CN118456419B - Self-adaptive finite time variable force tracking control method for mechanical arm - Google Patents

Abstract

A mechanical arm self-adaptive finite time variable force tracking control method relates to the technical field of mechanical arm control. According to the constructed interaction model between the mechanical arm end effector and the environment, a dynamic equation of force tracking error is obtained, then according to a position-based force tracking control thought, an adaptive control algorithm is designed on the basis of the dynamic equation to estimate unknown environment parameters, a control law is designed to enable the force tracking error to be converged, and finally limited time convergence is proved through a Lyapunov limited time stability theory. According to the force tracking control thought based on the position, the design framework of the traditional impedance control is jumped out, the control law is designed from the dynamic equation of the force tracking error, the estimation of the rigidity of an unknown environment is realized through designing the self-adaptive law, and on the basis, the tracking of the time-varying expected force with higher precision is realized through designing the finite time control law.

Description

An adaptive finite-time variable force tracking control method for a robotic arm

Technical Field

The invention relates to the technical field of mechanical arm control, and in particular to an adaptive finite-time variable force tracking control method for a mechanical arm.

Background Art

With the continuous development of science and technology, robotic arms can be used in a variety of fields to complete various complex tasks, such as grinding, polishing, cutting, etc., and effective control of the interaction force between the end effector of the robotic arm and the environment is the basis for completing the task.

The problem of controlling the interaction force between the end effector and the environment is essentially a problem of force tracking control. For the problem of force tracking control, the most classic method is position-based impedance control. The force controller of this control method can calculate the reference position of the end effector of the robot arm according to the force tracking error signal. The robot arm then tracks the reference position through the position loop, thereby achieving tracking control of the interaction force. Traditional impedance control requires accurate prior information about the environment, but in practical applications, this prior information is difficult to obtain in advance, which causes traditional impedance control to have large errors in some cases. On the other hand, due to the limitations of the controller form, traditional impedance control has a large tracking error for time-varying desired forces.

In order to overcome the limitations of traditional impedance control, researchers in the field have been working to seek innovative force tracking control solutions to address the above-mentioned drawbacks of impedance control.

Summary of the invention

In order to overcome the limitations of traditional impedance control, the present invention provides an adaptive finite-time variable force tracking control method for a robotic arm, which breaks away from the design framework of traditional impedance control and designs the control law starting from the dynamic equation of the force tracking error. By designing the adaptive law, the estimation of the unknown environmental stiffness is realized. On this basis, by designing the finite-time control law, the time-varying expected force is tracked with high precision.

To achieve the above object, the present invention adopts the following technical solution: a method for adaptive finite-time variable force tracking control of a robot arm, comprising the following steps:

Step 1: Model the interaction model between the end effector of the robot and the environment to obtain the force tracking control system block diagram

1.1. Assuming that the contact force generated by the environment is related to the distance that the end effector penetrates into the environment, we have:

Where, _Fe is the contact force generated between the end effector and the environment, _ke is the environment stiffness, X is the position of the end effector, and _Xe is the environment position;

The force tracking control system block diagram is obtained by using the position-based force tracking control method. The control idea is: the outer loop force controller is based on the desired force _Fd and its derivative with respect to time. And the feedback contact force information F _e , real-time calculation of the target reference position X _c of the end effector, the inner loop position controller controls the manipulator system to track the target reference position X _c , and then realizes the tracking of the desired force F _d ;

1.2. After the end effector contacts the environment, assuming that the position controller of the inner loop can reach X _c =X, the expression of the contact force is as follows:

Step 2: Design force tracking control law and adaptive law according to the control block diagram

2.1. After the end effector contacts the environment, let the force tracking error be ΔF, then:

ΔF＝F _d -F _e =F _{d -ke} (X _c -X _e )

set up The dynamic equation of the force tracking error is as follows:

2.2. Adaptive control method is introduced to estimate the environment stiffness _ke . Finite time control strategy is adopted to estimate the inverse of _ke . Let the inverse of _ke be _Be . Then the dynamic equation of force tracking error becomes:

2.21. Design the Lyapunov function V ₁ as follows:

Let the Lyapunov function V ₁ take the first-order derivative with respect to time:

Then the control law is designed as follows:

In the formula, is the estimated value of _Be , _k1 >0, _k2 >0, 0<β<1 are designable constants, ^sigβ (ΔF)=| ^ΔFβ ·sign(ΔF);

Substituting the control law into

In the formula, is the estimated error of the inverse of the ambient stiffness;

2.22. Design the Lyapunov function V ₂ as follows:

In the formula, Γ＞0 is a designable constant;

Let the Lyapunov function V ₂ take its first-order derivative with respect to time:

Then the adaptive law is designed as follows:

In the formula, _ka >0 is a designable constant;

Step 3: Based on Lyapunov stability theory, prove the finite time convergence of tracking error and estimation error

Substituting the adaptive law into

In the formula,

ΔF and The time T required to converge to the residual set Ω will have an upper bound T _max :

The residual set Ω is expressed as follows:

In the formula, 0＜v＜1.

Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention introduces adaptive control and finite-time stability theory into the force tracking control problem of the robotic arm, and realizes the estimation of the unknown environmental stiffness by designing the adaptive law. On this basis, by designing the finite-time control law, it is realized that for any expected force (including constant force and time-varying force), the force tracking error converges to a smaller value within a finite time, thereby realizing high-precision tracking of the time-varying expected force. The present invention is suitable for variable force tracking tasks with high requirements for control accuracy, and provides reliable technical support for the successful execution of high-precision force tracking tasks of the robotic arm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a block diagram of a force tracking control system in a control method of the present invention;

FIG2 is a simplified control block diagram of the outer loop in the control method of the present invention;

FIG3 is a schematic diagram of a two-degree-of-freedom robotic arm system in an embodiment;

4 is a schematic diagram of a force tracking control process of a two-degree-of-freedom robotic arm system in an embodiment;

FIG5 is a contact force variation curve of the three algorithms in the embodiment when the desired force is a sinusoidal signal;

FIG6 is a contact force error variation curve of the three algorithms in the embodiment when the desired force is a sinusoidal signal;

FIG. 7 is a diagram of the algorithm of the present invention when the desired force is a sinusoidal signal. The change curve of

FIG8 is a contact force variation curve of the three algorithms in the embodiment when the expected force is a ramp signal;

FIG9 is a contact force error variation curve of the three algorithms in the embodiment when the desired force is a ramp signal;

FIG. 10 is an example of an algorithm of the present invention for the desired force when the ramp signal is used. The change curve of

FIG11 is a contact force variation curve when the expected force of the three algorithms in the embodiment suddenly changes;

FIG12 is a contact force variation curve when the environmental stiffness of the three algorithms in the embodiment suddenly changes;

FIG. 13 is a contact force variation curve when the environmental positions of the three algorithms in the embodiment suddenly change.

DETAILED DESCRIPTION

The technical solution of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

As shown in FIG. 1 and FIG. 2, a method for adaptive finite-time variable force tracking control of a robot arm includes the following steps:

Step 1: Model the interaction model between the end effector of the robot and the environment to obtain the force tracking control system block diagram

1.1. In general, assuming that the contact force generated by the environment is similar to the spring model, that is, the contact force generated by the environment is related to the distance that the end effector penetrates into the environment, then:

Where _Fe is the contact force between the end effector and the environment, _ke is the environment stiffness, X is the position of the end effector, and _Xe is the environment position.

It can be seen from the above expression that the contact force exists only when the end effector penetrates deeply into the environment. Therefore, the control of the contact force is essentially to control the distance that the end effector penetrates deeply into the environment. Therefore, the present invention adopts a position-based force tracking control method. The obtained force tracking control system block diagram is shown in Figure 1. The control idea is: the outer loop force controller is based on the information of the desired force (the desired force _Fd and its derivative with respect to time) ) and the feedback contact force information F _e , calculate the target reference position X _c of the end effector in real time, and the inner loop position controller controls the manipulator system to track the target reference position X _c , thereby realizing the tracking of the desired force F _d ;

1.2. After the end effector contacts the environment, the running speed of the robot arm is generally very slow. Here, it is assumed that the position controller of the inner loop has a good tracking effect and can achieve X _c =X. Therefore, the control block diagram of the outer loop can be simplified. Combined with Figure 2, the expression of the contact force can be written as follows:

Step 2: Design force tracking control law and adaptive law according to the control block diagram

2.1. After the end effector contacts the environment, let the force tracking error be ΔF, then:

ΔF＝F _d -F _e =F _{d -ke} (X _c -X _e )

Here it is assumed that the contact environment is stationary, that is, Taking the first-order derivative of the above equation with respect to time, the dynamic equation of the force tracking error is as follows:

Among them, the target reference position X _c is the control law that needs to be designed;

2.2. In order to solve the problem of unknown environment stiffness _ke , an adaptive control method is introduced to estimate the environment stiffness. At the same time, a finite time control strategy is adopted to improve the convergence performance. Since the environment stiffness _ke is generally large, it is difficult to select the initial value when adaptively estimating it. Therefore, the inverse of _ke can be estimated instead. Assuming that the inverse of _ke is _Be , Then the dynamic equation of force tracking error can be changed to:

2.21. According to the dynamic equation of the force tracking error, the control law is designed through the following steps. The Lyapunov function V ₁ is designed as follows:

Let the Lyapunov function V ₁ take the first-order derivative with respect to time, and we can get:

Then the control law is designed as follows:

In the formula, is an estimated value of _Be , _k1 >0, _k2 >0, 0<β<1 are designable constants, ^sigβ (ΔF)=| ^ΔFβ ·sign(ΔF).

Substituting the control law into We can get:

In the formula, is the estimated error of the inverse of the ambient stiffness;

2.22. Based on the design of the control law, the adaptive law is designed to estimate _Be through the following steps. The Lyapunov function _V2 is designed as follows:

Wherein, Γ＞0 is a designable constant.

Taking the first-order derivative of the Lyapunov function V ₂ with respect to time, we can obtain:

Then the adaptive law is designed as follows:

In the formula, _ka >0 is a designable constant;

Step 3: Based on Lyapunov stability theory, prove the finite time convergence of tracking error and estimation error

Substituting the adaptive law into We can get:

In the formula,

According to the The expression can show that the tracking error of the force ΔF and the estimation error of the inverse of the environmental stiffness are The time T required to converge to the residual set Ω near 0 will have an upper bound T _max :

The residual set Ω is expressed as follows:

In the formula, 0＜v＜1. So far, the proof of finite time convergence is completed.

In summary, the control method of the present invention is based on the interaction model between the constructed robot end effector and the environment. The unknown environmental stiffness is estimated by designing an adaptive law, and the control law is designed on this basis to achieve the tracking of the time-varying desired force. Finally, based on Lyapunov's finite-time stability theory, it is proved that the estimation error of the environmental stiffness and the tracking error of the time-varying force can converge to a small neighborhood of the origin within a finite time.

Example

Considering the two-degree-of-freedom manipulator system shown in FIG. 3 , the control process is shown in FIG. 4 , and the goal is to control the contact between the end effector of the manipulator and the environment, and to enable the contact force _Fe to track the time-varying desired force _Fd .

In order to verify the effectiveness of the algorithm of the present invention, a force tracking comparison test is conducted on a two-degree-of-freedom manipulator system for the proposed method AFTC, traditional impedance control CIC, and adaptive hybrid impedance control AIC. The main test content is the tracking ability of the three algorithms for time-varying expected force and the robustness of the algorithms when the contact environment changes. In the simulation, the common parameters of the three algorithms will remain consistent.

Simulation 1: Time-varying desired force tracking control simulation

In the simulation, it is assumed that the environment contacted by the end effector is a plane, the environment stiffness is set to _ke = 1000N/m, and the environment position is set to _Xe = 1.1m. For the proposed method, the selected parameters are shown in Table 1 below:

Table 1 Selected parameters

The simulation time is 15s, and the simulation step is 0.001s. The tracked time-varying expected forces are the sinusoidal expected force F _d = (10 + sin (t)) N and the ramp expected force F _d = (10 + 0.4t) N. The simulation results are shown in conjunction with Figures 5 to 10. Among them, Figures 5 and 6 show the tracking of the three algorithms for the sinusoidal expected force and the error changes, and Figures 8 and 9 show the tracking of the three algorithms for the ramp expected force and the error changes. It can be seen from the figure that when the expected force is a time-varying force, both the CIC and AIC algorithms will have tracking errors to varying degrees, and the AFTC has better force tracking performance, which proves the effectiveness of the controller designed by the present invention. At the same time, compared with the other two algorithms, the AFTC has a faster response speed. Figures 7 and 10 show the estimation of the inverse of the environmental stiffness by AFTC when tracking the sinusoidal expected force and the ramp expected force. It can be seen from the figure that in both cases, the estimated value is close to the true value and the error is small, which shows that the adaptive law can estimate the unknown parameters more effectively. In order to further compare the tracking performance of the three algorithms for time-varying expected force, the three performance indicators of mean square error MSE, root mean square error RMSE, and mean absolute error MAE from 3 to 10 seconds are used for quantitative analysis. The comparison results are shown in Table 2 below:

Table 2 Comparison results

The data in the table prove that the MSE, RMSE and MAE performance indicators of AFTC are better than those of CIC and AIC. Taking the RMSE indicator of sinusoidal expected force tracking as an example, the tracking accuracy of AFTC is improved by 95.4% compared with CIC and 93.5% compared with AIC, which proves the effectiveness of the AFTC method.

Simulation 2: Control robustness test simulation

In the robustness test, three test environments are provided, namely: sudden change in expected constant force (case A), sudden change in environmental stiffness (case B), and sudden change in environmental position (case C). The test parameters of the three environments are selected as shown in Table 3 below:

Table 3 Test parameters of three environments

In the simulation, it is assumed that the environment contacted by the end effector is a plane, the environment stiffness is set to _ke = 1000N/m, and the environment position is set to _Xe = 1.1m. For the proposed method, the selected parameters are shown in Table 1. The simulation time is 30s and the simulation step is 0.001s. The simulation results are shown in Figures 11 to 13. Among them, as shown in Figure 11, when the expected force changes suddenly, the three algorithms can all track the expected force, but AIC has a large overshoot. In contrast, AFTC not only has a small overshoot, but also has a fast response speed. Similarly, as shown in Figures 12 and 13, when the environment stiffness or environment position changes suddenly, the convergence time of CIC and AIC is longer, while AFTC has the fastest response speed and almost no overshoot. Through simulation experiments in these three environments, it can be proved that AFTC has good robustness in sudden change environments.

It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other forms of assembly without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered in all respects as exemplary and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations within the meaning and range of equivalents of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims (1)

1. A mechanical arm self-adaptive finite time variable force tracking control method is characterized in that: the method comprises the following steps:

Step one: modeling is conducted on an interaction model between the mechanical arm end effector and the environment to obtain a force tracking control system block diagram

1.1, The contact force generated by the setting environment is related to the distance of the end effector going into the environment, and then:

Wherein F _e is the contact force generated between the end effector and the environment, k _e is the environmental stiffness, X is the end effector position, and X _e is the environmental position;

The method adopts a force tracking control method based on position, and the obtained force tracking control system block diagram has the following control thought: the force controller of the outer ring is based on the desired force F _d and its derivative with respect to time And the feedback contact force information F _e calculates a target reference position X _c of the end effector in real time, and a position controller of the inner ring controls the mechanical arm system to track the target reference position X _c, so that the tracking of the expected force F _d is realized;

1.2, after the end effector is contacted with the environment, if the position controller of the inner ring can reach X _c =x, the expression of the contact force is as follows:

Step two: designing force tracking control law and adaptive law according to control block diagram

2.1, After the end effector is in contact with the environment, let the tracking error of the force be Δf, there are:

ΔF＝F_d-F_e＝F_d-k_e(X_c-X_e)

Is provided with The dynamic equation for the tracking error of the force is obtained as follows:

2.2, an adaptive control method is introduced to estimate the environmental rigidity k _e, a finite time control strategy is adopted to estimate the reciprocal of k _e, and if the reciprocal of k _e is set as B _e, the dynamic equation of the tracking error of the force becomes:

2.21, design lyapunov function V ₁ as follows:

let Lyapunov function V ₁ solve for the first derivative over time:

the design control law is as follows:

In the formula, For the estimated value of B _e, k ₁＞0,k₂ > 0,0 < beta < 1 is a programmable constant, sig ^β(ΔF)＝|ΔF|^β. Sign (ΔF);

Substituting control law into

In the formula,An estimation error that is the inverse of the ambient stiffness;

2.22, design lyapunov function V ₂ as follows:

wherein Γ > 0 is a programmable constant;

let Lyapunov function V ₂ solve for the first derivative over time:

the adaptive law is further designed as follows:

Wherein k _a > 0 is a programmable constant;

step three: based on Lyapunov stability theory, the limited time convergence of tracking error and estimation error is proved

Substituting the adaptive law into

In the formula,

ΔF andThe time T required to converge to the residual set Ω will have an upper bound T _max:

The residue set Ω expression is as follows:

In the formula, v is more than 0 and less than 1.