CN114942593B - Mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation - Google Patents
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Abstract
The invention discloses a self-adaptive sliding mode control method of a mechanical arm based on disturbance observer compensation, and belongs to the technical field of tail end track tracking control of a mechanical arm of a robot. The method comprises the steps of establishing a kinematic model of the mechanical arm: establishing a coordinate system by using the D-H model, taking a base of the robot as a reference coordinate system, and taking each joint and the end effector as a motion reference system; establishing a kinetic model of the mechanical arm: modeling mechanical arm dynamics by using a Lagrangian method; designing an interference observer: the observation error is exponentially attenuated and finally converged by adjusting a gain matrix of the interference observer; sliding mode controller and novel control approach law: based on the constant velocity approach law and the exponential approach law, the buffeting is improved by applying the approach law of a special power function, and the convergence time and tracking error are reduced; and (3) designing a system self-adaption law: the non-observable part of the interference is compensated for by adaptive law control. The technical problem of large track tracking error of the tail end of the mechanical arm is solved.
Description
Technical Field
The invention relates to the technical field of tail end track tracking control of a robot mechanical arm, in particular to a mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation.
Background
With the progress and development of science and technology, the mechanical arm has a wide application prospect in the fields of industrial production, service robots, medical robots and the like, and a plurality of works need high-precision track tracking control. However, the mechanical arm is used as a complex multi-input multi-output system, and modeling errors and uncertainty disturbance exist in practical application, which are all uncertainty factors affecting the tracking accuracy of the system. Therefore, the realization of high-precision track tracking control of the tail end of the mechanical arm has become one of hot spots of the front-end technical research of robots. The sliding mode control is used as a hot spot for researching the track tracking control of the mechanical arm, has stronger robustness for the control of the mechanical arm with uncertain parameters, but has a buffeting phenomenon, so that how to reduce buffeting of the tail end is a key for the application of the sliding mode on the mechanical arm. The research of reducing the modeling error and external disturbance of the mechanical arm is also an important factor for ensuring high-precision control, and in recent years, a plurality of researchers at home and abroad apply a nonlinear disturbance observer and a self-adaptive control method to the track control of the mechanical arm, so that the real-time detection and processing of the disturbance error of the mechanical arm are realized, and the influence of disturbance on the accurate tracking of the track of the mechanical arm is reduced.
Aiming at the problems faced by the realization of the precise control of the tail end track of the mechanical arm, a method for dynamic modeling of delay estimation appears, but the method for estimating the dynamic model has a certain influence on the convergence of the control method because a plurality of uncertainty factors exist in practical application.
Disclosure of Invention
1. Technical problem to be solved
The present application aims to provide a method of adaptive sliding mode control of a robotic arm based on disturbance observer compensation, which is advantageous over the prior art described in the background art in at least one aspect.
2. Technical proposal
The aim of the invention is achieved by the following technical scheme:
a mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation comprises the following steps:
S1, establishing a kinematic model of the mechanical arm: establishing a coordinate system by using the D-H model, taking a base of the robot as a reference coordinate system, and taking each joint and the end effector as a motion reference system;
S2, establishing a kinetic model of the mechanical arm: modeling mechanical arm dynamics by using a Lagrangian method;
S3, designing an interference observer: on-line observation of an observable part of the modeling error of the mechanical arm system in the step S2, and exponential attenuation of the observed error is realized by adjusting a gain matrix of an interference observer, so that convergence is finally realized;
s4, a sliding mode controller and a novel control approach law: based on the constant velocity approach law and the exponential approach law, the buffeting is improved by applying the approach law of a special power function, and the convergence time and tracking error are reduced;
S5, designing a self-adaptive law of a mechanical arm system: the non-observable part of the interference is compensated for by adaptive law control.
The problem of large track error of the tail end track of the mechanical arm is solved by the scheme, buffeting of movement of each joint is obviously reduced, and the running stability of the mechanical arm system is proved.
The optimized technical scheme is detailed in specific embodiments, and is not described herein.
3. Advantageous effects
Compared with the prior art, the invention has the advantages that:
(1) According to the mechanical arm self-adaptive sliding mode control method based on interference observer compensation, the interference observer is introduced, the observable part of modeling error of the mechanical arm system is observed on line, and the observation error is attenuated exponentially and finally converged by adjusting the gain matrix of the interference observer. After the interference observer is adopted, the influence of interference on the mechanical arm system can be greatly improved;
(2) The design of the existing sliding mode controller can ensure that the mechanical arm system has better robustness, but the approach rule is not properly selected, so that the sliding mode is easy to have larger buffeting, and the tracking precision is reduced. The invention provides a novel special power function based on the constant velocity approach law and the exponential approach law, which can effectively improve buffeting and obviously reduce convergence time and tracking error.
(3) In the mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation, parameters can be reasonably selected by deriving the model, so that the mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation can ensureThe mechanical arm system is controlled to meet the Liapunov stability theory, the joint variable tracking error is approximately stable, and the stable tracking performance of the mechanical arm system can be ensured. MATLAB and SIMULINK are adopted as simulation experiment platforms, robotics System Toolbox and SimMechanics plug-in units carried by the mechanical arm system are combined, and simulation results verify the correctness of theory.
Drawings
FIG. 1 is a diagram of robot parameters and coordinates;
FIG. 2 is a graph showing the change in joint angle during movement of a robotic arm;
FIG. 3 is the error in the angle of the robotic arm joint;
Fig. 4 joint control moment.
Detailed Description
For a further understanding of the present invention, the present invention will be described in detail with reference to the drawings and examples.
Examples
The specific implementation manner of the mechanical arm self-adaptive sliding mode control method based on interference observer compensation in this embodiment is as follows:
(1) Establishing a coordinate system of the mechanical arm and establishing a kinematic model of the mechanical arm:
as shown in fig. 1, the parameters of each robot arm and the modified D-H coordinate method are defined as follows:
Link length a i: refers to the distance from z i to z i+1, along the x i axis, which is actually the length of the plumb line;
Link rotation angle α i: the angle of rotation in the direction from z i to z i+1 is positive, indicating positive rotation along the x i axis;
Link offset d i: defining a direction along the z i axis, the distance of movement of the x i-1 axis to the x i axis, actually describing the distance of movement between the two plumb lines;
Articulation angle θ i: the angle from x i-1 to x i is indicated, and rotation in the positive direction along the z i axis is indicated as positive.
The step of improving the establishment of the D-H coordinate system is to draw each joint axis:
z i axis: along the axial direction of the joint axis i;
Origin point: the common perpendicular to both the i+1th and i-th joint axes, or the intersection point of the intersection point with the i-th joint axis, is the origin of the coordinate system { l }. The direction of the joint axis i pointing to the joint axis i+1 is marked as the direction of the common vertical line of the x i axis along the common vertical line a i axis, and if the two joint axes intersect, the direction of the common vertical line of the plane where the x i axis is perpendicular to the two axes is the direction of the plane; the direction of y i is determined according to the right hand rule: if the variable of the first joint is 0, it is defined that the coordinate system {0} coincides with the coordinate system { l }, and the directions of other coordinate systems such as { n }, origin and x n are not specified and can be selected at will.
When the D-H model is used to build the coordinate system, the base of the robot is taken as the reference coordinate system, and each joint and the end effector are taken as the motion reference system. Then the pose conversion relation expression form between all adjacent two connecting rods is calculated, and the expression is:
the kinematic model of the tool coordinate system of the robot arm tip relative to the base coordinate system is described as:
Wherein θ= [ θ 1,θ2]T ] is the joint angle matrix, R tool is the terminal pose matrix, P tool is the terminal position vector, The unit coordinates x, y, z of the end effector are respectively the directional cosine of the base coordinates.
(2) Establishing a kinetic model of the mechanical arm:
The dynamic model is established on the premise of controlling the movement of the mechanical arm, and on the premise of establishing the coordinates of the mechanical arm, the mechanical arm dynamics is modeled by using a Lagrange method, and a Lagrange function is defined as follows:
L＝K-P(3)；
wherein L is Lagrangian function, K is total kinetic energy of the mechanical arm system, and P is total potential energy of the mechanical arm system, so that the method can obtain
Where θ i denotes a certain generalized coordinate in the robotic arm system,For the corresponding speed, τ i is the acting force or moment applied to the ith connecting rod, and n represents the number of the connecting rods.
Wherein the method comprises the steps ofIs an inertial matrix of the mechanical arm; Representing a matrix of centrifugal and inertial forces, For the gravity term matrix acting on the mechanical arm, τ e R 2×1 is the control moment acting on the joint, and the external disturbance signal μ e R 2×1 includes modeling errors, parameter transformations and other uncertain disturbance factors δ, namely
M21＝M12
C22＝0 (6)；
(3) Design of interference observer:
The observer predicts the estimated output by the difference between the estimated output and the actual output. The interference observer is introduced in the embodiment, the observable part of the system modeling error in the step (2) is observed online, and the observation error is attenuated exponentially and finally converged by adjusting the gain matrix of the interference observer. By adopting the interference observer, the influence of interference on the system can be greatly improved.
Without differential prior knowledge, it is assumed that the disturbance is slowly varying with respect to the disturbance observer, i.e.:
defining errors of an interference observer
Interference observations are defined as: the bonding (5) (7) (8) (9) formula can be obtained:
the expression of the disturbance observer is:
Wherein the method comprises the steps of Is an estimate of the external disturbance signal mu,The gain matrix of the nonlinear disturbance observer is defined, and z is a defined auxiliary vector.A nonlinear function to be designed.
After the interference observer is adopted, the interference of the system is changed from an interference signal mu to an error u of the observer, and the influence of the interference on the system is greatly reduced.
(4) Sliding mode controller and novel control approach law:
the design of the sliding mode controller can ensure that the system has better robustness, but the approach rule is selected improperly, so that the sliding mode is easy to have larger buffeting, and the tracking precision is reduced. The embodiment provides a novel special power function on the basis of a constant velocity approach law and an exponential approach law, and can obviously reduce convergence time and tracking error while effectively improving buffeting.
Defining joint position errors: e=q-q d (12), the joint velocity error isWherein q d is the desired joint position;
taking the sliding mode function as Wherein lambda is a coefficient matrix;
Take the novel control approach law as Function ofIs a novel nonlinear function, defined asTan h(s) is a hyperbolic tangent function with respect to the sliding mode function s, k1, k2, ε, δ, α, are constants ε >0,0< δ <1, α >0, k1>0, k2>0;
For the proposed new approach law, when |s| < delta, The important purpose is that when s.fwdarw.0,The existence of the system ensures that the approach speed of the system is not too small at the moment, and the approach speed is accelerated; when |s| > delta, k 1tanh(s)+k2 s plays a dominant role, and the function expression is continuous, so that discontinuity of a sign function is avoided, and buffeting of a system can be improved theoretically.
(5) Design of a system control law: from formula (14):
Let equation (16) equal zero, to achieve the approach law being the new power function approach law designed in (4), the control law of the system is selected as:
(6) And (3) designing a system self-adaption law:
The application of the system self-adaptive law is proposed for the unobservable part existing in the disturbance signal. On the basis of the sliding mode interference observer, adaptive control is introduced to estimate and compensate the unobservable part of the interference.
Definition of the definitionThe estimated value of the unobservable part is the estimated error
In order to derive the adaptive law of the system, the li-apunov function of the adaptive sliding mode control is defined as:
Order the Then:
The self-adaptive law is taken as
Where χ is a constant.
At this time, the control law of the system is:
(7) Analysis and simulation verification of system stability
Stability is the most basic requirement of a control system, an inversion control method is used, an observer and a self-adaptive sliding mode controller are combined, a stable Liapunov model of the whole system is designed and judged, proper constant values and constant value matrixes are selected, and gradual stability of the whole system with global significance is guaranteed.
For the stability judgment of the whole closed loop system, taking a Liidefenof function:
Deriving the model (22), and taking the formulas (10), (17) and (21) (22) into the following formula:
By selecting proper parameters, V 1 is less than or equal to 0.
In order to verify the feasibility and effectiveness of the method, MATLAB and SIMULINK are used as simulation experiment platforms, and Robotics System Toolbox and SimMechnics plug-in modules carried by the system are combined to be used as simulation platforms. The simulation experiment conditions were set as follows:
The initial joint angle (in radians) θ 0＝[0.7,-1.1]T of the mechanical arm moves for a total time period t=10.0 s, the time step dt=0.001 s, and the expected track is θ d = [0.1×sin (T), 0.1×sin (T) ], and the constant is set as follows: α＝0.25,δ＝0.05,
Simulation results of the mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation, which is provided by the embodiment, are shown in fig. 2-4. Fig. 2 illustrates the change in joint angle during movement of the robotic arm. The simulation result verifies the correctness of theory, and each joint angle is fast converged and can fast track the expected track.
Fig. 3 shows the error in the angle of the joints of the robot arm. The accuracy of track tracking is further shown, the tracking error of joints 1 and 2 of the mechanical arm is in the level of 10 -4 rad, and the tracking performance of the mechanical arm is improved.
Fig. 4 shows the control moment of the robot arm. In the diagram, a control moment curve is continuous and smooth, so that buffeting of sliding mode control is greatly reduced.
In summary, the mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation, which is designed by the invention, not only can accurately track the track, but also can better cope with the influence of disturbance caused by external disturbance and the change of parameters in the mechanical arm on the system performance.
The invention and its embodiments have been described above by way of illustration and not limitation, and the invention is illustrated in the accompanying drawings and described in the drawings in which the actual structure is not limited thereto. Therefore, if one of ordinary skill in the art is informed by this disclosure, the structural mode and the embodiments similar to the technical scheme are not creatively designed without departing from the gist of the present invention.
Claims (8)
1. A mechanical arm self-adaptive sliding mode control method based on disturbance observer compensation is characterized by comprising the following steps:
S1, establishing a kinematic model of the mechanical arm: establishing a coordinate system by using the D-H model, taking a base of the robot as a reference coordinate system, and taking each joint and the end effector as a motion reference system;
S2, establishing a kinetic model of the mechanical arm: modeling mechanical arm dynamics by using a Lagrangian method;
s3, designing an interference observer: on-line observation of the observable part of the system modeling error in the step S2, and exponential decay of the observation error is realized by adjusting a gain matrix of an interference observer, so that convergence is finally realized;
s4, a sliding mode controller and a novel control approach law: based on the constant velocity approach law and the exponential approach law, the buffeting is improved by applying the approach law of a special power function, and the convergence time and tracking error are reduced;
S5, designing a self-adaptive law of a mechanical arm system: estimating and compensating an unobservable part of the interference through adaptive law control;
In the step S4, a joint position error is defined: (12) The joint velocity error is （13）；
Wherein the method comprises the steps ofIs the desired joint position;
taking the sliding mode function as （14）；
Wherein the method comprises the steps ofIs a coefficient matrix;
The novel control approach law is taken as follows:
（15）；
Function of Is a novel nonlinear function, defined as，As to sliding mode functionsIs a function of the hyperbolic tangent of (c),Are all constant and are used for the preparation of the high-voltage power supply,；
For the approach law, whenIn the time-course of which the first and second contact surfaces,Plays an important role, namely，The existence of the device ensures that the approach speed of the mechanical arm system is not too small at the moment, and the approach speed is accelerated; when (when)In the time-course of which the first and second contact surfaces,The function expression is continuous, so that discontinuity of the sign function is avoided, and buffeting of the sliding mode controller is improved.
2. The method for controlling the adaptive sliding mode of the mechanical arm based on the disturbance observer compensation according to claim 1, wherein in the step S1:
S1-1, defining parameters of each mechanical arm and an improved D-H coordinate method as follows:
Length of connecting rod : Finger slaveMove toAlong the distance, directionA shaft, which is in fact the length of the plumb line;
Rotation angle of connecting rod : Finger slaveRotate toThe angle of rotation of the direction of (2) is recorded alongThe positive direction rotation of the shaft is positive;
Offset distance of connecting rod : Definition edgeThe direction of the axis is such that,Shaft-to-shaftIn fact describes the distance of movement between two plumb lines;
Articulation angle : Finger slaveRotate toAlong the angle of (a)The positive direction of the shaft rotates to be positive;
S1-2, the step of establishing an improved D-H coordinate system is that each joint axis is drawn:
And (3) a shaft: along the joint axis Is arranged in the axial direction of the cylinder;
Origin point: first, the And (b)Common perpendicular to both joint axes of the individual, or intersection point and the firstThe intersection point of the joint axes is a coordinate systemAn origin;
Joint shaft Directional joint axisThe direction of (2) is recorded asThe axis is along the public vertical lineThe direction of the common perpendicular to the axes, if the two joint axes intersect, thenThe plane where the axes are perpendicular to the two axes is the direction of the public perpendicular thereof; according to the right-hand ruleIs oriented in:
if the variable of the first joint is 0, a coordinate system is defined And a coordinate systemAre coincident with each other in other coordinate systemsOrigin andIs not specified.
3. The method for controlling the adaptive sliding mode of the mechanical arm based on the disturbance observer compensation according to claim 2, wherein the expression of the pose conversion relation between all the two adjacent connecting rods in the step S1 is calculated by:
（1）
the kinematic model of the tool coordinate system of the robot arm tip relative to the base coordinate system is described as:
（2）
Wherein the method comprises the steps of In the form of a matrix of joint angles,For the end pose matrix,As the end position vector of the object,Unit coordinates of the end effectors, respectivelyThe direction cosine with respect to the base coordinates.
4. The method for controlling the adaptive sliding mode of the mechanical arm based on the disturbance observer compensation according to claim 3, wherein the step S2 is to define a lagrangian function as follows:
（3）；
In the middle of Is a lagrangian function of the power,Is the total kinetic energy of the mechanical arm system,Is the total potential energy of the mechanical arm system, so can be obtained
， ，（4）；
Wherein the method comprises the steps ofRepresenting generalized coordinates in the robotic arm system,For the corresponding speed of the vehicle, the vehicle is at a speed,Is the firstThe force or moment to which the individual links are subjected,Representing the number of connecting rods;
the mechanical arm dynamics model is as follows:
,（5）；
Wherein the method comprises the steps of Is an inertial matrix of the mechanical arm;
Representing a matrix of centrifugal and inertial forces,
To act on the matrix of gravitational terms on the robotic arm,
In order to control the moment acting on the joint,
External disturbance signalIncluding modeling errors, parametric transformations, and other uncertain disturbancesI.e.；
（6）。
5. The method for controlling the adaptive sliding mode of the mechanical arm based on the compensation of the disturbance observer according to claim 4, wherein in the step S3, the disturbance is assumed to be slowly changed relative to the disturbance observer without differential prior knowledge, namely:
（7）；
defining errors of an interference observer (8)；
Interference observations are defined as:(9)
the following formulas (5), (7), (8) and (9) are combined:
（10）；
the expression of the disturbance observer is:
（11）；
Wherein the method comprises the steps of Is an external disturbance signalIs used for the estimation of the (c),Is a gain matrix of a nonlinear disturbance observer,Is defined as an auxiliary vector; A nonlinear function to be designed.
6. The adaptive sliding mode control method for a robot arm based on disturbance observer compensation according to claim 5, wherein the robot arm system control law is designed in the step S4: based on formula (14), it is possible to obtain:
（16）；
let (16) equal to zero, in order to achieve that the approach law is the special power approach law designed in step S4, the control law of the mechanical arm system is designed as follows:
（17）。
7. The method for controlling an adaptive sliding mode of a mechanical arm based on disturbance observer compensation according to claim 6, wherein the step S5 is defined as follows The estimated value of the unobservable part is the estimated error（18）；
In order to derive the adaptive law of the mechanical arm system, a self-adaptive sliding mode control Liapunov function is defined as follows:
（19）；
Order the (20) Then:
The self-adaptive law is taken as (21) WhereinIs a constant;
at this time, the control law of the mechanical arm system is:
（22）。
8. The method for controlling an adaptive sliding mode of a mechanical arm based on disturbance observer compensation according to claim 7, further comprising the steps of S6, simulation verification and result analysis: by using an inversion control method and combining an observer and a self-adaptive sliding mode controller, a Liapunov model for judging the stability of the whole mechanical arm system is designed, and proper constant values and constant value matrixes are selected to ensure the gradual stability of the whole mechanical arm system with global significance.
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