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

Multi-foot cooperative control method of underwater multi-foot robot system Download PDF

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CN110134018B
CN110134018B CN201910525253.3A CN201910525253A CN110134018B CN 110134018 B CN110134018 B CN 110134018B CN 201910525253 A CN201910525253 A CN 201910525253A CN 110134018 B CN110134018 B CN 110134018B
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秦洪德
李晓佳
孙延超
魏彤锦
李凌宇
牛广智
范金龙
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Harbin Engineering University
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Abstract

A multi-foot cooperative control method for an underwater multi-foot robot system belongs to the technical field of underwater multi-foot robot cooperative control. The invention aims to solve the problem that the multi-legged robot is influenced by the operation speed of a processor and a signal transmission path and communication delay exists between different mechanical legs. The invention introduces a logarithm-form barrier Lyapunov function to ensure that the trajectory tracking error of the system always meets the set error limit requirement; only the communication topology among different mechanical feet is required to be a directed graph, only part of followers can obtain the information of the pilot, and the communication burden caused by the overall information knowledge is avoided; an input signal source is selected as a virtual navigator, so that the navigator is more flexibly changed, and the requirement of the robot on the motion flexibility is met. The invention is suitable for cooperative motion control of the underwater multi-legged robot.

Description

Multi-foot cooperative control method of underwater multi-foot robot system
Technical Field
The invention belongs to the technical field of cooperative control of underwater multi-legged robots.
Background
As a marine big country, in recent years, China vigorously develops marine economy, and offshore development and utilization become more important. The offshore drilling platform is used as a carrier for ocean resource development and utilization, researches on safety of the offshore drilling platform are gradually deepened, daily inspection and maintenance of the offshore drilling platform are very important, but manual overhaul and maintenance are very inconvenient due to severe working environments. With the development of science and technology, the research and development work of underwater robots is more emphasized, the advantages of the multi-legged robot relative to the traditional roller or crawler robot are gradually shown, and the design of the underwater multi-legged robot becomes a hot spot. The research on the control problem of coordinated motion of the multi-legged bionic robot attracts the attention of more and more experts and scholars.
The theory currently used by the underwater multi-legged robot is the energy minimum assumption, namely, the energy consumed by the motion of the multi-legged robot is minimized by optimizing the shape structure and the control algorithm. The multi-body system dynamics is an important theoretical basis for the research and development of the multi-legged robot, so that the research result of the coordinated control of the multi-body system can be used for reference for the control problem of the underwater multi-legged robot. At present, the mode of tracking a navigator by a follower is mostly adopted for researching the multi-body system cooperative tracking problem, only the motion trail of the navigator needs to be accurately planned, other followers realize the tracking of the navigator by utilizing own or neighbor information under the action of an effective control algorithm, and the multi-legged robot system usually adopts the control mode to achieve the purposes of reducing energy consumption and saving cost. The signal source is virtualized to be a pilot, and each mechanical foot of the underwater multi-foot robot is regarded as a follower with multiple degrees of freedom.
However, in actual engineering projects, the requirement on control accuracy is often high, and particularly in a multi-legged robot system, if an error is too large, the multi-legged robot may turn on its side or even be paralyzed, resulting in irreparable consequences.
It is therefore necessary to limit the output error, a good way to limit the control error being the BLF (barrier lyapunov function) technique. At present, an asymmetric time-varying BLF (barrier Lyapunov function) is used for constructing an adaptive controller, and the Lyapunov stability theory is applied to ensure that an output error is in a set range. However, the multi-legged robot is controlled by a microcomputer network, and communication delay between different mechanical legs is often caused due to the influence of the operation speed of a processor and a signal transmission path.
Disclosure of Invention
The invention aims to solve the problems that the multi-legged robot is influenced by the operation speed and the signal transmission path of a processor and communication delay exists between different mechanical feet, and provides a multi-legged cooperative control method of an underwater multi-legged robot system.
The invention discloses a multi-foot cooperative control method of an underwater multi-foot robot system, which comprises the following specific steps:
firstly, establishing a dynamic model of all mechanical feet of the underwater multi-foot robot; obtaining an underwater multi-legged robot control system;
step two, establishing a communication relation directed topology structure diagram among all mechanical feet in the underwater multi-foot robot system in the step one; the root node of the directed topology structure chart is a pilot, each other node is a mechanical foot, the pilot is a control signal source, and one mechanical foot is a follower;
estimating the state information of the navigator, which is obtained by all the nodes, by using the distributed observer and the directed topology structure chart in the step two to obtain the state information of the navigator;
step four, carrying out error constraint on the state information of the navigator obtained in the step three by adopting an obstacle Lyapunov function; limiting the error variable within a specified range;
processing the nonlinear uncertainty in the underwater multi-legged robot system by utilizing a neural network technology; the estimation of unknown parameters is realized;
and step six, obtaining the self-adaptive control law of the underwater multi-foot robot control system according to the error compensation in the step four and the nonlinear uncertainty processing in the step five, and realizing the multi-foot cooperative control of the underwater multi-foot robot system.
The method comprehensively considers the conditions of constant communication delay among different mechanical feet of the multi-legged robot system and generalized interference of a multi-legged robot dynamics model, and simultaneously introduces a logarithm-form barrier Lyapunov function to ensure that the track tracking error of the system always meets the set error limit requirement; only the communication topology among different mechanical feet is required to be a directed graph, only part of followers can obtain the information of the pilot, and the communication burden caused by the overall information knowledge is avoided; an input signal source is selected as a virtual navigator, so that the navigator is more flexibly changed, and the requirement of the robot on the motion flexibility is met.
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FIG. 1 is a flow chart of the multi-foot cooperative control method of the underwater multi-foot robot system of the invention;
FIG. 2 is a schematic view of a mechanical foot of the underwater multi-legged robot;
FIG. 3 is a communication topology diagram of a mechanical foot of the underwater multi-foot robot;
FIG. 4 is a schematic structural view of a mechanical foot;
FIG. 5 is a graph of the motion trajectory of each mechanical foot joint 1 tracking the joint 1 of the navigator;
fig. 6 is a graph of the motion trajectory of each mechanical foot joint 2 tracking the navigator joint 2;
FIG. 7 is a graph of the trajectory tracking error of each mechanical foot joint 1 and the navigator joint 1;
fig. 8 is a graph of the trajectory tracking error of each mechanical foot joint 2 and the navigator joint 2;
FIG. 9 is a graph of input control torque at each mechanical foot joint 1;
FIG. 10 is a graph of input control torque at each mechanical foot joint 2;
FIG. 11 is the auxiliary variable Z11Graph with time varying error limit;
FIG. 12 is the auxiliary variable Z12Graph with time varying error limit;
FIG. 13 is the auxiliary variable Z21Graph with time varying error limit;
FIG. 14 is the auxiliary variable Z22Graph with time varying error limit;
FIG. 15 is the auxiliary variable Z31Relation to time-varying error limitsA graph;
FIG. 16 is the auxiliary variable Z32Graph with time varying error limit;
FIG. 17 is the auxiliary variable Z41Graph with time varying error limit;
FIG. 18 is the auxiliary variable Z42Graph with time varying error limit.
Detailed Description
The following detailed description of the embodiments of the present invention will be provided with reference to the accompanying drawings and examples, so that how to apply the technical means to solve the technical problems and achieve the corresponding technical effects can be fully understood and implemented. The embodiments and the features of the embodiments can be combined without conflict, and the technical solutions formed are all within the scope of the present invention.
The first embodiment is as follows: the present embodiment is described below with reference to fig. 1, and the method for cooperatively controlling multiple feet of an underwater multi-foot robot system in the present embodiment specifically includes the following steps:
firstly, establishing a dynamic model of all mechanical feet of the underwater multi-foot robot; obtaining an underwater multi-legged robot control system;
step two, establishing a communication relation directed topology structure diagram among all mechanical feet in the underwater multi-foot robot system in the step one; the root node of the directed topology structure chart is a pilot, each other node is a mechanical foot, the pilot is a control signal source, and one mechanical foot is a follower;
estimating the state information of the navigator, which is obtained by all the nodes, by using the distributed observer and the directed topology structure chart in the step two to obtain the state information of the navigator;
step four, carrying out error constraint on the state information of the navigator obtained in the step three by adopting an obstacle Lyapunov function; limiting the error variable within a specified range;
processing the nonlinear uncertainty in the underwater multi-legged robot system by utilizing a neural network technology; the estimation of unknown parameters is realized;
and step six, obtaining the self-adaptive control law of the underwater multi-foot robot control system according to the error compensation processing in the step four and the nonlinear uncertainty processing in the step five, and realizing the multi-foot cooperative control of the underwater multi-foot robot system.
The embodiment provides an observer considering communication delay among different mechanical feet, introduces a logarithm barrier Lyapunov function in the aspect of error limitation, and designs a distributed self-adaptive control algorithm by adopting a backstepping method to ensure that the track tracking error of the mechanical feet of the multi-legged robot to a pilot is converged to be near 0.
The topological structure among all mechanical feet of the underwater multi-foot robot is described by adopting a directed graph, and the research is mainly carried out on the motion conditions of n mechanical feet of the underwater multi-foot robot; since each mechanical foot of the underwater multi-foot robot can receive the control signal independently and has the independence of movement, the underwater multi-foot robot can be regarded as an independent individual. The underwater multi-legged robot system can be regarded as a multi-body system, and the motion situation of the mechanical foot of the underwater multi-legged robot is described by using the Euler-Lagrangian equation in consideration of the wide application of the Euler-Lagrangian equation in the non-linear multi-body system. Each mechanical foot of the underwater multi-legged robot is considered as a follower with p degrees of freedom, which is denoted by 1. Since the underwater multi-legged robot has high requirements for motion flexibility, and a pilot is usually added and arranged to realize additional flexible motion, the signal source is set as a virtual pilot, and the pilot is represented by n + 1.
The communication relation between 1 virtual pilot and n mechanical feet of the underwater multi-legged robot is represented by a directed graph ζ (υ, epsilon, A), wherein υ {1, 2., n +1} is the set of all vertexes,
Figure BDA0002097081550000041
denotes the set of all edges, A ═ aij}∈R(n+1)×(n+1)A adjacency matrix of representations; in the set of points v,υithe ith mechanical foot and side (upsilon) of the underwater multi-foot roboti,υj) Epsilon represents that the j th mechanical foot of the underwater multi-foot robot can obtain the information of the i th mechanical foot. Upsilon isiCalled parent node, upsilonjCalled child node, and viAnd upsilonjAre two adjacent nodes. The path of the directed graph is a finite sequence of vertices υi1,...,υinSatisfy ([ nu ])i,υj) E epsilon. A directed graph is called a directed tree if every node in the directed graph has one and only one parent node except for one node, called a root node (root), and there is a path from the root node to any of the other nodes. A directed spanning tree (directed spanning tree) of a directed graph is a directed tree that includes all nodes of the directed graph. If the directed graph has a subgraph which is a directed spanning tree, the directed graph is called to have the directed spanning tree. The adjacency matrix a is defined as a non-negative matrix. If and only if (v)i,υj) When epsilon is epsilon, element aij1, otherwise aij0. For all j 1, n +1, there is a(n+1)jDefine the matrix as 0
Figure BDA0002097081550000042
Wherein
Figure BDA0002097081550000043
i ∈ { 1., n }, j ═ 1., n + 1. When the directed graph ζ contains a directed spanning tree,
Figure BDA0002097081550000044
the denominator of (2) is not 0.
The second embodiment is as follows: in this embodiment, the method for the dynamic models of all the mechanical feet of the underwater multi-foot robot in the first step is the same, and the dynamic model of the ith mechanical foot is taken as an example for explanation, specifically:
Figure BDA0002097081550000051
wherein q isiThe joint rotation angle of the ith mechanical foot is i ═ 1, 2.. times, n }, n is a positive integer,
Figure BDA0002097081550000052
is the joint rotation angular velocity of the ith mechanical foot,
Figure BDA0002097081550000053
angular acceleration of joint of the ith mechanical foot, and qi,
Figure BDA0002097081550000054
RpIs a p-dimensional real column vector, τiIndicating control force rejection, τ, of the input ith mechanical footi∈Rp，Mi(qi) Representing a symmetric positive definite inertial matrix, Mi(qi)∈Rp×p，Rp×pIs a matrix of real numbers of p rows and p columns,
Figure BDA0002097081550000055
the eccentric force of the ith mechanical foot is shown,
Figure BDA0002097081550000056
gi(qi) Denotes the gravity of the ith mechanical foot, gi(qi)∈Rp，ωiRepresenting external disturbances, ωi∈RpThe external disturbance comprises environmental disturbance and external noise; wherein the inertia matrix M is positively determined symmetricallyi(qi) Eccentric force
Figure BDA0002097081550000057
And gravity gi(qi) Are unknown parameters.
The third concrete implementation mode: in the second embodiment, the method for estimating the state information of the navigator obtained by all the nodes by using the distributed observer and the directed topology structure diagram in the second step to obtain the state information of the navigator is specifically as follows:
since the kinetic model described by equation (1) satisfies the properties:
properties 1: matrix array
Figure BDA0002097081550000058
Is antisymmetric, then: for the
Figure BDA0002097081550000059
Figure BDA00020970815500000510
Properties 2: there are two positive numbers
Figure BDA00020970815500000511
AndBso that
Figure BDA00020970815500000512
Wherein IpRepresenting a p identity matrix.
Thus, the generalized coordinates q of the aircraftn+1Expressed as:
Figure BDA00020970815500000513
qn+1＝Fv (3)
wherein v is an auxiliary state variable of a pilot, and v belongs to Rm，
Figure BDA00020970815500000514
Is the derivative of v, S and F are constant real matrices, S ∈ Rm×m，F∈Rn×m；
The distributed observer estimates the state information of the navigator:
Figure BDA00020970815500000515
wherein,
Figure BDA0002097081550000061
Figure BDA0002097081550000062
is a matrix
Figure BDA0002097081550000063
The elements of (a) and (b),
Figure BDA0002097081550000064
is a contiguous matrix, ηjFor the estimate of the jth follower's position information to the pilot,
Figure BDA0002097081550000065
an estimate of the speed information of the jth follower over the pilot, j 1iRepresenting the estimate of the i-th follower for v, ηi∈Rm，
Figure BDA0002097081550000066
Is the estimated value of the speed information of i followers to the pilot, T represents the communication delay between adjacent followers i and j, and T is time.
In the present embodiment, the trajectory of the dynamic virtual navigator of the mechanical foot (follower) of the underwater robot is represented by equations (2) and (3). The communication delay between different mechanical feet of the underwater multi-foot robot is considered to be caused by different signal transmission paths. Under the condition that only part of mechanical feet (followers) can obtain information of a pilot, a distributed observer and an adaptive neural network control algorithm are designed to ensure that each mechanical foot is bounded to the track tracking error of the pilot. Meanwhile, in the multi-legged robot system, if the error between the actual motion trajectory and the expected trajectory is too large, the multi-legged robot may turn on its side or even be paralyzed, which may result in irreparable consequences. Therefore, the rotation angle of the mechanical foot joint of the underwater multi-foot robot is limited.
Since S and F are real matrices and the values of the elements in the real matrices are independent of time and the state quantity of the pilot, when all the mechanical feet can acquire the S and F information, it means that a part of the mechanical feet can obtain q by equation (3)n+1The purpose of obtaining the information of the pilot is achieved; one common method is to design a corresponding observer by using a dynamic matrix of a target object, and estimate the state information of a pilot by using the state information of adjacent mechanical feet through a distributed observer shown in formula (4). If:
Figure BDA0002097081550000067
obtaining an observation error eta of the observeriV is bounded, represented as:
Figure BDA0002097081550000068
wherein:
Figure BDA0002097081550000069
Figure BDA00020970815500000610
U0is a positive number less than 1, the positive number approaching 0, ReFor the communication delay gain matrix, 1nN being elements all 1 is a column vector,
Figure BDA00020970815500000611
is the estimated error of the nth follower to the pilot,
Figure BDA00020970815500000612
the observer is a neural operator, S and F are real number matrixes, A is an adjacent matrix with weight, the value of an element in the real number matrix is irrelevant to time and the state quantity of a pilot, and a corresponding observer is designed by utilizing a dynamic matrix of a target object.
The fourth concrete implementation mode: the embodiment further describes the multi-foot cooperative control method of the underwater multi-foot robot system in the third embodiment, and the fourth step of performing error constraint on the state information of the pilot obtained in the second step by adopting an obstacle Lyapunov function; the specific method for limiting the error variable within the specified range is as follows:
the rotation angle of the robot mechanical foot joint is limited, and a time-varying boundary is set as follows:kc(t)＝[kc1(t),kc2(t),...,kcn(t)]Tand
Figure BDA0002097081550000071
state quantity q for limiting output rotation angleiThe region of (t) is:
Figure BDA0002097081550000072
kci(t) and
Figure BDA0002097081550000073
respectively an expected motion trail q at the moment t of the ith mechanical footriThe upper and lower boundaries of the upper and lower,kcn(t) and
Figure BDA0002097081550000074
respectively an expected motion trail q at the nth mechanical foot t momentriUpper and lower boundaries of (q)i(t) is the joint rotation angle of the ith mechanical foot at the moment t; r is a real number matrix.
The fifth concrete implementation mode: the fourth embodiment further describes a multi-foot cooperative control method of an underwater multi-foot robot system, and the fifth embodiment uses a neural network technology to process the nonlinear uncertainty in the underwater multi-foot robot system, and the specific method for estimating the unknown parameters is as follows:
using the formula:
Figure BDA0002097081550000075
implementing a non-linear uncertainty quantity for the ith mechanical foot of a multi-legged robotic system
Figure BDA0002097081550000076
Compensation is carried out, wherein WiRepresenting an ideal weighting matrix, Wi TIs an ideal weighting matrix WiIs transferred, riIs a virtual controller of the ith mechanical foot,
Figure BDA0002097081550000077
is riA derivative of (a);
Figure BDA0002097081550000078
is the activation function, ΔiRepresenting an approximation error; and deltaiIs bounded, there is a positive number ΔMiMake | | | Δi||≤△Mi(ii) a For obtaining the ith mechanical foot of underwater multi-foot robot
Figure BDA0002097081550000079
Estimation of (2):
Figure BDA00020970815500000710
wherein,
Figure BDA00020970815500000711
is an ideal weighting matrix WiAn estimate of (d).
The sixth specific implementation mode: in the present embodiment, the method for cooperative control of the multiple legs of the underwater multi-leg robot system according to the fourth embodiment is further described, and in the present embodiment, the adaptive control law of the underwater multi-leg robot control system obtained by the error constraint processing according to the fourth step and the nonlinear uncertainty processing according to the fifth step is:
Figure BDA00020970815500000712
Figure BDA00020970815500000713
wherein, alpha, beta and mu are positive numbers, alpha, beta and mu are infinitely close to zero, K2iIs a symmetric positive definite matrix, Z2iFor virtual trajectory tracking error variable vector, Z2i TIs a vector Z2iThe transpose of (a) is performed,
Figure BDA0002097081550000081
is a matrix
Figure BDA0002097081550000082
Transpose of, Z1iIs the track tracking error variable vector and intermediate variable of the follower to the pilot
Figure BDA0002097081550000083
kaiAnd kbiUpper and lower bounds of the tracking error variable, respectively; wherein,
Figure BDA0002097081550000084
in the embodiment, the influence of the control error on the underwater multi-legged robot is considered, and a time-varying barrier Lyapunov function is used to ensure that the output state quantity meets the set time-varying limit requirement. In consideration of the problems of model uncertainty and unknown dynamic parameters of the underwater multi-legged robot system, the invention provides a distributed self-adaptive control method by using a step-back method. The method is realized based on mathematical lemmas 1 to 4;
introduction 1: if there is a continuous Lyapunov function V (L, t), satisfy
Figure BDA0002097081550000085
Satisfy the requirement of
Figure BDA0002097081550000086
Wherein
Figure BDA0002097081550000087
u is a positive number, and L (t) is given as bounded.
2, leading:
Figure BDA0002097081550000088
when E is a symmetric positive definite matrix, there is an inequality:
Figure BDA0002097081550000089
wherein λminDenotes the minimum eigenvalue, λ, of EmaxRepresents the maximum eigenvalue of E.
And 3, introduction: for a continuously differentiable equation Ψ (t), if Ψ (t) satisfies
Figure BDA00020970815500000810
L Ψ (t) | is less than or equal to phi, wherein phi is a positive number, for
Figure BDA00020970815500000811
Figure BDA00020970815500000812
Is bounded.
And (4) introduction: consider a positive number G ∈ R if x ∈ R and | x! R<If G |, then the following inequality holds:
Figure BDA00020970815500000813
considering an underwater multi-legged robot system containing model uncertainty and external disturbance, under the condition that constant communication delay exists among different mechanical feet of the underwater multi-legged robot, 1), 2) and 3) are established;
1) external disturbance omegaiIs bounded, i.e. there is a positive number γ, such that | | ωi||≤γ。
2) V and
Figure BDA00020970815500000814
are all provided withOf kingdom, S and F are known to all followers.
3) And the directed graph ζ comprises a directed spanning tree.
Firstly, defining a desired motion trail q of the ith mechanical footri：
qri＝Fηi (7)
Variable vector Z of track tracking error of follower to pilot1iComprises the following steps:
Z1i＝qi-qri (8)
virtual trajectory tracking error variable vector Z2iComprises the following steps:
Figure BDA0002097081550000091
wherein r isiIs a virtual controller;
variable vector Z of track tracking error of follower to pilot1iThe time-varying boundary of (1) is:
kai(t)＝qri(t)-kci(t) (10)
Figure BDA0002097081550000092
kaiand kbi(t) is the upper and lower bounds of the tracking error variable at time t,
Figure BDA0002097081550000093
the upper boundary of the expected motion trajectory at the moment t of the ith mechanical foot,kci(t) is the lower boundary of the expected motion trajectory of the ith mechanical foot at time t, qri(t) is the expected motion track of the ith mechanical foot at the time t;
wherein the Lyapunov function V1i(t)：
Figure BDA0002097081550000094
Wherein, kai(t) upper bound of tracking error variable at time t, kbi(t) lower bound for tracking error variable at time t, h (i) defining:
Figure BDA0002097081550000095
variable vector Z of track tracking error of follower to pilot1iCarrying out deformation to ensure that:
Figure BDA0002097081550000096
substitution of formula (14) into (12) yields:
Figure BDA0002097081550000097
obtaining V from formula (15)1i(t) at |. epsiloni|<1, is positive and continuously differentiable; to V1i(t) derivation may give:
Figure BDA0002097081550000098
Figure BDA0002097081550000101
tracking error variable upper bound kaiThe derivative of (a) of (b),
Figure BDA0002097081550000102
for tracking error variables the lower bound kbiA derivative of (a);
virtual controller r of ith mechanical footiComprises the following steps:
Figure BDA0002097081550000103
wherein,
Figure BDA0002097081550000104
comprises the following steps:
Figure BDA0002097081550000105
ensure that
Figure BDA0002097081550000106
In that
Figure BDA0002097081550000107
And
Figure BDA0002097081550000108
remains bounded when all are 0, K1＝diag[k11,k12,...,k1i，...,k1n]Is a symmetric positive definite matrix, k1iIs a gain matrix K1The ith non-0 element in (2) is obtained by substituting equations (17) and (18) into (16):
Figure BDA0002097081550000109
wherein, XiIs defined as:
Figure BDA00020970815500001010
and Xi∈X,X＝diag[X1,X2,....,Xi,....,Xn]；
And (3) obtaining a self-adaptive control law by utilizing a distributed self-adaptive control rule:
Figure BDA00020970815500001011
Figure BDA00020970815500001012
wherein, alpha, beta and mu are positive numbers, alpha, beta and mu are infinitely close to zero, K2iIs a symmetric positive definite matrix, Z2iFor virtual trajectory tracking error variable vector, Z2i TIs a vector Z2iThe transpose of (a) is performed,
Figure BDA00020970815500001013
is a matrix
Figure BDA00020970815500001014
Transpose of, Z1iIs the variable vector of the track tracking error of the follower to the pilot,
Figure BDA00020970815500001015
kaiand kbiUpper and lower bounds of the tracking error variable, respectively; wherein,
Figure BDA00020970815500001016
under the action of a distributed observer (5) and distributed adaptive control laws (21) and (22), an auxiliary variable Z1iThe final is consistently bounded and the tracking error between each mechanical foot and the pilot is bounded. At the same time, the state quantity q is outputi(t) satisfies a time-varying output constraint, i.e.
Figure BDA0002097081550000111
Proof of 1), 2) and 3): equation (9) is derived for t:
Figure BDA0002097081550000112
the formula (23) is substituted into (1) to obtain:
Figure BDA0002097081550000113
wherein,
Figure BDA0002097081550000114
tracking Z of error variable vector for virtual trajectory2iA derivative of (a);
Figure BDA0002097081550000115
because M isi(qi),
Figure BDA0002097081550000116
gi(qi) All unknown, there is a non-linear uncertainty in the underwater multi-legged robotic system represented by equation (1). Considering that neural networks have good approximation ability to unknown nonlinear functions, they are often used to deal with uncertainty problems in nonlinear systems.
Thus, non-linear uncertainty for underwater multi-legged robotic systems using neural network techniques
Figure BDA0002097081550000117
Carrying out self-adaptive compensation, and specifically comprising the following steps:
Figure BDA0002097081550000118
wherein WiRepresents an ideal weighting matrix, phiiThe function is the activation function, ΔiRepresents the approximation difference, ΔiIs bounded, i.e. there is a positive number ΔMiMake | | | Δi||≤△MiFor the ith mechanical foot of the underwater multi-foot robot,
Figure BDA0002097081550000119
the estimate of (d) is written as:
Figure BDA00020970815500001110
wherein,
Figure BDA00020970815500001111
is WiA matrix of estimated values of (a) is,
Figure BDA00020970815500001112
is a matrix
Figure BDA00020970815500001113
The transpose of (a) is performed,
Figure BDA00020970815500001114
is an activation function.
Based on neural network technology, selecting distributed adaptive control laws (21) and (22) and barrier Lyapunov function V2i；
Figure BDA00020970815500001115
Wherein,
Figure BDA00020970815500001116
Figure BDA00020970815500001117
is a matrix
Figure BDA00020970815500001118
Transpose of, Z1iError vector for tracking the trajectory of the follower to the pilot, Z2iFor virtual tracking error vectors, Z2i TIs a vector Z2iThe transpose of (a) is performed,
Figure BDA00020970815500001119
is composed of
Figure BDA00020970815500001120
The trace of (c).
To V2iDerivation, combining semi-symmetry and distributed adaptive control laws:
Figure BDA00020970815500001121
and weight estimation adaptation law:
Figure BDA0002097081550000121
K2iis a symmetric positive definite matrix, resulting in:
Figure BDA0002097081550000122
wherein phi isiIs an activation function, Z2i TTransposing of virtual tracking error matrix, phiiIs the activation function, ΔiIs the error of the estimation that is,
Figure BDA0002097081550000123
is to represent
Figure BDA0002097081550000124
Is that
Figure BDA0002097081550000125
The trace of (c). Because of DeltaiAnd ωiIs bounded, so there is a positive number
Figure BDA0002097081550000126
Such that:
Figure BDA0002097081550000127
simultaneously obtaining:
Figure BDA0002097081550000128
wherein,
Figure BDA0002097081550000129
is | | | Δi+ωiThe upper boundary of | is positive, and σ is infinite and approaches to 0;
in the formula (29), the reaction mixture is,
Figure BDA00020970815500001210
is a scalar, then the equation
Figure BDA00020970815500001211
If true;
the operational properties according to the matrix trace are:
Figure BDA00020970815500001212
substituting equations (19), (30) and (31) into (29) can yield:
Figure BDA00020970815500001213
λmin(K2i) Is a symmetric positive definite matrix K2iIs written as:
Figure BDA00020970815500001214
wherein:
Figure BDA00020970815500001215
Figure BDA00020970815500001216
Figure BDA00020970815500001217
is MiIs measured.
According to the introduction 1-4, V is2iSatisfying the full final consistent bounded, simultaneous integration on both sides of equation (33) can result:
Figure BDA00020970815500001218
let kappa1＞0，υ1> 0, giving:
Figure BDA0002097081550000131
V2i(t) is the Lyapunov function of the obstacle at time t, ρ is a constant less than 1, and is known from equation (28):
V1i≤V2i (38)
then there are:
Figure BDA0002097081550000132
substituting equations (13) and (14) into (39) can yield:
Figure BDA0002097081550000133
from (7), it can be seen that:
qi-qn+1＝qi-Fηi+Fηi-qn+1＝qi-Fηi+F(ηi-v) (41)
from formulas (6), (40) and (41):
Figure BDA0002097081550000134
the tracking error of the ith mechanical foot of the underwater multi-foot robot to a pilot is bounded, and the upper boundary value is shown as the formula (42).
From formulas (4), (8), (10), (11) and (40):
Figure BDA0002097081550000135
the output state quantity q is known from the equation (43)i(t) satisfying a time-varying output constraint.
The invention has the characteristics that: the method comprehensively considers the conditions of constant communication delay among different mechanical feet of the multi-legged robot system and generalized interference of a multi-legged robot dynamic model, and simultaneously introduces a logarithmic BLF (barrier Lyapunov function) to ensure that the track tracking error of the system always meets the set error limit requirement; only the communication topology among different mechanical feet is required to be a general directed graph, only part of followers can obtain the information of the pilot, and the communication burden caused by the overall information knowledge is avoided; in the research, an input signal source is selected as a virtual navigator, so that the navigator is more flexibly changed, and the requirement of the robot on the motion flexibility is met.
Concrete simulation example
Firstly, setting simulation parameters:
in order to verify the effectiveness of the distributed adaptive cooperative tracking control law provided by the invention. The simulation experiment is carried out by taking an 8-foot underwater multi-foot robot as an example. As the 8-foot underwater multi-foot robot uses double 4-foot gaits in motion, as shown in figure 2, the underwater multi-foot robot can ensure that the 8-foot underwater multi-foot robot can still provide four-foot support when half of the walking feet of the 8-foot underwater multi-foot robot are lifted off the ground. Meanwhile, because of the structural symmetry of the 8-foot underwater multi-foot robot, 4 mechanical feet which move together with the 8-foot underwater multi-foot robot can be researched, a directed communication network is formed by 1 two-degree-of-freedom virtual navigator and 4 two-degree-of-freedom underwater multi-foot robot mechanical feet (followers), wherein the numbers 1-4 represent four mechanical feet (followers) of the 8-foot underwater multi-foot robot shown in the figure 1, the number 5 represents a virtual navigator, and the communication topological relation is shown in the figure 3.
The kinetic equation of the ith mechanical foot of the 8-foot underwater multi-foot robot can be expressed as follows:
Figure BDA0002097081550000141
in the formula:
qi＝[qi1,qi2]T (45)
wherein q isi1,qi2Respectively represents the rotation angles of two joints of a mechanical foot of the underwater multi-foot robot.
Figure BDA0002097081550000142
Figure BDA0002097081550000143
Figure BDA0002097081550000144
Figure BDA0002097081550000145
Wherein: xii1＝Ji1+mi2li1 2,Ξi2＝0.25mi2li2 2+Ji2,Ξi3＝0.5mi2li1li2,Ξi4＝(0.5mi1+mi2)li1,
Ξi5＝0.5mi2li2,g＝9.8m/s2Representing the gravitational acceleration. As shown in FIG. 4, mi1And mi2Respectively representing the masses of the two links at the joint of the mechanical foot joint 2,/i1And li2Respectively, the length of the connecting rod of each mechanical foot of the underwater multi-foot robot. J. the design is a squarei1And Ji2Representing the moment of inertia at the joint. The specific parameters of the mechanical foot are shown in table 1:
table 1: specific parameters of the mechanical foot
Figure BDA0002097081550000146
Figure BDA0002097081550000151
The time-varying output limit is set as follows:
Figure BDA0002097081550000152
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)
wherein the tracking error Z1iThe boundary value of the tracking error of (2) is expressed as:
kai(t)＝qri(t)-k ci(t) (52)
Figure BDA0002097081550000153
therefore, the following steps are carried out:
-kai(t)<Z1i(t)<kbi(t) (54)
the angle of the mechanical foot of the underwater multi-foot robot is set as follows:
q11(0)＝π/5,q12(0)＝-π/3,q21(0)＝2π/5,q22(0)＝-π/6,q31(0)＝3π/5,
q32(0)＝π/6,q41(0)＝4π/5,q42(0)＝π/3,
Figure BDA0002097081550000154
for the ith follower (i ═ 1.., 4), the activation equation for the neural network system can be written as:
φi(z)＝[φi1(z),...,φi6(z)]T (55)
the gaussian equation is chosen as the activation function, which is of the form:
Figure BDA0002097081550000155
wherein,
Figure BDA0002097081550000156
it is assumed that all followers use the same activation equation. c. CijIs uniformly distributed in [ -5,5 [ ]]4×[-0.5,0.5]4Center of acceptance domain of (c), σijExpressing the width of the Gaussian equation, defining σ ij2, weighting matrix
Figure BDA0002097081550000157
Is set to
Figure BDA0002097081550000158
Target trajectory of virtual pilot:
Figure BDA0002097081550000159
Figure BDA00020970815500001510
wherein q is51_amp＝π/6,
Figure BDA00020970815500001511
q51_bias＝π/2,q52_amp＝2π/3,
Figure BDA00020970815500001512
q52_bias＝0，ω＝0.1π。
State quantity q of virtual pilot5Expressed as:
Figure BDA0002097081550000161
q5＝Fv (60)
wherein:
Figure BDA0002097081550000162
ω＝0.1π (62)
Figure BDA0002097081550000163
Figure BDA0002097081550000164
considering the requirements of the 8-foot underwater multi-legged robot in practical engineering, the adaptive control law tau is generally required to be controllediIs added with a boundary limit to achieve better control, using the following limit equation for τiLimiting the amplitude:
Figure BDA0002097081550000165
wherein, tauimaxIs a positive number, select τimax＝50。
The time for selecting the communication delay in the control algorithm is T0.2 s. And (3) analyzing a simulation result of the control algorithm: in the designed control algorithm, the control parameter is selected to be k1i＝10,K2i＝20I2,γ＝1,v＝10,I2Is an identity matrix. The simulation results are shown in FIGS. 5 to 18.
Fig. 5 and 6 show the state quantity change of the virtual pilot and each mechanical foot, from which it can be seen that each mechanical foot can effectively track the pilot after about 5 s. From fig. 7 and 8, the trajectory tracking error Z for the pilot at the two joints of each mechanical foot can be seen1iAnd Z2iConverged to a small region around 0 after about 3s, and stabilized Z1iDoes not exceed 0.1, Z2iDoes not exceed 0.05. FIGS. 8 and 9 show that the control laws for each mechanical foot input are continuousAnd the fluctuation range does not exceed 10, fig. 10 to 18 show that the trajectory tracking error of each mechanical foot to the pilot meets the set time-varying output limit condition. The simulation process effectively illustrates the effectiveness of the present invention.
Although the embodiments of the present invention have been described above, the descriptions are only for the convenience of understanding the present invention and are not intended to limit the present invention. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (2)

1. A multi-foot cooperative control method of an underwater multi-foot robot system is characterized by comprising the following specific steps:
firstly, establishing a dynamic model of all mechanical feet of the underwater multi-foot robot; obtaining an underwater multi-legged robot control system;
step two, establishing a communication relation directed topology structure diagram among all mechanical feet in the underwater multi-foot robot system in the step one; the root node of the directed topology structure chart is a pilot, each other node is a mechanical foot, the pilot is a control signal source, and one mechanical foot is a follower;
estimating the state information of the navigator, which is obtained by all the nodes, by using the distributed observer and the directed topology structure chart in the step two to obtain the state information of the navigator;
the specific method for obtaining the state information of the pilot comprises the following steps:
by the formula:
Figure FDA0003376754570000011
qn+1＝Fv (3)
obtaining generalized coordinates q of the pilotn+1Which isWhere v is the pilot's auxiliary state variable, v ∈ Rm，
Figure FDA0003376754570000012
Is the derivative of v, S and F are constant real matrices, S ∈ Rm×m，F∈Rn×m；
The distributed observer estimates the state information of the navigator:
Figure FDA0003376754570000013
wherein,
Figure FDA0003376754570000014
is a matrix
Figure FDA0003376754570000015
The elements of (a) and (b),
Figure FDA0003376754570000016
is a contiguous matrix, ηjFor the estimate of the jth follower's position information to the pilot,
Figure FDA0003376754570000017
an estimate of the speed information of the jth follower over the pilot, j 1iRepresenting the estimate of the i-th follower for v, ηi∈Rm，
Figure FDA0003376754570000018
Is the estimated value of the speed information of the ith follower to the pilot, T represents the communication delay between the adjacent followers i and j, and T is time;
step four, carrying out error constraint on the state information of the navigator obtained in the step three by adopting an obstacle Lyapunov function; limiting the error variable within a specified range;
the specific method for limiting the error variable within the specified range is as follows:
the rotation angle of the robot mechanical foot joint is limited, and a time-varying boundary is set as follows:
Figure FDA0003376754570000019
and
Figure FDA00033767545700000110
state quantity q for limiting output rotation angleiThe region of (t) is:
Figure FDA0003376754570000021
kci(t) and
Figure FDA0003376754570000022
respectively an expected motion trail q at the moment t of the ith mechanical footriThe upper and lower boundaries of the upper and lower,kcn(t) and
Figure FDA0003376754570000023
respectively an expected motion trail q at the nth mechanical foot t momentriUpper and lower boundaries of (q)i(t) is the joint rotation angle of the ith mechanical foot at the moment t; r is a real number matrix;
processing the nonlinear uncertainty in the underwater multi-legged robot system by utilizing a neural network technology; the estimation of unknown parameters is realized;
the specific method for estimating the unknown parameters comprises the following steps:
using the formula:
Figure FDA0003376754570000024
implementing a non-linear uncertainty quantity for the ith mechanical foot of a multi-legged robotic system
Figure FDA0003376754570000025
Compensation is carried out, wherein qiThe joint rotation angle of the ith mechanical foot is i ═ 1, 2.. times, n }, n is a positive integer,
Figure FDA0003376754570000026
the angular velocity of the joint of the ith mechanical foot, an
Figure FDA0003376754570000027
WiRepresenting an ideal weighting matrix, Wi TIs an ideal weighting matrix WiIs transferred, riIs a virtual controller of the ith mechanical foot,
Figure FDA0003376754570000028
is riA derivative of (a);
Figure FDA0003376754570000029
is an activation function, ΔiRepresenting an approximation error; and ΔiIs bounded and there is a positive number deltaMiSo that | | | Δi||≤ΔMi(ii) a For obtaining the ith mechanical foot of underwater multi-foot robot
Figure FDA00033767545700000210
Estimated value of (a):
Figure FDA00033767545700000211
wherein,
Figure FDA00033767545700000212
is an ideal weighting matrix WiIs determined by the estimated value of (c),
Figure FDA00033767545700000213
is a matrix
Figure FDA00033767545700000214
The transpose of (a) is performed,
Figure FDA00033767545700000215
is an activation function;
step six, obtaining the self-adaptive control law of the underwater multi-foot robot control system according to the error constraint in the step four and the nonlinear uncertainty processing in the step five, and realizing the multi-foot cooperative control of the underwater multi-foot robot system;
the self-adaptive control law for obtaining the underwater multi-legged robot control system specifically comprises the following steps:
Figure FDA00033767545700000216
Figure FDA00033767545700000217
wherein, alpha, beta and mu are positive numbers, alpha, beta and mu are infinitely close to zero, K2iIs a symmetric positive definite matrix, Z2iFor virtual trajectory tracking error variable vector, Z2i TIs a vector Z2iTranspose of, Z1iIs the track tracking error variable vector and intermediate variable of the follower to the pilot
Figure FDA0003376754570000031
kaiAnd kbiUpper and lower bounds of the tracking error variable, respectively; wherein,
Figure FDA0003376754570000032
2. the method for cooperative control of multiple feet of an underwater multi-foot robot system according to claim 1, wherein the method for dynamic models of all mechanical feet of the underwater multi-foot robot in the first step is the same, and the dynamic model of the ith mechanical foot is taken as an example for explanation, and specifically:
Figure FDA0003376754570000033
wherein q isiThe joint rotation angle of the ith mechanical foot is i ═ 1, 2.. times, n }, n is a positive integer,
Figure FDA0003376754570000034
is the joint rotation angular velocity of the ith mechanical foot,
Figure FDA0003376754570000035
is the angular acceleration of the joint of the ith mechanical foot, an
Figure FDA0003376754570000036
RpIs a p-dimensional real column vector, τiIndicating control force rejection, τ, of the input ith mechanical footi∈Rp，Mi(qi) Representing a symmetric positive definite inertial matrix, Mi(qi)∈Rp×p，Rp×pIs a matrix of real numbers of p rows and p columns,
Figure FDA0003376754570000037
the eccentric force of the ith mechanical foot is shown,
Figure FDA0003376754570000038
gi(qi) Denotes the gravity of the ith mechanical foot, gi(qi)∈Rp，ωiRepresenting external disturbances, ωi∈RpThe external disturbance comprises environmental disturbance and external noise; wherein the inertia matrix M is positively determined symmetricallyi(qi) Eccentric force
Figure FDA0003376754570000039
And gravity gi(qi) Are unknown parameters.
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