If a directed graph has a subgraph that is a directed spanning tree, the directed graph is said to have a directed spanning tree. Define the adjacency matrix A to be a non-negative matrix. The element a _ij =1 if and only if (υ _i ,υ _j )∈ε, otherwise a _ij =0. For all j=1,...,n+1, there is a _(n+1)j =0, defining the matrix

in

i∈{1,...,n}, j=1,...,n+1. When the directed graph ζ contains a directed spanning tree,

The denominator is not 0.

Specific embodiment 2: This embodiment further describes the multi-leg collaborative control method of the underwater multi-legged robot system described in the first embodiment. In this embodiment, all the mechanical mechanisms of the underwater multi-legged robot described in step The method of the dynamic model of the mechanical foot is the same, and the dynamic model of the i-th mechanical foot is taken as an example to illustrate, specifically:

为第i条机械足的关节转动角速度，

为第i条机械足的关节转动角加速度，且q_i,

R^p是p维实数列向量，τ_i表示输入第i条机械足的控制力拒，τ_i∈R^p，M_i(q_i)表示对称正定的惯性矩阵，M_i(q_i)∈R^p×p，R^p×p是p行p列的实数矩阵,

表示第i条机械足的偏心力，

g_i(q_i)表示第i条机械足的重力，g_i(q_i)∈R^p，ω_i表示外部扰动，ω_i∈R^p，所述外部扰动包括环境扰动和外部噪声；其中，对称正定的惯性矩阵M_i(q_i)、偏心力

和重力g_i(q_i)为未知参数。Among them, qi is the joint rotation angle of the _i -th mechanical foot, i={1,2,....,n}, n is a positive integer,

is the joint rotational angular velocity of the i-th mechanical foot,

is the joint rotational angular acceleration of the i-th mechanical foot, and q _i ,

R ^p is a p-dimensional real number sequence vector, τ _i represents the input control force of the i-th mechanical foot, τ _i ∈R ^p , M _i (q _i ) represents a symmetric positive definite inertia matrix, M _i (q _i )∈R ^p×p , R ^p×p is a real matrix with p rows and p columns,

represents the eccentric force of the i-th mechanical foot,

g _i (q _i ) represents the gravity of the i-th mechanical foot, g _i (q _i )∈R ^p , ω _i denotes the external disturbance, ω _i ∈ R ^p , the external disturbance includes environmental disturbance and external noise; wherein, Symmetric positive definite inertia matrix M _i (q _i ), eccentric force

and gravity g _i (q _i ) are unknown parameters.

Specific embodiment 3: This embodiment further describes a multi-legged cooperative control method for an underwater multi-legged robot system described in embodiment 2. In this embodiment, the distributed observer described in step 3 and step 2 The directed topology structure graph estimates the state information of the navigator obtained by all nodes, and the specific method for obtaining the state information of the navigator is:

Since the kinetic model described in equation (1) satisfies the property:

是反对称的，那么有：对于

Property 1: Matrix

is antisymmetric, then there are: for

和B使得

其中I_p表示p×p的单位矩阵。Property 2: There are two positive numbers

and B makes

where I _p represents the identity matrix of p × p.

Therefore, the generalized coordinate q _n+1 of the voyager is expressed as:

q _n+1 = Fv (3)

为v的导数，S和F为恒定实数矩阵， S∈R^m×m，F∈R^n×m；Among them, v is the auxiliary state variable of the leader, v∈R ^m ,

is the derivative of v, S and F are constant real number matrices, S∈R ^m×m , F∈R ^n×m ;

The distributed observer estimates the leader state information:

为矩阵

的元素，

为邻接矩阵，η_j为第j个跟随者对领航者的位置信息的估计值，

第j个跟随者对领航者的速度信息的估计值，j＝1,...,n+1，η_i表示第i个跟随者对于v的估计值，η_i∈R^m，

是i个跟随者对领航者的速度信息的估计值，T 表示相邻跟随者i和j之间的通讯延迟，t为时间。in,

is a matrix

Elements,

is the adjacency matrix, η _j is the estimated value of the position information of the jth follower to the leader,

The estimated value of the speed information of the leader by the jth follower, j=1,...,n+1, η _i represents the estimated value of v by the ith follower, η _i ∈R ^m ,

is the estimated value of the speed information of the leader by i followers, T represents the communication delay between adjacent followers i and j, and t is the time.

In this embodiment, the trajectory of the dynamic virtual leader of the mechanical feet (followers) of the underwater robot is represented by equations (2) and (3). Considering that the different signal transmission paths between different mechanical feet of an underwater multipedal robot will cause communication delays between different mechanical feet. In the case that only some of the robotic feet (followers) can obtain the leader information, a distributed observer and an adaptive neural network control algorithm are designed to ensure that each robotic foot has a bounded trajectory tracking error for the leader. At the same time, considering that in a multi-legged robot system, if the error between the actual motion trajectory and the expected trajectory is too large, it may cause the multi-legged robot to roll over or even become paralyzed, resulting in irreversible consequences. Therefore, the rotation angle of the mechanical foot joints of the underwater multipod robot is limited.

得到观测器的观测误差η_i-v是有界的，表示为：Because S and F are real number matrices, the value of the elements in the real number matrix has nothing to do with time and the state quantity of the pilot. When all the mechanical feet can obtain the information of S and F, it means that a part of the mechanical feet can pass the formula (3 ) to obtain the value of q _n+1 to achieve the purpose of obtaining the information of the leader; a common method is to use the dynamic matrix of the target object to design the corresponding observer, through the distributed observer shown in formula (4), use The state information of the adjacent mechanical feet is used to estimate the state information of the pilot. if:

It is obtained that the observation error η _i -v of the observer is bounded and expressed as:

in:

U₀是小于1的正数，该正数趋近于0，R_e为通讯延迟增益矩阵，1_n为元素都为1的n为列向量，

是第 n个跟随者对领航者的估计误差，

是神经操作器，因为S和F是实数矩阵，A为带权的邻接矩阵，实数矩阵中元素的值与时间和领航者的状态量无关，利用目标对象的动态矩阵设计相应的观测器。

U ₀ is a positive number less than 1, the positive number tends to 0, R _e is the communication delay gain matrix, 1 _n is a column vector with all elements of 1, n is a column vector,

is the estimated error of the nth follower to the leader,

is a neural operator, because S and F are real number matrices, A is a weighted adjacency matrix, and the value of the elements in the real number matrix is independent of time and the state quantity of the leader, and the corresponding observer is designed using the dynamic matrix of the target object.

Specific Embodiment 4: This embodiment further describes the multi-legged cooperative control method for an underwater multi-legged robot system described in Embodiment 3. In Step 4, the obstacle Lyapunov function is used to control the navigator obtained in Step 2. The state information is used to constrain the error; the specific method to limit the error variable within the specified range is as follows:

限定输出的转动角度的状态量q_i(t)的区域为：The rotation angle of the robot foot joint is limited, and the time-varying boundary is set as: k c(t)＝[ k c ₁ (t), k c ₂ (t),..., k c _n (t) ] ^T and

The region of the state quantity q _i (t) that defines the output rotation angle is:

分别为第i个机械足t时刻期望运动轨迹q_ri的上、下边界，kc_n(t)和

分别为第n个机械足t时刻期望运动轨迹q_ri的上、下边界，q_i(t)为t时刻第i条机械足的关节转动角度；R为实数矩阵。 k c _i (t) and

are the upper and lower boundaries of the expected motion trajectory q _ri of the i-th mechanical foot at time t, respectively, k c _n (t) and

are the upper and lower boundaries of the desired motion trajectory q _ri of the nth mechanical foot at time t, respectively, q _i (t) is the joint rotation angle of the ith mechanical foot at time t; R is a real matrix.

Specific embodiment 5: This embodiment further describes the multi-legged cooperative control method of the underwater multi-legged robot system described in the fourth embodiment. The linear uncertainty is dealt with, and the specific method to estimate the unknown parameters is as follows:

Use formula:

进行补偿，其中，W_i表示理想的加权矩阵，W_i ^T是理想的加权矩阵W_i的转置，r_i是第i个机械足的虚拟控制器，

为r_i的导数；

是激活函数，△_i表示逼近误差；且△_i是有界的，存在一个正数△_Mi使得||△_i||≤△_Mi；获得水下多足机器人的第i条机械足的

的估值：Realization of Nonlinear Uncertainty Quantity for the i-th Mechanical Foot of a Multi-legged Robot System

compensation, where Wi represents the ideal weighting matrix, Wi ^T is the transpose of the ideal weighting matrix Wi _{, ri} is the virtual controller of the _ith mechanical foot _,

is the derivative of _ri ;

is the activation function, △ _i represents the approximation error; and △ _i is bounded, there is a positive number △ _Mi such that ||△ _i ||≤△ _Mi ; obtain the ith mechanical foot of the underwater multipod robot

Valuation of:

是理想的加权矩阵W_i的估计值。in,

is an estimate of the ideal _weighting matrix Wi.

Embodiment 6: This embodiment further describes the multi-legged collaborative control method for an underwater multi-legged robot system described in Embodiment 4. In this embodiment, the error constraint described in step 6 is based on the error constraint described in step 4. After processing and processing the nonlinear uncertainty described in step 5, the adaptive control law of the control system of the underwater multi-legged robot is obtained as:

是矩阵

的转置，Z_1i是跟随者对领航者的轨迹跟踪误差变量向量，中间变量

ka_i和 kb_i分别是跟踪误差变量的上、下边界；其中，

Among them, α, β and μ are all positive numbers, and α, β and μ are infinitely close to zero, K _2i is a symmetric positive definite matrix, Z _2i is the virtual trajectory tracking error variable vector, Z _2i ^T is the vector Z the transpose of _2i ,

is the matrix

The transpose of , Z _1i is the trajectory tracking error variable vector of the follower to the leader, the intermediate variable

ka _i and kb _i are the upper and lower bounds of the tracking error variable, respectively; where,

In this embodiment, considering the influence of control errors on the underwater multipedal robot, a time-varying obstacle Lyapunov function is used to ensure that the output state quantity satisfies the set time-varying restriction requirements. Considering the model uncertainty and unknown dynamic parameters of the underwater multi-legged robot system, the present invention proposes a distributed adaptive control method by using the backstepping method. The method is implemented based on Mathematical Lemma 1 to 4;

满足

其中

u为正数，得出L(t)是有界的。Lemma 1: If there is a continuous Lyapunov function V(L,t), satisfying

Satisfy

in

u is a positive number, it is concluded that L(t) is bounded.

当E是对称正定矩阵时，有不等式：

其中λ_min表示E的最小特征值，λ_max表示E的最大特征值。Lemma 2:

When E is a symmetric positive definite matrix, there are inequalities:

where λ _min represents the minimum eigenvalue of E, and λ _max represents the maximum eigenvalue of E.

|Ψ(t)|≤Φ，其中Φ为一个正数，对于

是有界的。Lemma 3: For a continuously differentiable equation Ψ(t), if Ψ(t) satisfies for

|Ψ(t)|≤Φ, where Φ is a positive number, for

is bounded.

Lemma 4: Consider a positive number G∈R, if x∈R and |x|<|G|, the following inequality holds:

Considering the underwater multipedal robot system with model uncertainty and external disturbances, 1), 2) and 3) are established under the condition that there is a constant communication delay between different mechanical legs of the underwater multipedal robot;

1) The external disturbance ω _i is bounded, that is, there is a positive number γ such that ||ω _i ||≤γ.

都是有界的，S和F对于所有跟随者都是已知的。2), v and

are both bounded, and S and F are known to all followers.

3), the directed graph ζ contains a directed spanning tree.

First define the desired motion trajectory q _ri of the i-th mechanical foot:

q _ri =Fη _i (7)

The trajectory tracking error variable vector Z _1i of the follower to the leader is:

Z _1i = q _i -q _ri (8)

The virtual trajectory tracking error variable vector Z _2i is:

where _ri is the virtual controller;

The time-varying boundary of the trajectory tracking error variable vector Z _1i of the follower to the leader is:

ka _i (t) = q _ri (t) - k c _i (t) (10)

为第i个机械足t时刻的期望运动轨迹的上边界，kc_i(t)为t时刻第i个机械足的期望运动轨迹的下边界，q_ri(t)为t时刻第i个机械足的期望运动轨迹；ka _i and kb _i (t) are the upper and lower bounds of the tracking error variable at time t,

is the upper boundary of the expected motion trajectory of the i-th robotic foot at time t, k c _i (t) is the lower boundary of the expected motion trajectory of the i-th robotic foot at time t, and q _ri (t) is the i-th robotic foot at time t. the desired trajectory of the foot;

where, the Lyapunov function V _1i (t):

Among them, ka _i (t) is the upper boundary of the tracking error variable at time t, kb _i (t) is the lower boundary of the tracking error variable at time t, and h(i) is defined as:

Transform the track tracking error variable vector Z _1i of the follower to the leader, let:

Substitute equation (14) into (12) to get:

From equation (15), V _1i (t) is positive definite and continuously differentiable when |ε _i |<1; derivation of V _1i (t) can be obtained:

跟踪误差变量上边界ka_i的导数，

为跟踪误差变量下边界kb_i的导数；

the derivative of the upper bound ka _i of the tracking error variable,

is the derivative of the lower bound kb _i of the tracking error variable;

The virtual controller ri of the _i -th robotic foot is:

为：in,

for:

在

和

全为0时保持有界，K₁＝diag[k₁₁,k₁₂,...,k_1i，...,k_1n]是对称正定的矩阵，k_1i为增益矩阵K₁中的第i个非0元素，将式(17)和(18)代入(16)中，可得：make sure

exist

and

When all are 0, it remains bounded, K ₁ =diag[k ₁₁ ,k ₁₂ ,...,k _1i ,...,k _1n ] is a symmetric positive definite matrix, and k _1i is the ith in the gain matrix K ₁ There are non-zero elements, substituting equations (17) and (18) into (16), we can get:

where _Xi is defined as:

And X _i ∈X,X=diag[X ₁ ,X ₂ ,....,X _i ,....,X _n ];

Using the distributed adaptive control law, the adaptive control law is obtained:

是矩阵

的转置，Z_1i是跟随者对领航者的轨迹跟踪误差变量向量，

ka_i和kb_i分别是跟踪误差变量的上、下边界；其中，

Among them, α, β and μ are all positive numbers, and α, β and μ are infinitely close to zero, K _2i is a symmetric positive definite matrix, Z _2i is the virtual trajectory tracking error variable vector, Z _2i ^T is the vector Z the transpose of _2i ,

is the matrix

The transpose of , Z _1i is the trajectory tracking error variable vector of the follower to the leader,

ka _i and kb _i are the upper and lower bounds of the tracking error variable, respectively; where,

Under the action of distributed observer (5) and distributed adaptive control laws (21) and (22), the auxiliary variable Z _1i is eventually uniformly bounded and the tracking error between each robot foot and the pilot is bounded. At the same time, the output state quantity q _i (t) satisfies the time-varying output restriction condition, that is,

Prove 1), 2) and 3): Formula (9) has the following derivation with respect to t:

Substitute equation (23) into (1) to get:

为虚拟轨迹跟踪误差变量向量的Z_2i的导数；in,

is the derivative of Z _2i of the virtual trajectory tracking error variable vector;

g_i(q_i)全部未知，在式(1)表示的水下多足机器人系统中存在非线性不确定性。考虑到神经网络对未知非线性函数具有良好的逼近能力，常常用来处理非线性系统中的不确定性问题。Because M _i (q _i ),

All g _i (q _i ) are unknown, and there are nonlinear uncertainties in the underwater multi-legged robot system represented by equation (1). Considering that the neural network has a good approximation ability to unknown nonlinear functions, it is often used to deal with uncertainties in nonlinear systems.

进行自适应补偿，具体方法：Therefore, the nonlinear uncertainty of underwater multi-legged robot system using neural network technology

Perform adaptive compensation, the specific method:

的估计值写成：where Wi represents the ideal weighting matrix, the φ _i function is the activation function, △ _i represents the approximation difference, and △ _i is bounded, that is _, there is a positive number △ _Mi such that ||△ _i ||≤△ _Mi , for underwater For the i-th mechanical foot of a multi-legged robot,

The estimate of is written as:

是W_i的估计值矩阵，

为矩阵

的转置，

为激活函数。in,

is the estimated value matrix of _Wi ,

is a matrix

transpose of ,

is the activation function.

Based on the neural network technology, the distributed adaptive control laws (21) and (22) are selected, and the obstacle Lyapunov function V _2i is selected;

为矩阵

的转置，Z_1i为跟随者对领航者的轨迹跟踪误差向量， Z_2i为虚拟轨迹跟踪误差向量，Z_2i ^T为向量Z_2i的转置，

为

的迹。in,

is a matrix

The transpose of , Z _1i is the track tracking error vector of the follower to the leader, Z _2i is the virtual track tracking error vector, Z _2i ^T is the transpose of the vector Z _2i ,

for

trace.

和权值估计自适应律：

K_2i是一个对称正定的矩阵，得到：Derivative with respect to V _2i , combining semi-symmetry and distributed adaptive control law:

and weight estimation adaptive law:

K _2i is a symmetric positive definite matrix, we get:

是表示

是

的迹。因为△_i和ω_i是有界的，故存在一个正数

使得：

同时得到：where φ _i is the activation function, Z _2i ^T is the transpose of the virtual trajectory tracking error matrix, φ _i is the activation function, Δ _i is the estimation error,

is to indicate

Yes

trace. Since Δ _i and ω _i are bounded, there is a positive number

makes:

Also get:

是||△_i+ω_i||的上边界，σ为正数，且无限趋近于0；in,

is the upper boundary of ||△ _i +ω _i ||, σ is a positive number, and infinitely approaches 0;

是一个标量，那么等式

成立；In formula (29),

is a scalar, then the equation

established;

According to the operational properties of the matrix trace, there are:

Substitute equations (19), (30) and (31) into (29) to get:

λ _min (K _2i ) is the minimum value of the symmetric positive definite matrix K _2i , and equation (32) is written as:

in:

是M_i的最大值。

is the maximum value of _Mi.

According to Lemma 1-4, it can be seen that V _2i satisfies the eventually uniform and bounded condition, and simultaneously integrating both sides of Equation (33) can be obtained:

Let κ ₁ >0, υ ₁ >0, we get:

V _2i (t) is the obstacle Lyapunov function at time t, and ρ is a constant less than 1. It can be known from equation (28):

V _1i ≤V _2i (38)

Then there are:

Substitute equations (13) and (14) into (39) to get:

From (7) we know that:

q _i -q _n+1 =q _i -Fη _i +Fη _i -q _n+1 =q _i -Fη _i +F(η _i -v) (41)

From equations (6), (40) and (41), we can get:

The tracking error of the i-th mechanical foot of the underwater multipedal robot to the pilot is bounded, and the upper boundary value is shown in Eq. (42).

From equations (4), (8), (10), (11) and (40), it can be known that:

It can be known from equation (43) that the output state quantity q _i (t) satisfies the time-varying output restriction condition.

The characteristics of the invention are: comprehensively consider the existence of constant communication delay between different mechanical feet of the multi-legged robot system and the generalized interference of the dynamic model of the multi-legged robot, and at the same time introduce a logarithmic form of BLF (Barrier Lyapuno function) so that the trajectory tracking error of the system always meets the set error limit requirements; only the communication topology between different mechanical feet is required to be a general directed graph, and only some followers can obtain the information of the leader, avoiding the need for The communication burden brought by the global knowledge of information; the selection of the input signal source as the virtual navigator in the research makes the change of the navigator more flexible and meets the requirements of the robot for motion flexibility.

Specific simulation examples

First set the simulation parameters:

In order to verify the effectiveness of the distributed adaptive cooperative tracking control law proposed in the present invention. The simulation experiment is carried out by taking the 8-legged underwater multi-legged robot as an example. Since the 8-legged underwater multi-legged robot uses a bi-quadruped gait, as shown in Figure 2, it can ensure that the 8-legged underwater multi-legged robot can still provide quadruped support when half of its walking feet are lifted off the ground. At the same time, because of the structural symmetry of the 8-legged underwater multi-legged robot, the 4 mechanical feet that move together with the 8-legged underwater multi-legged robot can be studied. The directed communication network composed of the mechanical feet (followers) of the underwater multipod robot, in which the numbers 1-4 represent the four mechanical feet (followers) of the 8-legged underwater multipod robot shown in Figure 1, and the number 5 represents the virtual pilot Or, the communication topology relationship is shown in Figure 3.

The dynamic equation of the i-th mechanical foot of the 8-legged underwater multipod robot can be expressed as:

where:

q _i =[q _i1 ,q _i2 ] ^T (45)

Among them, q _i1 and q _i2 respectively represent the rotation angles of the two joints of the mechanical foot of the underwater multi-legged robot.

Wherein: Ξ _i1 =J _i1 +m _i2 l _i1 ² , Ξ _i2 =0.25m _i2 l _i2 ² +J _i2 , Ξ _i3 =0.5m _i2 l _i1 l _i2 , Ξ _i4 =(0.5m _i1 +m _i2 )l _i1 ,

Ξ _i5 =0.5m _i2 l _i2 , g=9.8m/s ² represents the acceleration of gravity. As shown in Figure 4, m _i1 and m _i2 respectively represent the mass of the two links at the joint 2 of the mechanical foot, and l _i1 and l _i2 respectively represent the length of each link of the mechanical foot of the underwater multipedal robot. J _i1 and J _i2 represent the moments of inertia at the joints. The specific parameters of the mechanical foot are shown in Table 1:

Table 1: Specific parameters of the mechanical foot

Set the limits for the time-varying output as follows:

k _c (t)=[1.8+15e ^-0.8t ,1.8+15e ^-0.5t ,1.8+15e ^-0.5t ,1.8+15e ^-0.3t ] ^T (51)

The boundary value of the tracking error of the tracking error Z _1i is expressed as:

k _ai (t) = q _ri (t) - k _ci (t) (52)

It is known that:

-k _ai (t)<Z _1i (t)<k _bi (t) (54)

The angles of the mechanical feet of the underwater multipedal robot are set as follows:

q ₁₁ (0) = π/5, q ₁₂ (0) = -π/3, q ₂₁ (0) = 2π/5, q ₂₂ (0) = -π/6, q ₃₁ (0) = 3π/ 5,

q ₃₂ (0)=π/6, q ₄₁ (0)=4π/5, q ₄₂ (0)=π/3,

For the ith follower (i=1,...,4), the activation equation of the neural network system can be written as:

φ _i (z)=[φ _i1 (z),...,φ _i6 (z)] ^T (55)

The Gaussian equation is chosen as the activation function, and its form is:

假设所有的跟随者都用一样的激活方程。c_ij表示均匀分布在[-5,5]⁴×[-0.5,0.5]⁴上的接受域的中心，σ_ij表示高斯方程的宽度，定义σ_ij＝2，加权矩阵

的初始值设定为

in,

Assume that all followers use the same activation equation. c _ij represents the center of the receptive field uniformly distributed on [-5,5] ⁴ ×[-0.5, 0.5] ⁴ , σ _ij represents the width of the Gaussian equation, defines σ _ij =2, the weighting matrix

The initial value is set to

The target trajectory of the virtual pilot:

q_{51_bias}＝π/2,q_{52_amp}＝2π/3,

q_{52_bias}＝0，ω＝0.1π。Wherein, q _{51_amp} =π/6,

q _{51_bias} =π/2,q _{52_amp} =2π/3,

q _{52_bias} = 0, ω = 0.1π.

_The state quantity q5 of the virtual pilot is expressed as:

q ₅ =Fv (60)

in:

ω=0.1π (62)

Considering the needs of the 8-legged underwater multi-legged robot in practical engineering, it is usually necessary to add a boundary limit to the amplitude of the adaptive control law τ _i to achieve a better control effect _. Amplitude limit:

where τ _imax is a positive number, and τ _imax =50 is selected.

In this control algorithm, the time of communication delay is selected as T=0.2s. Analysis of the simulation results of the control algorithm: In the designed control algorithm, the control parameters are selected as k _1i =10, K _2i =20I ₂ , γ=1, v=10, and I ₂ is the unit matrix. The simulation results are shown in Figures 5-18.

Figures 5 and 6 show the changes of the state quantities of the virtual navigator and each mechanical foot, from which it can be seen that each mechanical foot can effectively track the navigator after about 5s. It can be seen from Figure 7 and Figure 8 that the trajectory tracking errors Z _1i and Z _2i for the pilot at the two joints of each mechanical foot converge to a small area near 0 after about 3s, and the fluctuation of Z _1i after stabilization The amplitude will not exceed 0.1, and the Z _2i will not fluctuate more than 0.05. Figures 8 and 9 show that the input control law of each mechanical foot is continuous and the fluctuation range does not exceed 10. Figures 10 to 18 show that the trajectory tracking error of each mechanical foot to the pilot meets the set time-varying output limit constraints. The simulation process effectively demonstrates the effectiveness of the present invention.

Although the embodiments of the present invention are described above, the content described is only an embodiment adopted to facilitate understanding of the present invention, and is not intended to limit the present invention. Any person skilled in the art to which the present invention belongs, without departing from the spirit and scope of the present invention, can make any modifications and changes in the form and details of the implementation, but the scope of patent protection of the present invention, The scope as defined by the appended claims shall still prevail.

Claims (2)

1. a multi-legged collaborative control method of an underwater multi-legged robot system, is characterized in that, the concrete steps of this method comprise: Step 1. Establish the dynamic model of all the mechanical feet of the underwater multi-legged robot; obtain the control system of the underwater multi-legged robot; Step 2, establishing a directed topology structure diagram of the communication relationship between the mechanical feet in the underwater multi-legged robot system described in step 1; the root node of the directed topology structure diagram is the navigator, and each other node is a Mechanical feet, the leader is the control signal source, and a mechanical foot is a follower; Step 3, using the distributed observer and the directed topology diagram described in step 2 to estimate the state information of the leader obtained by all nodes, and obtain the state information of the leader; The specific method to obtain the status information of the navigator is as follows: Via the formula:

q _n+1 = Fv (3) Obtain the generalized coordinate q _n+1 of the navigator, where v is the auxiliary state variable of the navigator, v∈R ^m ,

is the derivative of v, S and F are constant real number matrices, S∈R ^m×m , F∈R ^n×m ; The distributed observer estimates the leader state information:

in,

is a matrix

Elements,

is the adjacency matrix, η _j is the estimated value of the position information of the jth follower to the leader,

The estimated value of the velocity information of the leader by the jth follower, j=1,...,n+1, η _i represents the estimated value of v by the ith follower, η _i ∈R ^m ,

is the estimated value of the speed information of the leader by the ith follower, T represents the communication delay between adjacent followers i and j, and t is the time; Step 4: Use the obstacle Lyapunov function to constrain the error of the pilot's state information obtained in step 3; limit the error variable within a specified range; The specific method to limit the error variable within the specified range is as follows: The rotation angle of the robot foot joint is limited, and the time-varying boundary is set as:

and

The region of the state quantity q _i (t) that defines the output rotation angle is:

k c _i (t) and

are the upper and lower boundaries of the expected motion trajectory q _ri of the i-th mechanical foot at time t, respectively, k c _n (t) and

are the upper and lower boundaries of the expected motion trajectory q _ri of the nth mechanical foot at time t, respectively, q _i (t) is the joint rotation angle of the ith mechanical foot at time t; R is a real matrix; Step 5, using neural network technology to deal with the nonlinear uncertainty in the underwater multi-legged robot system; realize the estimation of unknown parameters; The specific method to realize the estimation of the unknown parameters is as follows: Use the formula:

Realization of nonlinear uncertainty quantities for the i-th mechanical foot of a multi-legged robot system

Compensation is performed, where qi is the joint rotation angle of the _i -th mechanical foot, i={1,2,....,n}, n is a positive integer,

is the joint rotational angular velocity of the i-th mechanical foot, and

Wi represents the ideal weighting matrix, Wi ^T is the transpose of the ideal weighting matrix Wi _{, ri} is the virtual controller of the _ith mechanical foot _,

is the derivative of _ri ;

is the activation function, Δ _i represents the approximation error; and Δ _i is bounded, there is a positive number Δ _Mi such that ||Δ _i ||≤Δ _Mi ; obtain the ith mechanical foot of the underwater multipod robot

Estimated value of :

in,

is an estimate of the ideal _weighting matrix Wi,

is a matrix

transpose of ,

is the activation function; Step 6: According to the error constraint described in step 4 and the processing of nonlinear uncertainty described in step 5, the adaptive control law of the underwater multi-legged robot control system is obtained, and the multi-legged control of the underwater multi-legged robot system is realized. collaborative control; The adaptive control law to obtain the control system of the underwater multi-legged robot is as follows:

Among them, α, β and μ are all positive numbers, and α, β and μ are infinitely close to zero, K _2i is a symmetric positive definite matrix, Z _2i is the virtual trajectory tracking error variable vector, Z _2i ^T is the vector Z The transpose of _2i , Z _1i is the track tracking error variable vector of the follower to the leader, the intermediate variable

ka _i and kb _i are the upper and lower bounds of the tracking error variable, respectively; where,

2. the multi-legged cooperative control method of a kind of underwater multi-legged robot system according to claim 1, is characterized in that, the method of the dynamic model of all mechanical feet of the underwater multi-legged robot described in the step 1 is all the same , take the dynamic model of the i-th mechanical foot as an example to illustrate, specifically:

Among them, qi is the joint rotation angle of the _i -th mechanical foot, i={1,2,....,n}, n is a positive integer,

is the joint rotational angular velocity of the i-th mechanical foot,

is the angular acceleration of the joint rotation of the i-th mechanical foot, and

R ^p is a p-dimensional real number sequence vector, τ _i represents the input control force of the i-th mechanical foot, τ _i ∈R ^p , M _i (q _i ) represents a symmetric positive definite inertia matrix, M _i (q _i )∈R ^p×p , R ^p×p is a real matrix with p rows and p columns,

represents the eccentric force of the i-th mechanical foot,

g _i (q _i ) represents the gravity of the i-th mechanical foot, g _i (q _i )∈R ^p , ω _i denotes the external disturbance, ω _i ∈ R ^p , the external disturbance includes environmental disturbance and external noise; wherein, Symmetric positive definite inertia matrix M _i (q _i ), eccentric force

and gravity g _i (q _i ) are unknown parameters.

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