CN112684709B - Cluster tracking kinematics modeling method, device and storage medium - Google Patents

Cluster tracking kinematics modeling method, device and storage medium Download PDF

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CN112684709B
CN112684709B CN202011565823.0A CN202011565823A CN112684709B CN 112684709 B CN112684709 B CN 112684709B CN 202011565823 A CN202011565823 A CN 202011565823A CN 112684709 B CN112684709 B CN 112684709B
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赵睿英
惠记庄
王杰
张雅倩
张�浩
李梦
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Changan University
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Abstract

The invention discloses a modeling method, a system, equipment and a storage medium for cluster tracking kinematics, which comprises the following steps of describing a dynamic tracking process of a multi-mobile robot as a mathematical expression; secondly, constructing a potential field function according to an artificial potential field method based on mathematical description of a dynamic tracking process; and step three, establishing a multi-mobile cluster tracking kinematics model according to the potential field function. The robot has the biological clustering characteristic while completing the tracking task in a complex environment.

Description

Cluster tracking kinematics modeling method, device and storage medium
Technical Field
The invention belongs to the field of robots, and relates to a cluster tracking kinematics modeling method, a system, equipment and a storage medium.
Background
Real-time tracking of dynamic targets by collision-free completion in a work environment with obstacles is a basic requirement for mobile robots to accomplish work tasks. Common algorithms for path planning of a single mobile robot are based on a behavior method, a fuzzy control method, an artificial potential field method and the like. The coordinated tracking of the multi-mobile robot cluster system can be understood as the cluster tracking problem of the system, the cluster tracking of the multi-mobile robot can realize dynamic target tracking, and the multi-mobile robot cluster system has biological cluster behavior characteristics, such as: consistency, formation retention, collision avoidance, aggregation, and the like. As the cluster tracking is a complex cluster behavior, in order to realize the cluster tracking task of the mobile robot, the cluster behavior of the robot is firstly analyzed, and a mathematical model of the cluster behavior is established.
At present, common modeling methods for cluster behaviors of a multi-mobile robot system include: the cluster behavior modeling method based on the gravitation/repulsion effect, the Eulerian method and the simulation-based modeling method mostly enable the system to have good stability. At present, research scholars in various countries propose a plurality of modeling and analyzing methods aiming at cluster behaviors of a multi-mobile robot system, and the common methods comprise the following steps: a cluster behavior modeling method based on an attractive/repulsive force (Attraction/replication) effect, an Euler (Euler) method, and a simulation-based modeling method. The cluster behavior modeling method based on the attraction/repulsion action follows the basic principle of attraction in the long-range and repulsion in the near-range, and is a common method for ensuring cluster separation and aggregation. The developed A/R model expressing the interaction relationship between individuals based on the artificial potential field theory can coordinate the whole group system by applying the interaction force between individuals and groups in the system, and systematically research the cluster motion of the multi-mobile robot according to the interaction principle between the individuals. The cluster tracking of multiple mobile robots can realize dynamic target tracking, and different from the target tracking of a single mobile robot, the cluster tracking of multiple mobile robots must complete a tracking task and keep certain characteristics of biological cluster behaviors, such as convergence, anti-collision performance, formation behaviors and the like. The existing multi-mobile robot modeling method is difficult to ensure that robots with various structures have biological clustering characteristics while completing tracking tasks in a complex environment.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a clustering tracking kinematics modeling method, a system, equipment and a storage medium, wherein a robot has a biological clustering characteristic while completing a tracking task in a complex environment.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
a cluster tracking kinematics modeling method comprises the following steps;
describing a dynamic tracking process of a plurality of mobile robots as a mathematical expression;
secondly, constructing a potential field function according to an artificial potential field method based on mathematical description of a dynamic tracking process;
and step three, establishing a multi-mobile cluster tracking kinematics model according to the potential field function.
Preferably, in the first step, the dynamic tracking process of the multiple mobile robots is as follows: the central position of the multi-mobile robot group continuously approaches to the motion track of the mobile target until the central position of the multi-mobile robot is superposed with the mobile target.
Preferably, the mathematical expression in the step one is as follows:
Figure BDA0002860630730000021
wherein the set
Figure BDA0002860630730000022
Indicating the desired position of the robot cluster center in the cluster tracking control,
Figure BDA0002860630730000023
Figure BDA0002860630730000024
for a cluster of mobile robots, a central location, qi(t)∈RnA position vector representing the mobile robot;
Figure BDA0002860630730000025
a position vector representing the dynamic moving object, i ∈ {1,2, …, N }.
Preferably, in the second step, according to the principle of the artificial potential field method, the moving target has an attraction force to the robot, the magnitude of the attraction force is in direct proportion to the distance, a repulsive force exists between the moving robots, the magnitude of the repulsive force is in inverse proportion to the distance, and the constructed potential field function is
Figure BDA0002860630730000031
Wherein G (q) is a function of the i total potential field of the mobile robot, Gij(qi,qj) Indicating a mobile robot i and a mobile robotA potential field function between j;
Figure BDA0002860630730000032
representing the potential field function between the mobile robot i and the moving object.
Preferably, in step three, the multi-mobile cluster tracking kinematics model is
Figure BDA0002860630730000033
Wherein
Figure BDA0002860630730000034
Is the derivative of the position vector of the dynamically moving object.
A cluster tracking kinematics modeling system comprising:
the conversion module is used for describing the dynamic tracking process of the multi-mobile robot into a mathematical expression;
the potential field function building module is used for building a potential field function according to an artificial potential field method based on mathematical description of a dynamic tracking process;
and the kinematic model building module is used for building a multi-mobile cluster tracking kinematic model according to the potential field function.
A computer device comprising a memory, a processor and a computer program stored in the memory and executable on the processor, the processor implementing the steps of the cluster tracking kinematics modeling method as described in any of the above when executing the computer program.
A computer-readable storage medium, having stored thereon a computer program which, when being executed by a processor, carries out the steps of the cluster tracking kinematics modeling method according to any of the previous claims.
Compared with the prior art, the invention has the following beneficial effects:
the method is used for establishing a multi-mobile-robot cluster tracking kinematics model based on an artificial potential field method. Dynamic tracking and gathering performance of the robot are achieved by endowing dynamic tracking target attractive force, meanwhile, the mobile robots in the potential field function cluster have repulsive force, anti-collision performance in the robot tracking process can be achieved, a required formation mode such as a circle can be formulated by adjusting parameters of a final multi-mobile cluster tracking kinematic model, and the robot has biological cluster characteristics while completing tracking tasks in a complex environment.
Drawings
Fig. 1 is a track diagram of a wheeled mobile robot and a mobile object in an unlimited environment according to the present invention;
FIG. 2 shows the central position of a robot cluster in an unrestricted environment according to the present invention
Figure BDA0002860630730000041
And an object
Figure BDA0002860630730000042
Error between
Figure BDA0002860630730000043
A simulation result graph;
fig. 3 is a track diagram of a wheeled mobile robot and a mobile target in a confined environment according to the present invention;
FIG. 4 illustrates a central position of a robot cluster in a constrained environment according to the present invention
Figure BDA0002860630730000044
And an object
Figure BDA0002860630730000045
Error between
Figure BDA0002860630730000046
And (5) a simulation result graph.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
the modeling method comprises the following steps:
(1) the tracking problem is described mathematically: when the task is completed, the central position of the robot cluster and the motion trail of the moving target are not connected continuouslyNear coincidence state, i.e., when time t → + ∞, the robot cluster center position
Figure BDA0002860630730000047
Will arrive at the set
Figure BDA0002860630730000048
Within an arbitrarily small neighborhood of the cell. And when tracking the moving target, the multi-mobile robot system forms an expected formation and has the biological clustering behavior characteristics. The mathematical expression for this process is:
Figure BDA0002860630730000049
wherein the set
Figure BDA00028606307300000410
Indicating the desired position of the robot cluster center in the cluster tracking control,
Figure BDA00028606307300000411
Figure BDA00028606307300000412
for a cluster of mobile robots, a central location, qi(t)∈RnA position vector representing the mobile robot;
Figure BDA00028606307300000413
a position vector representing the dynamic moving object, i ∈ {1,2, …, N }.
(2) Selecting a potential field function: according to the principle of an artificial potential field method, a moving target has attraction force on robots, the attraction force is in direct proportion to the distance, repulsion force exists among the moving robots, and the repulsion force is in inverse proportion to the distance, so that a special potential field function is selected, the attraction force of the moving target is ensured to be large enough, and the repulsion force among the moving robots is moderate. The potential field function is as follows:
Figure BDA0002860630730000051
wherein G isij(qi,qj) Comprises the following steps:
Figure BDA0002860630730000052
selected by
Figure BDA0002860630730000053
Comprises the following steps:
Figure BDA0002860630730000054
wherein, a, bij,cijAnd s are potential field coefficients and are both positive numbers, G (q) is a function of the total potential field formed by the mobile robot i and the mobile robot, Gij(qi,qj) Representing a potential field function between mobile robot i and mobile robot j;
Figure BDA00028606307300000511
representing the potential field function between the mobile robot i and the moving object.
(3) Establishing a kinematic model of each mobile robot: in the cluster tracking problem, considering that a multi-mobile robot tracks a moving target, an ideal kinematic model of the mobile robot i e {1,2, …, N } can be described as (where
Figure BDA00028606307300000512
Is qiAnd q isJDerivative of (d):
Figure BDA0002860630730000055
order to
Figure BDA0002860630730000056
According to a function Gij(qi,qj) And
Figure BDA0002860630730000057
can be obtained by the expression (c):
Figure BDA0002860630730000058
Figure BDA0002860630730000059
to simplify the subsequent proof process, the arguments of the function are omitted without generating ambiguity. According to the basic principle of the artificial potential field method, the potential field function G (q) is at qiAnd
Figure BDA00028606307300000510
the derivatives at (a) are:
Figure BDA0002860630730000061
thus, the kinematic model (7) of the system can be rewritten as
Figure BDA0002860630730000062
Analyzing the tracking performance of the cluster: firstly, a Lyapunov function of the error between the center position of the multi-mobile robot cluster and the distance of a mobile target is constructed, and then derivation and deformation are carried out on the Lyapunov function. Finally, according to the LaSalle Yoshizawa theory, when the expression of the derivative of the Lyapunov function is the expression (17), namely the derivative of the Lyapunov function of the error between the center position of the multi-mobile-robot cluster and the distance of the mobile target is a constant negative value, the Lyapunov function finally tends to zero. The fact that the central position of the robot cluster is overlapped with the motion track of the moving target under the condition that the time is continuously increased proves that the central position of the robot cluster can track the moving target.
Defining a tracking error: multiple movementMoving robot cluster center position
Figure BDA0002860630730000063
And moving objects
Figure BDA0002860630730000064
And a position vector q of the mobile robot iiAnd cluster center location
Figure BDA0002860630730000065
The error between is defined as:
Figure BDA0002860630730000066
then error is generated
Figure BDA0002860630730000067
The derivative with respect to time t is:
Figure BDA0002860630730000068
constructing a lyapunov function: known mobile robot cluster center position
Figure BDA0002860630730000069
Capable of tracking moving objects and entering the collection over time t → ∞
Figure BDA00028606307300000610
Clustering central positions using multiple mobile robots
Figure BDA00028606307300000611
And the error vector of the moving target and the transpose thereof, a lyapunov function can be constructed:
Figure BDA00028606307300000612
derivation and modification of the Lyapunov function: the derivative of the lyapunov function V with respect to time t is:
Figure BDA00028606307300000613
to describe the robot cluster center position
Figure BDA00028606307300000614
And moving object
Figure BDA00028606307300000615
The relation between the central positions of the robot clusters is firstly deduced according to the expression (8) and the expression of the central positions of the robot clusters
Figure BDA0002860630730000071
Derivative with respect to time:
Figure BDA0002860630730000072
substituting equation (7) into the above equation can result in:
Figure BDA0002860630730000073
for simplicity, by assuming giiExcluding the case where j ≠ i at 0, and depends on gij(qi,qj)＝ -gji(qi,qj) And
Figure BDA0002860630730000074
derivation of
Figure BDA0002860630730000075
The expression of (c) can be simplified as:
Figure BDA0002860630730000076
finally, the derivative of the Lyapunov function is known by taking the formula (16) into the formula (13)
Figure BDA0002860630730000077
The expression of (a) is:
Figure BDA0002860630730000078
analyzing aggregation behavior: firstly, a mobile robot i and a robot cluster center position are constructed
Figure BDA0002860630730000079
The convergence error (i ═ 1,2, …, N.) between. And secondly, carrying out derivation and deformation on the linear derivative, wherein according to the LaSalle Yoshizawa theory, when the derivative expression of the Lyapunov function of the convergence error is constant negative, the convergence error Lyapunov function finally tends to zero. And finally, proving that the multi-mobile-robot system has always-bounded property and finally-bounded property according to a bounded theory. Thus proving that the mobile robots always gather near the center position of the robot cluster and can gather each other while tracking the dynamic moving target.
a. Defining the convergence error: the convergence error between the position vector of the mobile robot and the cluster center position is:
Figure BDA00028606307300000710
error eicThe derivative with respect to time t is:
Figure BDA00028606307300000711
b. by using the convergence error vector between the position vector of the mobile robot and the cluster center position and the transposition thereof, a Lyapunov function can be constructed and selected as follows:
Figure BDA0002860630730000081
c. lyapunov function on convergence error
Figure BDA00028606307300000810
And (3) carrying out derivation and deformation, wherein the derivation result is as follows:
Figure BDA0002860630730000082
the simplification process is as follows: the formula (14) and the formula (16) are introduced into the formula (20),
Figure BDA0002860630730000083
firstly, according to the formula
Figure BDA0002860630730000084
The first term in analytical formula (22),
Figure BDA0002860630730000085
then, the second term in equation (22) is analyzed according to equation (7),
Figure BDA0002860630730000086
wherein k isi＝∑jkij. Finally, when the formula (22) and the formula (23) are brought into the formula (20), the
Figure BDA0002860630730000087
Lyapunov function available according to equation (24)
Figure BDA0002860630730000088
The derivative of (c) is:
Figure BDA0002860630730000089
for the first term to the left of equation (25), equation (7) is taken to be available:
Figure BDA0002860630730000091
at the same time, the user can select the desired position,
Figure BDA0002860630730000092
from the expressions (26) and (27), the Lyapunov function can be simplified
Figure BDA0002860630730000093
Derivative of (a):
Figure BDA0002860630730000094
wherein the content of the first and second substances,
Figure BDA0002860630730000095
for equation (28), if satisfied
Figure BDA0002860630730000096
Conditional, derivative of the Lyapunov function
Figure BDA0002860630730000097
Negative values:
Figure BDA0002860630730000098
d. proving that the multi-mobile robot system is bounded: when the derivative of the Lyapunov function satisfies the formula (29) Time, error vector ecWith consistent bounding, i.e. for any given r>0, and | ec(t0)‖<r when t>t0When there is a positive real number d (r) as shown in equation (30) such that there is a positive real number m | ec(t)‖≤d(r)。
Figure BDA0002860630730000099
Wherein the content of the first and second substances,
Figure BDA00028606307300000910
at the same time, the error vector ecThere is also a consistent and ultimately bounded nature,
d＝R， (32)
Figure BDA00028606307300000911
analysis of formation performance: based on the analysis of the system aggregation performance, the formation behavior of the multi-mobile robot cluster system will be studied next. After the Lyapunov function is established and subjected to derivative transformation, the result of equation (39) means that according to the LaSalle Yoshizawa theory
Figure BDA0002860630730000101
For any given i, when
Figure BDA0002860630730000102
Description of the invention
Figure BDA0002860630730000103
This is true. Meanwhile, according to the formulas (34) to (36), it is possible to obtain
Figure BDA0002860630730000104
In summary, the set Γ is
Figure BDA0002860630730000105
When time t → ∞, makes the trajectory q (t) invariant) Tending to assemble Γ. In performing the cluster tracking task, equation (35) illustrates that the multi-mobile-robot system will form a certain formation in a sufficient time. And the formation of the formation shape is related to the design parameters of the set Γ in equation (33). In practical application, the function G can be designedijAnd
Figure BDA0002860630730000106
is used to formulate the required formation pattern (also parameter of Γ).
a. Determining a Lyapunov function by a potential field function between the mobile robot i and the mobile robot j and a potential field function between the mobile robot i and a moving target
Figure BDA0002860630730000107
Wherein
Figure BDA0002860630730000108
Let psihRepresents a set of generalized coordinates q such that
Figure BDA0002860630730000109
b. Defining a motion track set Γ (i epsilon {1,2, …, N }) of the mobile robot i: suppose for some h>0, set ψhIs bounded. Order to
Figure BDA00028606307300001010
With increasing time, the multi-mobile robot system and the mobile target are in the region psihEach trace q (t) within tends towards the set Γ.
c. Derivation lyapunov function: the function is known from formula (33)
Figure BDA00028606307300001011
About the locus qiThe derivative of (c) is:
Figure BDA00028606307300001012
according to a function GijAnd
Figure BDA00028606307300001016
is a function of (9)
Figure BDA00028606307300001013
With respect to moving objects
Figure BDA00028606307300001014
The derivative of (c) is:
Figure BDA00028606307300001015
in bringing equation (34) into equation (35), equation (36) may be rewritten as:
Figure BDA0002860630730000111
according to
Figure BDA0002860630730000112
The above formula can be organized as:
Figure BDA0002860630730000113
combining the results of equations (35) to (38), functions
Figure BDA0002860630730000114
The derivative of (d) can be written as:
Figure BDA0002860630730000115
simulation research is carried out on a system formed by ten Irobot wheel type mobile robots in matlab: fig. 1 and fig. 2 in simulation results show that ten Irobot wheel-type mobile robots using the modeling method can be gathered in the center of multiple mobile robots to complete high-precision tracking of dynamic targets in a circular queue in an unlimited environment. Fig. 3 and 4 in simulation results show that ten Irobot wheel-type mobile robots using the modeling method can be gathered in the center of multiple mobile robots to complete high-precision tracking of dynamic targets in a circular queue in a limited environment.
The invention discloses a cluster tracking kinematics modeling system, which comprises:
the conversion module is used for describing the dynamic tracking process of the multi-mobile robot into a mathematical expression;
the potential field function building module is used for building a potential field function according to an artificial potential field method based on mathematical description of a dynamic tracking process;
and the kinematic model building module is used for building a multi-mobile cluster tracking kinematic model according to the potential field function.
The computer device of the present invention includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and the processor implements the steps of the cluster tracking kinematics modeling method according to any one of the above items when executing the computer program.
The computer-readable storage medium of the present invention stores a computer program, which when executed by a processor implements the steps of the cluster tracking kinematics modeling method as described in any of the above.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (5)

1. A cluster tracking kinematics modeling method is characterized by comprising the following steps;
describing a dynamic tracking process of a multi-mobile robot as a mathematical expression;
secondly, constructing a potential field function according to an artificial potential field method based on mathematical description of a dynamic tracking process;
according to the principle of an artificial potential field method, a moving target has attraction force to robots, the attraction force is in direct proportion to the distance, repulsive force exists among the robots, the repulsive force is in inverse proportion to the distance, and the constructed potential field function is
Figure FDA0003594551460000011
Selected by
Figure FDA0003594551460000012
Comprises the following steps:
Figure FDA0003594551460000013
wherein a, bij,cijAnd s is a potential field coefficient and is both a positive number, G (q) is a total potential field function of the mobile robot i, Gij(qi,qj) Representing a potential field function between mobile robot i and mobile robot j;
Figure FDA0003594551460000014
representing a potential field function between the mobile robot i and a mobile target;
step three, establishing a multi-mobile cluster tracking kinematics model according to the potential field function;
the ideal kinematic model of the mobile robot i e {1,2, …, N } is described as:
Figure FDA0003594551460000015
wherein
Figure FDA0003594551460000016
And
Figure FDA0003594551460000017
is qiAnd q isJDerivative of (1), order
Figure FDA0003594551460000018
Figure FDA0003594551460000019
According to a function Gij(qi,qj) And
Figure FDA00035945514600000110
can be obtained by the expression (c):
Figure FDA00035945514600000111
Figure FDA00035945514600000112
according to the basic principle of the artificial potential field method, the potential field function G (q) is at qiAnd
Figure FDA00035945514600000113
the derivatives at (a) are:
Figure FDA00035945514600000114
the multi-mobile cluster tracking kinematics model is as follows:
Figure FDA00035945514600000115
2. the modeling method of cluster tracking kinematics according to claim 1, wherein in step one, the dynamic tracking process of the multiple mobile robots is as follows: the central position of the multi-mobile robot group continuously approaches to the motion track of the mobile target until the central position of the multi-mobile robot is superposed with the mobile target.
3. The modeling method for cluster tracking kinematics according to claim 1 wherein the mathematical expression in step one is:
Figure FDA0003594551460000021
wherein the set
Figure FDA0003594551460000022
Indicating the desired position of the robot cluster center in the cluster tracking control,
Figure FDA0003594551460000023
Figure FDA0003594551460000024
for a cluster of mobile robots, a central location, qi(t)∈RnA position vector representing the mobile robot;
Figure FDA0003594551460000025
a position vector representing the dynamic moving object, i ∈ {1,2, …, N }; n is the number of robots and N is the dimension of the position vector.
4. A computer device comprising a memory, a processor and a computer program stored in the memory and executable on the processor, characterized in that the processor implements the steps of the cluster tracking kinematics modeling method according to any of the claims 1 to 3 when executing the computer program.
5. A computer-readable storage medium, in which a computer program is stored which, when being executed by a processor, carries out the steps of a cluster tracking kinematics modeling method according to any of the claims 1 to 3.
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