CN113848905A - Mobile robot trajectory tracking method based on neural network and adaptive control - Google Patents

Mobile robot trajectory tracking method based on neural network and adaptive control Download PDF

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CN113848905A
CN113848905A CN202111121940.2A CN202111121940A CN113848905A CN 113848905 A CN113848905 A CN 113848905A CN 202111121940 A CN202111121940 A CN 202111121940A CN 113848905 A CN113848905 A CN 113848905A
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CN113848905B (en
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谭明虎
柴斌
张科
王靖宇
苏雨
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0214—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory in accordance with safety or protection criteria, e.g. avoiding hazardous areas
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
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Abstract

The invention discloses a mobile robot trajectory tracking method based on a neural network and self-adaptive control, which comprises the steps of firstly carrying out dynamics and kinematics modeling on WMRs under the conditions of incomplete constraint and speed limitation constraint; next, adopting an MPC method to realize track tracking under speed limitation; then fitting a kinetic equation of WMRs parameters by using an RBF network, and designing a control law based on an MRAC method; and finally, realizing accurate track tracking of the WMRs according to a designed control law. The invention can realize the motion control of the WMRs under specific conditions, improves the track tracking and anti-interference capability of the WMRs under special working conditions, and is beneficial to expanding the working space of the WMRs.

Description

Mobile robot trajectory tracking method based on neural network and adaptive control
Technical Field
The invention belongs to the technical field of robots, and particularly relates to a mobile robot trajectory tracking method.
Background
Robotics is an important branch in the field of automation and control, and has found wide application in industrial production and daily life. Robots can be divided into two major categories, namely fixed robots and mobile robots, wherein the mobile robots move in the environment by using wheels or 'simulated legs', and the application range and the working radius of the robot are greatly improved compared with those of the fixed robots. Wheeled Mobile Robots (WMRs) are the most common type of Mobile Robots in daily life, play an important role in military, civil and scientific exploration, and can perform information acquisition and processing tasks instead of people even in some severe environments (such as planet exploration, earthquake relief and the like). Currently, exploring the autonomous movement of WMRs in a complex dynamic environment is a hot topic of research.
The trajectory tracking is the basis of autonomous movement, and aims to control the WMRs to quickly and stably track one or more curves which are planned in advance and take time as a variable function by a reasonably designed controller. The traditional method for solving WMRs trajectory tracking comprises PID control, backstepping control, sliding mode control or self-adaptive control and the like, which all achieve good tracking effect, but when the conditions of non-integrity constraint, unknown dynamic model parameters, limited motion speed and the like exist, the ideal tracking effect is often not achieved. In recent years, with the development of artificial intelligence, a Model-Reference Adaptive Control (MRAC) technology which integrates the strong function fitting capability of a neural network has a good application effect in the Control field, realizes end-to-end information perception and intelligent Control facing complex environments, and provides a new idea for solving the problem of trajectory tracking of complex nonlinear systems.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a mobile robot track tracking method based on a neural network and self-adaptive control, firstly, the WMRs are subjected to dynamics and kinematics modeling under the conditions of incomplete constraint and speed limitation constraint; next, adopting an MPC method to realize track tracking under speed limitation; then fitting a kinetic equation of WMRs parameters by using an RBF network, and designing a control law based on an MRAC method; and finally, realizing accurate track tracking of the WMRs according to a designed control law. The invention can realize the motion control of the WMRs under specific conditions, improves the track tracking and anti-interference capability of the WMRs under special working conditions, and is beneficial to expanding the working space of the WMRs.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: performing dynamics and kinematics modeling on the WMRs under the conditions of incomplete constraint and speed limitation constraint;
setting the radius of a wheel of WMRs as r, the distance between two driving wheels as 2b, the center of the distance between the two wheels as a point G, the center of mass of the WMRs is concentrated on a point C, the mass of the WMRs is m, the moment of inertia of the WMRs is J, and the distance between the point C and the point G is d; the global coordinate system xoy is a Cartesian inertial coordinate system for WMRs to move in, is used for representing environmental information and corresponds to a global environment map; local coordinate system xccycThe center of mass C of the WMRs is used as an origin, and the direction with the forward linear velocity of the movement of the center of mass of the WMRs as upsilon is cxcA shaft; the rotation angular velocity of the mass center motion of WMRs is omega, the direction of upsilon and cxcThe axes are always in the same direction, and the included angle between the axes and the ox axis is defined as a forward angle theta, and omega represents that WMRs can rotate for 360 degrees around a point C;
step 1-1: the kinetic equation for WMRs is expressed as:
Figure BDA0003277555560000021
wherein p represents a system pose vector, M (p) represents a system symmetric positive definite inertia matrix,
Figure BDA0003277555560000022
a matrix of centripetal and coriolis forces is represented,
Figure BDA0003277555560000023
representing the surface dynamic friction vector, G (p) representing the gravity vector, τdRepresenting finite uncertainty factor, b (p) representing input transformation matrix, τ ═ τr,τl]TRepresenting input torque vector, τr,τlA component representing τ, a (p) represents an incomplete constraint matrix, and λ represents a constraint force vector;
when the wheel-ground contact surface has no sliding or sideslip phenomenon, the motion state of the WMRs meets the non-integrity constraint condition:
Figure BDA0003277555560000024
constructing an incomplete constraint matrix A (p) and an arbitrary tensor S (p) in a non-zero state space, and satisfying the relation:
ST(p)A(p)＝0 (3)
the forward linear velocity of the mass center motion of the WMRs is upsilon, the rotation angular velocity is omega, and a control vector u of the motion of the WMRs is formed, namely u is [ upsilon, omega)]TAnd satisfies the relation:
Figure BDA0003277555560000025
by substituting equation (1) after deriving equation (4) and then combining equations (2) and (3), the following relation (5) is obtained in the absence of surface dynamic friction and neglecting gravity:
Figure BDA0003277555560000026
wherein
Figure BDA0003277555560000027
Figure BDA0003277555560000028
Step 1-2: position coordinate (x) at centroid C position for pose of WMRsc,yc) And a forward direction angle θ, the coordinate positional relationship satisfies:
Figure BDA0003277555560000031
wherein v iscRepresenting the linear velocity of the forward motion of the center of mass;
namely:
Figure BDA0003277555560000032
simultaneously obtaining:
A(p)＝[cosθ,-sinθ,0]T (8)
the parameters are as follows:
Figure BDA0003277555560000033
G(p)＝O3×1，
Figure BDA0003277555560000034
step 1-3: given the desired tracking trajectory vector p for WMRsr＝[xr,yr,θr]TAnd a desired motion velocity vector ur＝[υr,ωr]TAnd it satisfies the relation:
Figure BDA0003277555560000035
Figure BDA0003277555560000036
assuming that the velocity vector u satisfies the relation:
0<u≤umax (11)
therefore, the objective of trajectory tracking is to solve the control moment τ so that p → p as the moving time of WMRs increases, while satisfying the relations (5), (7) to (11)rAnd u → urNamely:
Figure BDA0003277555560000037
defining the pose error as follows:
Figure BDA0003277555560000041
wherein
Figure BDA0003277555560000042
Local coordinate system x representing slave WMRsccycA transformation matrix to the global coordinate system xoy; e.g. of the typep1、ep2、ep3Respectively representing error components of the pose error along the x direction, the y direction and the forward angle direction;
derivation of equation (13) yields:
Figure BDA0003277555560000043
equation (12) is equivalent to:
Figure BDA0003277555560000044
step 2: the method adopts an MPC method to realize the track tracking under the speed limitation;
step 2-1: carrying out linearization and discretization on the formula (14), and taking the time k as the current time;
step 2-2: the design objective function is:
Figure BDA0003277555560000045
where Q and R are both weight matrices, N1Representing a prediction time domain, p (k + i | t) represents a value of a system actual motion pose vector for predicting k + i moment forward by taking a t moment value as a reference, and pr(k + i | t) represents the value of the reference trajectory pose vector at the k + i moment, t represents the current moment, N represents the forward prediction of the reference trajectory pose vector based on the t moment value2Represents a control time domain, and delta u (k + i | t) represents a value of a control vector increment for predicting k + i time forward by taking a t time value as a reference;
step 2-3: solving an optimization problem that satisfies an objective function (16) and a velocity constraint (11)To obtain the time domain [ k, k + N ] in the control time domain2]A series of control sequences are included, the first element of the control sequence is used as the actual control quantity of the controlled object, the process is repeated at the moment of k +1, the whole solving process is finished in a rolling way, and the optimal control quantity is obtained
Figure BDA0003277555560000046
Figure BDA0003277555560000047
Wherein k is1、k2And k3Are all different parameters;
and step 3: fitting a kinetic equation of WMRs parameters by using an RBF network, and designing a control law based on an MRAC method;
step 3-1: transforming equation (5) yields:
Figure BDA0003277555560000051
in the formula (18)
Figure BDA0003277555560000052
And
Figure BDA0003277555560000053
are all related to the structural parameters of WMRs, there are unmeasured or unknown items,
Figure BDA0003277555560000054
the method belongs to unknown items, and therefore, a radial basis function RBF is used for carrying out approximation processing on the unknown items;
step 3-2: network input vector of RBF
Figure BDA0003277555560000055
The RBF network approximation algorithm is as follows:
Figure BDA0003277555560000056
Figure BDA0003277555560000057
wherein upsilon is*Represents the optimal value of the linear velocity of the forward motion of the system, omega*Representing the optimum value of the angular velocity of rotation of the system, hiFor the output of a single neuron, g (.) represents the mapping relation of a radial basis kernel function, l is the weight of the neural network, and h (z) represents hiA finite linear combination of; by setting a parameter ciAnd biSetting learning rate and momentum factor to obtain new weight l when initial weight l is 0
Figure BDA0003277555560000058
Infinitely approaches u satisfying equation (18);
step 3-3: suppose that
Figure BDA0003277555560000059
Defining:
Figure BDA00032775555600000510
wherein k is4Is a control rate parameter;
based on the MRAC method, a reference self-adaptive control law is designed as follows:
Figure BDA00032775555600000511
wherein:
Figure BDA00032775555600000512
Figure BDA00032775555600000513
Figure BDA00032775555600000514
represents the Kronecker product; v. ofmaxRepresents the maximum value of the linear velocity of the forward motion of the system, omegamaxRepresenting the maximum value, x, of the angular velocity of rotation of the systemmaxDenotes vmaxOr ωmax；
And 4, step 4: and 3, realizing accurate track tracking of the WMRs according to the control law designed in the step 3.
Preferably, said τ isdThe finite uncertainty factors represented include unmodeled errors and bounded disturbances.
The invention has the following beneficial effects:
the invention can realize the motion control of the WMRs under specific conditions (non-integrity constraint, limited speed, unknown or partially unknown structural parameters), improves the track tracking and anti-interference capability of the WMRs under special working conditions, and is beneficial to expanding the working space of the WMRs.
Drawings
Fig. 1 is a planar structural model of typical WMRs of the present invention.
FIG. 2 is a path for solving the trajectory tracking problem of the present invention.
FIG. 3 is a control law block diagram of the present invention.
FIG. 4 illustrates the tracking effect of an embodiment of the present invention on circular and linear trajectories, wherein (a) is a circle and (b) is a straight line.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
When the wheeled mobile robot is driven in a differential mode, the wheeled mobile robot is simple in structure and flexible in steering, and is a good platform for verifying various advanced control algorithms. The invention aims at a Differential Drive (Differential Drive) wheel type mobile robot, and solves the technical problems that: when the WMRs move on the hard ground, the non-integrity constraint is not damaged because no sliding or sliding phenomenon exists between the wheel lands; for stability and safety of the movement, the movement speed of the WMRs must be properly limited; structural parameters of WMRs are unknown or vary due to limitations in measurement means or increased wear with increasing service life; therefore, in the presence of non-integrity constraints, limited motion speed and unknown or partially unknown structural parameters, how to control the motion state of WMRs enables tracking a pre-planned time-varying trajectory, thereby stably traveling according to the expected state.
A mobile robot track tracking method based on a neural network and self-adaptive control comprises the following steps:
step 1: performing dynamics and kinematics modeling on the WMRs under the conditions of incomplete constraint and speed limitation constraint;
a planar structural model of typical WMRs is shown in fig. 1. In the figure, WMRs are generally powered by two differential driving wheels coaxially mounted as rear wheels, and by universal wheels as front wheels to control direction and control balance, and can realize arbitrary movement and rotation in the horizontal plane (but only longitudinal movement and rotation back and forth, but not lateral movement left and right). Setting the radius of a wheel of WMRs as r (the diameter is 2r correspondingly), the distance between two driving wheels as 2b, the center of the distance between the two wheels as a point G, the center of mass of the WMRs is concentrated on a point C, the mass of the WMRs is m (the mass of the wheel is not contained), the moment of inertia is J, and the distance between the point C and the point G is d (in the actual design process, the center of mass point C can be coincided with the center point G through a method of measuring the center of mass, so that d can be determined as 0 in calculation); the Global coordinate system xoy is a cartesian inertial coordinate system for the WMRs to move therein for representing environmental information, corresponding to a Global environment Map (GM); local coordinate system xccycThe center of mass C of the WMRs is used as an origin, and the direction with the forward linear velocity of the movement of the center of mass of the WMRs as upsilon is cxcAxial, local coordinate system xccycThe motion coordinate system is fixedly connected on WMRs to represent the motion state of the WMRs and corresponds to a Local environment Map (LM); the rotation angular velocity of the mass center motion of WMRs is omega, the direction of upsilon and cxcThe axes are always in the same direction, and the included angle between the axes and the ox axis is defined as a forward angle theta, and omega represents that WMRs can rotate for 360 degrees around a point C;
step 1-1: according to the relevant knowledge of robot kinematics, the kinetic equation of the WMRs is expressed as:
Figure BDA0003277555560000071
wherein p represents a system pose vector, M (p) represents a system symmetric positive definite inertia matrix,
Figure BDA0003277555560000072
a matrix of centripetal and coriolis forces is represented,
Figure BDA0003277555560000079
representing the surface dynamic friction vector, G (p) representing the gravity vector, τdRepresenting finite uncertainty factor, b (p) representing input transformation matrix, τ ═ τr,τl]TRepresenting an input torque vector, A (p) representing an incomplete constraint matrix, and lambda representing a constraint force vector;
when the wheel-ground contact surface has no sliding or sideslip phenomenon, the motion state of the WMRs meets the non-integrity constraint condition:
Figure BDA0003277555560000073
constructing an incomplete constraint matrix A (p) and an arbitrary tensor S (p) in a non-zero state space, and satisfying the relation:
ST(p)A(p)＝0 (3)
the forward linear velocity of the mass center motion of the WMRs is upsilon, the rotation angular velocity is omega, and a control vector u of the motion of the WMRs is formed, namely u is [ upsilon, omega)]TAnd satisfies the relation:
Figure BDA0003277555560000074
after the derivation of the formula (4), the formula (1) is substituted, then the formulas (2) and (3) are combined, and in consideration of the actual situation (the non-integrity constraint conditions determine that no surface dynamic friction exists, and gravity can be ignored during plane motion), the following relation (5) is obtained:
Figure BDA0003277555560000075
wherein
Figure BDA0003277555560000076
Figure BDA0003277555560000077
Step 1-2: in fig. 1, the position coordinates (x) of WMRs at the centroid C position for posec,yc) And a forward direction angle θ, the coordinate positional relationship satisfies:
Figure BDA0003277555560000078
namely:
Figure BDA0003277555560000081
simultaneously obtaining:
A(p)＝[cosθ,-sinθ,0]T (8)
the parameters are as follows:
Figure BDA0003277555560000082
G(p)＝O3×1，
Figure BDA0003277555560000083
step 1-3: given the desired tracking trajectory vector p for WMRsr＝[xr,yr,θr]TAnd a desired motion velocity vector ur＝[υr,ωr]TAnd it satisfies the relation:
Figure BDA0003277555560000084
Figure BDA0003277555560000085
in order to ensure the stability of WMRs during motion, the actual motion velocity needs to be limited, assuming that the velocity vector u satisfies the relation:
0<u≤umax (11)
therefore, the objective of trajectory tracking is to solve the control moment τ so that p → p as the moving time of WMRs increases, while satisfying the relations (5), (7) to (11)rAnd u → urNamely:
Figure BDA0003277555560000086
the solution idea is shown in fig. 2.
Defining the pose error as follows:
Figure BDA0003277555560000087
wherein
Figure BDA0003277555560000091
Local coordinate system x representing slave WMRsccycA transformation matrix to the global coordinate system xoy;
derivation of equation (13) yields:
Figure BDA0003277555560000092
equation (12) is equivalent to:
Figure BDA0003277555560000093
step 2: adopting a Model Predictive Control (MPC) method to realize the track tracking under the speed limitation;
step 2-1: carrying out linearization and discretization on the formula (14), and taking the time k as the current time;
step 2-2: the design objective function is:
Figure BDA0003277555560000094
wherein Q and R are both weight matrices;
step 2-3: on the basis of the current pose state measurement value and the speed control quantity measurement value, a future section of time domain [ k, k + N ] of the system is predicted by combining a formula (4)1](prediction time domain) output, and solving the optimization problem which meets the objective function (16) and the speed constraint formula (11) to obtain the optimization problem in the control time domain [ k, k + N ]2]A series of control sequences are included, the first element of the control sequence is used as the actual control quantity of the controlled object, the process is repeated at the moment of k +1, the whole solving process is finished in a rolling way, and the optimal control quantity is obtained
Figure BDA0003277555560000095
Figure BDA0003277555560000096
Wherein k is1、k2And k3Are all different parameters;
and step 3: as shown in fig. 3, fitting a kinetic equation of WMRs parameters by using RBF network, and designing a control law based on the MRAC method;
step 3-1: transforming equation (5) yields:
Figure BDA0003277555560000097
in the formula (18)
Figure BDA0003277555560000101
And
Figure BDA0003277555560000102
are all related to the structural parameters of WMRs, there are unmeasured or unknown items,
Figure BDA0003277555560000103
the method belongs to unknown items, and therefore, a radial basis function RBF is used for carrying out approximation processing on the unknown items;
step 3-2: network input vector of RBF
Figure BDA0003277555560000104
The RBF network approximation algorithm is as follows:
Figure BDA0003277555560000105
Figure BDA0003277555560000106
wherein h isiIs the output of a single neuron, and l is the weight of the neural network; by setting a parameter ciAnd biSetting learning rate and momentum factor to obtain new weight l when initial weight l is 0
Figure BDA0003277555560000107
Infinitely approaches u satisfying equation (18);
step 3-3: suppose that
Figure BDA0003277555560000108
Defining:
Figure BDA0003277555560000109
wherein k is4Is a control rate parameter;
based on the MRAC method, a reference self-adaptive control law is designed as follows:
Figure BDA00032775555600001010
wherein:
Figure BDA00032775555600001011
Figure BDA00032775555600001012
represents the Kronecker product;
the reference adaptive control law meets the Lyapunov global stability law, so that the WMRs can be ensured to realize stable track tracking under the conditions of meeting non-integrity constraint and speed constraint;
when calculating, the unknown parameter k in the control law1～k4The Optimization can be performed by using Particle Swarm Optimization (PSO) algorithm. The PSO algorithm is a powerful tool for function optimization, the problem is intelligently solved by utilizing a possible solution of each particle representative problem and by means of simple behaviors of individual particles and information interaction in particle swarm, and the optimal parameter k is repeatedly iteratediThe searching and positioning of (2) has the advantages of high convergence rate and less parameters to be set.
And 4, step 4: and 3, realizing accurate track tracking of the WMRs according to the control law designed in the step 3.
From this control rate, the WMRs trajectory tracking effect is shown in fig. 4.
The method for tracking the WMRs track can realize accurate track tracking of the WMRs with unknown structural parameters or partially unknown structural parameters in a short time under the conditions of non-integrity constraint and speed limitation, has good tracking effect, and realizes motion control of the WMRs.

Claims (2)

1. A mobile robot track tracking method based on a neural network and self-adaptive control is characterized by comprising the following steps:
step 1: performing dynamics and kinematics modeling on the WMRs under the conditions of incomplete constraint and speed limitation constraint;
setting the radius of a wheel of WMRs as r, the distance between two driving wheels as 2b, the center of the distance between the two wheels as a point G, the center of mass of the WMRs is concentrated on a point C, the mass of the WMRs is m, the moment of inertia of the WMRs is J, and the distance between the point C and the point G is d; the global coordinate system xoy is a Cartesian inertial coordinate system for WMRs to move in, is used for representing environmental information and corresponds to a global environment map; local coordinate system xccycThe center of mass C of the WMRs is used as an origin, and the direction with the forward linear velocity of the movement of the center of mass of the WMRs as upsilon is cxcA shaft; the rotation angular velocity of the mass center motion of WMRs is omega, the direction of upsilon and cxcThe axes are always in the same direction, and the included angle between the axes and the ox axis is defined as a forward angle theta, and omega represents that WMRs can rotate for 360 degrees around a point C;
step 1-1: the kinetic equation for WMRs is expressed as:
Figure FDA0003277555550000011
wherein p represents a system pose vector, M (p) represents a system symmetric positive definite inertia matrix,
Figure FDA0003277555550000012
a matrix of centripetal and coriolis forces is represented,
Figure FDA0003277555550000013
representing the surface dynamic friction vector, G (p) representing the gravity vector, τdRepresenting finite uncertainty factor, b (p) representing input transformation matrix, τ ═ τr，τl]TRepresenting input torque vector, τr，τlA component representing τ, a (p) represents an incomplete constraint matrix, and λ represents a constraint force vector;
when the wheel-ground contact surface has no sliding or sideslip phenomenon, the motion state of the WMRs meets the non-integrity constraint condition:
Figure FDA0003277555550000014
constructing an incomplete constraint matrix A (p) and an arbitrary tensor S (p) in a non-zero state space, and satisfying the relation:
ST(p)A(p)＝0 (3)
the forward linear velocity of the mass center motion of the WMRs is upsilon, the rotation angular velocity is omega, and a control vector u of the motion of the WMRs is formed, namely u is [ upsilon, omega)]TAnd satisfies the relation:
Figure FDA0003277555550000015
by substituting equation (1) after deriving equation (4) and then combining equations (2) and (3), the following relation (5) is obtained in the absence of surface dynamic friction and neglecting gravity:
Figure FDA0003277555550000016
wherein
Figure FDA0003277555550000021
Figure FDA0003277555550000022
Step 1-2: position coordinate (x) at centroid C position for pose of WMRsc，yc) And a forward direction angle θ, the coordinate positional relationship satisfies:
Figure FDA0003277555550000023
wherein upsilon iscRepresenting the linear velocity of the forward motion of the center of mass;
namely:
Figure FDA0003277555550000024
simultaneously obtaining:
A(p)＝[cosθ，-sinθ，0]F (8)
the parameters are as follows:
Figure FDA0003277555550000025
G(p)＝Q3×1，
Figure FDA0003277555550000026
step 1-3: given the desired tracking trajectory vector p for WMRsr＝[xr，yr，θr]TAnd a desired motion velocity vector ur＝[υr，ωr]TAnd it satisfies the relation:
Figure FDA0003277555550000027
Figure FDA0003277555550000028
assuming that the velocity vector u satisfies the relation:
0＜u≤umax (11)
therefore, the objective of trajectory tracking is to solve the control moment τ so that p → p as the moving time of WMRs increases, while satisfying the relations (5), (7) to (11)rAnd u → urNamely:
Figure FDA0003277555550000031
defining the pose error as follows:
Figure FDA0003277555550000032
wherein
Figure FDA0003277555550000033
Local coordinate system x representing slave WMRsccycA transformation matrix to the global coordinate system xoy; e.g. of the typep1、ep2、ep3Respectively representing error components of the pose error along the x direction, the y direction and the forward angle direction;
derivation of equation (13) yields:
Figure FDA0003277555550000034
equation (12) is equivalent to:
Figure FDA0003277555550000035
step 2: the method adopts an MPC method to realize the track tracking under the speed limitation;
step 2-1: carrying out linearization and discretization on the formula (14), and taking the time k as the current time;
step 2-2: the design objective function is:
Figure FDA0003277555550000036
where Q and R are both weight matrices, N1Representing a prediction time domain, p (k + i | t) represents a value of a system actual motion pose vector for predicting k + i moment forward by taking a t moment value as a reference, and pr(k + i | t) represents the value of the reference trajectory pose vector at the k + i moment, t represents the current moment, N represents the forward prediction of the reference trajectory pose vector based on the t moment value2Represents a control time domain, and delta u (k + i | t) represents a value of a control vector increment for predicting k + i time forward by taking a t time value as a reference;
step 2-3: solving for fullnessThe optimization problem of the foot objective function (16) and the speed constraint formula (11) is obtained in a control time domain [ k, k + N ]2]A series of control sequences are included, the first element of the control sequence is used as the actual control quantity of the controlled object, the process is repeated at the moment of k +1, the whole solving process is finished in a rolling way, and the optimal control quantity is obtained
Figure FDA0003277555550000037
Figure FDA0003277555550000038
Wherein k is1、k2And k3Are all different parameters;
and step 3: fitting a kinetic equation of WMRs parameters by using an RBF network, and designing a control law based on an MRAC method;
step 3-1: transforming equation (5) yields:
Figure FDA0003277555550000041
in the formula (18)
Figure FDA0003277555550000042
And
Figure FDA0003277555550000043
are all related to the structural parameters of WMRs, there are unmeasured or unknown items,
Figure FDA0003277555550000044
the method belongs to unknown items, and therefore, a radial basis function RBF is used for carrying out approximation processing on the unknown items;
step 3-2: network input vector of RBF
Figure FDA0003277555550000045
RBF network approximation algorithmComprises the following steps:
Figure FDA0003277555550000046
Figure FDA0003277555550000047
wherein upsilon is*Represents the optimal value of the linear velocity of the forward motion of the system, omega*Representing the optimum value of the angular velocity of rotation of the system, hiFor the output of a single neuron, g (.) represents the mapping relation of a radial basis kernel function, l is the weight of the neural network, and h (z) represents hiA finite linear combination of; by setting a parameter ciAnd biSetting learning rate and momentum factor to obtain new weight l when initial weight l is 0
Figure FDA00032775555500000414
Infinitely approaches u satisfying equation (18);
step 3-3: suppose that
Figure FDA0003277555550000048
Defining:
Figure FDA0003277555550000049
wherein k is4Is a control rate parameter;
based on the MRAC method, a reference self-adaptive control law is designed as follows:
Figure FDA00032775555500000410
wherein:
Figure FDA00032775555500000411
Figure FDA00032775555500000412
Figure FDA00032775555500000413
represents the Kronecker product; v. ofmaxRepresents the maximum value of the linear velocity of the forward motion of the system, omegamaxRepresenting the maximum value, x, of the angular velocity of rotation of the systemmaxDenotes vmaxOr ωmax；
And 4, step 4: and 3, realizing accurate track tracking of the WMRs according to the control law designed in the step 3.
2. The method for tracking the trajectory of the mobile robot based on the neural network and the adaptive control as claimed in claim 1, wherein τ is greater than or equal to τdThe finite uncertainty factors represented include unmodeled errors and bounded disturbances.
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