An Adaptive Non-Singular Terminal Sliding Mode Control Method for Robotic Arms Based on Fixed-Time Observers

Abstract

This invention relates to an adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer, belonging to the field of robot control technology. The invention includes: constructing a fixed-time extended state observer (FxESO) with a double power structure by establishing a dynamic model of the robotic arm that includes parameter uncertainties and external disturbances, for real-time and accurate capture of the comprehensive disturbances experienced by the system; designing a non-singular terminal sliding surface and combining it with an adaptive control law, incorporating a zero-crossing buffer constant and a smoothing function in the control law to address gain abrupt changes and torque chattering problems when the velocity crosses zero; and using an update algorithm with a leakage term to adjust the robust gain online to compensate for the residual error of the observer. This invention enables the robotic arm to quickly return to zero error even under inaccurate modeling and unknown disturbances, ensuring a smooth and continuous control process, and significantly improving the tracking accuracy and response speed of the robotic arm in complex working environments.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to an adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer. Background Technology

[0002] With the continuous improvement of industrial automation, robotic arms are increasingly widely used in precision assembly, medical surgery, and collaborative processing. Robotic arm systems inherently possess highly nonlinear, strongly coupled, and time-varying dynamic characteristics. In actual operation, they are often subject to interference from model parameter variations, joint friction, and complex external loads; these factors are collectively referred to as lumped disturbances in the system. To ensure operational accuracy, a trajectory tracking controller with extremely strong anti-interference capabilities must be designed.

[0003] Sliding mode control (SMC) has attracted widespread attention due to its strong robustness to disturbances. However, traditional linear sliding mode can only achieve asymptotic convergence of the error, with a slow convergence speed. To shorten the convergence time, terminal sliding mode control (TSMC) has been proposed and applied to robotic arm control, but when the system state is close to the equilibrium point, this method is prone to the singularity problem of infinite control input. Although non-singular terminal sliding mode avoids this risk to some extent, its performance is highly dependent on the prediction of the upper limit of disturbance when facing dynamic environments with uncertainties. If the control gain is blindly increased to counteract unknown disturbances, it will cause severe high-frequency torque chattering, which will not only affect tracking accuracy but may also accelerate the physical wear and tear of the motor and reducer.

[0004] To compensate for disturbances in real time, extended state observers (ESOs) are a good choice. However, conventional observers can usually only guarantee asymptotic or finite-time convergence of the observation error, which is difficult to meet the requirement of "fixed-time convergence" in modern high-dynamic operations, where the convergence time is greatly affected by the initial state. Therefore, how to design a control scheme that can overcome control singularities and torque chattering, and achieve fixed-time tracking without needing to know the upper limit of the disturbance, has become a key technical challenge that urgently needs to be addressed in the field of robotic arm control. Summary of the Invention

[0005] This invention proposes an adaptive non-singular terminal sliding mode control method for robotic arms based on a fixed-time observer. The core objective is to address the problem of decreased tracking accuracy in industrial robotic arms under complex environments due to parameter variations, joint friction, and unknown loads. This invention addresses technical bottlenecks in traditional control algorithms, such as convergence time dependence on initial deviation, singularity traps at velocity zero crossings, and high-frequency torque chattering caused by the sign function. By introducing an observation mechanism with homogeneous characteristics and a non-singular adaptive compensation control law, this invention ensures that the system achieves global fixed-time rapid convergence of trajectory tracking errors without requiring prior knowledge of the upper limit of disturbance.

[0006] The technical solution adopted in this invention is: an adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer, comprising the following specific steps: Step 1: Establish a dynamic model of the rigid robotic arm and transform the parameter uncertainties, friction, and external environmental disturbances contained in the model into lumped disturbances under the state-space expression. Step 2: Using the real-time state variables of the robotic arm and the feedback system control torque, construct a double power fixed-time extended state observer to accurately estimate the lumped disturbance mentioned in Step 1 in real time and within a fixed time. Step 3: Design a non-singular terminal sliding mode surface based on the tracking error of the robotic arm, and combine it with the lumped disturbance estimate output in Step 2 to construct an adaptive non-singular terminal sliding mode control law, which introduces a zero-crossing buffer constant and a smoothing function to suppress torque chattering and singularity. Step 4: Design an adaptive gain update law with a leakage term, adjust the robust gain in real time according to the feedback of the sliding surface variables, compensate for the residual estimation error of the observer in Step 2, and then calculate and output the final joint control torque. Step 5: Apply the final joint control torque output from Step 4 to the robotic arm actuator to achieve convergence of the robotic arm's motion trajectory to the desired trajectory within a fixed time. At the same time, feed this control torque back to Step 2 as a known control input to participate in the disturbance estimation of the next control cycle, thereby realizing continuous closed-loop control of the system.

[0007] Specifically, the process of establishing the dynamic model of the rigid robotic arm described in Step 1 is as follows:

[0008] in, , , These represent the joint angular position, angular velocity, and angular acceleration vectors of the robotic arm, respectively. Here is the system inertia matrix. For the centripetal force and Coriolis force matrix, It is the gravitational torque vector. The joint control torque output by the actuator. This refers to the external disturbance torque.

[0009] The nominal values ​​and uncertainties of the above dynamic model are separated, and the lumped disturbance term is... The synthesis incorporates modeling uncertainties and external perturbation moments, and assumes their derivatives. It is bounded, and the specific formula is as follows:

[0010] Define system state variables , , The dynamic model of the robotic arm is transformed into the following nonlinear system state-space expression:

[0011] in, For joint angle position variables, Its derivative, For joint angular velocity variables, Its derivative, For equivalent lumped disturbance variables Its derivative, For system control input gain, For the system control law, For aggregated disturbance The derivative of .

[0012] Specifically, the dynamic equations of the double-power fixed-time extended state observer FxESO described in Step 2 are as follows:

[0013] in, For observation error, For the observed values ​​of the state variables, For observer gain parameters, and These are the exponential terms that rapidly approach large errors and precisely lock onto small errors, respectively. .

[0014] To ensure that the observation error of the fixed-time extended state observer (FxESO) converges within a fixed time, its power parameter must strictly satisfy the homogeneity principle, specifically defined as: setting a small constant. The exponent for the fast approach term with large errors is: , , The index for the small error precision locking term is: , , Simultaneously, the observer gain parameter The value of makes the characteristic polynomial of the observer error dynamic system satisfy the Hurwitz stability criterion.

[0015] Specifically, in Step 3, the non-singular terminal sliding surface The specific design process for the relevant error variables is shown below: Define position tracking error Speed ​​tracking error ,in The desired angular position of the robotic arm joint. For the desired angular velocity; construct a non-singular terminal sliding surface. The formula is:

[0016] in Here are the parameters for the sliding surface, and these parameters satisfy: , and All are positive odd numbers, and their ratio satisfies This ensures the boundedness and non-singularity of the control quantity as the system state approaches the equilibrium point.

[0017] Specifically, the system control law solution formula for the adaptive non-singular terminal sliding mode control law described in Step 3 is as follows:

[0018] in, For the desired angular acceleration, For speed tracking error, For the approach law parameters, For adaptive robust gain.

[0019] At the same time, control law In subsequent simulations, in order to eliminate discontinuous sign functions The resulting torque chattering phenomenon is addressed using a continuous hyperbolic tangent function. Boundary layer replacement is performed, where This is a smoothing constant; it is used to solve the control law. middle The system crosses the velocity zero point (i.e. The gain surge singularity problem caused by this issue is addressed by introducing a zero-crossing buffer constant. Correct the singularity to This ensures the smooth and continuous control torque.

[0020] Specifically, the adaptive gain update law with leakage term described in Step 4 adjusts the adaptive robust gain in real time based on feedback from the sliding surface variables. The specific formula is as follows:

[0021] in, for The derivative of For adaptive learning rate, The leakage dissipation coefficient is used. This adaptive adjustment mechanism compensates for residual observation errors online, ultimately outputting control torque to drive each joint of the robotic arm, achieving high-precision, fixed-time tracking of the desired trajectory.

[0022] The analytical process for verifying the actual fixed-time stability of the closed-loop system is as follows: Define the adaptive estimation error. ,in It is a positive real constant containing an unknown upper limit of perturbation. Because... It is a constant and has an adaptive estimation error. derivative Choose the extended Lyapunov function:

[0023] For Lyapunov functions By differentiating and applying Young's inequality to the nonlinear control term, the Lyapunov function can be derived. derivative Simplified to:

[0024] make The above equation shows that when the system trajectory is outside the decision boundary (i.e. )hour, The result is strictly true. According to the theory of practical fixed-time stability, the tracking error of the closed-loop system can converge to the smallest neighborhood of the origin in a fixed time independent of the initial state, thus proving that the proposed method possesses practical fixed-time stability.

[0025] The beneficial effects of this invention are: it constructs a closed-loop control architecture that integrates a fixed-time extended state observer with a double-power structure and a non-singular terminal sliding surface. This mechanism enables the robotic arm to rapidly and accurately converge its trajectory tracking error to a minimal neighborhood of the origin within a rigorously proven fixed time when facing parameter uncertainties and strong external disturbances, greatly improving the system's transient response speed and time predictability. Relying on the extended state observer's active feedforward compensation for lumped disturbances, the control system can accurately cancel out unknown dynamics without prior information on the upper limit of disturbances.

[0026] This invention also addresses key pain points in practical engineering applications. By innovatively introducing a zero-crossing buffer constant in the control law design, it mitigates the gain surge and actuator torque chatter problems that are easily triggered when the system state crosses the zero velocity point. Combined with the boundary layer substitution of the sign function by the continuous hyperbolic tangent function, it effectively reduces the occurrence of high-frequency torque chattering in traditional sliding mode control. While ensuring high-precision trajectory tracking, it reduces the losses of the robotic arm joint motors. To enhance system reliability, this invention designs an adaptive robust gain dynamic adjustment mechanism with a leakage term. This mechanism can adaptively track unknown residual observation errors online and dynamically generate compensating control quantities; its built-in leakage dissipation coefficient can force the control gain to automatically and smoothly decrease when external disturbances weaken or disappear. This ensures the fixed-time stability and robust performance of the robotic arm control system throughout the entire cycle. Attached Figure Description

[0027] Figure 1 This is a flowchart of an adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer, according to the present invention. Figure 2 This is a comparison chart of the tracking trajectory curves of joint 1 under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 3 This is a comparison chart of the tracking trajectory curves of joint two under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 4 This is a comparison chart of the tracking trajectory error curves of joint one and joint two under the traditional SMC_ESO control method and the control method of the present invention (AFXNTSMC); Figure 5 This is a comparison chart of the speed tracking curves of joint 1 under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 6 This is a comparison chart of the speed tracking curves of joint 2 under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 7 A comparison chart of the speed tracking error curves of joint one and joint two under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 8 This is a comparison chart of the control input curves of joint 1 under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 9 This is a comparison chart of the control input curves of joint two under the traditional SMC-ESO control method and the control method of the present invention (AFXNTSMC); Figure 10 Adaptive parameters designed for the control method of this invention , Numerical variation curves at joint one and joint two, respectively. Detailed Implementation

[0028] The present invention will be further described below with reference to embodiments and accompanying drawings, but the scope of protection of the present invention is not limited to the described scope. The overall flow of a robotic arm control method based on a fixed-time observer and adaptive sliding mode according to the present invention is as follows: Figure 1 As shown, the specific steps are as follows: An adaptive nonsingular terminal sliding mode control method for a robotic arm based on a fixed-time observer includes the following specific steps: Step 1: Establish a dynamic model of the rigid robotic arm and transform the parameter uncertainties, friction, and external environmental disturbances contained in the model into lumped disturbances under the state-space expression. Step 2: Using the real-time state variables of the robotic arm and the feedback system control torque, construct a double power fixed-time extended state observer to accurately estimate the lumped disturbance mentioned in Step 1 in real time and within a fixed time. Step 3: Design a non-singular terminal sliding mode surface based on the tracking error of the robotic arm, and combine it with the lumped disturbance estimate output in Step 2 to construct an adaptive non-singular terminal sliding mode control law, which introduces a zero-crossing buffer constant and a smoothing function to suppress torque chattering and singularity. Step 4: Design an adaptive gain update law with a leakage term, adjust the robust gain in real time according to the feedback of the sliding surface variables, compensate for the residual estimation error of the observer in Step 2, and then calculate and output the final joint control torque. Step 5: Apply the final joint control torque output from Step 4 to the robotic arm actuator to achieve convergence of the robotic arm's motion trajectory to the desired trajectory within a fixed time. At the same time, feed this control torque back to Step 2 as a known control input to participate in the disturbance estimation of the next control cycle, thereby realizing continuous closed-loop control of the system.

[0029] Furthermore, the specific process of Step 1 is as follows: Step 1.1, Obtain the actual Physical parameters of a rigid robotic arm with two degrees of freedom. Taking a two-degree-of-freedom robotic arm as an example, its physical parameters are set as follows: link mass... , Link length Moment of inertia Gravitational acceleration .

[0030] Step 1.2: Establish the specific dynamic matrix equations for this two-degree-of-freedom robotic arm:

[0031] in, , , These represent the joint angular position, angular velocity, and angular acceleration vectors of the robotic arm, respectively. The joint control torque output by the actuator is to absorb external environmental disturbances. A strong time-varying sinusoidal external disturbance is defined as: System inertia matrix The elements are: , , Centripetal-Coriolis matrix The elements are: The matrix structure is as follows: Gravitational torque vector The elements are: , The matrix structure is as follows: .

[0032] Step 1.3: Define system state variables , , The system state-space model is established as follows:

[0033] in, For joint angle position variables, Its derivative, For joint angular velocity variables, Its derivative, For equivalent lumped disturbance variables Its derivative, For system control input gain, For the system control law, For aggregated disturbance The derivative of .

[0034] The specific process for Step 2 is as follows: Step 2.1: Construct a fixed-time extended state observer (FxESO) for lumped disturbances. The dynamic equation for real-time estimation is shown below:

[0035] in, For observation error, For the observed values ​​of the state variables, For observer gain parameters, and These are the exponential terms that rapidly approach large errors and precisely lock onto small errors, respectively. ; Step 2.2: Set the observer exponential parameters. Select a small constant. The exponent of the fast approach term with large error is set to... , , The index for the small error precise locking term is: , , .

[0036] The specific process for Step 3 is as follows: Step 3.1: Define the desired trajectory Set the system's large initial deviation state as , To verify the global fixed-time convergence performance of the controller under large initial deviations.

[0037] Step 3.2, Define position error With speed error The design of the non-singular terminal sliding surface is shown below:

[0038] The sliding surface parameters are configured as follows: , , .at this time It satisfies the non-singularity condition.

[0039] Step 3.3: Construct observations incorporating FxESO perturbations Adaptive fixed-time nonsingular terminal sliding mode control law As shown below:

[0040] in, For the desired angular acceleration, For speed tracking error, For sliding surface parameters, The zero-crossing buffer constant, For adaptive robust gain; the reaching law parameters are configured as follows: , , , Control input gain set to .

[0041] Step 3.4: In the actual output, the zero-crossing buffer constant is taken as... Using the continuous hyperbolic tangent function Alternative To reduce high-frequency vibration of torque.

[0042] The specific process for Step 4 is as follows: Step 4.1: Construct Adaptive Robust Gain As shown below:

[0043] in for The derivative of the adaptive learning rate is taken. Leakage dissipation coefficient Set the initial value to .

[0044] Step 4.2: Apply the calculated ideal control torque to the robotic arm solid model and set the physical limit. The effectiveness of this invention is verified through simulation comparison with traditional sliding mode control methods and extended state observer (SMC-ESO).

[0045] The specific process for Step 5 is as follows: The control simulation of the method of this invention was performed using the Simulink module in MATLAB. The main focus was on comparing the control effects of the traditional SMC-ESO control method and the control method of this invention (AFXNTSMC) on the aforementioned two-degree-of-freedom robotic arm, generating... Figures 2 to 10 The graph is shown below. Its specific representation and effect are as follows: Figure 2 The figure shows the tracking trajectory curves of joint one under the SMC-ESO control method and the control method of the present invention. In the figure, q1d is the ideal position tracking curve of joint one, and AFXNTSMC-q1 and SMC-ESO-q1 are respectively the position curves of joint one under the control method of the present invention and the SMC-ESO control method. Figure 3 The figure shows the tracking trajectory curves of joint two under the traditional SMC_ESO control method and the control method of the present invention. In the figure, q2d is the ideal position tracking curve of joint two, and AFXNTSMC-q2 and SMC-ESO-q2 represent the position curves of joint two under the control method of the present invention and the SMC-ESO control method, respectively. Figure 2 and Figure 3 It can be seen that the present invention has superior dynamic response and fitting speed in position tracking. The corresponding tracking trajectory error results are as follows: Figure 4As shown, the diagram consists of two smaller graphs, one above the other. The upper graph's AFXNTSMC-q1e and AFXNTSMC-q2e curves represent the position tracking errors of joint one and joint two under the control method of this invention, respectively. The lower graph's SCM-ESO-q1e and SMC-ESO-q2e curves represent the position tracking errors of joint one and joint two under the SMC-ESO control method, respectively. Figure 4 It can be seen that the position convergence speed of the method of the present invention under large initial deviation is significantly better than that of the SMC-ESO control method, and it can still achieve accurate convergence in a fixed time.

[0046] Figure 5 The figure shows the speed tracking curves of joint one under the traditional SMC-ESO control method and the control method of the present invention. In the figure, dq1d is the ideal speed tracking curve of joint one, and AFXNTSMC-dq1 and SCM-ESO-dq1 are respectively the speed curves of joint one under the control method of the present invention and the SMC-ESO control method. Figure 6 The figure shows the speed tracking curves of joint two under the traditional SMC-ESO control method and the control method of the present invention. In the figure, dq2d is the ideal speed tracking curve of joint two, and AFXNTSMC-dq2 and SCM-ESO-dq2 represent the speed curves of joint two under the control method of the present invention and the SMC-ESO control method, respectively. The corresponding... Figure 7 The speed tracking error curves of joint one and joint two under the SMC-ESO control method and the control method of the present invention are composed of two smaller graphs, one above the other. The AFXNTSMC-dq1e curve and the AFXNTSMC-dq2e curve in the upper graph represent the speed tracking errors of joint one and joint two under the control method of the present invention, respectively; the SCM-ESO-dq1e curve and the SMC-ESO-dq2e curve in the lower graph represent the speed tracking errors of joint one and joint two under the SMC-ESO control method, respectively. Figures 5 to 7 This indicates that the method of the present invention is significantly superior to traditional sliding mode control in terms of speed and accuracy in speed tracking.

[0047] Figure 8 The figure shows the control input curves of joint one under the conventional SMC-ESO control method and the control method of the present invention. In the figure, AFXNTSMC-u and SCM-ESO-u represent the input torque curves of joint one under the control method of the present invention and the SMC-ESO control method, respectively. Figure 9 The figure shows the control input curves of joint two under the traditional SMC-ESO control method and the control method of the present invention. In the figure, AFXNTSMC-u and SCM-ESO-u represent the input torque curves of joint two under the control method of the present invention and the SMC-ESO control method, respectively. Figure 8 and Figure 9This indicates that the present invention can effectively suppress high-frequency chattering in the control input torque.

[0048] Figure 10 The figure shows the numerical variation curves of the adaptive parameters at joint one and joint two. K1 and K2 in the figure represent the changes in parameter K values ​​at joint one and joint two, respectively, proving the parameter... It can adaptively adjust according to the sliding surface deviation to counteract strong external disturbances and avoid excessive gain expansion after the disturbance stabilizes.

[0049] This invention is illustrated through specific implementation processes. Various modifications and equivalent substitutions can be made to this invention without departing from its scope. Therefore, this invention is not limited to the disclosed specific implementation processes, but should include all embodiments falling within the scope of the claims.

Claims

1. A sliding mode control method for an adaptive non-singular terminal block of a robotic arm based on a fixed-time observer, characterized in that: The specific steps include the following: Step 1: Establish a dynamic model of the rigid robotic arm and transform the parameter uncertainties, friction, and external environmental disturbances contained in the model into lumped disturbances under the state-space expression. Step 2: Using the real-time state variables of the robotic arm and the feedback system control torque, construct a double power fixed-time extended state observer to accurately estimate the lumped disturbance mentioned in Step 1 in real time and within a fixed time. Step 3: Design a non-singular terminal sliding mode surface based on the tracking error of the robotic arm, and combine it with the lumped disturbance estimate output in Step 2 to construct an adaptive non-singular terminal sliding mode control law, which introduces a zero-crossing buffer constant and a smoothing function to suppress torque chattering and singularity. Step 4: Design an adaptive gain update law with a leakage term, adjust the robust gain in real time according to the feedback of the sliding surface variables, compensate for the residual estimation error of the observer in Step 2, and then calculate and output the final joint control torque. Step 5: Apply the final joint control torque output from Step 4 to the robotic arm actuator to achieve convergence of the robotic arm's motion trajectory to the desired trajectory within a fixed time. At the same time, feed this control torque back to Step 2 as a known control input to participate in the disturbance estimation of the next control cycle, thereby realizing continuous closed-loop control of the system.

2. The adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer according to claim 1, characterized in that: The specific process for establishing the dynamic model of the rigid robotic arm as described in Step 1 is as follows: ； in, , , These represent the joint angular position, angular velocity, and angular acceleration vectors of the robotic arm, respectively. Here is the system inertia matrix. For the centripetal force and Coriolis force matrix, It is the gravitational torque vector. The joint control torque output by the actuator. For external disturbance torque, lumped disturbance term The synthesis incorporates modeling uncertainties and external perturbation moments, and assumes their derivatives. Bounded; Define system state variables , , The dynamic model of the robotic arm is transformed into the following nonlinear system state-space expression: ； in, For joint angle position variables, Its derivative, For joint angular velocity variables, Its derivative, For equivalent lumped disturbance variables Its derivative, For system control input gain, For the system control law, For aggregated disturbance The derivative of .

3. The adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer according to claim 2, characterized in that: The specific observer dynamic equations for constructing the double power fixed-time extended state observer FxESO described in Step 2 are as follows: ； in, For observation error, For the observed values ​​of the state variables, For observer gain parameters, and These are the exponential terms that rapidly approach large errors and precisely lock onto small errors, respectively. .

4. The adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer according to claim 1, characterized in that: The design of the non-singular terminal sliding surface described in Step 3 The equation for the sliding surface is as follows: ； in, For position tracking error, Here are the parameters of the sliding surface, and the parameters satisfy... , and All are positive odd numbers, and their ratio satisfies .

5. The adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer according to claim 4, characterized in that: The system control law solution formula for the adaptive non-singular terminal sliding mode control law described in Step 3 is as follows: ； in, For the desired angular acceleration, For speed tracking error, For the approach law parameters, For adaptive robust gain.

6. The adaptive non-singular terminal sliding mode control method for a robotic arm based on a fixed-time observer according to claim 1, characterized in that: The adaptive gain update law with leakage term described in Step 4 adjusts the adaptive robust gain in real time based on feedback from the sliding surface variables. The specific formula is as follows: ； in, for The derivative of For adaptive learning rate, This is the leakage dissipation coefficient.

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