CN117032293A - Random tracking control method for service robot motion environment data driving observation - Google Patents
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Abstract
The invention discloses a random tracking control method for service robot motion environment data driving observation, which is characterized by comprising the following steps: separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing data driving observation on the man-machine motion environment change; based on a service robot dynamics model containing a man-machine motion environment, converting eccentricity into a random variable, establishing a random dynamics model of a man-machine system, designing a random controller to inhibit the influence of the environment on the motion of the robot, and simultaneously enabling a tracking error system to realize random stability; and an output PWM signal is provided for a motor driving module based on an MSP340 series singlechip, so that the robot can restrain the influence of a motion environment and realize the tracking of a given track signal. The method ingeniously utilizes a data driving method to observe the motion environment information of the man-machine system.
Description
Technical field:
the invention relates to the field of control of service robots, in particular to the field of control of data-driven observation of a motion environment.
The background technology is as follows:
because the elderly people and the disabled lower limbs cannot finish daily independent life, a heavy burden is brought to family care personnel and society, a sitting service robot can replace walking functions of the elderly and disabled lower limbs, and can finish simple daily life activities by only sitting on the robot and utilizing healthy upper limbs, a brand new life paradigm is brought to the elderly and disabled lower limbs, so that the study of the service robot is widely focused by students. As the motion environment of the man-machine system is changed in the process of operating various life actions by a user, and the eccentricity is generated in different sitting postures, the tracking motion of the robot is seriously influenced, even larger tracking errors are generated to collide with surrounding objects, and the safety of the user is threatened. Therefore, the method solves the problem of observation and random tracking control of the motion environment of the service robot and has important significance for improving the tracking precision and the safety of a man-machine system. There are many research results about the service robot tracking control, however, most of these results neglect the influence of the motion environment on the robot tracking motion, and more importantly, the existing results do not solve the data-driven observation problem of the motion environment, resulting in non-ideal tracking precision. In fact, the data driving observation does not depend on a mathematical model of the man-machine system, and can greatly improve the accuracy of the motion environment observation by completely depending on the system measurable data, thereby laying a foundation for designing a tracking control algorithm of the man-machine system; in addition, in the process of the human-computer system cooperation completing life actions, different sitting postures can generate eccentricity, and the tracking precision of the human-computer system is affected, however, the problem of random change of the eccentricity is not considered in the existing achievements. Up to now, there is no study on a method of motion environment data-driven observation and random tracking control of a suppression environment. Therefore, research on how to accurately obtain the motion environment of the man-machine system and restrain the influence of the environment on the tracking precision of the robot has great significance for improving the tracking performance and the safety of the man-machine system.
The invention comprises the following steps:
the invention aims to:
in order to solve the problems, the invention provides a random tracking control method for service robot motion environment data-driven observation
The technical scheme is as follows:
the invention is realized by the following technical scheme:
a random tracking control method for service robot motion environment data driving observation is characterized in that:
1) Separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing data driving observation on the man-machine motion environment change;
2) Based on a service robot dynamics model containing a man-machine motion environment, the eccentricity is converted into a random variable, a random dynamics model of a man-machine system is established, a random controller is designed to inhibit the influence of the environment on the motion of the robot, and meanwhile, a tracking error system is enabled to realize random stability.
The method comprises the following steps:
step 1) separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing the data driving observation on the man-machine motion environment change, and is characterized in that: the system kinematics model is described as follows:
wherein the method comprises the steps of
V (t) represents the movement speed of each wheel of the robot,representing the movement speed, K of the robot in three directions of x, y and rotation angle G (t) represents a coefficient matrix, θ represents an angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, λ i The distance of the center of gravity of the robot to the center of each wheel is expressed, i=1, 2,3.
Separating coefficient matrix K by using model (1) G The physical quantities of the motion environment of the man-machine system in (t) are as follows
K G (t)＝K(t)+ΔK(t) (2)
Wherein the method comprises the steps of
The model (1) is then converted into the following form:
wherein the method comprises the steps of
Order theCan be obtained from the model (3)
X(k+1)＝TV(k)+X(k)+Tω(k) (4)
Where T represents the sampling time. X (k) = [ X ] 1 (k) X 2 (k) X 3 (k)] T ，V(k)＝[V 1 (k) V 2 (k) V 3 (k)] T ，ω(k)＝[ω 1 (k) ω 2 (k) ω 3 (k)] T And respectively representing the motion position, the motion speed and the motion environment physical quantity sampling values of the man-machine system at the moment k.
Order theThe system model for describing the change of the human-machine motion environment is obtained by using the formula (4) and is as follows:
wherein DeltaX (k) =X (k) -X (k-1), deltaV (k) =V (k) -V (k-1),
meanwhile, as can be seen from the expression form of the system model (4), the service robot can be represented as a nonlinear system as follows:
X(k+1)＝Ψ(X(k),...,X(k-n X ),V(k),...,V(k-n V ),ω(k)) (6)
wherein ψ (= (ψ) 1 (·),Ψ 2 (·),Ψ 3 (·)) T Representing a nonlinear vector function, n X And n V Is a given constant, and ψ i (. Cndot.) the partial derivatives of V (k) and ω (k) are continuous.
From the model (6), it can be seen that:
equation (7) can be written in the following form according to the Cauchy's median theorem:
wherein the method comprises the steps of
And->Respectively represent ψ i Partial derivative of (-).
Further, there is a numerical matrix γ (k) for equation (11), converting equation (11) into the following form:
Γ(k)＝γ(k)ΔV(k) (12)
since Δv (k) represents the difference between two different time samples and the two time samples are typically not equal, Δv (k) noteq0. Then from equation (12) it is known that there is a solution gamma * (k) And makeThe model (8) can thus be given the following form:
wherein the method comprises the steps ofSince the model (8) is built from the model (7), and the model (7) is described by the system data, a data driven model (13) of the service robot system (5) is obtained.
Next, the motion environment in the data-driven model (13) is changedObservation is carried out to make kappa 1 (k) =x (k) represents the robot motion trajectory, +.>Representing the man-machine movement environment variation, available based on a model (13):
order theAnd->The estimated values respectively represent the motion trail and the human-machine motion environment variation and are provided with(sigma=1, 2) represents an estimation error, and the man-machine movement environment variation is designed +.>The data driven observer of (a) is as follows:
wherein delta 1 ＝diag[δ 11 ,δ 12 ,δ 13 ]And delta 2 ＝diag[δ 21 ,δ 22 ,δ 23 ]Respectively represent the adjustment matrix of the observer, and delta 1i ＝δ 2i ＝δ 0 ＞0。
The expressions according to formulae (14) and (15) are available:
order theRepresenting the motion trail of each axis and the motion environment estimation error,representing the motion environment variation of each axis, +.>The observed error system using equations (14) - (16) is as follows:
wherein the method comprises the steps of
The nominal error system obtainable according to equation (17) is:
the characteristic polynomial is further obtained as follows:
wherein ρ is a characteristic polynomial factor and I is an identity matrix.
Using transformationsχ is the transform factor, then the characteristic polynomial equation of equation (19) is:
T 2 δ 1i δ 2i χ 2 +(2Tδ 1i +2Tδ 2i -2T 2 δ 1i δ 2i )χ+(T 2 δ 1i δ 2i -2Tδ 1i -2Tδ 2i +4)＝0 (20)
select parameter delta 0 Satisfy 0<Tδ 0 <2, obtaining:
due to delta 1i ＝δ 2i ＝δ 0 > 0, obtainable by:
thus, the system (18) is stabilized according to the formula (22). At the same time, the man-machine system is subjected to a motion environmentThe effect of (2) is to cause the system (17) to produce steady state errors as follows:
due to environmental variationsIs bounded and requires to observe the change of the environment at any time in practical application, so the sampling time is usually small, thus making +.>γ i Representing a small positive number. Thus, the +.A.can be approximated from formula (23)>And->And then the observation error system (17) is stabilized. Thus, the observer (15) can realize the motion environment change quantity of the man-machine system>Data-driven observations of (a) thereby obtaining a man-machine motion environment such asThe following steps:
step 2) based on a service robot dynamics model containing a man-machine motion environment, converting eccentricity into a random variable, establishing a random dynamics model of a man-machine system, designing a random controller to inhibit the influence of the environment on the robot motion, and enabling a tracking error system to realize random stability, and the method is characterized in that: the kinetic model containing the human-machine motion environment is described as follows
Wherein the method comprises the steps of
Wherein M represents the quality of the service robot, M represents the quality of the user, M 0 Representing coefficient matrix, X (t) represents motion track of robot in three directions of X, y and rotation angle, u (t) is control input force of three wheels of robot, r 0 Indicating eccentricity, I 0 Representing the moment of inertia of the robot,representing the moment of inertia of the user, θ represents the angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, and l represents the distance from the center of gravity of the robot to the center of each wheel.
Decomposing the eccentricity r in the coefficient matrix M0 by using the model (25) 0 The system (25) is converted into the following form
Wherein the method comprises the steps of
In the formula (26), Θ (t) has random noise characteristics, which are expressed asWherein Ω represents an independent random process, available
Order theAnd calculate
From equation (28), the model (27) can be expressed as follows:
let the spectral density of random noise Θ (t) beI.e. dΩ=Σdψ, where Σ represents a spectral density matrix, ψ represents a random process with a spectral density distribution, then a service robot random dynamics model containing a man-machine motion environment can be obtained as follows:
further, let theThe model (30) is transformed into the following form:
let the designated movement track of the service robot be X d And (t), the actual motion trail is X (t), and the trail tracking error and the speed tracking error are designed as follows:
e 1 (t)＝X(t)-X d (t) (32)
wherein the method comprises the steps ofRepresenting the parameters to be designed.
The random tracking error system obtained by combining equations (31) - (33) is as follows:
in order to analyze the stability of the tracking error system, the Lyapunov function was designed as follows:
based on the random stabilization theory, obtain
Wherein I represents an identity matrix.
According to Young's inequality, for a given constant y 1 ＞0,γ 2 > 0, have
Wherein the method comprises the steps ofRepresents the F-norm of the matrix and +.>The upper boundary is upsilon.
Substituting formulas (38) - (39) into formula (37) to obtain:
the controller u (t) is designed for the error systems (34) - (35) as follows:
substituting the controller (41) into equation (40) yields:
wherein the method comprises the steps ofμ＝min{μ 1 ,μ 2 }，/>Therefore, as can be seen from the formula (42), the tracking error systems (34) - (35) are randomly stable, and the service robot can inhibit the influence of the environment on the motion of the man-machine system, so that stable tracking is realized.
Step 3) providing an output PWM signal for a motor driving module based on an MSP340 serial singlechip, so that the robot suppresses a motion environment and system eccentricity, and realizes tracking of a reference track signal, and the method is characterized in that: taking an MSP430 series singlechip as a main controller, wherein the input of the main controller is connected with a motor speed measuring module, and the output of the main controller is connected with a motor driving module; the motor driving module is connected with the direct current motor; the power supply system supplies power to the respective electrical devices. The control method of the main controller is to read the feedback signal of the motor encoder and the control command signal X given by the main controller d (t) andan error signal is calculated. According to the error signal, the main controller calculates the control quantity of the motor according to a preset control algorithm and sends the control quantity to the motor driving module, and the motor rotates to drive the wheels to maintain self balance and move in a specified mode.
The advantages and effects:
the invention relates to a random tracking control method for service robot motion environment data driving observation, which has the following advantages: the invention skillfully utilizes a service robot kinematic model and a data driving method to obtain a human-machine system kinematic environment, and applies the kinematic environment to a dynamic model, and simultaneously separates the eccentricity in the dynamic model to establish a random dynamic model containing the kinematic environment; on the basis, the random controller is designed to inhibit the influence of the motion environment on the tracking performance, so that the tracking precision of the robot is improved, surrounding obstacles are avoided from being collided, and the safety of a man-machine system is ensured.
Description of the drawings:
FIG. 1 is a block diagram of the operation of a controller according to the present invention;
FIG. 2 is a structural graph of a service robot
FIG. 3 is a schematic diagram of a MSP430 single-chip microcomputer minimal system according to the present invention;
FIG. 4 is a main controller peripheral expansion circuit of the present invention;
fig. 5 is a circuit of the general principles of the hardware of the present invention.
The specific embodiment is as follows:
the present invention will be further described with reference to the accompanying drawings, but the scope of the present invention is not limited by the examples.
A random tracking control method for service robot motion environment data driving observation is characterized in that:
1) Separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing data driving observation on the man-machine motion environment change;
2) Based on a service robot dynamics model containing a man-machine motion environment, the eccentricity is converted into a random variable, a random dynamics model of a man-machine system is established, a random controller is designed to inhibit the influence of the environment on the motion of the robot, and meanwhile, a tracking error system is enabled to realize random stability.
The method comprises the following steps:
step 1) separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing the data driving observation on the man-machine motion environment change, and is characterized in that: the system kinematics model is described as follows:
wherein the method comprises the steps of
V (t) represents the movement speed of each wheel of the robot,representing the movement speed, K of the robot in three directions of x, y and rotation angle G (t) represents a coefficient matrix, θ represents an angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, λ i The distance of the center of gravity of the robot to the center of each wheel is expressed, i=1, 2,3.
Separating coefficient matrix K by using model (1) G The physical quantities of the motion environment of the man-machine system in (t) are as follows
K G (t)＝K(t)+ΔK(t) (2)
Wherein the method comprises the steps of
The model (1) is then converted into the following form:
wherein the method comprises the steps of
Order theCan be obtained from the model (3)
X(k+1)＝TV(k)+X(k)+Tω(k) (4)
Where T represents the sampling time. X (k) = [ X ] 1 (k) X 2 (k) X 3 (k)] T ，V(k)＝[V 1 (k) V 2 (k) V 3 (k)] T ，ω(k)＝[ω 1 (k) ω 2 (k) ω 3 (k)] T And respectively representing the motion position, the motion speed and the motion environment physical quantity sampling values of the man-machine system at the moment k.
Order theRepresenting human-machine transportationThe dynamic environment, the system model for describing the change of the human-machine motion environment is obtained by using the formula (4) as follows:
wherein DeltaX (k) =X (k) -X (k-1), deltaV (k) =V (k) -V (k-1),
meanwhile, as can be seen from the expression form of the system model (4), the service robot can be represented as a nonlinear system as follows:
X(k+1)＝Ψ(X(k),...,X(k-n X ),V(k),...,V(k-n V ),ω(k)) (6)
wherein ψ (= (ψ) 1 (·),Ψ 2 (·),Ψ 3 (·)) T Representing a nonlinear vector function, n X And n V Is a given constant, and ψ i (. Cndot.) the partial derivatives of V (k) and ω (k) are continuous.
From the model (6), it can be seen that:
equation (7) can be written in the following form according to the Cauchy's median theorem:
wherein the method comprises the steps of
And->Respectively represent ψ i Partial derivative of (-).
Further, there is a numerical matrix γ (k) for equation (11), converting equation (11) into the following form:
Γ(k)＝γ(k)ΔV(k) (12)
since Δv (k) represents the difference between two different time samples and the two time samples are typically not equal, Δv (k) noteq0. Then from equation (12) there is a solution γ (k), and letThe model (8) can thus be given the following form:
wherein the method comprises the steps ofSince the model (8) is built from the model (7), and the model (7) is described by the system data, a data driven model (13) of the service robot system (5) is obtained.
Next, the motion environment in the data-driven model (13) is changedObservation is carried out to make kappa 1 (k) =x (k) represents the robot motion trajectory, +.>Representation ofThe human-machine motion environment variation is obtained based on a model (13):
order theAnd->The estimated values respectively represent the motion trail and the human-machine motion environment variation and are provided with(sigma=1, 2) represents an estimation error, and the man-machine movement environment variation is designed +.>The data driven observer of (a) is as follows:
wherein delta 1 ＝diag[δ 11 ,δ 12 ,δ 13 ]And delta 2 ＝diag[δ 21 ,δ 22 ,δ 23 ]Respectively represent the adjustment matrix of the observer, and delta 1i ＝δ 2i ＝δ 0 ＞0。
The expressions according to formulae (14) and (15) are available:
order theRepresenting the motion trail of each axis and the motion environment estimation error,representing the motion environment variation of each axis, +.>The observed error system using equations (14) - (16) is as follows:
wherein the method comprises the steps of
The nominal error system obtainable according to equation (17) is:
the characteristic polynomial is further obtained as follows:
wherein ρ is a characteristic polynomial factor and I is an identity matrix.
Using transformationsχ is the transform factor, then the characteristic polynomial equation of equation (19) is:
T 2 δ 1i δ 2i χ 2 +(2Tδ 1i +2Tδ 2i -2T 2 δ 1i δ 2i )χ+(T 2 δ 1i δ 2i -2Tδ 1i -2Tδ 2i +4)＝0 (20)
select parameter delta 0 Satisfy 0<Tδ 0 <2, obtaining:
due to delta 1i ＝δ 2i ＝δ 0 > 0, obtainable by:
thus, the system (18) is stabilized according to the formula (22). At the same time, the man-machine system is subjected to a motion environmentThe effect of (2) is to cause the system (17) to produce steady state errors as follows:
due to environmental variationsIs bounded and requires to observe the change of the environment at any time in practical application, so the sampling time is usually small, thus making +.>γ i Representing a small positive number. Thus, the +.A.can be approximated from formula (23)>And->And then the observation error system (17) is stabilized. Thus, the observer (15) can realize the motion environment change quantity of the man-machine system>The man-machine motion environment is thus obtained as follows:
step 2) based on a service robot dynamics model containing a man-machine motion environment, converting eccentricity into a random variable, establishing a random dynamics model of a man-machine system, designing a random controller to inhibit the influence of the environment on the robot motion, and enabling a tracking error system to realize random stability, and the method is characterized in that: the kinetic model containing the human-machine motion environment is described as follows
Wherein the method comprises the steps of
Wherein M represents the quality of the service robot, M represents the quality of the user, M 0 Representing coefficient matrix, X (t) represents motion track of robot in three directions of X, y and rotation angle, u (t) is control input force of three wheels of robot, r 0 Indicating eccentricity, I 0 Representing the moment of inertia of the robot,representing the moment of inertia of the user, θ represents the angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, and l represents the distance from the center of gravity of the robot to the center of each wheel.
Decomposing coefficient matrix M by using model (25) 0 The eccentricity r of (b) 0 The system (25) is converted into the following form
Wherein the method comprises the steps of
In the formula (26), Θ (t) has random noise characteristics, which are expressed asWherein Ω represents an independent random process, available
Order theAnd calculate +.>
From equation (28), the model (27) can be expressed as follows:
let the spectral density of random noise Θ (t) beI.e. dΩ=Σdψ, where Σ represents a spectral density matrix, ψ represents a random process with a spectral density distribution, then a service robot random dynamics model containing a man-machine motion environment can be obtained as follows:
further, let theThe model (30) is transformed into the following form:
let the designated movement track of the service robot be X d And (t), the actual motion trail is X (t), and the trail tracking error and the speed tracking error are designed as follows:
e 1 (t)＝X(t)-X d (t) (32)
wherein the method comprises the steps ofRepresenting the parameters to be designed.
The random tracking error system obtained by combining equations (31) - (33) is as follows:
in order to analyze the stability of the tracking error system, the Lyapunov function was designed as follows:
based on the random stabilization theory, obtain
Wherein I represents an identity matrix.
According to Young's inequality, for a given constant y 1 ＞0,γ 2 > 0, have
/>
Wherein the method comprises the steps ofRepresents the F-norm of the matrix and +.>The upper boundary is upsilon.
Substituting formulas (38) - (39) into formula (37) to obtain:
the controller u (t) is designed for the error systems (34) - (35) as follows:
substituting the controller (41) into equation (40) yields:
wherein the method comprises the steps ofμ＝min{μ 1 ,μ 2 }，/>Therefore, as can be seen from the formula (42), the tracking error systems (34) - (35) are randomly stable, and the service robot can inhibit the influence of the environment on the motion of the man-machine system, so that stable tracking is realized.
Step 3) providing an output PWM signal for a motor driving module based on an MSP340 serial singlechip, so that the robot suppresses a motion environment and system eccentricity, and realizes tracking of a reference track signal, and the method is characterized in that: taking an MSP430 series singlechip as a main controller, wherein the input of the main controller is connected with a motor speed measuring module, and the output of the main controller is connected with a motor driving module; the motor driving module is connected with the direct current motor; the power supply system supplies power to the respective electrical devices. The control method of the main controller is to read the feedback signal of the motor encoder and the control command signal X given by the main controller d (t) andan error signal is calculated. According to the error signal, the main controller calculates the control quantity of the motor according to a preset control algorithm and sends the control quantity to the motor driving module, and the motor rotates to drive the wheels to maintain self balance and move in a specified mode.
The invention skillfully utilizes a service robot kinematic model and provides a data driving method, so as to obtain a human-machine system kinematic environment, and simultaneously, the kinematic environment is acted on a dynamic model and the eccentricity of the system is separated, so that a random dynamic model containing the human-machine kinematic environment is established; the random tracking control method is provided, the influence of the environment on the tracking motion of the robot is restrained, the random stability of a tracking error system is guaranteed, the tracking precision of a man-machine system is improved, the collision risk of an overlarge tracking error is avoided, and the safety of the man-machine system is guaranteed.
Claims (3)
1. The random tracking control method for the service robot motion environment data driving observation is characterized by comprising the following steps of: separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing data driving observation on the man-machine motion environment change; based on a service robot dynamics model containing a man-machine motion environment, converting eccentricity into a random variable, establishing a random dynamics model of a man-machine system, designing a random controller to inhibit the influence of the environment on the motion of the robot, and simultaneously enabling a tracking error system to realize random stability; the method comprises the following steps:
1) Separating the physical quantity of the motion environment of the man-machine system by utilizing the kinematic model of the service robot, establishing a data driving model for describing the motion environment change, designing a data driving observer, and realizing data driving observation on the man-machine motion environment change;
2) Based on a service robot dynamics model containing a man-machine motion environment, the eccentricity is converted into a random variable, a random dynamics model of a man-machine system is established, a random controller is designed to inhibit the influence of the environment on the motion of the robot, and meanwhile, a tracking error system is enabled to realize random stability.
2. The random tracking control method for service robot motion environment data-driven observation according to claim 1, wherein: the system kinematics model is described as follows:
wherein the method comprises the steps of
V (t) represents the movement speed of each wheel of the robot,representing the movement speed, K of the robot in three directions of x, y and rotation angle G (t) represents a coefficient matrix, θ represents an angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, λ i Representing the distance of the center of gravity of the robot to the center of each wheel, i=1, 2,3;
separating coefficient matrix K by using model (1) G The physical quantities of the motion environment of the man-machine system in (t) are as follows
K G (t)＝K(t)+ΔK(t) (2)
Wherein the method comprises the steps of
The model (1) is then converted into the following form:
wherein the method comprises the steps of
Order theCan be obtained from the model (3)
X(k+1)＝TV(k)+X(k)+Tω(k) (4)
Wherein T represents the sampling time; x (k) = [ X ] 1 (k) X 2 (k) X 3 (k)] T ，V(k)＝[V 1 (k) V 2 (k) V 3 (k)] T ，ω(k)＝[ω 1 (k) ω 2 (k) ω 3 (k)] T Sampling values respectively representing the motion position, the motion speed and the motion environment physical quantity of the man-machine system at the moment k; order theThe system model for describing the change of the human-machine motion environment is obtained by using the formula (4) and is as follows:
wherein DeltaX (k) =X (k) -X (k-1), deltaV (k) =V (k) -V (k-1),
meanwhile, as can be seen from the expression form of the system model (4), the service robot can be represented as a nonlinear system as follows:
X(k+1)＝Ψ(X(k),...,X(k-n X ),V(k),...,V(k-n V ),ω(k)) (6)
wherein ψ (= (ψ) 1 (·),Ψ 2 (·),Ψ 3 (·)) T Representing a nonlinear vector function, n X And n V Is a given constant, and ψ i (. Cndot.) the partial derivatives of V (k) and ω (k) are continuous;
from the model (6), it can be seen that:
equation (7) can be written in the following form according to the Cauchy's median theorem:
wherein the method comprises the steps of
And->Respectively represent ψ i Partial derivative of (-);
further, there is a numerical matrix for equation (11)The equation (11) is converted into the following form:
since Δv (k) represents the difference between two different time samples and the two time samples are typically not equal, Δv (k) noteq0; then from equation (12) it can be seen that there is a solutionAnd let->The model (8) can thus be given the following form:
wherein the method comprises the steps ofSince the model (8) is built from the model (7), and the model (7) is described by the system data, a data driven model (13) of the service robot system (5) is obtained;
next, the motion environment in the data-driven model (13) is changedObservation is carried out to make kappa 1 (k) =x (k) represents the robot motion trajectory, +.>Representing the man-machine movement environment variation, available based on a model (13):
order theAnd->The estimated values respectively represent the motion trail and the human-machine motion environment variation and are provided with Representing estimation error, designing human-machine motion environment variation amount +.>The data driven observer of (a) is as follows:
wherein delta 1 ＝diag[δ 11 ,δ 12 ,δ 13 ]And delta 2 ＝diag[δ 21 ,δ 22 ,δ 23 ]Respectively represent the adjustment matrix of the observer, and delta 1i ＝δ 2i ＝δ 0 ＞0；
The expressions according to formulae (14) and (15) are available:
order theRepresenting the motion trail of each axis and the motion environment estimation error,representing the motion environment variation of each axis, +.>The observed error system using equations (14) - (16) is as follows:
wherein the method comprises the steps of
The nominal error system obtainable according to equation (17) is:
the characteristic polynomial is further obtained as follows:
wherein ρ is a characteristic polynomial factor, and I is an identity matrix;
using transformationsχ is the transform factor, then the characteristic polynomial equation of equation (19) is:
T 2 δ 1i δ 2i χ 2 +(2Tδ 1i +2Tδ 2i -2T 2 δ 1i δ 2i )χ+(T 2 δ 1i δ 2i -2Tδ 1i -2Tδ 2i +4)＝0 (20)
select parameter delta 0 Satisfy 0<Tδ 0 <2, obtaining:
due to delta 1i ＝δ 2i ＝δ 0 > 0, obtainable by:
thus, the system (18) is stabilized according to the formula (22); at the same time, the man-machine system is subjected to a motion environmentThe effect of (2) is to cause the system (17) to produce steady state errors as follows:
due to environmental variationsIs bounded and requires to observe the change of the environment at any time in practical application, so the sampling time is usually small, thus making +.>γ i Representing a small positive number, such that +.A.about.can be approximated by formula (23)>And->Then, the observation error system (17) is stabilized; thus, the observer (15) can realize the motion environment change quantity of the man-machine system>The man-machine motion environment is thus obtained as follows:
3. the random tracking control method for service robot motion environment data-driven observation according to claim 1, wherein: the kinetic model containing the human-machine motion environment is described as follows
Wherein the method comprises the steps of
Wherein M represents the quality of the service robot, M represents the quality of the user, M 0 Representing coefficient matrix, X (t) represents motion track of robot in three directions of X, y and rotation angle, u (t) is robotControl input force of three wheels r 0 Indicating eccentricity, I 0 Representing the moment of inertia of the robot,representing the moment of inertia of the user, θ representing the angle between the horizontal axis and the connection between the center of the robot and the center of the first wheel, and l representing the distance from the center of gravity of the robot to the center of each wheel;
decomposing coefficient matrix M by using model (25) 0 The eccentricity r of (b) 0 The system (25) is converted into the following form
Wherein the method comprises the steps of
In the formula (26), Θ (t) has random noise characteristics, which are expressed asWherein Ω represents an independent random process, available
Order theAnd calculate
From equation (28), the model (27) can be expressed as follows:
let the spectral density of random noise Θ (t) beI.e. dΩ=Σdψ, where Σ represents a spectral density matrix, ψ represents a random process with a spectral density distribution, then a service robot random dynamics model containing a man-machine motion environment can be obtained as follows:
further, let theThe model (30) is transformed into the following form:
let the designated movement track of the service robot be X d And (t), the actual motion trail is X (t), and the trail tracking error and the speed tracking error are designed as follows:
e 1 (t)＝X(t)-X d (t) (32)
wherein the method comprises the steps ofRepresenting parameters to be designed;
the random tracking error system obtained by combining equations (31) - (33) is as follows:
in order to analyze the stability of the tracking error system, the Lyapunov function was designed as follows:
based on the random stabilization theory, obtain
Wherein I represents an identity matrix;
according to Young's inequality, for a given constant y 1 ＞0,γ 2 > 0, have
Wherein the method comprises the steps ofRepresents the F-norm of the matrix and +.>The upper boundary is upsilon;
substituting formulas (38) - (39) into formula (37) to obtain:
the controller u (t) is designed for the error systems (34) - (35) as follows:
substituting the controller (41) into equation (40) yields:
wherein the method comprises the steps ofμ＝min{μ 1 ,μ 2 }，/>Thus, as can be seen from the equation (42), the tracking error systems (34) - (35) are randomly stable, and the service robot can suppress the influence of the environment on the motion of the man-machine system and realize stable tracking.
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