CN114488790B - Omnidirectional mobile robot self-adaptive sliding film control method based on nominal model - Google Patents
Omnidirectional mobile robot self-adaptive sliding film control method based on nominal model Download PDFInfo
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Abstract
The invention discloses an omnidirectional mobile robot self-adaptive synovial membrane control method based on a nominal model, which belongs to the technical field of tracking system control, and comprises the following steps: establishing an omnidirectional mobile robot kinematics equation; establishing an omnidirectional mobile robot dynamics equation according to the omnidirectional mobile robot dynamics equation; calculating to obtain a slip form surface s (t) and a slip film control rate tau based on an omnidirectional mobile robot kinematic equation and a dynamic equation; setting a controller for adjusting the self-adaptive gain of the nominal model according to the sliding mode surface and the sliding film control rate, obtaining a dynamic equation of the controller omnidirectional mobile robot based on the self-adaptive gain adjustment of the nominal model, and completing the self-adaptive sliding film control of the omnidirectional mobile robot based on the nominal model; the problem that the track tracking error is large because the omnidirectional mobile robot can not completely eliminate interference under the condition of interference is solved.
Description
Technical Field
The invention belongs to the technical field of tracking system control, and particularly relates to an omnidirectional mobile robot self-adaptive synovial membrane control method based on a nominal model.
Background
With the rapid development of industry, research on omni-directional mobile robots is also receiving more and more attention, and omni-directional wheel mobile robots are used in various scenes due to advantages of better operability, zero turning radius in any direction and the like; compared with the traditional omni-directional wheel type mobile robot, the Mecanum wheel type mobile robot can move transversely and even can move along a curved path and rotate in situ; however, with the use of independent motor control, its motion control becomes with uncertainty when there is a disturbance; many scholars have derived kinematic and kinetic modeling about omni-wheel mobile robots; mechanical characteristics of a single wheel are studied in detail from the incision of the single omnidirectional wheel, the pose error of the robot is obtained by establishing an omnidirectional mobile robot kinematics equation, and a dynamics equation is established by analyzing influence factors of friction force on the four wheels; however, the simplest PD control is used in the control process, so that the shake of the robot cannot be eliminated; many scholars deduce a dynamic model of the omnidirectional wheel mobile robot under the condition of friction force, an active disturbance rejection controller with friction compensation is designed by establishing the dynamic model of the friction force, a friction compensation item is firstly carried out, then a nonlinear controller is adopted for track tracking, and the influence of dead zone nonlinearity of the friction force on the control precision of a system is greatly reduced; however, only the limited input condition is considered, and the conditions such as gradual stability, exponential stability and the like are not considered, so that the stability is poor, and meanwhile, the influence of dead zone nonlinearity of friction force on a system cannot be completely eliminated.
The motion of the omni-wheel mobile robot under the disturbance condition has better tracking effect compared with a linear controller (PID) by using a nonlinear controller; the existing slide film control research on track tracking of the wheeled mobile robot aims at the problem of external interference track tracking control of the wheeled mobile robot, a fuzzy rapid double-power approach law slide film control strategy is designed, the robustness of a track tracking control system is improved, the buffeting phenomenon of the robot is eliminated, and the self-adaptive slide film control is realized on the wheeled mobile robot; in order to improve the robustness of path tracking control of the wheeled mobile robot, an inversion self-adaptive sliding mode control strategy is provided, a method of inversion control is adopted by a kinematic control law, linear speed and angular speed are used as auxiliary quantity control input, so that the track approaches to an ideal track as much as possible, a dynamic model adopts a self-adaptive sliding film control law, an input torque controller of the mobile robot is obtained, and the speed approaches to an expected speed as much as possible, but the controller is not well applied to the Mecanum wheel mobile robot.
Disclosure of Invention
Aiming at the defects in the prior art, the self-adaptive sliding film control method of the omnidirectional mobile robot based on the nominal model combines the advantages of a sliding mode control system and a self-adaptive control system, and solves the problem that the omnidirectional mobile robot cannot completely eliminate interference under the condition of interference, so that the track tracking error is larger.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
the invention provides an omnidirectional mobile robot self-adaptive synovial membrane control method based on a nominal model, which comprises the following steps:
s1, establishing an omnidirectional mobile robot kinematics equation;
s2, establishing an omnidirectional mobile robot dynamics equation according to the omnidirectional mobile robot dynamics equation;
s3, calculating to obtain a sliding mode surface S (t) and a sliding film control rate tau based on an omnidirectional mobile robot kinematics equation and a dynamics equation;
and S4, setting a controller for adjusting the self-adaptive gain of the nominal model according to the sliding mode surface and the sliding film control rate, obtaining a dynamic equation of the controller omnidirectional mobile robot based on the self-adaptive gain adjustment of the nominal model, and completing the self-adaptive sliding film control of the omnidirectional mobile robot based on the nominal model.
The beneficial effects of the invention are as follows: according to the self-adaptive synovial membrane control method of the omnidirectional mobile robot based on the nominal model, which is provided by the invention, the robustness is very strong under the interference condition according to the synovial membrane control, the self-adaptive control can enable the system to work in an optimal or near-optimal motion state, and the accurate control performance is obtained.
Further, the step S1 includes the steps of:
s11, setting the geometric center of the omnidirectional mobile robot to coincide with the mass center, and defining a global coordinate system O f X f Y f Moving coordinate system O m X m Y m And a Mecanum wheel coordinate system O wi X wi Y wi Wherein i is equal to 1,2,3,4;
s12, decomposing the axle center speed of each Mecanum wheel of the omnidirectional mobile robot to obtain the speed of each Mecanum wheel respectively;
s13, obtaining the relation between the axle center speed of each Mecanum wheel and the main body center speed of the omnidirectional mobile robot;
s14, obtaining an inverse kinematics equation of the omnidirectional mobile robot according to the relation between the axle center speed of each Mecanum wheel and the central speed of the main body of the omnidirectional mobile robot and the structural characteristics of the omnidirectional mobile robot;
and S15, obtaining an omnidirectional mobile robot kinematics equation according to the omnidirectional mobile robot inverse kinematics equation, and completing establishment of the omnidirectional mobile robot kinematics equation.
The beneficial effects of adopting the further scheme are as follows: the kinematics equation of the omnidirectional mobile robot is established by analyzing the omnidirectional mobile robot main body and the Mecanum wheel, and a foundation is provided for establishing the dynamics equation of the omnidirectional mobile robot.
Further, the expression of the kinematic equation of the omnidirectional mobile robot in the step S15 is as follows:
wherein,,
first derivative representing lateral displacement of omni mobile robot,/->
First derivative representing longitudinal displacement of omni mobile robot,/->
Representing the first derivative of the rotation angle of an omnidirectional mobile robot, R representing the radius of the Mecanum wheel, l 1 And l 2 Respectively represent a Mecanum wheel center distance movement coordinate system x m Axes and y m Distance of axis>
And
the angular velocities of the first, second, third and fourth mecanum wheels of the omnidirectional mobile robot are represented, respectively.
The beneficial effects of adopting the further scheme are as follows: the omnidirectional mobile robot kinematics equation is provided, and a foundation is provided for establishing the omnidirectional mobile robot kinematics equation.
Further, the step S2 includes the steps of:
s21, constructing an omnidirectional mobile robot kinetic energy equation according to the omnidirectional mobile robot kinetic energy equation;
s22, calculating to obtain an omnidirectional mobile robot kinetic equation according to the Lagrangian kinematic equation and the omnidirectional mobile robot kinetic equation, and completing establishment of the omnidirectional mobile robot kinetic equation.
The beneficial effects of adopting the further scheme are as follows: when the robot dynamics equation is established, the condition of external disturbance is considered, and a better accurate dynamics model can be obtained under the condition of disturbance.
Further, the expression of the kinetic equation of the omnidirectional mobile robot in the step S22 is as follows:
wherein M represents an inertial matrix of the omnidirectional mobile robot,
represents the angular acceleration of the wheels of the Mecanum robot,/->
The friction moment of the Mecanum wheel is represented, tau represents the total moment of the omnidirectional mobile robot, and d represents the external disturbance moment.
The beneficial effects of adopting the further scheme are as follows: and establishing an omnidirectional mobile robot dynamics equation under the condition of considering external disturbance factors, providing an omnidirectional mobile robot dynamics equation, and providing a foundation for the omnidirectional mobile robot to run on a sliding mode surface by using a controller.
Further, the step S3 includes the following steps:
s31, defining a tracking error e (t) of the omnidirectional mobile robot according to a kinematic equation and a dynamic equation of the omnidirectional mobile robot;
s32, calculating to obtain a sliding mode surface and a sliding film control rate according to the tracking error of the omnidirectional mobile robot.
The beneficial effects of adopting the further scheme are as follows: the linear sliding hyperplane is selected, so that the tracking error is gradually converged to zero after the system reaches the sliding mode, and the convergence speed can be adjusted by selecting a control rate matrix of the sliding mode surface.
Further, the expression of the tracking error e (t) of the omnidirectional mobile robot in the step S31 is as follows:
wherein,,
indicating the desired angular velocity of the Mecanum wheel, < >>
The actual angular velocities of the respective mecanum wheels are represented, where i=1, 2,3,4, respectively represent the angular velocities of the first, second, third and fourth mecanum wheels.
The beneficial effects of adopting the further scheme are as follows: the tracking error equation of the omnidirectional mobile robot is provided, and error reference is provided for controlling and adjusting the omnidirectional mobile robot to run on the sliding mode surface.
Further, the expressions of the slip plane S (t) and the slip film control rate τ' in the step S32 are as follows:
wherein,,
derivative, κ, representing tracking error 4×4 Representing a constant positive definite matrix, e (t) representing tracking error,/->
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Represents the angular acceleration of the wheels of the Mecanum robot,/->
Representing the derivative of the slide face,
Nominal parameter representing moment of inertia,/->
Representing the angular velocity of the Mecanum wheel, ρ representing the slip-mode control rate gain and sgn representing the step function.
The beneficial effects of adopting the further scheme are as follows: the sliding film control convergence speed is high, the sliding film control convergence speed can be adjusted by selecting a sliding mode surface parameter matrix, but the stability is poor, so that the controller for nominal model self-adaptive gain adjustment is adopted to control the omnidirectional mobile robot.
Further, the step S4 includes the steps of:
s41, setting a nominal model self-adaptive gain adjustment controller according to a sliding mode surface and a sliding film control rate to obtain a control rate tau' of the nominal model self-adaptive gain adjustment controller;
s42, obtaining a controller omnidirectional mobile robot dynamic model based on the nominal model adaptive gain adjustment according to the nominal parameters of the inertia matrix of the omnidirectional mobile robot, the nominal parameters of the friction coefficient of the omnidirectional mobile robot, the control rate tau' of the controller of the nominal model adaptive gain adjustment and the omnidirectional mobile robot dynamic model, and completing the omnidirectional mobile robot adaptive synovial membrane control based on the nominal model.
The beneficial effects of adopting the further scheme are as follows: because the controlled object is greatly influenced by external disturbance, the system can work in an optimal or near-optimal motion state by adopting self-adaptive control, and the accurate control performance is obtained.
Further, the controller omnidirectional mobile robot dynamics model expression based on the nominal model adaptive gain adjustment in the step S42 is as follows:
u r ＝-ρsgn(s(t))
wherein M represents an inertial matrix of the omnidirectional mobile robot,
represents the angular acceleration of the wheels of the Mecanum robot,/->
Representing the friction moment of the Mecanum wheel, +.>
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Nominal parameters representing moment of inertia, d representing external disturbance torque, τ' representing slip film control rate, +.>
Indicating jerk, κ of the wheels of a Mecanum robot 4×4 Representing a constant positive definite matrix, ">
Representing the derivative of the tracking error, e (t) representing the tracking error,/and (c)>
Represents the angular velocity of the Mecanum wheel, +.>
Representing the nominal value of the external disturbance moment, u θ Representing the friction compensation term of the omnidirectional mobile robot, u r The slip film approach law is represented, T represents time, T represents period, ρ represents slip mode control rate gain, sgn represents step function.
The beneficial effects of adopting the further scheme are as follows: the controller omnidirectional mobile robot dynamics model based on the nominal model self-adaptive gain adjustment is provided, and is verified according to the Leiprov function derivation.
Drawings
Fig. 1 is a flowchart of steps of a method for controlling an adaptive synovial membrane of an omni-directional mobile robot based on a nominal model according to an embodiment of the present invention.
Fig. 2 is a schematic structural diagram of an omnidirectional mobile robot in an embodiment of the present invention.
Fig. 3 (a) is a position tracking graph of the tracking process of the omni-directional mobile robot in the embodiment of the present invention.
Fig. 3 (b) is a partial enlarged view of a position tracking curve of the tracking process of the omni-directional mobile robot in the embodiment of the present invention.
Fig. 4 (a) is a graph comparing X-axis displacement errors in the tracking process of the omni-directional mobile robot in the embodiment of the present invention.
Fig. 4 (b) is a graph comparing X-axis direction speed errors in the tracking process of the omni-directional mobile robot in the embodiment of the present invention.
Fig. 5 (a) is a graph comparing Y-axis displacement errors in the tracking process of the omni-directional mobile robot in the embodiment of the present invention.
Fig. 5 (b) is a graph comparing Y-axis directional velocity errors in the tracking process of the omni-directional mobile robot in accordance with the embodiment of the present invention.
Fig. 6 is an input torque diagram of a first mecanum wheel in the process of tracking the track of the omnidirectional mobile robot in an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and all the inventions which make use of the inventive concept are protected by the spirit and scope of the present invention as defined and defined in the appended claims to those skilled in the art.
Example 1
As shown in fig. 1 and fig. 2, the invention provides an omnidirectional mobile robot self-adaptive synovial membrane control method based on a nominal model, which comprises the following steps:
s1, establishing an omnidirectional mobile robot kinematics equation;
the step S1 includes the steps of:
s11, setting the geometric center of the omnidirectional mobile robot to coincide with the mass center, and defining a global coordinate system O f X f Y f Moving coordinate system O m X m Y m And a Mecanum wheel coordinate system O wi X wi Y wi Wherein i is equal to 1,2,3,4;
s12, decomposing the axle center speed of each Mecanum wheel of the omnidirectional mobile robot to obtain the speed of each Mecanum wheel respectively;
the omnidirectional mobile robot Mecanum wheel comprises a first Mecanum wheel, a second Mecanum wheel, a third Mecanum wheel and a fourth Mecanum wheel;
the speed expressions of the first Mecanum wheel and the third Mecanum wheel are as follows:
wherein v is ix Representing the speed of the Mecanum wheel of the omnidirectional mobile robot along the x direction, v iy Representing the speed of the Mecanum wheel of the omnidirectional mobile robot along the y direction, R represents the radius of the Mecanum wheel, phi represents the axis of the Mecanum robot body and the coordinate system x of the wheels wi An included angle between the shafts;
representing the angular velocity of the wheels of the mic robot; v oi Representing the speed of the mic robot body; i is equal to 1,3, wherein i represents the first Mecanum wheel when i is equal to 1 and the third Mecanum wheel when i is equal to 3;
the speed expressions of the second wheel and the fourth wheel are as follows:
wherein i is equal to 2,4, wherein i is equal to 2 for the second Mecanum wheel and i is equal to 4 for the fourth Mecanum wheel;
s13, obtaining the relation between the axle center speed of each Mecanum wheel and the main body center speed of the omnidirectional mobile robot;
in the step S13, the expression of the relationship between the axle center speed of the wheel of the omnidirectional mobile robot and the center speed of the main body of the omnidirectional mobile robot is as follows:
q m ＝[x m y m φ] T
wherein x is m Represents the transverse displacement of the omnidirectional mobile robot, y m Representing the longitudinal displacement of the omni-directional mobile robot,
represents the first derivative of the rotation angle phi, J represents a Jacobian matrix,/and>
represents a microphone robot body center position vector q m First derivative of l 1 And l 2 Respectively represent a coordinate system X for the movement of the center distance of wheels of the Mecanum robot m Axes and Y m The distance of the axis;
s14, obtaining an inverse kinematics equation of the omnidirectional mobile robot according to the relation between the axle center speed of each Mecanum wheel and the central speed of the main body of the omnidirectional mobile robot and the structural characteristics of the omnidirectional mobile robot;
the inverse kinematics model expression of the mic robot in the step S14 is as follows:
wherein J represents a Jacobian matrix,
represents a microphone robot body center position vector q m First derivative of l 1 And l 2 Respectively represent a coordinate system X for the movement of the center distance of wheels of the Mecanum robot m Axes and Y m The distance of the axis;
s15, obtaining an omnidirectional mobile robot kinematics equation according to the omnidirectional mobile robot inverse kinematics equation, and completing establishment of the omnidirectional mobile robot kinematics equation;
the expression of the kinematic equation of the omnidirectional mobile robot in the step S15 is as follows:
wherein,,
first derivative representing lateral displacement of omni mobile robot,/->
First derivative representing longitudinal displacement of omni mobile robot,/->
Representing the first derivative of the rotation angle of an omnidirectional mobile robot, R representing the radius of the Mecanum wheel, l 1 And l 2 Respectively represent a Mecanum wheel center distance movement coordinate system x m Axes and y m Distance of axis>
And
respectively representing the angular speeds of a first Mecanum wheel, a second Mecanum wheel, a third Mecanum wheel and a fourth Mecanum wheel of the omnidirectional mobile robot;
s2, establishing an omnidirectional mobile robot dynamics equation according to the omnidirectional mobile robot dynamics equation;
the step S2 includes the steps of:
s21, constructing an omnidirectional mobile robot kinetic energy equation according to the omnidirectional mobile robot kinetic energy equation;
the kinetic energy model expression of the omnidirectional mobile robot in the step S21 is as follows:
wherein E represents the kinetic energy of the omnidirectional mobile robot, m represents the mass of the omnidirectional mobile robot, v x Representing the transverse displacement speed, v, of an omnidirectional mobile robot y Indicating the longitudinal displacement speed of the omnidirectional mobile robot, I z Representing moment of inertia of the omnidirectional mobile robot around the z axis, I o Representing the moment of inertia of the omni-directional mobile robot about a center point,
representing the angular velocity of the wheels of the mic robot;
s22, calculating to obtain an omnidirectional mobile robot kinetic equation according to the Lagrangian kinematic equation and the omnidirectional mobile robot kinetic equation, and completing establishment of the omnidirectional mobile robot kinetic equation;
the expression of the dynamic equation of the omnidirectional mobile robot in the step S22 is as follows:
wherein M represents an inertial matrix of the omnidirectional mobile robot,
represents the angular acceleration of the wheels of the Mecanum robot,/->
The method comprises the steps of representing the friction moment of a Mecanum wheel, tau representing the total moment of the omnidirectional mobile robot, d representing the external disturbance moment, alpha representing a first inertia matrix parameter, and beta representing a second inertia matrix parameter.
S3, calculating to obtain a sliding mode surface S (t) and a sliding film control rate tau based on an omnidirectional mobile robot kinematics equation and a dynamics equation;
the step S3 includes the steps of:
s31, defining a tracking error e (t) of the omnidirectional mobile robot according to a kinematic equation and a dynamic equation of the omnidirectional mobile robot;
the expression of the tracking error e (t) of the omnidirectional mobile robot in the step S31 is as follows:
wherein,,
indicating the desired angular velocity of the Mecanum wheel, < >>
The actual angular velocities of the respective mecanum wheels are represented, where i=1, 2,3,4, respectively represent the angular velocities of the first, second, third and fourth mecanum wheels.
S32, calculating to obtain a sliding mode surface and a sliding film control rate according to the tracking error of the omnidirectional mobile robot;
the expressions of the slip-form surface S (t) and slip-film control rate τ' in step S32 are as follows:
wherein,,
derivative, κ, representing tracking error 4×4 Representing a constant positive definite matrix, e (t) representing tracking error,/->
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Indicating the angular acceleration of the wheels of the mecanum robot,
representing the derivative of the slide face,
Nominal parameter representing moment of inertia,/->
Representing the angular velocity of the Mecanum wheel, ρ representing the slip-mode control rate gain, sgn representing the step function;
defining a first Lyapunov function, deriving the function, and verifying the input torque jitter of the omnidirectional mobile robot;
the expression of the first lyapunov function is as follows:
wherein V is 1 Representing a first lyapunov function value;
the expression of the first Lyapunov function after derivation is as follows:
wherein,,
representing the function value of the first Lyapunov function value after derivation, <>
Representing the difference value between the nominal parameter of the inertia matrix of the omnidirectional mobile robot and the inertia matrix of the omnidirectional mobile robot;
after the first lyapunov function is derived, when the sliding mode control rate gain rho is large enough, the stability of the robot can be ensured, which can cause serious input torque jitter and requires the controller to adjust;
s4, setting a controller for adjusting the self-adaptive gain of the nominal model according to the sliding mode surface and the sliding film control rate, obtaining a dynamic equation of the controller omnidirectional mobile robot based on the self-adaptive gain adjustment of the nominal model, and completing the self-adaptive sliding film control of the omnidirectional mobile robot based on the nominal model;
the step S4 includes the steps of:
s41, setting a nominal model self-adaptive gain adjustment controller according to a sliding mode surface and a sliding film control rate to obtain a control rate tau' of the nominal model self-adaptive gain adjustment controller;
s42, obtaining a controller omnidirectional mobile robot dynamic model based on the nominal model adaptive gain adjustment according to the nominal parameters of the inertia matrix of the omnidirectional mobile robot, the nominal parameters of the friction coefficient of the omnidirectional mobile robot, the control rate tau' of the controller of the nominal model adaptive gain adjustment and the omnidirectional mobile robot dynamic model, and completing the omnidirectional mobile robot adaptive synovial membrane control based on the nominal model;
the dynamic model expression of the controller omnidirectional mobile robot based on the nominal model adaptive gain adjustment in the step S42 is as follows:
u r ＝-ρsgn(s(t))
wherein M represents an inertial matrix of the omnidirectional mobile robot,
represents the angular acceleration of the wheels of the Mecanum robot,/->
Representing the friction moment of the Mecanum wheel, +.>
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Nominal parameters representing moment of inertia, d representing external disturbance torque, τ' representing slip film control rate, +.>
Indicating jerk, κ of the wheels of a Mecanum robot 4×4 Representing a constant positive definite matrix, ">
Representing the derivative of the tracking error, e (t) representing the tracking error,/and (c)>
Representing Mecanum wheelsAngular velocity (V/V)>
Representing the nominal value of the external disturbance moment, u θ Representing the friction compensation term of the omnidirectional mobile robot, u r Representing a slip film approach law, T representing time, T representing a period, ρ representing a slip mode control rate gain, sgn representing a step function;
designing a second Lyapunov function, deriving the second Lyapunov function, and verifying convergence of a dynamic model of the controller omnidirectional mobile robot based on the nominal model adaptive gain adjustment:
the expression of the second lyapunov function is as follows:
wherein V represents the value of the second lyapunov function;
the expression of the second Lyapunov function after derivation is as follows:
wherein,,
a derivative value representing the second lyapunov function,/->
Representing the difference between the nominal parameter of the external disturbance torque and the external disturbance torque;
the derivative value of the second Lyapunov function
The controller based on the nominal model adaptive gain adjustment has good convergence, obtains high tracking precision, and simultaneously suppresses jitter of control input torque.
Example 2
As shown in fig. 3 (a) and 3 (b), in a practical example of the present invention, the present embodiment adopts a circular model as the trajectory tracking, and the circular model trajectory tracking equation expression is as follows:
wherein x represents lateral position tracking and y represents longitudinal position tracking;
setting the omnidirectional mobile robot to run from the origin, wherein the expression of the initial position of the omnidirectional mobile robot is as follows:
setting the disturbance d of each Mecanum wheel i Differences between nominal parameters of inertial matrix of omnidirectional mobile robot and inertial matrix of omnidirectional mobile robot
Nominal parameter of moment of inertia->
Difference between sense parameter and external disturbance moment +.>
The expressions are as follows:
d i ＝2sin(t+1)
setting a nominal model-based adaptive synovial membrane control related parameter: omnidirectional mobile robot mass m=6kg, moment of inertia I about z axis z ＝6.9210kg·m 2 Moment of inertia I about a point of center o ＝0.0945kg·m 2 Radius r=0.075 m of the mecanum wheel, center distance of the mecanum wheel moves coordinate system x m Distance of axis l 1 =0.26m, mecanum wheel center distance movement coordinate system y m Distance of axis l 2 =0.135 m, coefficient of friction β of each Mecanum wheel i Constant positive definite matrix κ=0.02 4×4 =diag (5 5 5 5), sliding mode control rate gain ρ=diag (30 30 30 30);
the error comparison of the PID controller, the traditional synovial membrane TSC and the NMASC controller under the conditions of sinusoidal interference, step interference and no interference is obtained through simulation, and the error comparison is shown in the table 1:
TABLE 1
The NMASC controller adopted by the scheme has the minimum error in track tracking effect.
As shown in fig. 4 (a), 4 (b), 5 (a) and 5 (b), in order to analyze the track tracking effect of the three controllers more clearly, tracking simulation is performed on the condition that sinusoidal interference exists on the omnidirectional mobile robot, according to fig. 4 (a) and 5 (a), the error of the PID controller in the X axis and the Y axis is obviously larger than the error of the adaptive slip film controller NMASC and the conventional slip film controller TSC based on the nominal model, and the PID controller generates larger jitter in the X axis and the Y axis directions, and the PID controller cannot have better convergence in the X axis and the Y axis; according to fig. 4 (b) and fig. 5 (b), the conventional sliding film TSC can achieve a better tracking effect on the desired track, but the conventional sliding film TSC generates obvious jitter on the X and Y axes, and the convergence speed of the NMASC model proposed herein is obviously better than that of the other two controllers.
As shown in fig. 6, the input torque of the wheel 1 at the initial time is greatly changed, and the input torque gradually becomes stable at [ -20 ] but after the adaptive adjustment, and remains within the range of [ -5 5 ].
Claims (5)
1. The self-adaptive synovial membrane control method of the omnidirectional mobile robot based on the nominal model is characterized by comprising the following steps of:
s1, establishing an omnidirectional mobile robot kinematics equation;
s2, establishing an omnidirectional mobile robot dynamics equation according to the omnidirectional mobile robot dynamics equation;
s3, calculating to obtain a sliding mode surface S (t) and a sliding film control rate tau' based on an omnidirectional mobile robot kinematic equation and a dynamic equation;
s4, setting a controller for adjusting the self-adaptive gain of the nominal model according to the sliding mode surface and the sliding film control rate, obtaining a dynamic equation of the controller omnidirectional mobile robot based on the self-adaptive gain adjustment of the nominal model, and completing the self-adaptive sliding film control of the omnidirectional mobile robot based on the nominal model;
the step S3 includes the steps of:
s31, defining a tracking error e (t) of the omnidirectional mobile robot according to a kinematic equation and a dynamic equation of the omnidirectional mobile robot;
s32, calculating to obtain a sliding mode surface and a sliding film control rate according to the tracking error of the omnidirectional mobile robot;
the expression of the tracking error e (t) of the omnidirectional mobile robot in the step S31 is as follows:
wherein,,
indicating the desired angular velocity of the Mecanum wheel, < >>
Representing the actual angular velocity of each mecanum wheel, wherein i=1, 2,3,4, represents the angular velocities of the first, second, third and fourth mecanum wheels, respectively;
the expressions of the slip-form surface S (t) and slip-film control rate τ' in step S32 are as follows:
wherein,,
derivative, κ, representing tracking error 4×4 Representing a constant positive definite matrix, e (t) representing tracking error,/->
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Represents the angular acceleration of the wheels of the Mecanum robot,/->
Representation ofDerivative of the sliding surface,
Nominal parameter representing moment of inertia,/->
Representing the angular velocity of the Mecanum wheel, ρ representing the slip-mode control rate gain, sgn representing the step function;
the step S4 includes the steps of:
s41, setting a nominal model self-adaptive gain adjustment controller according to a sliding mode surface and a sliding film control rate to obtain a control rate tau' of the nominal model self-adaptive gain adjustment controller;
s42, obtaining a controller omnidirectional mobile robot dynamic model based on the nominal model adaptive gain adjustment according to the nominal parameters of the inertia matrix of the omnidirectional mobile robot, the nominal parameters of the friction coefficient of the omnidirectional mobile robot, the control rate tau' of the controller of the nominal model adaptive gain adjustment and the omnidirectional mobile robot dynamic model, and completing the omnidirectional mobile robot adaptive synovial membrane control based on the nominal model;
the dynamic model expression of the controller omnidirectional mobile robot based on the nominal model adaptive gain adjustment in the step S42 is as follows:
u r ＝-ρsgn(s(t))
wherein M represents an omnidirectional mobile robotThe inertia matrix is used to determine the mass of the material,
represents the angular acceleration of the wheels of the Mecanum robot,/->
Representing the friction moment of the Mecanum wheel, +.>
Nominal parameters representing the inertial matrix of an omnidirectional mobile robot,/->
Nominal parameters representing moment of inertia, d representing external disturbance torque, τ' representing slip film control rate, +.>
Indicating jerk, κ of the wheels of a Mecanum robot 4×4 Representing a constant positive definite matrix, ">
Representing the derivative of the tracking error, e (t) representing the tracking error,/and (c)>
Represents the angular velocity of the Mecanum wheel, +.>
Representing the nominal value of the external disturbance moment, u θ Representing the friction compensation term of the omnidirectional mobile robot, u r The slip film approach law is represented, T represents time, T represents period, ρ represents slip mode control rate gain, sgn represents step function.
2. The method for controlling the adaptive synovial membrane of the omni-directional mobile robot based on the nominal model according to claim 1, wherein said step S1 comprises the steps of:
s11, setting the geometric center of the omnidirectional mobile robot to coincide with the mass center, and defining a global coordinate system O f X f Y f Moving coordinate system O m X m Y m And a Mecanum wheel coordinate system O wi X wi Y wi Wherein i is equal to 1,2,3,4;
s12, decomposing the axle center speed of each Mecanum wheel of the omnidirectional mobile robot to obtain the speed of each Mecanum wheel respectively;
s13, obtaining the relation between the axle center speed of each Mecanum wheel and the main body center speed of the omnidirectional mobile robot;
s14, obtaining an inverse kinematics equation of the omnidirectional mobile robot according to the relation between the axle center speed of each Mecanum wheel and the central speed of the main body of the omnidirectional mobile robot and the structural characteristics of the omnidirectional mobile robot;
and S15, obtaining an omnidirectional mobile robot kinematics equation according to the omnidirectional mobile robot inverse kinematics equation, and completing establishment of the omnidirectional mobile robot kinematics equation.
3. The method for adaptive synovial membrane control of an omni mobile robot based on a nominal model according to claim 2, wherein the expression of the kinematic equation of the omni mobile robot in step S15 is as follows:
wherein,,
first derivative representing lateral displacement of omni mobile robot,/->
First derivative representing longitudinal displacement of omni mobile robot,/->
Representing the first derivative of the rotation angle of an omnidirectional mobile robot, R representing the radius of the Mecanum wheel, l 1 And l 2 Respectively represent a Mecanum wheel center distance movement coordinate system x m Axes and y m Distance of axis>
And->
The angular velocities of the first, second, third and fourth mecanum wheels of the omnidirectional mobile robot are represented, respectively.
4. The method for controlling the adaptive synovial membrane of the omni-directional mobile robot based on the nominal model according to claim 1, wherein said step S2 comprises the steps of:
s21, constructing an omnidirectional mobile robot kinetic energy equation according to the omnidirectional mobile robot kinetic energy equation;
s22, calculating to obtain an omnidirectional mobile robot kinetic equation according to the Lagrangian kinematic equation and the omnidirectional mobile robot kinetic equation, and completing establishment of the omnidirectional mobile robot kinetic equation.
5. The method for adaptive synovial membrane control of an omni mobile robot based on a nominal model according to claim 4, wherein the expression of the kinetic equation of the omni mobile robot in step S22 is as follows:
wherein M represents an inertial matrix of the omnidirectional mobile robot,
represents the angular acceleration of the wheels of the Mecanum robot,/->
The friction moment of the Mecanum wheel is represented, tau represents the total moment of the omnidirectional mobile robot, and d represents the external disturbance moment.
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