CN117075604A - Wheel type mobile robot control method based on generalized performance measurement - Google Patents
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Abstract
The invention discloses a wheel type mobile robot control method based on generalized performance measurement, which comprises the following steps: establishing a WMR dynamic model according to a WMR system of the wheeled mobile robot; on the basis of the established dynamic model, taking system constraint conditions into consideration, and designing a robust controller based on Udwadia-Kalaba theory and generalized performance measurement; aiming at the designed robust controller, an intelligent optimization method is adopted to carry out the optimization design of the controller parameters so as to achieve the purpose of reducing the control force under the condition of better control performance. The controller designed by the invention has excellent performance, can obviously improve the external interference resistance of the system and ensures the stability and reliability of control performance. In addition, the method can realize the tracking of the ideal track in a shorter time, thereby improving the performance and efficiency of the system.
Description
Technical Field
The invention belongs to the technical field of wheeled mobile robot control, and particularly relates to a wheeled mobile robot control method based on generalized performance measurement.
Background
The wheel type mobile robot (WMR) is a robot with high practical value, has wide application in various fields of military, ocean and human life, and can perform tasks such as mine removal, submarine survey, unmanned vehicle driving, coal mine underground operation and the like. However, the wheel type mobile robot is affected by complete constraint, incomplete constraint, uncertainty and the like in the track tracking control process, and affects the control precision and stability. Aiming at the problem that students at home and abroad adopt different methods to process, or inconvenient to process incomplete constraint, or the design of the controller is inflexible, or the controller is not optimized, the relation between control precision and control cost cannot be well balanced, the flexibility of the design of the controller is limited, and the invention provides a wheel type mobile robot control method based on generalized performance measurement for solving the problem. The method can effectively improve the control precision of the wheeled mobile robot, reduce the control cost and remarkably improve the flexibility of control design. In addition, the controller design method can also provide a new idea for the design and optimization of a control system.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a wheel type mobile robot control method based on generalized performance measurement, which is reasonable in design, overcomes the defects in the prior art and has good effect.
A wheel type mobile robot control method based on generalized performance measurement comprises the following steps:
step 1: establishing a WMR dynamic model according to a WMR system of the wheeled mobile robot;
step 2: based on the established dynamics model, considering the constraint condition of the system, and based on Udwadia-Kalaba theory and generalized performance measurementDesigning a robust controller;
step 3: aiming at the designed robust controller, an intelligent optimization method is adopted to carry out the optimization design of the controller parameters so as to achieve the purpose of reducing the control force under the condition of better control performance.
Further, the specific process of the step 1 is as follows:
the structural constraints of building WMR are:
in the method, in the process of the invention,for the Y-axis position component Y 0 First derivative of>For the X-axis position component X 0 θ is the included angle between the connecting line of the centroid C and the origin of coordinates O and the X-axis direction;
establishing a WMR dynamic model containing uncertainty as follows:
wherein X is E R n As a coordinate vector of the system,for velocity vector, +.>As the acceleration vector, the acceleration vector is calculated,uncertainty parameters representing the system, +.>Representing boundaries of uncertainty parameters, t.epsilon.R n For time, τ ε R n For control input, M (x, σ, t) ∈R n×n For the quality matrix of the system, < > for>Is a matrix of the coriolis force,f (x, sigma, t) ε R is gravity n×1 Other external disturbance forces;
in the formula (2), the amino acid sequence of the compound,
wherein m is the mass of the mobile robot, and the unit is kg; j is the moment of inertia of the mobile robot, and the unit is kg.m 2 。
Further, the step 2 includes the following substeps:
step 2.1: m (-) containing uncertainty factors is decomposed into:
in the middle ofIs the nominal part of the mass matrix M (,), ΔM (,) is the uncertainty part of the mass matrix M (,), let ∈ -> Then->Order theThus Δd (x, σ, t) =d (x, t) E (x, σ, t);
decomposing C (-) into nominal fractionsAnd an indeterminate moiety ΔC (·), decomposing G (·) into a nominal moiety +.>And an indeterminate moiety ΔG (·) decomposing F (·) into a nominal moiety +.>And an uncertainty fraction Δf (·);
step 2.2: establishing a WMR track constraint;
the mobile robot moves along a circle with the radius of 1m, the circle center is the origin of coordinates O (0, 0), and the expected track equation is x 0d =cost,y 0d =sint, the error between the actual trajectory and the desired trajectory is:
e 1 =x 0 -cost, (4)
e 2 =y 0 -sint, (5)
combining equations (4) and (5), establishing a trajectory constraint as:
wherein i=1, 2, k i Is a control parameter;
differentiating equation (6) once with respect to time is:
expressed as a second order matrix form:
wherein,
step 2.3: designing a controller τ comprising the sub-steps of:
step 2.3.1: based on Udwadia-Kalaba theory, a controller tau of a nominal part of a processing system is designed 1 :
Designing generalized performance metrics
Wherein,for describing the dynamic function of the performance metric dynamics, +.>Wherein ψ is a control parameter;
for all ofThe method meets the following conditions:
wherein the function gamma j (. Cndot.) is a function related to the Lyapunov function, j=1, 2,3, satisfying γ j (0)=0,
Step 2.3.2: design controller tau 2 ;
There is an unknown constant vector alpha epsilon (0, ++) k And a known function pi (·): r is R n ×R n ×R→R n + So that
Wherein ρ is E In order to control the parameters of the device,σ∈Ξ;
definition:
wherein x, t E R n X R, V is Lyapunov function, the control parameter ρ in equation (14) E > -1, and satisfy:
wherein lambda is m Is the minimum characteristic value;
in the absence of an uncertainty as to whether or not there is any uncertainty,thus selecting ρ E =0;
Thus, a controller τ of a processing system is designed that contains an uncertainty portion 2 The method comprises the following steps:
wherein, kappa is a control parameter;
step 2.3.3: the controller is as follows:
further, the step 3 includes the following substeps:
step 3.1: design of WMR System control Performance index J 1 (κ):
In the method, in the process of the invention,is the maximum value of alpha;
step 3.2: design of WMR System control cost index J 2 (κ):
J 2 (κ)=h 1 (κ), (20)
In the formula, h 1 (·):R + →R + Is a strictly increasing function of κ;
step 3.3: design of WMR system performance index J (κ):
J(κ)=a 1 J 1 (κ)+a 2 J 2 (κ), (21)
wherein a is 1 、a 2 > 0 is a weight factor, a k The larger the value of (1), k=1, 2, giving the performance index J m The greater the weight of (a), m=1, 2, a is set 1 And a 2 The relation between is a 1 +a 2 =1;
Step 3.4: the optimal solution is solved and the method comprises the steps of,
finding kappa to minimize J (kappa) is a constraint optimization problem, the requirements of which are:
the full conditions are as follows:
assuming that the solution κ of the conditional expression (22) exists, it is a candidate solution to the optimization problem, if κ * Satisfying the sufficient condition (23), it is the optimal solution of the optimization problem;
let gamma 3 =l 1 r p ,h(κ)=l 2 κ q Wherein l 1 ,l 2 >0,p,q>0,l 1 、l 2 P and q are all design parameters, then there is always one unique solution for the requirement, and the unique solution is:
the invention has the beneficial technical effects that:
the controller designed by the invention has excellent performance, can obviously improve the external interference resistance of the system and ensures the stability and reliability of control performance. In addition, the method can realize the tracking of the ideal track in a shorter time, thereby improving the performance and efficiency of the system. Simulation and experimental results prove the effectiveness and feasibility of the control method provided by the invention. The controller design method also provides a new idea for the design and optimization of a control system.
Drawings
FIG. 1 is a flow chart of a control method of a wheeled mobile robot according to the present invention;
FIG. 2 is a schematic view of a wheeled mobile robot according to the present invention;
FIG. 3 is a schematic diagram of a wheel type mobile robot system for simulating stable circular track tracking within 0-8 seconds in the invention;
FIG. 4 is a schematic diagram of a wheel type mobile robot system for simulating stable circular track tracking within 0-20 seconds in the invention;
FIG. 5 is a graph of constraint following error β over time for a wheeled mobile robotic system of the present invention;
FIG. 6 is a graph showing performance metrics of a wheeled mobile robotic system in accordance with the present inventionA graph of change over time;
Detailed Description
The following is a further description of embodiments of the invention, in conjunction with the specific examples:
a wheel type mobile robot control method based on generalized performance measurement, as shown in figure 1, comprises the following steps:
step 1: establishing a WMR dynamic model according to a WMR system of the wheeled mobile robot;
step 1 comprises the following sub-steps:
step 1.1: the dynamic model of the automatic driving automobile system without constraint is established according to Newton mechanics and is as follows:
wherein M is an inertial mass matrix,the second derivative of the X-axis position component X, Q is the given force of the autopilot; m, x, Q are defined as follows:
step 1.2: establishing structural constraint of WMR;
WMR moves on a plane, during which forces perpendicular to the wheels are balanced, structurally constrained by the structural knowledge of WMR:
in the method, in the process of the invention,for the Y-axis position component Y 0 First derivative of>For the X-axis position component X 0 θ is the included angle between the connecting line of the centroid C and the origin of coordinates O and the X-axis direction;
equation (2) performs a primary differentiation of time t:
the structural constraint (3) can be expressed in matrix form:
wherein A is s =[-sinθ cosθ 0],
Step 1.3: establishing a WMR dynamic model containing uncertainty;
according to Udwadia-Kalaba theory, the WMR is subjected to structural constraint force Q s The method comprises the following steps:
Q s =M 1/2 A s M -1/2 (b s -A s M -1 Q), (5)
thus, the autopilot dynamics equation with structural constraints is:
taking the uncertainty of the system into consideration, integrating the formula (6), wherein a dynamics model of the WMR system containing uncertainty is as follows:
wherein x epsilon Rn is the coordinate vector of the system,for velocity vector, +.>As the acceleration vector, the acceleration vector is calculated, uncertainty parameters representing the system, +.>Representing boundaries of uncertainty parameters, t.epsilon.R n For time, τ ε R n Is a control input; m (x, sigma, t) ∈R n×n For the quality matrix of the system, < > for>Is a matrix of the coriolis force, f (x, sigma, t) ε R is gravity n×1 Other external disturbance forces;
in the formula (7):
wherein m is the mass of the mobile robot, and the unit is kg; j is the moment of inertia of the mobile robot, and the unit is kg.m 2 In this embodiment, m=50 kg, j=100 kg·m 2 。
Step 2: based on the established dynamics model, considering the constraint condition of the system, and based on Udwadia-Kalaba theory and generalized performance measurementDesigning a robust controller;
step 2 comprises the following sub-steps:
step 2.1: decomposing system parameters;
m (-) containing uncertainty factors is decomposed into:
in the middle ofIs the nominal part of the mass matrix M (·), Δm (·) is the uncertainty part of the mass matrix M (·), in this example Δm=10sin0.1t, let ∈ ->ThenLet->Thus Δd (x, σ, t) =d (x, t) E (x, σ, t);
similarly, C (-) is decomposed into nominal fractionsAnd an indeterminate moiety ΔC (·), decomposing G (·) into a nominal moiety +.>And an indeterminate moiety ΔG (·) decomposing F (·) into a nominal moiety +.>And an uncertainty fraction Δf (·).
Step 2.2: establishing a WMR track constraint;
the mobile robot moves along a circle with the radius of 1m, the circle center is the origin of coordinates O (0, 0), and the expected track equation is x 0d =cost,y 0d =sint, the error between the actual trajectory and the desired trajectory is:
e 1 =x 0 -cost, (9)
e 2 =y 0 -sint, (10)
combining equations (9) and (10), establishing a trajectory constraint as:
wherein i=1, 2, k i For controlling parameters, in the present embodiment, k i =1;
Differentiating equation (11) once with respect to time is:
expressed as a second order matrix form:
wherein,
step 2.3: designing a controller τ comprising the sub-steps of:
step 2.3.1: based on Udwadia-Kalaba theory, a controller tau of a nominal part of a processing system is designed 1 :
Designing generalized performance metrics
In previous studies, control performance was expressed by β for an uncertainty system:
differentiating the time t to obtain:
the performance metric β represents a "static" metric that follows the constraint, however, the performance metric of the present inventionThe "dynamic" metric is a generalization of the "static" metric β. />Is novel in that a combination of +.>The "dynamic" function describing the performance metric dynamics increases the flexibility of the control design. In the newly proposed robust control method, < >>The definition is as follows:
wherein,for describing the dynamic function of the performance metric dynamics, +.>Wherein ψ is the control parameter, in this embodiment +.>
For all ofThe method meets the following conditions:
wherein the function gamma j (. Cndot.) is a function related to the Lyapunov function, j=1, 2,3, satisfying γ j (0)=0,
Step 2.3.2: design controller tau 2 :
There is an unknown constant vector alpha epsilon (0, ++) k And a known function pi (·): r is R n ×R n ×R→R n + So that
Wherein ρ is E In order to control the parameters of the device,σ∈Ξ:
definition:
wherein x, t E R n X R, V is Lyapunov function, the control parameter ρ in equation (14) E > -1, and satisfy:
wherein lambda is m Is the minimum characteristic value;
in the absence of an uncertainty as to whether or not there is any uncertainty,thus selecting ρ E =0;
Thus, a controller τ of a processing system is designed that contains an uncertainty portion 2 The method comprises the following steps:
wherein, kappa is a control parameter;
step 2.3.3: the controller is as follows:
step 3: aiming at the designed robust controller, an intelligent optimization method is adopted to carry out the optimization design of the controller parameters so as to achieve the purpose of reducing the control force under the condition of better control performance.
Step 3 comprises the following sub-steps:
step 3.1: design of WMR System control Performance index J 1 (κ):
Wherein J is 1 Reflecting the control performance of the device,is the maximum value of alpha;
step 3.2: design of WMR System control cost index J 2 (κ):
J 2 (κ)=h 1 (κ), (27)
Wherein J is 2 Reflecting the control cost, h1 (·): r is R + →R + Is a strictly increasing function of κ;
step 3.3: design of WMR system performance index J (κ):
in order to obtain the optimal choice of kappa, an intelligent optimization method is used for searching the optimal solution of the problem, and a system performance index J (kappa) is defined as:
J(κ)=a 1 J 1 (κ)+a 2 J 2 (κ), (28)
wherein a is 1 、a 2 > 0 is a weight factor, a k The larger the value of (k=1, 2), the performance index J is given m The greater the weight of (2), m=1. Setting lambda 1 And lambda (lambda) 2 The relation between is a 1 +a 2 =1, in this example a 1 =0.6;a 2 =0.4;
Step 3.4: solving an optimal solution;
finding kappa to have J (kappa) minimum is a constraint optimization problem, and in order to solve the optimization problem, the necessary condition and sufficient condition of solving are needed to be considered;
the requirements for this optimization problem are:
the full conditions are as follows:
assuming that the solution κ of the conditional expression (29) exists, it is a candidate solution to the optimization problem, if κ * Satisfying the sufficient condition (30), it is the optimal solution of the optimization problem;
let gamma 3 =l 1 r p ,h(κ)=l 2 κ q Wherein l 1 ,l 2 >0,p,q>0,l 1 、l 2 P and q are all design parameters, then there is always one unique solution for the requirement, and the unique solution is:
in this embodiment, κ * =3.748. On the one hand, the requirements provide candidate solutions for the optimization problem. On the other hand, the candidate solution is filtered under sufficient conditions, and finally the optimal solution is obtained.
FIG. 2 is a schematic diagram of a motion model of a wheeled mobile robot; the wheel type mobile robot moves on a horizontal plane, the motion track is a circle with a radius d of 1m by taking the origin of coordinates as a circle center, and the coordinate system is a geodetic coordinate system XOY. Wherein (x) 0 ,y 0 ) The coordinates of a centroid C point of the wheeled mobile robot are shown, and theta is an included angle between the centroid C and the X-axis direction; in this embodiment, d=1 m, x 0 =cost,y 0 =sint。
Fig. 3 and 4 are comparisons of an actual track of a target circle with an expected track in the case that WMR is incompatible in an initial state, where a solid line is the actual track and a dotted line is the expected track. Fig. 3 is a graph of an actual trajectory of WMR versus an expected trajectory in 0-8 seconds, and fig. 4 is a graph of an actual trajectory of WMR versus an expected trajectory in 0-20 seconds. As seen from the figure, the system motion gradually tends to the actual track with the passage of time, so that the control system is excellent in track tracking control, and can track the target track well.
Fig. 5 is a graph showing the variation of the "static" constraint error of WMR with time, and simulation results show that the WMR can stabilize the "static" constraint error of the system in about 6 seconds under the condition that the initial states are incompatible. FIG. 6 is a "dynamic" constraint error for WMRThe change curve with time, the simulation result shows that under the new servo robust control condition, the 'dynamic' constraint error of the system can be controlled in a smaller range rapidly and stably.
By analyzing the simulation result, the method can enable the control system to quickly realize real-time tracking control of the target track under the incompatibility of the initial state, has a high-precision control effect, and simultaneously provides more ideas for the design and optimization of the control method.
It should be understood that the above description is not intended to limit the invention to the particular embodiments disclosed, but to limit the invention to the particular embodiments disclosed, and that the invention is not limited to the particular embodiments disclosed, but is intended to cover modifications, adaptations, additions and alternatives falling within the spirit and scope of the invention.
Claims (4)
1. The wheel type mobile robot control method based on the generalized performance measurement is characterized by comprising the following steps of:
step 1: establishing a WMR dynamic model according to a WMR system of the wheeled mobile robot;
step 2: based on the established dynamics model, considering the constraint condition of the system, and based on Udwadia-Kalaba theory and generalized performance measurementDesigning a robust controller;
step 3: aiming at the designed robust controller, an intelligent optimization method is adopted to carry out the optimization design of the controller parameters so as to achieve the purpose of reducing the control force under the condition of better control performance.
2. The method for controlling the wheeled mobile robot based on the generalized performance metric according to claim 1, wherein the specific process of step 1 is as follows:
the structural constraints of building WMR are:
in the method, in the process of the invention,for the Y-axis position component Y 0 First derivative of>For the X-axis position component X 0 θ is the included angle between the connecting line of the centroid C and the origin of coordinates O and the X-axis direction;
establishing a WMR dynamic model containing uncertainty as follows:
wherein X is E R n As a coordinate vector of the system,for velocity vector, +.>For acceleration vector +.>Uncertainty parameters representing the system, +.>Representing boundaries of uncertainty parameters, t.epsilon.R n For time, τ ε R n For control input, M (x, σ, t) ∈R n×n For the quality matrix of the system, < > for>Is a coriolis force matrix,/->F (x, sigma, t) ε R is gravity n×1 Other external disturbance forces;
in the formula (2), the amino acid sequence of the compound,
wherein m is the mass of the mobile robot, and the unit is kg; j is the moment of inertia of the mobile robot, and the unit is kg.m 2 。
3. A method of controlling a wheeled mobile robot based on generalized performance metrics according to claim 1, characterized in that said step 2 comprises the sub-steps of:
step 2.1: m (-) containing uncertainty factors is decomposed into:
in the middle ofIs the nominal part of the mass matrix M (,), ΔM (,) is the uncertainty part of the mass matrix M (,), let Then->Order theThus Δd (x, σ, t) =d (x, t) E (x, σ, t);
decomposing C (-) into nominal fractionsAnd an indeterminate moiety ΔC (·), decomposing G (·) into a nominal moiety +.>And an indeterminate moiety ΔG (·) decomposing F (·) into a nominal moiety +.>And an uncertainty fraction Δf (·);
step 2.2: establishing a WMR track constraint;
the mobile robot moves along a circle with the radius of 1m, the circle center is the origin of coordinates O (0, 0), and the expected track equation is x 0d =cost,y 0d =sint, the error between the actual trajectory and the desired trajectory is:
e 1 =x 0 -cost, (4)
e 2 =y 0 -sint, (5)
combining equations (4) and (5), establishing a trajectory constraint as:
wherein i=1, 2, k i Is a control parameter;
differentiating equation (6) once with respect to time is:
expressed as a second order matrix form:
wherein,
step 2.3: designing a controller v, comprising the sub-steps of:
step 2.3.1: based on Udwadia-Kalaba theory, a controller tau of a nominal part of a processing system is designed 1 :
Designing generalized performance metrics
Wherein,for describing the dynamic function of the performance metric dynamics, +.>Wherein ψ is a control parameter;
for all ofThe method meets the following conditions:
wherein the function gamma j (. Cndot.) is a function related to the Lyapunov function, j=1, 2,3, satisfying γ j (0)=0,
Step 2.3.2: design controller tau 2 ;
There is an unknown constant vector alpha epsilon (0, ++) k And a known function pi (·) R n ×R n ×R→R n + So that
Wherein ρ is E In order to control the parameters of the device,σ∈Ξ;
definition:
in the middle of,x,t∈R n X R, V is Lyapunov function, the control parameter ρ in equation (14) E >-1, and satisfies:
wherein lambda is m Is the minimum characteristic value;
in the absence of an uncertainty as to whether or not there is any uncertainty,thus selecting ρ E =0;
Thus, a controller τ of a processing system is designed that contains an uncertainty portion 2 The method comprises the following steps:
wherein, kappa is a control parameter;
step 2.3.3: the controller is as follows:
4. a method of controlling a wheeled mobile robot based on generalized performance metrics according to claim 1, characterized in that said step 3 comprises the sub-steps of:
step 3.1: design of WMR System control Performance index J 1 (κ):
In the method, in the process of the invention,is a maximum value of II alpha II;
step 3.2: design of WMR System control cost index J 2 (κ):
J 2 (κ)=h 1 (κ), (20)
In the formula, h 1 (·):R + →R + Is a strictly increasing function of κ;
step 3.3: design of WMR system performance index J (κ):
J(κ)=a 1 J 1 (κ)+a 2 J 2 (κ), (21)
wherein a is 1 、a 2 >0 is a weight factor, a k The larger the value of (1), k=1, 2, giving the performance index J m The greater the weight of (a), m=1, 2, a is set 1 And a 2 The relation between is a 1 +a 2 =1;
Step 3.4: the optimal solution is solved and the method comprises the steps of,
finding kappa to minimize J (kappa) is a constraint optimization problem, the requirements of which are:
the full conditions are as follows:
assuming that the solution κ of the conditional expression (22) exists, it is a candidate solution to the optimization problem, if κ * Satisfying the sufficient condition (23), it is the optimal solution of the optimization problem;
let gamma 3 =l 1 r p ,h(κ)=l 2 κ q Wherein l 1 ,l 2 >0,p,q>0,l 1 、l 2 P and q are all design parameters, then there is always one unique solution for the requirement, and the unique solution is:
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