CN110687870A - Mechanical arm tracking controller and system based on nonlinear extended state observer - Google Patents

Mechanical arm tracking controller and system based on nonlinear extended state observer Download PDF

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CN110687870A
CN110687870A CN201910802822.4A CN201910802822A CN110687870A CN 110687870 A CN110687870 A CN 110687870A CN 201910802822 A CN201910802822 A CN 201910802822A CN 110687870 A CN110687870 A CN 110687870A
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唐文杰
宗西举
程新功
陈镇镇
刘大志
衣文雨
刘依明
刘蕊
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Abstract

The present disclosure provides a mechanical arm tracking controller and system based on a nonlinear extended state observer, including: a nonlinear tracking differentiator, a nonlinear extended state observer and a nonlinear state error feedback control law; the nonlinear tracking differentiator tracks the expected position signal of the joint angle of the mechanical arm to obtain a tracking signal of the expected position signal and a differential signal thereof and outputs the tracking signal and the differential signal; the nonlinear extended state observer is used for obtaining an estimated value with system time delay and disturbance as uncertainty factors; the nonlinear state error feedback control law utilizes the difference between the output of the nonlinear tracking differentiator and the output of the nonlinear extended state observer to form a feedback control component, and the estimated value of the nonlinear extended state observer is added into the system control law in a feedforward compensation mode to obtain the total control law of the system. The nonlinear active disturbance rejection control method has the advantages that disturbance can be suppressed, and rapid convergence of system tracking errors can be achieved.

Description

Mechanical arm tracking controller and system based on nonlinear extended state observer
Technical Field
The present disclosure relates to the field of automation and mechanical arm control technologies, and in particular, to a mechanical arm tracking controller, system, and method based on a nonlinear extended state observer.
Background
In recent years, robot arms have been widely used in industrial production and also play an increasingly important role. However, with the complication of the application environment of the robot arm and the enlargement of industrial production, the traditional control method is difficult to meet the increasing industrial application requirements of people, and the research on the control method of the robot arm has become a popular development trend. However, due to the limitation of network resources and the influence of transmission distance, the problems of data loss, data retransmission and the like when each node in the mechanical arm control system exchanges information are inevitable, so that time delay occurs in the industrial mechanical arm control system, control information in the system cannot be updated timely, the control performance of the system is reduced, and the system is even unstable under severe conditions.
In addition, in a network control system of a mechanical arm, besides being affected by time delay, factors such as parameter perturbation and external disturbance of the mechanical arm control system can also have adverse effects on the tracking control performance of the system.
Disclosure of Invention
The purpose of the embodiments of the present description is to provide a robot arm tracking controller based on a nonlinear extended state observer, which can well implement control on an industrial robot arm system with interference and time delay, and after the interference and time delay in the system are estimated and compensated on line by the algorithm, the system output can stably track an expected position signal, and the response time is short and the steady-state error is small.
The embodiment of the specification provides a mechanical arm tracking controller based on a nonlinear extended state observer, which is realized by the following technical scheme:
the method comprises the following steps: a nonlinear tracking differentiator, a nonlinear extended state observer and a nonlinear state error feedback control law;
the nonlinear tracking differentiator tracks the expected position signal of the joint angle of the mechanical arm to obtain a tracking signal of the expected position signal and a differential signal thereof and outputs the tracking signal and the differential signal;
the nonlinear extended state observer is used for obtaining an estimated value with system time delay and disturbance as uncertainty factors;
the nonlinear state error feedback control law utilizes the difference between the output of the nonlinear tracking differentiator and the output of the nonlinear extended state observer to form a feedback control component, and the estimated value of the nonlinear extended state observer is added into the system control law in a feedforward compensation mode to obtain the total control law of the system.
In a further aspect, the nonlinear tracking differentiator:
Figure BDA0002182812200000021
in the formula (7), 0<α<1，xir(k) Desired position signal, x, for the i-th joint angleid1(k),xid2(k) Respectively a tracking signal of the expected position signal at the time k and a differential signal thereof; r is a fast factor of the tracking differentiator; t is the step size of the tracking differentiator.
In a further technical solution, the nonlinear tracking differentiator is:
wherein the fal function is defined as:
discretizing equation (9), which is expressed as:
Figure BDA0002182812200000031
in the formula, epsiloni1(k) Observing error of the system state variable by the linear extended state observer; z is a radical ofi1(k),zi2(k),zi3(k) Are respectively xi1(k),xi2(k),xi3(k) Estimate of beta1,β2,β3Is the observer gain to be designed.
In a further technical scheme, the total control law of the system is as follows:
Figure BDA0002182812200000032
wherein, beta1,β2,β3Is the observer gain to be designed.
The embodiment of the specification provides a mechanical arm tracking control system based on a nonlinear extended state observer, which comprises a controller, wherein the controller issues a command to an actuator through a network, and the actuator controls the action of an industrial mechanical arm according to a control law;
and the action signals of the industrial mechanical arm are simultaneously collected by the actuator and the running state is fed back to the controller through the network.
The embodiment of the present specification provides a mechanical arm tracking control method based on a nonlinear extended state observer, including:
establishing a mathematical model of an industrial mechanical arm control system containing time delay and interference;
the unmodeled dynamic state and various disturbances of a system mathematical model are regarded as total disturbance, observation and compensation of the total disturbance are realized by establishing a nonlinear extended state observer, an industrial mechanical arm control system is decoupled into an integral series system, decoupling of the system is realized, and then each joint is controlled respectively.
According to the further technical scheme, when the mathematical model of the industrial mechanical arm control system containing time delay and interference is established, the dynamic model of the joint mechanical arm system is described by considering the influence of unmodeled dynamic state, external bounded interference and friction of the system.
Compared with the prior art, the beneficial effect of this disclosure is:
aiming at the problems of time delay, perturbation of model parameters and tracking control of the industrial mechanical arm under the influence of external bounded disturbance, the method adopts an active disturbance rejection control method to design a nonlinear tracking differentiator, a nonlinear state observer and a controller so as to improve the tracking control performance and the disturbance rejection capability of the system.
The nonlinear active disturbance rejection control method has the advantages that disturbance can be suppressed, and rapid convergence of system tracking errors can be achieved. The rapidity and the reliability of the system are improved. The controller is designed by utilizing the nonlinear state error feedback control rate, and the active disturbance rejection capability of the controller can be realized to a certain degree.
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The accompanying drawings, which are included to provide a further understanding of the disclosure, illustrate embodiments of the disclosure and together with the description serve to explain the disclosure and are not to limit the disclosure.
FIG. 1 is a block diagram of a network control system according to an exemplary embodiment of the present disclosure;
FIG. 2 is a block diagram of an industrial robot linear active disturbance rejection tracking control according to an exemplary embodiment of the present disclosure;
FIG. 3 is a plot comparing NLESO and LESO error curves for an example embodiment of the present disclosure;
FIG. 4 is a schematic diagram of tracking a trajectory using LESO in an example embodiment of the present disclosure;
FIG. 5 is a schematic diagram of tracking traces using NLESO in accordance with an exemplary embodiment of the present disclosure;
FIG. 6 is a schematic diagram of tracking error using LESO in an example embodiment of the present disclosure;
fig. 7 is a graphical illustration of tracking error using NLESO in an example embodiment of the disclosure.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present disclosure. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
Example of implementation 1
The embodiment discloses a mechanical arm tracking control method based on a nonlinear extended state observer, which comprises the following steps:
(1) establishing a mathematical model of an industrial mechanical arm control system containing time delay and interference
(1.1) for the joint mechanical arm system, considering the unmodeled dynamic state of the system, the influence of external bounded interference and friction force, the dynamic model can be described as the following equation:
in the formula (I), the compound is shown in the specification,
Figure BDA0002182812200000052
respectively representing the position vector, the velocity vector and the acceleration vector of each joint angle of the mechanical arm, wherein R represents a real number domain; m (theta) belongs to Rn×nA positive definite symmetric inertia matrix of the system;
Figure BDA0002182812200000053
is the centrifugal and coriolis force matrices of the system; g (theta) ∈ RnIs a gravity term matrix acting on the mechanical arm joint; u is an element of RnThe input vector of the model represents the driving moment acting on the joint of the mechanical arm; d (t) ε RnRepresenting a perturbation, t is time, and n is any positive integer.
Definition of
Figure 1
The system dynamics model can be written as:
in the formula (I), the compound is shown in the specification,
Figure BDA0002182812200000061
as can be seen from the equation (1.2) and (2), the model of the ith joint of the industrial robot arm is:
Figure BDA0002182812200000062
in the formula (I), the compound is shown in the specification,
Figure BDA0002182812200000063
b is a compensation coefficient; u. ofi,ujControl signals representing the i-th and j-th joints, respectively, qiIs the ith value in Q.
Referring to fig. 1, in the structure of a delay network control system, a controller, a sensor and an actuator are connected through a network, and the bandwidth constraint and the time division multiplexing mechanism of the network lead to the inevitable time delay of network control system information in the transmission process, and tau is used1Representing the time delay, τ, of transmission of network information between the controller and the actuator2The network information transmission time delay between the sensor and the controller is represented, and the total system time delay can be represented as tau-tau1+τ2；
Assuming that a sensor in the system is in a time driving mode and a sampling period is T, a controller and an actuator both adopt an event driving mode, namely when new data arrives at a node, relevant operations are immediately executed to consider the situation that the network has uncertain time delay and short time delay which changes in one sampling period, and the time sequence of a control signal of the network control system is as follows:
in the formula, tkAt the kth sampling instant, T is the sampling period: u. ofi(k-1)A control signal of the ith joint of the industrial mechanical arm at the moment of k-1 is obtained; u. ofikThe control signal of the ith joint of the industrial mechanical arm at the time k is recorded
Figure BDA0002182812200000065
The discretization model of system equation (3) including the network-induced delay can be expressed as:
Figure BDA0002182812200000071
to facilitate the design of the extended state observer, equation (5) can be rewritten as:
Figure BDA0002182812200000072
in the formula (I), the compound is shown in the specification,which may be seen as a total disturbance of the industrial robot system.
(1.3) and the formula (6) show that the industrial mechanical arm control system belongs to a strong nonlinear and strong coupling system, the conventional control method hardly meets the high-performance control requirement, the active disturbance rejection technology does not depend on a system model and is only related to the order of the system, unmodeled dynamic state and various disturbances of the system can be regarded as sum disturbance, observation and compensation of the sum disturbance of the system can be realized by designing an extended state observer, the industrial mechanical arm control system is decoupled into an integral series system, decoupling of the system is realized, and then each joint can be controlled respectively.
Example II
Referring to fig. 2, the controller used in this embodiment is composed of a nonlinear tracking differentiator, a nonlinear extended state observer, and a nonlinear state error feedback control law.
(1) Firstly, the input signal with noise is passed through the non-linear tracking differentiator to obtain high-quality output signal xid1And its differential signal xid2。
(2) Estimating the state z of the industrial mechanical arm according to the output y and the input signal u of the industrial mechanical armi1， zi2And total disturbance z acting on the industrial roboti3。
(3) Nonlinear state error feedback control rate. The state error of the system is e1＝xid1-zi1， e2＝xid2-zi2The error feedback rate is based on the error e1，e2To determine a control law u for controlling a pure integrator tandem type object0。
(4) For error feedback control quantity u0Using disturbance estimate z3Determines the final control quantity.
Figure BDA0002182812200000081
Or
Figure BDA0002182812200000082
Here, the parameter b0The compensation factor is used as an adjustable parameter for determining the strength of compensation.
Designing a nonlinear active disturbance rejection controller:
(2.1), designing a nonlinear tracking differentiator:
by designing the tracking differentiator, on one hand, a high-quality input signal and a differentiated signal thereof can be extracted, on the other hand, the contradiction between rapidity and overshoot in conventional industrial control can be solved, the violent change of a control quantity is avoided, and the anti-interference capability of the controller is improved.
The tracking differentiator for the ith joint of the system may be designed as:
Figure BDA0002182812200000083
in the formula (7), 0<α<1，xir(k) Desired position signal, x, for the i-th joint angleid1(k),xid2(k) Respectively a tracking signal of the expected position signal at the time k and a differential signal thereof; r being a tracking differentiatorA fast factor; t is the step length of the tracking differentiator;
(2.2) design of nonlinear extended State observer
In practical application, because the system delay and the disturbance are uncertainty factors and cannot be accurately obtained, but the estimated value can be obtained by designing the extended state observer, so that the sum disturbance term in the formula (6) is extended to a new state, and then the sum disturbance term in the formula (6) is obtained
Figure BDA0002182812200000085
Expand to a new state x13Then the equivalent system of equation (6) can be expressed as:
Figure BDA0002182812200000084
in the formula, h (k) ═ (ψ (k +1) - ψ (k))/T; x is the number ofi3(k +1) is in an expanded state xi3(k) The value of the next time instant; according to the design theory of the extended state observer, the nonlinear extended state observer of the mechanical arm system formula is designed as follows:
Figure BDA0002182812200000091
wherein, beta1,β2To set the parameters, δ is the interval length of the linear segment, xiRepresents the system state, ziRepresenting observer state, b is a constant, uikFor control rate, t represents time, and the fal function is defined as:
Figure BDA0002182812200000092
wherein, 0<α<1,β1,β2To set the parameters.
Discretizing equation (9) can be represented as:
in the formula, epsiloni1(k) Observing error of the system state variable by the linear extended state observer; z is a radical ofi1(k),zi2(k),zi3(k) Are respectively xi1(k),xi2(k),xi3(k) Estimate of beta1,β2,β3Is the observer gain to be designed.
(2.3) design of nonlinear state error feedback control law
Nonlinear state error feedback control law forms feedback control component u using the difference between the output of a tracking differentiator and the output of an extended state observer0Comprises the following steps:
u0＝β1ei1+β2fe+β3fe1(12)
in order to overcome the adverse effect of system time delay and disturbance factors on the tracking control performance, the estimated value of the nonlinear extended state observer needs to be added into a system control law in a feedforward compensation mode, so that the total control law of the system is obtained as follows:
Figure BDA0002182812200000101
wherein, beta1,β2,β3Is the observer gain to be designed.
In order to more intuitively illustrate the technical solutions and technical advantages of the present disclosure, the technical solutions of the present disclosure are further described below with reference to specific embodiments, with reference to fig. 3 to 5.
The two-joint mechanical arm is taken as a verification object, and the dynamic model is as follows:
Figure BDA0002182812200000102
in equation (14), the positive fixed inertia coefficient is:
Figure BDA0002182812200000103
the centrifugal and coriolis forces are:
Figure BDA0002182812200000104
the gravity term vector is: g (theta) ═ m1+m2)l1gcos(θ2)+m2l2gcos(θ1+θ2)。
To better illustrate the effectiveness of the control method provided by the present disclosure, the nonlinear active disturbance rejection control method provided by the present disclosure is compared with the linear active disturbance rejection control method in a simulation experiment.
The object parameter is set to m1＝1kg，m2＝0.6kg，l1＝0.61m，l2＝0.4m。
Desired trajectory is set to xir＝2sint-0.45cos(2πt)+1,t∈[0,40]。
The fast factor r of the tracking differentiator is 50.
The observer parameters were: beta is a1＝100,β2＝300,β3＝1000。
Assume that the system disturbance signal is taken as:
Figure BDA0002182812200000105
the experimental results of the observation error curves of the nonlinear extended state observer and the linear extended state observer are as follows
As shown in fig. 3, it is obvious from the figure that the nonlinear extended state observer has obvious advantages and small error.
Fig. 4 and 5 are schematic diagrams of a robot arm tracking trajectory using the LESO control method and the NLESO control method, respectively. The nonlinear active disturbance rejection control method can well realize the control of the industrial mechanical arm system with disturbance and time delay, and after the disturbance and the time delay in the system are estimated and compensated on line through the algorithm, the system output can stably track the expected position signal.
Although the tracking control of the system can be completed by utilizing the linear active disturbance rejection control method, and the effect of suppressing disturbance can also be achieved, the tracking effect and the disturbance suppression effect are not good compared with the nonlinear active disturbance rejection control.
Fig. 6 and 7 are schematic diagrams of the robot arm tracking trajectory error by using the LESO control method and the NLESO control method, respectively. Therefore, the response time of the nonlinear active disturbance rejection control method is short, and the steady-state error is small.
The present disclosure is directed to the problem of industrial robot tracking control with time delay, external disturbance.
The nonlinear active disturbance rejection control method has the advantages that disturbance can be suppressed, and rapid convergence of system tracking errors can be achieved. The rapidity and the reliability of the system are improved. The controller is designed by utilizing the nonlinear state error feedback control rate, and the active disturbance rejection capability of the controller can be realized to a certain degree.
The method adopts an active disturbance rejection control method to design a nonlinear tracking differentiator, a nonlinear state observer and a nonlinear state error feedback control law so as to improve the tracking control performance and the disturbance rejection capability of the system. The nonlinear active disturbance rejection control method is utilized to enable the system output to stably track the expected position signal, and the response time is short, and the steady-state error is small.
Example III
The embodiment of the specification provides a mechanical arm tracking control system based on a nonlinear active disturbance rejection controller. Analog signals such as temperature, pressure and the like are obtained by a sensor of the industrial mechanical arm, the obtained analog signals are uploaded to a network, and then a reference signal is transferred to the nonlinear active disturbance rejection controller by the network. The controller issues commands to the actuator through a network so as to control the action of the industrial mechanical arm;
and the action signals of the industrial mechanical arm are simultaneously collected by the actuator and the running state is fed back to the controller through the network.
It is to be understood that throughout the description of the present specification, reference to the term "one embodiment", "another embodiment", "other embodiments", or "first through nth embodiments", etc., is intended to mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present disclosure. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, or materials described may be combined in any suitable manner in any one or more embodiments or examples.
The above description is only a preferred embodiment of the present disclosure and is not intended to limit the present disclosure, and various modifications and changes may be made to the present disclosure by those skilled in the art. Any modification, equivalent replacement, improvement and the like made within the spirit and principle of the present disclosure should be included in the protection scope of the present disclosure.

Claims (7)

1. A mechanical arm tracking controller based on a nonlinear extended state observer is characterized in that,
the method comprises the following steps: a nonlinear tracking differentiator, a nonlinear extended state observer and a nonlinear state error feedback control law;
the nonlinear tracking differentiator tracks the expected position signal of the joint angle of the mechanical arm to obtain a tracking signal of the expected position signal and a differential signal thereof and outputs the tracking signal and the differential signal;
the nonlinear extended state observer is used for obtaining an estimated value with system time delay and disturbance as uncertainty factors;
the nonlinear state error feedback control law utilizes the difference between the output of the nonlinear tracking differentiator and the output of the nonlinear extended state observer to form a feedback control component, and the estimated value of the nonlinear extended state observer is added into the system control law in a feedforward compensation mode to obtain the total control law of the system.
2. The non-linear extended state observer-based robotic arm tracking controller of claim 1, wherein the non-linear tracking differentiator:
Figure FDA0002182812190000011
in the formula (7), xir(k) Desired position signal, x, for the i-th joint angleid1(k),xid2(k) Are respectively the desired positionTracking signals of the signals at the time k and differential signals of the signals; r is a fast factor of the tracking differentiator; t is the step size of the tracking differentiator.
3. The non-linear extended state observer-based robotic arm tracking controller of claim 1, wherein the non-linear tracking differentiator is:
Figure FDA0002182812190000012
wherein the fal function is defined as:
Figure FDA0002182812190000021
discretizing equation (9), which is expressed as:
Figure FDA0002182812190000022
in the formula, epsiloni1(k) Observing error of the system state variable by the linear extended state observer;
zi1(k),zi2(k),zi3(k) are respectively xi1(k),xi2(k),xi3(k) Estimate of beta1,β2,β3Is the observer gain to be designed.
4. The non-linear extended state observer-based robotic arm tracking controller of claim 1, wherein the overall control law of the system is:
Figure FDA0002182812190000023
wherein, beta1,β2,β3Is the observer gain to be designed.
5. A mechanical arm tracking control system based on a nonlinear extended state observer is characterized by comprising the controller of any one of claims 1 to 4, wherein the controller issues commands to an actuator through a network, and the actuator controls the action of an industrial mechanical arm according to a control law;
and the action signals of the industrial mechanical arm are simultaneously collected by the actuator and the running state is fed back to the controller through the network.
6. The mechanical arm tracking control method based on the nonlinear extended state observer is characterized by comprising the following steps of:
establishing a mathematical model of an industrial mechanical arm control system containing time delay and interference;
the unmodeled dynamic state and various disturbances of a system mathematical model are regarded as total disturbance, the controller of any one of claims 1 to 5 is designed to establish a nonlinear extended state observer to realize observation and compensation of the total disturbance, an industrial mechanical arm control system is decoupled into an integral series system to realize decoupling of the system, and then each joint is controlled respectively.
7. The method for tracking and controlling the mechanical arm based on the nonlinear extended state observer as claimed in claim 6, wherein when a mathematical model of an industrial mechanical arm control system containing time delay and interference is established, a dynamic model of the joint mechanical arm system is described by considering the unmodeled dynamic state of the system, the external bounded interference and the influence of friction.
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