CN112148036A - Bilateral tracking control method of fixed time estimator of networked robot system - Google Patents

Bilateral tracking control method of fixed time estimator of networked robot system Download PDF

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CN112148036A
CN112148036A CN202010953886.7A CN202010953886A CN112148036A CN 112148036 A CN112148036 A CN 112148036A CN 202010953886 A CN202010953886 A CN 202010953886A CN 112148036 A CN112148036 A CN 112148036A
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robot
fixed time
time
estimator
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CN112148036B (en
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梁昌铎
苏鹏
葛明峰
丁腾飞
葛子月
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China University of Geosciences
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Abstract

The invention discloses a bilateral tracking control method of a fixed time estimator of a networked robot system, which has the beneficial effect of realizing bilateral tracking control of the networked robot system in fixed time. Firstly, the bilateral tracking control method is popularized to a networked robot system, and is mainly applied to the practical problems that the networked robot system is divided into two groups to execute opposite tasks and the like; secondly, the invention provides a fixed time control framework for researching a complex robot system, so that the time required for completing a control target has a clear and knowable range, and the control margin is increased; finally, a robot dynamic model with external disturbance and uncertain control parameters is considered, and the condition that a networked robot system is possibly interfered under the working condition is simulated, so that the control effect is more practical.

Description

Bilateral tracking control method of fixed time estimator of networked robot system
Technical Field
The invention relates to the field of intelligent control of robots, in particular to a bilateral tracking control method of a fixed time estimator of a networked robot system.
Background
In recent years, research on cooperative control of networked robot systems has been rapidly progressing. Networked robotic systems are a group of controllable autonomous robots that can accomplish single or multiple complex tasks through local communication. Networked robotic systems can accomplish more complex tasks and use a more efficient and flexible approach than a single robot. Therefore, the control of the networked robot system is widely applied to various fields such as industrial production, medical treatment, aerospace, deep sea exploration, military and the like.
In addition, the distributed control algorithm of multi-agent using networked robot system as carrier has also been widely used, among which the control methods widely studied by the scholars include single-target and multi-target tracking control method, formation and contained control method, bilateral tracking control method including competition and cooperation relation, etc. In a real networked robot system, the connection between the robots is in a cooperative and competitive relationship. Therefore, the control method considered by the patent effectively solves the bilateral tracking problem and has better practical application value.
In a real networked system, a robot is usually required to complete a task within a set time range to improve the efficiency and accuracy of task completion. Therefore, the convergence time required for control is also an important control performance. Among them, the limited time control and the fixed time control are developed most rapidly. The difference between the two is that the convergence time of the finite time control depends on the initial value, whereas the convergence time of the fixed time control is not affected by the initial value. This patent uses a control method based on a fixed time estimator. A method of time base signal generator is used in the estimation layer. Convergence over a fixed time is achieved by adjusting the gain of the generator with the time base signal. The method can control the robot to complete the task within a determined time range, and increases the control margin.
Therefore, in combination with the above requirements of industrial practical application, it is of great significance to design a bilateral tracking control method based on a fixed time estimator for a networked robot system.
Disclosure of Invention
The technical problem to be solved by the present invention is to provide a bilateral tracking control method for a fixed time estimator of a networked robot system, aiming at the bilateral tracking problem of the cooperation and competition relationship between robots which are less considered in the prior art and the problem of completing a control task within a fixed time.
The technical scheme adopted by the invention for solving the technical problems is as follows: a bilateral tracking control method of a fixed time estimator of a networked robot system is constructed, and the bilateral tracking control method comprises the following steps:
s1, performing dynamic modeling on the ith robot in the networked robot system comprising N robots, and constructing a dynamic model:
Figure BDA0002677943040000021
wherein M isi(qi) A positive definite inertial matrix is represented,
Figure BDA0002677943040000022
denotes the Coriolis centrifuge matrix, gi(qi) Representing a matrix of gravitation, di(t) denotes input disturbance, τiRepresents a control input; q. q.si、
Figure BDA0002677943040000023
And
Figure BDA0002677943040000024
respectively representing the position, the speed and the acceleration of the ith robot in a joint space; t represents time; i represents the number of the robot;
setting the (i + 1) th robot as a tracking target, wherein the state information of the tracking target comprises a position q0Velocity v0And acceleration u0Satisfy the following requirements
Figure BDA0002677943040000025
S2, K. FDescribing information interaction relations among robots by using a symbolic topological graph G { V, E, A }, wherein V { 1., N } represents a vertex set consisting of N robots, and the vertices are used for representing the robots;
Figure BDA0002677943040000031
representing a set of edges connecting any two robots, and being used for representing the direction of information interaction between the robots; a ═ aij]A weight matrix composed of adjacent elements and used for representing the value of mutual information between robots; if the j-th robot and the i-th robot have interaction and cooperative relationship, aijIs greater than 0; if the j-th robot and the i-th robot have interaction and are in competition relationship, aij< 0, otherwise, aij0; and the ith robot itself has no self-circulation connection, i.e. aii＝0；
Define diagonal matrix D ═ diag (D)1,d2,...dN),diThe method belongs to +/-1 }, meets the condition that DAD is a semi-positive definite matrix, and divides all robots into two parts to respectively execute opposite tasks; when the robot belongs to the first group, d i1 is ═ 1; when the robot belongs to the second group, di-1; adding a tracking target robot into the directed symbol topological graph G, and defining the graph as an augmented graph G; using the matrix B ═ ai0]A value representing the information interaction between the virtual leader and the robot, a if there is an interaction between the virtual leader and the roboti0Is greater than 0; if there is no interaction between the virtual leader and the robot, then ai0＝0。
S3, designing a distributed controller based on a fixed time estimator according to the robot dynamics model established in the step S1 and the directed symbolic topological graph established in the step S2;
the distributed controller comprises a local control layer with fixed time and an estimation layer with fixed time;
the estimation layer of the fixed time is used for obtaining estimation values of the joint position and the joint speed of the ith robot, and the estimation values are obtained through the position information, the movement speed information and the information interaction numerical value of the ith robot and the adjacent robot;
the estimated values of the joint position and the velocity of the ith robot are further input into a local control layer of the fixed time, and the control layer is used for enabling the actual joint position and the actual velocity of the ith robot to converge to the estimated values obtained by the fixed time estimation layer.
And S4, a double-layer control structure consisting of an estimation layer with fixed time and a local control layer with fixed time, and controlling the actual joint position and speed of the ith robot to reach the expected joint position and speed, thereby realizing the bilateral tracking control of the networked robot system.
Further, for the estimation layer of the fixed time in step S3, the corresponding control gain η of the fixed time is designed1(t)、η2(t), the fixed time is mainly realized by a time-base signal generator TBG, and the method comprises the following two steps:
designing a TBG function:
Figure BDA0002677943040000041
the TBG function is a piecewise time varying function, tfIs an artificially set convergence time point, and the convergence time required by the function control is designed to be tfWithin;
designing a control gain function comprising a TBG function:
Figure BDA0002677943040000042
Figure BDA0002677943040000043
wherein the content of the first and second substances,1，2is a normal number and satisfies 0<1,2≤1，MD＝DAD+B，InIs an n-dimensional identity matrix, H is a positive definite diagonal matrix, such that MDH is a positive definite matrix, HmaxRepresents the maximum of the diagonal elements of the H matrix,
Figure BDA0002677943040000044
representation matrix
Figure BDA0002677943040000045
The minimum eigenvalue of (c).
Further, for the position q of the i-th robot in the joint space required in step S1iAnd velocity
Figure BDA0002677943040000046
Designing a fixed-time state estimator in the mathematical form:
Figure BDA0002677943040000047
wherein the content of the first and second substances,
Figure BDA0002677943040000048
estimates of positions in joint space for robots i and j;
Figure BDA0002677943040000049
estimates of the velocity in joint space for robots i and j, c1、c2Constant control gain, control gain of fixed time being eta respectively1(t) and η2(t)；
Position error of state estimator combined with fixed time
Figure BDA0002677943040000051
And speed error
Figure BDA0002677943040000052
Wherein
Figure BDA0002677943040000053
The mathematical form of the fixed-time state estimator translates into:
Figure BDA0002677943040000054
further, a slip form surface S of a fixed time is designed for the dynamic model obtained in step S1iController tau to assist in designing joint spaceiSaid slip form surface s of fixed timeiThe mathematical expression of (a) is:
Figure BDA0002677943040000055
wherein the position error between the estimated value and the true value
Figure BDA0002677943040000056
And speed error
Figure BDA0002677943040000057
Are respectively as
Figure BDA0002677943040000058
Figure BDA0002677943040000059
K1i、K2iRespectively, are positive definite diagonal matrices,
Figure BDA00026779430400000510
n is the number of joints of the robot,
Figure BDA00026779430400000511
0<Δ≤1，0<p<1，
Figure BDA00026779430400000512
Figure BDA00026779430400000513
l>1。
further, the controller τ of the joint space of the fixed time is designed for the dynamic model obtained in step S1iThe mathematical expression of (a) is:
Figure BDA00026779430400000514
wherein, tauiIs composed of three parts: tau isi0An equivalent control term, τ, associated with the robot dynamics modeli1Non-linear term, τ, representing the effect of fixed time controli2Representing a control term for counteracting the external disturbance;
Figure BDA0002677943040000061
an estimated value of joint velocity obtained for a fixed time controller; mi0、Ci0、gi0Are respectively Mi、Ci、giThe expected value of (d); k. b0、b1、b2Is a normal number;
Figure BDA0002677943040000062
is composed of
Figure BDA0002677943040000063
Function to variable
Figure BDA0002677943040000064
And converts the form of the column vector into the form of a diagonal matrix,
Figure BDA0002677943040000065
is composed of
Figure BDA0002677943040000066
Function to variable
Figure BDA0002677943040000067
And converts the form of the column vector into the form of a diagonal matrix.
Further, in step S3, specifically, the method includes:
s31, and controller tau of joint space with fixed timeiSubstituting the kinetic model constructed in S1, and adding the fixed-time state estimator, a closed-loop system is obtained as follows:
Figure BDA0002677943040000068
wherein, K0iIs a positive definite diagonal matrix, and r is a constant satisfying r>1, rho (t) is a disturbance term and an uncertain term in a robot dynamic model,
Figure BDA0002677943040000069
ΔMi(qi)、
Figure BDA00026779430400000610
Δgi(qi) Are respectively Mi(qi)、
Figure BDA00026779430400000611
gi(qi) Uncertainty of term, di(t) is a perturbation term;
s32, analyzing the stability of the state estimator with fixed time in the closed loop system in S31;
constructing a Lyapunov function for errors in position and velocity of the robot in the closed-loop system:
Figure BDA00026779430400000612
Figure BDA00026779430400000613
wherein the content of the first and second substances,
Figure BDA0002677943040000071
Figure BDA0002677943040000072
n is the number of robots in the networked robot system;
and deriving the time of the Lyapunov by:
Figure BDA0002677943040000073
Figure BDA0002677943040000074
therefore, the error of the position and the speed obtained by the correlation theory of the TBG function can be converged into an adjustable boundary within a set time range, and the following conditions are met:
Figure BDA0002677943040000075
Figure BDA0002677943040000076
Figure BDA0002677943040000077
Figure BDA0002677943040000078
wherein the content of the first and second substances,1、2for autonomous setting of a constant, T, for adjusting the error rangef1、Tf2Respectively setting time in a TBG function in the position estimator and the speed estimator, so that the control time is in a set time range, namely the stability analysis of the fixed time state estimator is completed;
s33, analyzing the stability of a robot dynamic state controller (a controller of a joint space with fixed time) in the closed-loop system S31;
for a fixed time synovial surface siConstructing the Lyapunov function
Figure BDA0002677943040000079
And derived along the trajectory of said closed loop system
Figure BDA00026779430400000710
Reissue to order
Figure BDA00026779430400000711
To obtain
Figure BDA00026779430400000712
Substituted into the above formula to obtain
Figure BDA0002677943040000081
According to the theory of fixed time, obtain
Figure BDA0002677943040000082
Within, s i0; the dynamics of the error system of joint position and velocity are translated into
Figure BDA0002677943040000083
Further, the discussion is divided into two cases
Figure BDA00026779430400000819
The time required to converge to the set small constant Δ.
When in use
Figure BDA0002677943040000084
Selecting a Lyapunov function
Figure BDA0002677943040000085
Derived from time
Figure BDA0002677943040000086
When in use
Figure BDA0002677943040000087
Selecting the Lyapunov function Vca2In the same way, the time is derived
Figure BDA0002677943040000088
Thus, it is derived from
Figure BDA0002677943040000089
Within time, error
Figure BDA00026779430400000810
Further, in step S4, the analysis of the fixed time state estimator and the analysis of the robot dynamic state controller (i.e. the controller of the joint space at the fixed time) are respectively performed on the closed-loop system according to the double-layer control structure composed of the fixed time estimation layer and the fixed time local control layer, so as to obtain the fixed time T at the set time Tf1+Tf2+Ts+TeWhen the robot i belongs to the first group: position state
Figure BDA00026779430400000811
I.e. qiApproach to
Figure BDA00026779430400000812
Approaches to q0Velocity state
Figure BDA00026779430400000813
I.e. viApproach to
Figure BDA00026779430400000814
Approaches to v0(ii) a When robot i belongs to the second group: position state
Figure BDA00026779430400000815
I.e. qiApproach to
Figure BDA00026779430400000816
Approaches to-q0Velocity state
Figure BDA00026779430400000817
I.e. viApproach to
Figure BDA00026779430400000818
Approaches to-v0Therefore, bilateral tracking control of the networked robot system is realized.
In summary, the bilateral tracking control method of the fixed time estimator of the networked robot system according to the present invention is implemented by a layered control structure, and defines an estimation layer of fixed time and a local control layer for the robot, and in consideration of the relationship between the robots, which is both cooperative and competitive, in the estimation layer of fixed time, the estimation layer is used to simplify the complex data required by the control process and to achieve convergence within the fixed time, thereby increasing the control margin; the local control layer is designed, the robot can track an estimated value within a fixed time through the auxiliary design of the slide film surface, the dynamic model of the corresponding robot is accurately measured under the condition of combined action of the estimation layer and the control layer, and the measurement accuracy is further improved through the stability analysis of the steps S32 and S33, the control cost of a multi-robot system is reduced, and the working efficiency is improved.
The bilateral tracking control method based on the fixed time estimator and suitable for the networked robot system has the following beneficial effects that:
1. the practical problems of change of model parameters, external disturbance and the like in a robot dynamics model are considered, and the practicability of research results is higher;
2. the coexistence of cooperation and competition is considered in the information interaction between the robots, so that the bilateral tracking control method has universal applicability;
3. the designed control algorithm comprises two layers of control, the control cost of the multi-robot system is reduced, and the working efficiency is improved.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a control flow diagram of a control method proposed by the present invention;
FIG. 2 is a directed topology network of a networked robotic system to which the present invention relates;
FIG. 3 is a simplified model of the physical structure of a robot contemplated by the present invention;
FIG. 4 is a trajectory tracking diagram of actual and estimated positions of a networked robotic system in accordance with the present invention;
FIG. 5 is a trajectory tracking diagram of actual and estimated velocities of a networked robotic system in accordance with the present invention;
FIG. 6 is a graph of actual position tracking errors for a networked robotic system in accordance with the present invention;
FIG. 7 is a graph of position tracking error for an estimator of a networked robotic system in accordance with the present invention;
FIG. 8 is a diagram of position tracking errors at the local control layer of a networked robotic system in accordance with the present invention;
fig. 9 is a graph of actual velocity tracking error for the networked robotic system of the present invention.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
Please refer to fig. 1, which is a control flow chart of the control method of the present invention, specifically including the following steps:
s1, performing dynamics modeling on the ith robot (specifically, refer to fig. 3) in the networked robot system including N robots, and constructing a dynamics model:
Figure BDA0002677943040000101
wherein M isi(qi) A positive definite inertial matrix is represented,
Figure BDA0002677943040000105
denotes the Coriolis centrifuge matrix, gi(qi) Representing a matrix of gravitation, di(t) denotes input disturbance, τiRepresents a control input; q. q.si、
Figure BDA0002677943040000102
And
Figure BDA0002677943040000103
respectively representing the position, the speed and the acceleration of the ith robot in a joint space; t represents time; i represents the ith robot which is the index mark of the robot;
the following table shows the physical parameters of the arm, k ∈ {1,2 }.
Figure BDA0002677943040000104
Figure BDA0002677943040000111
These physical parameters are substituted into the intermediate variables of the kinetic model, the expressions for the intermediate variables are given as follows:
Figure BDA0002677943040000112
p2＝m2r1r2，
Figure BDA0002677943040000113
p4＝p3+I2，p5＝(m1+m2)r1g，p6＝m2r2g。
further solving a matrix required by the dynamic model, and giving an expression of the matrix:
Figure BDA0002677943040000114
Figure BDA0002677943040000115
Figure BDA0002677943040000116
the state information of the tracking target includes a position q0Velocity v0And acceleration u0Satisfy the following requirements
Figure BDA0002677943040000117
Figure BDA0002677943040000118
Selecting
Figure BDA0002677943040000119
S2, establishing a directed symbol topological graph G ═ { V, E, A } to describe information interaction relations among the robots, wherein V ═ 1.. and N } represents a vertex set consisting of N robots, and the vertices are the robots;
Figure BDA00026779430400001110
representing a set of edges connecting any two robots, and being used for representing the direction of information interaction between the robots; a ═ aij]A weight matrix composed of adjacent elements and used for representing the value of mutual information between robots; if the j-th robot and the i-th robot have interaction and cooperative relationship, aijIs greater than 0; if the j-th robot and the i-th robot have interaction and are in competition relationship, aij< 0, otherwise, aij0; and the ith robot itself has no self-circulation connection, i.e. aii0. Define diagonal matrix D ═ diag (D)1,d2,...dN),diE { ± 1}, making DAD a semi-positive definite matrix, and dividing all robots into two parts to perform the opposite task, respectively. When the robot belongs to the first group, d i1 is ═ 1; when the robot belongs to the second group, diIs-1. Defined as an augmented graph
Figure BDA0002677943040000121
And adding a virtual leader, namely a tracking target, into the directed symbol topological graph G. Using the matrix B ═ ai0]A numerical value representing the information interaction between the virtual leader and the robot. If there is an interaction between the virtual leader and the robot, then ai0Is greater than 0; if there is no interaction between the virtual leader and the robot, then ai00. Referring to FIG. 2, robots 1-8 are divided into two groups and join virtual leader robot 0, robots 1-4 belong to sub-network A, robots 5-8 belong to sub-network B, robots are in cooperative relationship within the group and in competitive relationship between the groups.
S3, designing a distributed controller based on a fixed time estimator according to the robot dynamics model established in the step S1 and the directed symbolic topological graph established in the step S2;
the distributed controller comprises a local control layer with fixed time and an estimation layer with fixed time;
the estimation layer of the fixed time is used for obtaining estimation values of the joint position and the joint speed of the ith robot, and the estimation values are obtained through the position information, the movement speed information and the information interaction numerical value of the ith robot and the adjacent robot;
the estimated values of the joint position and the velocity of the ith robot are further input into a local control layer of the fixed time, and the control layer is used for enabling the actual joint position and the actual velocity of the ith robot to converge to the estimated values obtained by the fixed time estimation layer.
And S4, a double-layer control structure consisting of an estimation layer with fixed time and a local control layer with fixed time, and controlling the actual joint position and speed of the ith robot to reach the expected joint position and speed, thereby realizing the bilateral tracking control of the networked robot system.
Designing the corresponding fixed-time control gain eta for the fixed-time estimation layer in the step S31(t)、η2(t) implementation by means of a time-base signal generator TBGThe timing is specifically divided into two steps:
designing a TBG function:
Figure BDA0002677943040000131
the TBG function is a piecewise time varying function, tfIs an artificially set convergence time point, and the convergence time required by the function control is designed to be tfWithin;
designing a control gain function comprising a TBG function:
Figure BDA0002677943040000132
Figure BDA0002677943040000133
wherein the content of the first and second substances,1，2is a normal number and satisfies 0<1,2≤1，MD＝DAD+B，InIs an n-dimensional identity matrix, H is a positive definite diagonal matrix, such that MDH is a positive definite matrix, HmaxRepresents the maximum of the diagonal elements of the H matrix,
Figure BDA0002677943040000134
representation matrix
Figure BDA0002677943040000135
The minimum eigenvalue of (c).
For the position q of the i-th robot in the joint space required in step S1iAnd velocity
Figure BDA00026779430400001312
Designing a fixed-time state estimator in the mathematical form:
Figure BDA0002677943040000136
wherein the content of the first and second substances,
Figure BDA0002677943040000137
estimates of positions in joint space for robots i and j;
Figure BDA0002677943040000138
estimates of the velocity in joint space for robots i and j, c1、c2Constant control gain, control gain of fixed time being eta respectively1(t) and η2(t)；
Position error of state estimator combined with fixed time
Figure BDA0002677943040000139
And speed error
Figure BDA00026779430400001310
Wherein
Figure BDA00026779430400001311
The mathematical form of the fixed-time state estimator translates into:
Figure BDA0002677943040000141
designing a sliding mode surface S with fixed time for the dynamic model obtained in step S1iController tau to assist in designing joint spaceiSaid slip form surface s of fixed timeiThe mathematical expression of (a) is:
Figure BDA0002677943040000142
wherein the position error between the estimated value and the true value
Figure BDA0002677943040000143
And speed error
Figure BDA0002677943040000144
Are respectively as
Figure BDA0002677943040000145
Figure BDA0002677943040000146
K1i、K2iRespectively, are positive definite diagonal matrices,
Figure BDA0002677943040000147
n is the number of joints of the robot,
Figure BDA0002677943040000148
0<Δ≤1，0<p<1，
Figure BDA0002677943040000149
Figure BDA00026779430400001410
l>1。
design of a constant-time joint space controller τ for the dynamic model obtained in step S1iThe mathematical expression of (a) is:
Figure BDA00026779430400001411
wherein, tauiIs composed of three parts: tau isi0An equivalent control term, τ, associated with the robot dynamics modeli1Non-linear term, τ, representing the effect of fixed time controli2Representing a control term for counteracting the external disturbance;
Figure BDA00026779430400001412
an estimated value of joint velocity obtained for a fixed time controller; mi0、Ci0、gi0Are respectively Mi、Ci、giThe expected value of (d); k. b0、b1、b2Is a normal number;
Figure BDA0002677943040000151
is composed of
Figure BDA0002677943040000152
Function to variable
Figure BDA0002677943040000153
And converts the form of the column vector into the form of a diagonal matrix,
Figure BDA0002677943040000154
is composed of
Figure BDA0002677943040000155
Function to variable
Figure BDA0002677943040000156
And converts the form of the column vector into the form of a diagonal matrix.
Step S3 specifically includes:
s31, and controller tau of joint space with fixed timeiSubstituting the kinetic model constructed in S1, and adding the fixed-time state estimator, a closed-loop system is obtained as follows:
Figure BDA0002677943040000157
wherein, K0iIs a positive definite diagonal matrix, and r is a constant satisfying r>1, rho (t) is a disturbance term and an uncertain term in a robot dynamic model,
Figure BDA0002677943040000158
ΔMi(qi)、
Figure BDA0002677943040000159
Δgi(qi) Are respectively Mi(qi)、
Figure BDA00026779430400001510
gi(qi) Uncertainty of term, di(t) is a perturbation term;
s32, analyzing the stability of the state estimator with fixed time in the closed loop system in S31;
constructing a Lyapunov function for errors in position and velocity of the robot in the closed-loop system:
Figure BDA00026779430400001511
Figure BDA00026779430400001512
wherein the content of the first and second substances,
Figure BDA00026779430400001513
Figure BDA00026779430400001514
n is the number of robots in the networked robot system;
and deriving the time of the Lyapunov by:
Figure BDA0002677943040000161
Figure BDA0002677943040000162
therefore, the error of the position and the speed obtained by the correlation theory of the TBG function can be converged into an adjustable boundary within a set time range, and the following conditions are met:
Figure BDA0002677943040000163
Figure BDA0002677943040000164
Figure BDA0002677943040000165
Figure BDA0002677943040000166
wherein the content of the first and second substances,1、2for autonomous setting of a constant, T, for adjusting the error rangef1、Tf2Respectively setting time in a TBG function in the position estimator and the speed estimator, so that the control time is in a set time range, namely the stability analysis of the fixed time state estimator is completed;
s33, analyzing the stability of a robot dynamic state controller (a controller of a joint space with fixed time) in the closed-loop system S31;
construction of Lyapunov function for fixed-time sliding mode surface
Figure BDA0002677943040000167
And derived along the trajectory of said closed loop system
Figure BDA0002677943040000168
Reissue to order
Figure BDA0002677943040000169
To obtain
Figure BDA00026779430400001610
Substituted into the above formula to obtain
Figure BDA00026779430400001611
According to the theory of fixed time, obtain
Figure BDA00026779430400001612
Within, s i0; so the dynamics of the error system of position and velocity are transformed into
Figure BDA0002677943040000171
Further, the discussion is divided into two cases
Figure BDA0002677943040000172
The time required to converge to the set small constant Δ.
When in use
Figure BDA0002677943040000173
Error in selecting joint position
Figure BDA0002677943040000174
Lyapunov function
Figure BDA0002677943040000175
Derived from time
Figure BDA0002677943040000176
When in use
Figure BDA0002677943040000177
Error in selecting joint position
Figure BDA0002677943040000178
Lyapunov function Vca2In the same way, the time is derived
Figure BDA0002677943040000179
Thus, it is derived from
Figure BDA00026779430400001710
Within time, error
Figure BDA00026779430400001711
In step S4, the analysis of the fixed time state estimator and the analysis of the robot dynamic state controller (i.e. the controller of the joint space at the fixed time) are respectively performed on the closed-loop system according to the double-layer control structure composed of the fixed time estimation layer and the fixed time local control layer, so as to obtain the fixed time T at the set time Tf1+Tf2+Ts+TeWhen the robot i belongs to the first group: position state
Figure BDA00026779430400001712
I.e. qiApproach to
Figure BDA00026779430400001713
Approaches to q0Velocity state
Figure BDA00026779430400001714
I.e. viApproach to
Figure BDA00026779430400001715
Approaches to v0(ii) a When robot i belongs to the second group: position state
Figure BDA00026779430400001716
I.e. qiApproach to
Figure BDA00026779430400001717
Approaches to-q0Velocity state
Figure BDA00026779430400001718
I.e. viApproach to
Figure BDA00026779430400001719
Approaches to-v0Therefore, bilateral tracking control of the networked robot system is realized.
The required control parameters are as follows: p is 0.5, q is 1.2, r is 1.5, k is 1, b0＝12，b1＝2.8，b2＝0.5，1＝0.01，2＝0.01，c1＝0.01，c2＝1，K0i＝diag(4,3)，K1i＝diag(2,2)，K2i＝diag(2,2)，
Figure BDA00026779430400001720
Please refer to fig. 4 to fig. 9 for simulation results.
Wherein, fig. 4 is a track tracing diagram of the actual position and the estimated position of the networked robot system according to the present invention, which respectively shows that the proposed control method enables the position of the robot joint to reach the expected position within a fixed time and the proposed estimation layer of the fixed time enables the estimated position of the robot joint to reach the expected position within the fixed time;
FIG. 5 is a trajectory tracking diagram of the actual and estimated velocities of the networked robotic system in accordance with the present invention, respectively illustrating that the proposed control method enables the velocity of the robotic joint to reach the desired velocity in a fixed time and the proposed estimation horizon of the fixed time enables the estimated velocity of the robotic joint to reach the desired velocity in a fixed time;
FIG. 6 is a graph of actual position tracking error for a networked robotic system in accordance with the present invention, showing that the actual position tracking error can trend toward 0 over a fixed time;
FIG. 7 is a position tracking error plot of an estimator of a networked robotic system in accordance with the present invention, showing that the position tracking error of a fixed time estimation layer can go to 0 over a fixed time;
FIG. 8 is a position tracking error map of the local control layer of the networked robotic system in accordance with the present invention, illustrating that the position tracking error of the local control layer can go to 0 in a fixed time;
fig. 9 is a graph of the actual speed tracking error of the networked robotic system of the present invention, showing that the actual speed tracking error can go to 0 over a fixed time.
The invention provides a bilateral tracking control algorithm based on a fixed time estimator, which is suitable for a networked robot system, wherein the actual conditions of cooperation and competition among robots, external disturbance existing in a robot model and model parameter change are considered at the same time. The bilateral tracking control method based on the fixed time estimator has the following advantages:
1. the practical problems of change of model parameters, external disturbance and the like in a robot dynamics model are considered, and the practicability of research results is higher;
2. the coexistence of cooperation and competition is considered in the information interaction between the robots, so that the bilateral tracking control method has universal applicability;
3. the designed control algorithm comprises two layers of control, the control cost of the multi-robot system is reduced, and the working efficiency is improved.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (7)

1. The bilateral tracking control method of the fixed time estimator of the networked robot system is characterized in that: the method specifically comprises the following steps:
s1, performing dynamic modeling on the ith robot in the networked robot system comprising N robots, and constructing a dynamic model:
Figure FDA0002677943030000011
wherein M isi(qi) A positive definite inertial matrix is represented,
Figure FDA0002677943030000012
denotes the Coriolis centrifuge matrix, gi(qi) Representing a matrix of gravitation, di(t) denotes input disturbance, τiRepresents a control input; q. q.si、
Figure FDA0002677943030000013
And
Figure FDA0002677943030000014
respectively representing the position, the speed and the acceleration of the ith robot in a joint space; t represents time; i represents the number of the robot;
setting the (i + 1) th robot as a tracking target, wherein the state information of the tracking target comprises a position q0Velocity v0And acceleration u0Satisfy the following requirements
Figure FDA0002677943030000015
S2, establishing a directed symbol topological graph G { V, E, A } to describe information interaction relations among robots, wherein V { 1.... N } represents a vertex set consisting of N robots, and the vertices are used for representing the robots;
Figure FDA0002677943030000016
representing a set of edges connecting any two robots, and being used for representing the direction of information interaction between the robots; a ═ aij]A weight matrix composed of adjacent elements and used for representing the value of mutual information between robots; if the j-th robot and the i-th robot have interaction and cooperative relationship, aijIs greater than 0; if the j-th robot and the i-th robot have interaction and are in competition relationship, aij< 0, otherwise, aij0; and the ith robot itself has no self-circulation connection, i.e. aii＝0；
Define diagonal matrix D ═ diag (D)1,d2,...dN),diBelongs to { +/-1 }, satisfies DAD as a semi-positive definite matrix, and divides all robots into two parts to respectively execute opposite tasks(ii) a When the robot belongs to the first group, di1 is ═ 1; when the robot belongs to the second group, di-1; adding a tracking target robot to the directed symbol topological graph G and defining the graph as an augmented graph
Figure FDA0002677943030000017
Using the matrix B ═ ai0]A value representing the information interaction between the virtual leader and the robot, a if there is an interaction between the virtual leader and the roboti0Is greater than 0; if there is no interaction between the virtual leader and the robot, then ai0＝0；
S3, designing a distributed controller based on a fixed time estimator according to the robot dynamics model established in the step S1 and the directed symbolic topological graph established in the step S2;
the distributed controller comprises a local control layer with fixed time and an estimation layer with fixed time;
the estimation layer of the fixed time is used for obtaining estimation values of the joint position and the joint speed of the ith robot, and the estimation values are obtained through the position information, the movement speed information and the information interaction numerical value of the ith robot and the adjacent robot;
the estimated values of the joint position and the speed of the ith robot are further input into a local control layer of the fixed time, and the control layer is used for enabling the actual joint position and the speed of the ith robot to converge to the estimated values obtained by the fixed time estimation layer;
and S4, a double-layer control structure consisting of an estimation layer with fixed time and a local control layer with fixed time, and controlling the actual joint position and speed of the ith robot to reach the expected joint position and speed, thereby realizing the bilateral tracking control of the networked robot system.
2. The bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 1, wherein:
designing the corresponding fixed-time control gain eta for the fixed-time estimation layer in the step S31(t)、η2(t), the fixed time is mainly realized by a time-base signal generator TBG, and the method comprises the following two steps:
designing a TBG function:
Figure FDA0002677943030000021
the TBG function is a piecewise time varying function, tfIs an artificially set convergence time point, and the convergence time required by the function control is designed to be tfWithin;
designing a control gain function comprising a TBG function:
Figure FDA0002677943030000031
Figure FDA0002677943030000032
wherein the content of the first and second substances,1，2is a normal number and satisfies 0<1,2≤1，MD＝DAD+B，InIs an n-dimensional identity matrix, H is a positive definite diagonal matrix, such that MDH is a positive definite matrix, HmaxRepresents the maximum of the diagonal elements of the H matrix,
Figure FDA0002677943030000033
representation matrix
Figure FDA0002677943030000034
Is determined by the minimum characteristic value of (c),
Figure FDA0002677943030000035
representing the kronecker product.
3. The bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 2, wherein:
for the position q of the i-th robot in the joint space required in step S1iAnd velocity
Figure FDA0002677943030000036
Designing a fixed-time state estimator in the mathematical form:
Figure FDA0002677943030000037
wherein the content of the first and second substances,
Figure FDA0002677943030000038
estimates of positions in joint space for robots i and j;
Figure FDA0002677943030000039
estimates of the velocity in joint space for robots i and j, c1、c2Constant control gain, control gain of fixed time being eta respectively1(t) and η2(t)；
Position error of state estimator combined with fixed time
Figure FDA00026779430300000310
And speed error
Figure FDA00026779430300000311
Wherein
Figure FDA00026779430300000312
The mathematical form of the fixed-time state estimator translates into:
Figure DEST_PATH_BDA0002677943040000054
4. the bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 1, wherein:
designing a sliding mode surface S with fixed time for the dynamic model obtained in step S1iController tau to assist in designing joint spaceiSaid slip form surface s of fixed timeiThe mathematical expression of (a) is:
Figure FDA0002677943030000042
wherein the position error between the estimated value and the true value
Figure FDA0002677943030000043
And speed error
Figure FDA0002677943030000044
Are respectively as
Figure FDA0002677943030000045
Figure FDA0002677943030000046
K1i、K2iRespectively, are positive definite diagonal matrices,
Figure FDA0002677943030000047
n is the number of joints of the robot,
Figure FDA0002677943030000048
0<Δ≤1，0<p<1，
Figure FDA0002677943030000049
Figure FDA00026779430300000410
Figure FDA00026779430300000411
l>1。
5. the bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 4, wherein:
controller τ for designing joint space of fixed time for the kinetic model described in step S1iThe mathematical expression is as follows:
Figure FDA0002677943030000051
wherein, tauiIs composed of three parts: tau isi0An equivalent control term, τ, associated with the robot dynamics modeli1Non-linear term, τ, representing the effect of fixed time controli2Representing a control term for counteracting the external disturbance;
Figure FDA0002677943030000052
an estimated value of joint velocity obtained for a fixed time controller; mi0、Ci0、gi0Are respectively Mi、Ci、giThe expected value of (d); k. b0、b1、b2Is a normal number;
Figure FDA0002677943030000053
is composed of
Figure FDA0002677943030000054
Function to variable
Figure FDA0002677943030000055
And converts the form of the column vector into the form of a diagonal matrix,
Figure FDA0002677943030000056
is composed of
Figure FDA0002677943030000057
Function to variable
Figure FDA0002677943030000058
And converts the form of the column vector into the form of a diagonal matrix.
6. The bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 3, wherein:
step S3 specifically includes:
s31, and controller tau of joint space with fixed timeiSubstituting the kinetic model constructed in S1, and adding the fixed-time state estimator, a closed-loop system is obtained as follows:
Figure DEST_PATH_BDA0002677943040000068
wherein, K0iIs a positive definite diagonal matrix, and r is a constant satisfying r>1, rho (t) is a disturbance term and an uncertain term in a robot dynamic model,
Figure FDA0002677943030000061
ΔMi(qi)、
Figure FDA0002677943030000062
Δgi(qi) Are respectively Mi(qi)、
Figure FDA0002677943030000063
gi(qi) Uncertainty of term, di(t) is a perturbation term;
s32, analyzing the stability of the state estimator with fixed time in the closed loop system in S31;
constructing a Lyapunov function for errors in position and velocity of the robot in the closed-loop system:
Figure FDA0002677943030000064
Figure FDA0002677943030000065
wherein the content of the first and second substances,
Figure FDA0002677943030000066
Figure FDA0002677943030000067
n is the number of robots in the networked robot system;
and the lyapunov is subjected to derivation on time to obtain:
Figure FDA0002677943030000068
Figure FDA0002677943030000069
by the relevant theorem of the TBG function, the position and speed errors of the robot in the closed-loop system can be converged into an adjustable boundary within a set time range, and the following requirements are met:
Figure FDA00026779430300000610
Figure FDA00026779430300000611
Figure FDA00026779430300000612
Figure FDA00026779430300000613
wherein the content of the first and second substances,1、2for autonomous setting of a constant, T, for adjusting the error rangef1、Tf2Respectively time set in the TBG function in the position estimator and the speed estimator;
s33, analyzing the stability of the controller of the joint space with fixed time in the closed loop system in S31;
for a fixed time synovial surface siConstructing the Lyapunov function
Figure FDA0002677943030000071
And along siIs derived by
Figure FDA0002677943030000072
Reissue to order
Figure FDA0002677943030000073
To obtain
Figure FDA0002677943030000074
Substituted into the above formula to obtain
Figure FDA0002677943030000075
According to the theory of fixed time, it is obtained that
Figure FDA0002677943030000076
Within, si0; the dynamics of the error system of joint position and velocity are translated into
Figure FDA0002677943030000077
Further, the two cases are dividedDiscussion of the related Art
Figure FDA0002677943030000078
The time required to converge to a set constant Δ that approaches zero;
when in use
Figure FDA0002677943030000079
Error to joint position
Figure FDA00026779430300000710
Selecting a Lyapunov function
Figure FDA00026779430300000711
Derived from time
Figure FDA00026779430300000712
When in use
Figure FDA00026779430300000713
Error to joint position
Figure FDA00026779430300000714
Selecting the Lyapunov function Vca2In the same way, the time is derived
Figure FDA00026779430300000715
Thus, it is derived from
Figure FDA00026779430300000716
Within time, error
Figure FDA00026779430300000717
7. The bilateral tracking control method of a fixed time estimator of a networked robot system according to claim 1, wherein:
in step S4, the analysis of the fixed time state estimator and the analysis of the controller of the joint space at the fixed time are respectively performed on the closed-loop system according to the double-layer control structure composed of the fixed time estimation layer and the fixed time local control layer to obtain the fixed time Tf1+Tf2+Ts+TeWhen the robot i belongs to the first group: position state
Figure FDA0002677943030000081
I.e. qiApproach to
Figure FDA0002677943030000082
Approaches to q0Velocity state
Figure FDA0002677943030000083
I.e. viApproach to
Figure FDA0002677943030000084
Approaches to v0(ii) a When robot i belongs to the second group: position state
Figure FDA0002677943030000085
I.e. qiApproach to
Figure FDA0002677943030000086
Approaches to-q0Velocity state
Figure FDA0002677943030000087
I.e. viApproach to
Figure FDA0002677943030000088
Approaches to-v0Therefore, bilateral tracking control of the networked robot system is realized.
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