CN110554607B - Cooperative control method and system with obstacle avoidance and navigation protection tasks for multi-Euler-Lagrange system - Google Patents

Cooperative control method and system with obstacle avoidance and navigation protection tasks for multi-Euler-Lagrange system Download PDF

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CN110554607B
CN110554607B CN201910877456.9A CN201910877456A CN110554607B CN 110554607 B CN110554607 B CN 110554607B CN 201910877456 A CN201910877456 A CN 201910877456A CN 110554607 B CN110554607 B CN 110554607B
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宋锐
高嵩
李贻斌
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Shandong University
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Abstract

The invention discloses a cooperative control method and a system with an obstacle avoidance and navigation protection task for a multi-Euler-Lagrange system, wherein the cooperative control method comprises the following steps: describing a physical model of the convoy task by adopting a multi-Euler-Lagrange system; the controller adopts an inner and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and expected speed and expected movement track required by an inner ring physical model are generated; and the inner ring applies an adaptive proportional-differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion trail. The invention has the beneficial effects that: in the presence of model uncertainty, interference and obstacles, a robust hierarchical control structure is established. A vacant space behavior controller is put forward in an outer ring to avoid obstacles and protect and send targets; while producing the required velocity of the inner ring. An adaptive proportional differential-sliding mode control law is adopted in the inner ring to track the required speed and ensure fast convergence and robustness.

Description

Cooperative control method and system with obstacle avoidance and navigation protection tasks for multi-Euler-Lagrange system
Technical Field
The invention relates to the technical field of cooperative control in an obstacle environment, in particular to a cooperative control method and system with an obstacle avoidance and navigation protection task for a multi-Euler-Lagrange system.
Background
The statements in this section merely provide background information related to the present disclosure and may not necessarily constitute prior art.
In recent years, the control of multi-robot systems has attracted considerable attention because these systems can overcome the limitation of using only a single robot. Many scholars are concerned with studying the coordinated control of multiple agents, have uncertain nonlinear dynamics, non-integrity mobile robots, and generally multi-mechanical systems. More and more research has found that modeling some practical applications using only single or double integral dynamics, ignoring non-linear dynamics, is not acceptable in reality. Thus, many researchers have begun studying the euler-lagrange system because it can represent many mechanical systems, such as rigid body pose dynamics, robotic arms, mobile robots, and walking robots, among others. However, most of the existing research results can only be applied to ideal environments, and they are not suitable for being applied to complex environments, such as the situation of obstacles. Therefore, it is crucial to study the coordinated control of the multi-euler-lagrange system, especially when obstacles, model uncertainty, external disturbances, parameter uncertainty and unknown non-linear dynamics need to be considered.
The task of convoying can be viewed as a task of surrounding objects and maintaining formation. When the target moves, the formation moves along with the target and ensures that the target is at the center of the formation, a specified distance is kept between the robot and the target, and the robot is evenly distributed around the target. A number of documents have previously studied the convoy mission of many robots. However, most of the strategies proposed by previous studies can only be applied to planar cases, and they cannot be simply extended to the multi-euler-lagrange system in the three-dimensional case either. Therefore, in the case of the p-dimension (p ≧ 3 is a positive integer), it is more meaningful to design a controller for a suitable convoy mission while avoiding obstacles and providing strict proof of convergence and stability.
Behavior-based approaches have been extensively studied in a number of multi-robot system coordinated control frameworks. Space-time behavior (NSB) control is a task-first inverse kinematics control method; the main feature of space-behavior (NSB) control is that basic tasks are prioritized and combined by a space-space projection matrix of high-priority tasks to form a final behavior, so that multiple tasks are activated simultaneously. Furthermore, unlike the classical behavior control method, the space-time behavior (NSB) control method has an analytical structure, allowing the stability of the task to be elaborated. The present study discusses the convoy mission in detail in the space-behavior (NSB) control scheme, which has the advantages of flexibility and versatility. However, the space-time behavior (NSB) control method still has some unsolved problems, such as how to realize more accurate nonlinear dynamic control under the behavior control architecture.
Sliding Mode Control (SMC) is considered as a powerful technique to help nonlinear systems handle uncertainties, external disturbances, etc. in the system; the Sliding Mode Control (SMC) method includes an equivalent control term that requires some knowledge of the dynamics of the system, and a robust term, which is difficult to obtain in practice for complex systems. There is a high frequency oscillation, known as buffeting, of the control signal of a Sliding Mode Controller (SMC), which causes the signal to oscillate rapidly around the sliding mode face.
The prior art proposes a hybrid control law that can switch between proportional-derivative control (PD) and Sliding Mode Control (SMC) for tracking control of a robot arm. It is proposed to apply proportional derivative control (PD) instead of the equivalent part of Sliding Mode Control (SMC) and the proposed control scheme does not require an accurate dynamic model of the robot arm, which means that the proposed scheme is model independent, although it shows quite good performance, the switching between Proportional Derivative (PD) and Sliding Mode Control (SMC) is indeed a matter that needs to be carefully considered in practice. The prior art proposes a proportional-derivative controller (PD) and sliding-mode controller (SMC) based approach for linear robot systems and assumes that the gain of the sliding-mode controller (SMC) is larger than the upper limit of the dynamical system, however, in most cases this gain will be overestimated because it is difficult to obtain an upper limit of uncertainty of the dynamical system.
Disclosure of Invention
In order to solve the problems, the invention provides a cooperative control method and a cooperative control system with an obstacle avoidance and navigation protection task for a multi-Euler-Lagrange system, wherein an inner-outer loop control structure is adopted, and an empty space behavior (NSB) control is adopted in an outer loop to generate a reference speed vector required by the inner loop; adaptive proportional differential-sliding mode control (APD-SMC) techniques are used for the design of the dynamic structure of the robot to follow the reference speed in the inner ring.
In some embodiments, the following technical scheme is adopted:
the cooperative control method of the multi-Euler-Lagrange system with the obstacle avoidance and navigation protection task comprises the following steps:
describing a physical model of the convoy task by adopting a multi-Euler-Lagrange system;
the controller adopts an inner and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and expected speed and expected movement track required by an inner ring physical model are generated; and the inner ring applies an adaptive proportional-differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion trail.
In other embodiments, the following technical solutions are adopted:
the multi-Euler-Lagrange system is provided with a cooperative control system for avoiding obstacles and protecting navigation tasks, and is characterized by comprising the following steps: the controller adopts an inner ring and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and the expected speed and the expected motion track of the physical model required by the inner ring are generated; and the inner ring applies an adaptive proportional-differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion trail.
In some embodiments, the following technical scheme is adopted:
a terminal device comprising a processor and a computer-readable storage medium, the processor being configured to implement instructions; a computer readable storage medium stores a plurality of instructions adapted to be loaded by a processor and executed as described above
The Lagrange system is provided with a cooperative control method of an obstacle avoidance and navigation protection task.
Compared with the prior art, the invention has the beneficial effects that:
the invention establishes a hierarchical control structure with robustness under the condition of model uncertainty, interference and obstacles. A space-time behavior (NSB) controller is proposed in the outer loop to avoid obstacles and escort targets; while producing the required velocity of the inner ring. An adaptive proportional differential-sliding mode control law (APD-SMC) is employed in the inner loop to track the required speed and ensure fast convergence and robustness. The feasibility of the control scheme is illustrated by Lyapunov theorem and simulation experiment results.
Drawings
Fig. 1 is a block diagram of cooperative control with an obstacle avoidance and navigation protection task in a multi-euler-lagrange system according to an embodiment of the present invention;
FIG. 2 is a block diagram of the control of the empty space of three tasks according to a first embodiment of the present invention;
FIG. 3 is a block diagram of the combined total speed of two task speeds according to one embodiment of the present invention;
FIG. 4 is a diagram of the positions of six robots and a target in a three-dimensional space according to a first embodiment of the present invention;
FIGS. 5(a) - (c) are respectively a distance between a robot and an object, a distance between adjacent robots, and a position tracking error when APD-SMC is used in a two-dimensional space for a scene 1 according to an embodiment of the present invention;
fig. 6(a) - (c) are diagrams illustrating the distance between a robot and a target, the distance between adjacent robots, and the position tracking error, respectively, in a two-dimensional space using ASMC in scene 1 according to a first embodiment of the present invention;
FIG. 7 is a diagram illustrating a trajectory of a robot and an object in a three-dimensional space using APD-SMC according to a first embodiment of the present invention;
FIGS. 8(a) - (c) are diagrams illustrating the use of APD-SMC in a two-dimensional space for the distance between a robot and a target, the distance between adjacent robots, and the position tracking error in scene 2, respectively, according to a first embodiment of the present invention;
FIG. 9 illustrates the use of APD-SMC in a two-dimensional space for the location of five robots and objects when scene 2 encounters an obstacle in accordance with a first embodiment of the present invention;
FIG. 10 is a diagram illustrating the use of APD-SMC in a two-dimensional space for the relative distance between a robot and an obstacle when scene 2 encounters an obstacle according to a first embodiment of the present invention;
FIG. 11 is a diagram illustrating a trajectory of a robot and an object in scene 2 using APD-SMC in three-dimensional space according to a first embodiment of the present invention;
FIGS. 12(a) - (c) are diagrams illustrating the use of APD-SMC in three-dimensional space for the distance between a robot and a target, the distance between adjacent robots, and the position tracking error in scene 2, respectively, according to a first embodiment of the present invention;
fig. 13(a) - (c) are diagrams illustrating the distance between a robot and a target, the distance between adjacent robots, and the position tracking error, respectively, in a three-dimensional space using ASMC according to a first embodiment of the present invention;
FIG. 14 is a diagram illustrating a trajectory of a robot and an object in a two-dimensional space using APD-SMC according to a first embodiment of the present invention;
15(a) - (c) are respectively the distance between a robot and a target, the distance between adjacent robots and the position tracking error when APD-SMC is used in a three-dimensional space for a scene 3 according to a first embodiment of the present invention;
FIG. 16 illustrates the use of APD-SMC in three-dimensional space for the location of six robots and objects when an obstacle is encountered in scene 3 in accordance with a first embodiment of the present invention;
fig. 17 illustrates the use of APD-SMC in three dimensions for the relative distance between the robot and the object when scene 3 encounters an obstacle in accordance with a first embodiment of the present invention.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
Example one
In one or more embodiments, a cooperative control method with an obstacle avoidance and navigation protection task for a multi-Euler-Lagrange system is disclosed, which includes the following steps:
describing a physical model of the convoy task by adopting a multi-Euler-Lagrange system; the physical model may be: multi-robot arms, multi-mobile robots, multi-step robots, multi-spacecraft, and the like; in this embodiment, a multi-euler-lagrange system is used to describe the multi-mobile robot system.
The controller adopts an inner and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and expected speed and expected movement track required by an inner ring physical model are generated; the inner ring applies an adaptive proportional-derivative sliding mode control method, so that each physical model tracks the expected speed and the expected motion track (the total expected speed is integrated to obtain the expected track).
1. Consider a set of n mobile robots whose dynamics can be described as a multi-Euler-Lagrangian system, expressed as:
Figure BDA0002204776870000041
wherein M isi(qi)∈Rp×pIs a positive definite inertia matrix, qi∈RpIs a vector of coordinates in a broad sense,
Figure BDA0002204776870000042
is the vector and centrifugal moment of Coriolis, gi(qi) Is the moment of gravity, τiIs the control input vector for robot i,
Figure BDA0002204776870000043
representing an unknown disturbance force.
Assume that the multi-euler-lagrange system has the following properties.
Property 1.1 boundedness:
for any one i, there is a normal number
Figure BDA0002204776870000044
m i,kCiAnd kgiSo that
Figure BDA0002204776870000045
And Ci(x，y)||≤kCiFor all vectors, | y | |
Figure BDA0002204776870000051
And gi(qi)||≤kgiThis is true.
Property 1.2 skew symmetry:
Figure BDA0002204776870000052
is diagonally symmetrical.
Property 1.3 dynamic parameter linearization:
for all vectors
Figure BDA0002204776870000053
Figure BDA0002204776870000054
Is a regression vector, and ΘiIs a common parameter vector associated with the ith robot.
Properties 1.4: assumption of disturbance force
Figure BDA0002204776870000055
Is that the material is bounded by the surface,
Figure BDA0002204776870000056
in which ξi＞0。
2. To perform the convoy and obstacle avoidance tasks, an empty space behavior (NSB) control merging behavior is proposed in the outer loop to define the final motion of the robot. Adaptive proportional differential-sliding mode control (APD-SMC) is used for an inner loop of a multi-Euler-Lagrange system to compensate unknown disturbance, uncertain parameters and the like. A block diagram of the entire control system is shown in fig. 1.
2.1 space-time behavior (NSB) control
Aiming at the navigation task with the obstacle avoidance requirement, three different tasks are considered, namely the obstacle avoidance task, the subtask of the robot around the target and the subtask of the robot maintained on the surface of a sphere or a hyper-sphere taking the target as the center are uniformly distributed, and the two next subtasks belong to the navigation task. The expected speed of the convoy mission with the obstacle avoidance requirement is designed as
Figure BDA0002204776870000057
Wherein the content of the first and second substances,
Figure BDA0002204776870000058
and
Figure BDA0002204776870000059
representing the desired speed required for the obstacle avoidance task,
Figure BDA00022047768700000510
and
Figure BDA00022047768700000511
is the desired speed required for the convoy mission. J. the design is a square1Is a Jacobian matrix of the obstacle avoidance task,
Figure BDA00022047768700000512
Is the pseudo-inverse, J of the jacobian matrix of the obstacle avoidance task2Is a Jacobian matrix that evenly distributes the subtasks of the robot around the target,
Figure BDA00022047768700000513
Is the pseudo-inverse of the jacobian matrix of the uniformly distributed robot subtasks around the target, and n is the number of robots.
The priority of the task depends on the actual considered factors (e.g. safety actions, such as obstacle avoidance generally having the highest priority) or on the actions that have to be implemented in case of a conflict).
Fig. 2 depicts a space-time behavior (NSB) control architecture with three different tasks, each of which may be prioritized by the administrator depicted in fig. 2. If new requirements are given, the administrator can change the priorities (and weights) of the different tasks during the whole task.
To eliminate conflicting elements, different task speeds need to be projected into the empty space behavior (NSB) control framework, created by the jacobian matrix of the high priority tasks. As shown in figure 3 of the drawings,
Figure BDA00022047768700000514
needs to be projected at
Figure BDA00022047768700000515
In the empty space of (a). This means that the active elements of each task combine to build an integrated overall speed command to drive the robot.
2.1.1 obstacle avoidance task
The obstacle avoidance task maintains a specified distance between the robot and the obstacle. Each one of which isRobot virtual space
Figure BDA0002204776870000061
Is surrounded by
Figure BDA0002204776870000062
Represents the position of the current obstacle for the ith robot, and Bi,oRepresents sigmai,1d＝diRegion of where diIs the minimum allowable safe distance between the ith robot and the obstacle.
The obstacle avoidance task function and the task function error are expressed as:
Figure BDA0002204776870000063
Figure BDA0002204776870000064
wherein q isiIs the position, σ, of the ith roboti,1Is a task function, sigma, of the ith robot obstacle avoidance taski,1dIs an expected task function of the ith robot obstacle avoidance task,
Figure BDA0002204776870000065
Is the task function error of the ith robot obstacle avoidance task.
Definition of
Figure BDA0002204776870000066
Wherein
Figure BDA0002204776870000067
In the form of a jacobian matrix,
Figure BDA0002204776870000068
is a unit vector, λ, pointing to the nearest obstaclei,1> 0 is a state dependent gain, and
Figure BDA0002204776870000069
if and only if the robot is close enough to the obstacle,
Figure BDA00022047768700000610
and when the task is not active,
Figure BDA00022047768700000611
this task is built independently for each robot, rather than a cumulative task function.
2.1.2 convoy mission
In the convoy mission, all robots are evenly distributed around the target. The motion of the target is not known in advance, but can be measured in real time. The robot convinces the target in a p-dimensional space at a fixed distance (where p ≧ 2 is a positive integer). Two subtasks are defined here, a subtask that evenly distributes the robot around the target and a subtask that is maintained on a spherical or hypersphere surface centered on the target.
2.1.2.1 evenly distributing the subtasks of the robot around the target
In the case of a reference plane, the requirement that n robots are evenly distributed around the target is met by staying n robots at the vertices of a regular n polygon, wherein the distances between adjacent vertices of the regular n polygon are equal.
In the planar case, the vector k is defined as an identifying index of the circle position. Thus k does not necessarily refer to the kth robot but only to an index, kjAnd kj+1Are meant to refer to two consecutive positions along the loop. Obviously, the position of any robot on the circle can be set to k1And k is1And knIs a continuous position index.
The task function, the desired task function and the task function error are respectively expressed as:
Figure BDA0002204776870000071
Figure BDA0002204776870000072
Figure BDA0002204776870000073
wherein liIs a suitable distance between two adjacent robots.
The desired speed for this task is
Figure BDA0002204776870000074
The corresponding jacobian matrix is:
Figure BDA0002204776870000075
the pseudo-inverse is:
Figure BDA0002204776870000076
Figure BDA0002204776870000077
representing the gain of a constant positive definite matrix.
2.1.2.2 subtasks maintained on a spherical or hypersphere surface centred on the target
The task function of the subtask, the expected task function and the task function error are respectively expressed as:
Figure BDA0002204776870000078
Figure BDA0002204776870000079
Figure BDA00022047768700000710
wherein q is1Is the position of the 1 st robot, qnIs the location of the nth robot;
Figure BDA00022047768700000711
is the ideal velocity vector on the sphere/hypersphere surface that maintains the robot at a distance R from the target c, the corresponding jacobian matrix is:
Figure BDA00022047768700000712
and the pseudo-inverse is:
Figure BDA00022047768700000713
Figure BDA00022047768700000714
and Λ2Similarly, it is also a constant positive definite matrix of the gain.
In the case of a plane, the robot is distributed over the vertices of a positive polymorphism, liCan be arranged as
Figure BDA00022047768700000715
However, the problem of how to distribute points on spheres and hyperspheres is considered as a thomson problem in three dimensions and in p-dimensions (p > 3). Many scholars have studied this problem and there are indeed suitable distances.
If any robot is out of control and enters the virtual area B of another roboti,oIt will be seen as an obstacle that the rest of the robots must avoid. If two or more obstacles are considered simultaneously, the nearest obstacle will be processed first.
2.2 inner Loop controller design
The inner loop is a combination of linear proportional-derivative control (PD) and non-linear sliding-mode control (SMC) with adaptive control. A Proportional Derivative (PD) control part is used for stabilizing the equivalent part, a Sliding Mode Controller (SMC) part is used for compensating external interference and uncertainty of the system, and an adaptive control part is used for estimating unknown system parameters.
2.2.1 adaptive proportional differential-sliding mode control law (APD-SMC)
The control problem involves designing the controller so that each robot tracks the desired trajectory
Figure BDA0002204776870000081
Can be obtained by comparing the equation (2)
Figure BDA0002204776870000082
Is obtained by the integration of (a) to (b),
Figure BDA0002204776870000083
are similarly defined, they are all bounded functions.
Figure BDA0002204776870000084
And
Figure BDA0002204776870000085
respectively tracking error and tracking velocity error.
The equivalent reference state is defined as follows:
Figure BDA0002204776870000086
Figure BDA0002204776870000087
wherein the content of the first and second substances,
Figure BDA0002204776870000088
and γ ═ diag (γ)1,γ2...γn) Is a positive diagonal matrix defined as a sliding mode constant, and a sliding mode surface is defined as:
Figure BDA0002204776870000089
wherein the content of the first and second substances,
Figure BDA00022047768700000810
based on Properties 1.3, ginsengThe reference torque is described as follows:
Figure BDA00022047768700000811
based on the cooperative control of the multi-Euler-Lagrange system, the following adaptive proportional differential sliding mode control law (APD-SMC) is provided
Figure BDA00022047768700000812
Wherein k ispiAnd kdiProportional and differential control gains, k, respectivelyiIs a robust control gain. All three gain matrices are set to be positive,
Figure BDA00022047768700000813
is the reference torque ρiAnd updated with the adaptive law. Due to the use of the adaptive proportional differential-sliding mode control law (APD-SMC), the large control gain required in the proportional differential controller to compensate for the unknown term is avoided.
Figure BDA00022047768700000814
Figure BDA00022047768700000815
Wherein, muiIs a diagonal matrix, representing the adaptive update rate,
Figure BDA0002204776870000091
an initial value representing the estimated torque vector.
It can be seen that the adaptive proportional differential-sliding mode control law (APD-SMC) in equation (9) is only related to the tracking error eiAnd tracking velocity error
Figure BDA0002204776870000092
It is related. Thus, the control laws are modeless and easily implemented in practice.
3. Stability analysis
From equation (7), one can obtain:
Figure BDA0002204776870000093
Figure BDA0002204776870000094
with respect to equations (1), (8), and (11), it can be found that:
Figure BDA0002204776870000095
theorem 3.1 consider the mission function error in equations (3) - (5), the multi-Euler-Lagrangian system equation (1) and the adaptive proportional-derivative sliding-mode control law (APD-SMC) (9) and adaptive update rate (10). Assuming that properties 1.1-1.4 hold, one can obtain:
(1) if there are no conflicts between the three tasks, they can be completed simultaneously. The system is globally asymptotically stable and the tracking error eiMay converge to zero.
(2) If the obstacle avoidance task is active and conflicts with the convoy task, the gain may be set to
Figure BDA0002204776870000096
Wherein
Figure BDA00022047768700000912
Is based on robustness against noise, such as measurement noise. The obstacle avoidance task is completed first, and in addition, the system can be obtained to be globally asymptotically stable, and the tracking error eiMay converge to zero.
Proves that the Lyapunov function V of the whole system is designed as
V＝V1+V2
Figure BDA0002204776870000097
Figure BDA0002204776870000098
Wherein the content of the first and second substances,
Figure BDA0002204776870000099
η1,η2and η3Is a design parameter, and
Figure BDA00022047768700000910
inertia matrix MiIs positive, kpi,kdi,η1,η2And η3Is set to positive. It can easily be demonstrated that V is a positive function. Based on equations (9) and (12), the following equation can be obtained
Figure BDA00022047768700000911
Will V1Is differentiated with respect to time to obtain
Figure BDA0002204776870000101
Substituting equations (7), (10) and (14) into (15) can result in
Figure BDA0002204776870000102
Lyapunov function V1Is positive, the time derivative thereof
Figure BDA0002204776870000103
Is negative. It can be obtained that the inner loop system is globally asymptotically stable and the tracking error eiConvergence to zero can be obtained according to the Lyapunov method. In the above equation, only
Figure BDA0002204776870000104
Equation of time
Figure BDA0002204776870000105
Is satisfied.
In practice, the discontinuous term function sign () in equation (9) may cause buffeting problems. To avoid the chattering problem, a saturation function tanh (-) is introduced instead of the discontinuous function sign (-).
Control law modification in equation (9)
Figure BDA0002204776870000106
Wherein the content of the first and second substances,
Figure BDA0002204776870000107
using equation (13), one can obtain:
Figure BDA0002204776870000108
in the following, two cases are discussed separately: conflicting tasks and non-conflicting tasks. Assuming that there is no conflict between every two tasks, an interesting property of the jacobian matrix can be exploited, i.e. the non-conflicting relation between three tasks can be expressed as
Figure BDA0002204776870000111
Figure BDA0002204776870000112
Figure BDA0002204776870000113
This means that the three tasks project to the robot velocity space are orthogonal; thus, they can be implemented simultaneously. Substituting equation (19) into equation (18) may result in
Figure BDA0002204776870000114
Thus, from the above analysis, it was found that the outer ring subsystem is asymptotically stable if there is no conflict between each pair of tasks. At the same time, if the obstacle avoidance task is activated and conflicts with the convoy task
Figure BDA0002204776870000115
It is in this form
Figure BDA0002204776870000116
Wherein
Figure BDA0002204776870000117
And P ═ Pij]I, j is 1,2,3, the submatrices are p respectively11＝η1λ1,
Figure BDA0002204776870000118
Figure BDA0002204776870000119
Figure BDA00022047768700001110
For any one
Figure BDA00022047768700001111
Application of 2| ab | < a2+b2To obtain
Figure BDA00022047768700001112
Wherein p isij,mAnd pij,MRespectively represent the Induction subblocks pijUpper and lower limits of (d).
Figure BDA00022047768700001113
It can be seen that because | | J1||＝||J2||＝||J3||＝1，p22,m＝p 22,M0 and p33,m＝p 33,M0, loss
Figure BDA00022047768700001114
And
Figure BDA00022047768700001115
in this case V2Should be reset to
Figure BDA00022047768700001116
To obtain
Figure BDA00022047768700001117
Similarly, the outer ring subsystem is globally asymptotically stable.
In this case, the conclusion of theorem 4.1 is reached, which means that the obstacle avoidance task has a higher priority and is first achieved when it conflicts with the convoy task. And when no conflict exists, the convoy mission is executed again. Since the space-time behavior (NSB) control method is a kinematics that reacts to dynamics with a desired velocity rather than a desired position, λ is appropriately designedi,1So that the velocity error dominates the position error.
Sliding-mode equation (7) contains position and velocity errors by substituting equation (2) into equation (7) and taking into account the fact that if every two tasks collide
Figure BDA0002204776870000121
And
Figure BDA0002204776870000122
can be used forIs removed, therefore
Figure BDA0002204776870000123
Can obtain
Figure BDA0002204776870000124
By using inequality (23) as an equation and taking norm on both sides, it is possible to obtain
Figure BDA0002204776870000125
Figure BDA0002204776870000126
Figure BDA0002204776870000127
Figure BDA0002204776870000128
Thus, by selecting
Figure BDA0002204776870000129
Wherein
Figure BDA00022047768700001219
Is based on robustness to noise, which constant ensures that the robot is far from the obstacle.
It should be noted that if only two subtasks in the convoy mission conflict, each robot only completes the first two higher-level tasks. No collision will occur, wherein
Figure BDA00022047768700001210
If a robot collides with an obstacle in the front,and the projection in the tangential direction is empty, the robot will stop moving. Measuring noise
Figure BDA00022047768700001220
Local minima caused by this situation can be avoided.
4. Simulation verification
The simulation verification proves that the proposed algorithm is more effective than an Adaptive Sliding Mode Control (ASMC) method in two-dimensional and three-dimensional spaces under three scenes respectively.
The kinetic equation of each robot is set as
Figure BDA00022047768700001211
Where the parameters are defined similarly to equation (1), the noise is considered to be contained in a tight set
Figure BDA00022047768700001212
And the measurement state is
Figure BDA00022047768700001213
And
Figure BDA00022047768700001214
and assuming the control force | | τi||≤10N。
Consider five robots in two-dimensional space, assuming that the system parameters, controller parameters, and adaptive law parameters are M, respectivelyi＝1，Ci＝0，kpi＝30,kdi＝50,ki＝1,γ＝diag(20,...,20),
Figure BDA00022047768700001215
μ i20, where i 1. The task parameters are R-5 and d respectivelyi＝2,
Figure BDA00022047768700001216
Λ2＝Λ3＝0.8I,
Figure BDA00022047768700001217
The parameter of the external disturbance is
Figure BDA00022047768700001218
The initial positions of the five robots are q1(0)＝[5,10]T,q2(0)＝[-5,5]T,q3(0)＝[-5,-5]T,q4(0)＝[5,-10]T,q5(0)＝[5,0]T。
In three-dimensional space, six robots are considered during simulation, and corresponding parameters and initial positions are set
Figure BDA0002204776870000131
Figure BDA0002204776870000132
q1(0)＝[-10,1,0]T,q2(0)＝[-1,-10,0.3]T,q3(0)＝[10,0,1]T,q4(0)＝[0,0.5,10]T,q5(0)＝[0,10,0.3]T,q6(0)＝[0,0,-10]T. The other parameters are similar to the parameter settings in the two-dimensional case.
ASMC is designed according to an adaptive control law (10)
Figure BDA0002204776870000133
Wherein the controller parameter is kai＝80。
In three-dimensional control, six robots eventually reach the apex position of the octahedron, where the target is the center of the octahedron, the lenz of which is the distance between adjacent robots, as shown in fig. 4. Thus, the pseudo-inverse of the task function and the Jacobian matrix can be rewritten as
Figure BDA0002204776870000134
Figure BDA0002204776870000135
Figure BDA0002204776870000136
Figure BDA0002204776870000137
Equation (2) can be rewritten as
Figure BDA0002204776870000138
All simulation experiments are carried out with
Figure BDA0002204776870000139
Runs under MATLAB R2016a installed by a computer of Core I5-4590, a 3.6-GHz CPU,12GB of RAM,1000-GB solid-state disk drive.
4.1 convoying a stationary target
4.1.1 two-dimensional case
Five robots convoy a stationary object c ═ 3,0]TFig. 5(a) - (c) show the distance between the robot and the target, the distance between adjacent robots and the position tracking error, respectively, under adaptive proportional differential-sliding mode control (APD-SMC).
The results show that all robots can maintain a fixed distance from the target and surround it evenly. Fig. 6(a) - (c) show poor control performance compared to using adaptive proportional differential-sliding mode control (APD-SMC). This further verifies that adaptive proportional differential-sliding mode control (APD-SMC) achieves convergence faster and with higher control accuracy than Adaptive Sliding Mode Control (ASMC).
4.1.2 three-dimensional case
Fig. 7 shows the trajectory of the robot and the target in three-dimensional space. This verifies the effectiveness of adaptive proportional differential-sliding mode control (APD-SMC) in three dimensions.
4.2 convoying a constant speed target
4.2.1 two-dimensional case
Fig. 8 shows the proposed control algorithm for convoy c ═ 3+0.1t,0]TThe performance of the target.
The validity of the proposed control laws when there are two obstacles is verified below. As a result of the simulation, as shown in fig. 9 and 10, when two obstacles enter the threshold circles of the robot 1 and the robot 4, respectively, all the robots change their positions to avoid the collision between the obstacle and the robots.
5.2.2 three-dimensional case
Fig. 11 and fig. 12(a) - (c) show convoying target c ═ 0.1t,0 in three-dimensional space using the proposed law]TThe simulation experiment result of (2). FIGS. 13(a) - (c) show the results when ASMC is used; and slower convergence speed and lower control accuracy are observed compared to using APD-SMC.
4.3 convoying a target of varying speed
4.3.1 two-dimensional case
Fig. 14 and 15(a) - (c) depict convoying a target c ═ 3+0.1t, sin (0.1t) in two-dimensional space]TAnd (4) obtaining a simulation result.
5.3.2. Three dimensional situation
In three dimensions, there is an obstacle q0＝[15,3,-4.5]TWhen the convoy target c is [0.1t, sin (0.1t),0]TAs shown in fig. 16 and 17.
In summary, the simulation results show that the control strategy proposed by the present embodiment has faster and higher control performance than Adaptive Sliding Mode Control (ASMC), and also shows good performance when avoiding obstacles.
Example two
In one or more embodiments, a cooperative control system with an obstacle avoidance and navigation protection task for a multi-euler-lagrange system is disclosed, which includes: the controller adopts an inner ring and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and expected speed and expected motion trail required by an inner ring physical model are generated; and the inner ring applies an adaptive proportional-differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion trail.
EXAMPLE III
In one or more embodiments, a terminal device is disclosed that includes a processor and a computer-readable storage medium, the processor to implement instructions; the computer-readable storage medium is used for storing a plurality of instructions, and the instructions are suitable for being loaded by a processor and executing the cooperative control method with the obstacle avoidance and navigation protection task of the euler-lagrange system in the embodiment one.
Although the embodiments of the present invention have been described with reference to the accompanying drawings, it is not intended to limit the scope of the present invention, and it should be understood by those skilled in the art that various modifications and variations can be made without inventive efforts by those skilled in the art based on the technical solution of the present invention.

Claims (9)

1. The cooperative control method of the multi-Euler-Lagrange system with the obstacle avoidance and navigation protection task is characterized by comprising the following steps:
describing a physical model of the convoy task by adopting a multi-Euler-Lagrange system;
the controller adopts an inner and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and expected speed and expected movement track required by an inner ring physical model are generated; the inner ring adopts a self-adaptive proportional differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion track;
the space behavior control framework considering obstacles specifically includes:
the following are respectively described by an empty space behavior control method: the obstacle avoidance task comprises subtasks which are uniformly distributed around a target and subtasks which are maintained on the surface of a sphere or a hyper-sphere which takes the target as a center, the subtasks are subtasks of a convoy task, and the priority of the obstacle avoidance convoy task is defined;
the desired velocities of the individual tasks are projected into the empty space created by the Jacobian matrix of high priority tasks, building an integrated total desired velocity command to drive the physical model.
2. The cooperative control method with obstacle avoidance and navigation protection task for multi-Euler-Lagrange system of claim 1, wherein the physical model comprises: any one of a multi-robot arm, a multi-mobile robot, a multi-step robot, and a multi-spacecraft.
3. The cooperative control method with the obstacle avoidance and navigation protection task for the multi-Euler-Lagrange system according to claim 1, wherein the integrated total desired speed is specifically as follows:
Figure FDA0002607219680000011
wherein the content of the first and second substances,
Figure FDA0002607219680000012
a desired speed output for the outer loop;
Figure FDA0002607219680000013
the expected speed required by the obstacle avoidance task;
Figure FDA0002607219680000014
and
Figure FDA0002607219680000015
desired speeds required for evenly distributing the robot's subtasks around the target and the subtasks maintained on the surface of a sphere or hyper-sphere centered on the target, respectively; j. the design is a square1Is a Jacobian matrix of the obstacle avoidance task,
Figure FDA0002607219680000016
Is the pseudo-inverse, J of the jacobian matrix of the obstacle avoidance task2Is a Jacobian matrix that evenly distributes the subtasks of the robot around the target,
Figure FDA0002607219680000017
Is the pseudo-inverse of the jacobian matrix of the uniformly distributed robot subtasks around the target, and n is the number of robots.
4. The cooperative control method with the obstacle avoidance and navigation task for the multi-Euler-Lagrange system as claimed in claim 1, wherein the task function and the task function error of the obstacle avoidance task are respectively expressed as:
Figure FDA0002607219680000018
Figure FDA0002607219680000019
wherein the content of the first and second substances,
Figure FDA0002607219680000021
representing the current obstacle position for the ith robot, diIs the minimum allowable safe distance between the ith robot and the obstacle, qiIs the position, σ, of the ith roboti,1Is a task function, sigma, of the ith robot obstacle avoidance taski,1dIs an expected task function of the ith robot obstacle avoidance task,
Figure FDA0002607219680000022
Is the task function error of the ith robot obstacle avoidance task.
5. The cooperative control method with the obstacle avoidance and navigation protection task for the multi-Euler-Lagrange system according to claim 1, wherein the task functions of the subtasks uniformly distributed around the target, the expected task functions and the task function errors are respectively expressed as:
Figure FDA0002607219680000023
Figure FDA0002607219680000024
Figure FDA0002607219680000025
wherein liIs the distance between two adjacent robots, the vector k is defined as the identification index of the circle position, pkjIndicating the position of the robot on the circle.
6. The cooperative control method with obstacle avoidance and navigation task for multi-Euler-Lagrange system as claimed in claim 1, wherein the subtask maintained on the surface of a sphere or a hyper-sphere centered on the target has a task function, an expected task function and a task function error respectively expressed as:
Figure FDA0002607219680000026
Figure FDA0002607219680000027
Figure FDA0002607219680000028
wherein the content of the first and second substances,
Figure FDA0002607219680000029
is to maintain the ideal velocity vector q of the robot on the surface of a sphere or a hyper-sphere with a radius R and a target c as the center1Is the position of the 1 st robot, qnIs the position of the nth robot.
7. The cooperative control method with the obstacle avoidance and navigation protection task for the multi-Euler-Lagrange system according to claim 1, wherein the inner loop applies an adaptive proportional differential sliding mode control method, which specifically comprises:
Figure FDA00026072196800000210
wherein k ispiAnd kdiProportional and differential control gains, k, respectivelyiIs a robust control gain;
Figure FDA00026072196800000211
is the reference torque ρiAnd updated with an adaptive law; e.g. of the typeiAnd
Figure FDA0002607219680000031
respectively tracking error and tracking velocity error, siIs a slip form surface.
8. The multi-Euler-Lagrange system is provided with a cooperative control system for avoiding obstacles and protecting navigation tasks, and is characterized by comprising the following steps: the controller adopts an inner ring and outer ring control structure, the outer ring adopts a space behavior control framework considering obstacles, and the expected speed and the expected motion track of the physical model required by the inner ring are generated; the inner ring adopts a self-adaptive proportional differential sliding mode control method, so that each physical model tracks the expected speed and the expected motion track;
the space behavior control framework considering obstacles specifically includes:
the following are respectively described by an empty space behavior control method: the obstacle avoidance task comprises subtasks which are uniformly distributed around a target and subtasks which are maintained on the surface of a sphere or a hyper-sphere which takes the target as a center, the subtasks are subtasks of a convoy task, and the priority of the obstacle avoidance convoy task is defined;
the desired velocities of the individual tasks are projected into the empty space created by the Jacobian matrix of high priority tasks, building an integrated total desired velocity command to drive the physical model.
9. A terminal device comprising a processor and a computer-readable storage medium, the processor being configured to implement instructions; the computer-readable storage medium is configured to store a plurality of instructions, wherein the instructions are adapted to be loaded by a processor and to execute the cooperative control method with obstacle avoidance and navigation task for the multi-Euler-Lagrangian system according to any one of claims 1-7.
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