Trajectory Control Method Based on High-Order Finite-Time Observer

Abstract

The present invention relates to a trajectory control method based on a high-order finite-time observer. The method includes: establishing a system dynamics model, designing a high-order finite-time homogeneous observer, restricting parameter drift through a projection operator, and introducing a fractional-order time-varying gain adaptive law to cope with sudden perturbations; reconstructing the observation error using an asymmetric barrier function to improve the convergence speed under asymmetric dynamics; injecting a time-varying scaling term into the observer to balance the initial fast response and steady-state noise suppression; constructing a backstepping sliding-mode differential controller, designing a fractional-order stepped sliding-mode surface, combining multi-scale weighted gain and a nonlinear continuous operator to dynamically adjust the control parameters, suppressing high-frequency chattering and eliminating steady-state phase lag. The present invention significantly improves the trajectory tracking accuracy, anti-disturbance ability, and system robustness through finite-time observation, adaptive disturbance compensation, and fractional-order sliding-mode control.

Description

Technical Field

[0001] The present invention belongs to the field of motion control, and particularly relates to a trajectory control method based on a high-order finite-time observer. Background Art

[0002] In existing robot trajectory control methods, traditional sliding mode control has the problem of high-frequency chattering, and it is difficult for a fixed-time observer to accurately estimate the state and disturbance within a finite time. Fixed gain design cannot effectively cope with sudden disturbances (such as impact loads), resulting in compensation lag or over-compensation; the convergence speed of a symmetric observer decreases under unilateral disturbances (such as unilateral friction); the proportional-integral characteristics of a traditional sliding mode surface are prone to introducing phase lag, affecting the steady-state accuracy. In the prior art, although fractional-order differentiation is used to adjust the dynamic response, it is not combined with multi-scale gain, and it is impossible to take into account both high-frequency and low-frequency error suppression; most adaptive laws adopt integer-order design, with a relatively high risk of parameter drift, and lack a time-varying gain mechanism to balance dynamic response and steady-state performance. In addition, the fixed convergence mechanism of the observer is vulnerable to high-frequency noise interference in the initial stage and cannot effectively suppress residual oscillations in the steady state. Summary of the Invention

[0003] The present invention provides a trajectory control method based on a high-order finite-time observer, which includes the following steps:

[0004] Step 1, establish a dynamic model of a multi-degree-of-freedom robot, and define an extended state vector to describe the position, velocity, acceleration, and lumped disturbance;

[0005] Step 2, design a high-order finite-time homogeneous observer, which consists of a recursive form of an observation error equation, restricts the drift of adaptive parameters through a projection operator, or uses a fractional-order time-varying gain adaptive law to compensate for disturbances in real time;

[0006] Step 3, design a fractional-order step-shaped sliding mode surface and a control law, where the sliding mode surface dynamically adjusts the error weighting through a fractional-order differential operator, and the control law includes an integral term and a time-varying smooth nonlinear damping term.

[0007] Furthermore, the high-order finite-time homogeneous observer configures the observer gain coefficient through a Hurwitz polynomial and designs the gain parameter in combination with the upper bound of the Lipschitz constant of the unmodeled dynamics of the system.

[0008] Furthermore, the lumped disturbance estimation uses a projection operator to constrain the parameter range, and the projection operator dynamically truncates or adjusts the parameter update direction according to the preset upper limit of the disturbance amplitude.

[0009] Furthermore, the disturbance compensation term is updated through a fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding mode surface error.

[0010] Furthermore, the observer introduces an asymmetric barrier function to independently set the convergence exponent in the positive and negative error directions.

[0011] Furthermore, a time-varying scaling term is injected into the observer equation to retain the fast response characteristic in the initial stage and gradually suppress the high-frequency noise component over time.

[0012] Furthermore, the sliding surface and the control law are respectively:

[0013] , where D is the fractional-order differential, ζ is the order of the fractional-order differential, m is the order of the sliding surface, Λ k is the gain matrix, M is the inertia matrix, K ω is the integral gain matrix, ω is the power parameter of the integral term, Η(s) is the non-linear continuous operator, and ρ is the robust gain.

[0014] Furthermore, the non-linear continuous operator is:

[0015] , where δ is the smoothing factor, δ0 is the initial value of the smoothing factor, η is the power parameter, and δ ∞ is the steady-state value.

[0016] The present invention also provides a trajectory control system based on a high-order finite-time observer, which is used to execute the foregoing method.

[0017] Beneficial technical effects: By constraining the parameter range through the projection operator and combining the fractional-order time-varying gain adaptive law to enhance the robustness of disturbance compensation; designing an asymmetric barrier function to optimize the convergence characteristic of the observation error; introducing a time-varying scaling term to achieve dynamic attenuation of the observer; constructing a fractional-order stepped sliding surface and multi-scale gains to dynamically adjust the control law, eliminating chattering and improving the tracking accuracy within a finite time. This method can significantly suppress sudden disturbances and improve the smoothness of control. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Att Figure 1 is the control schematic diagram of the trajectory control method based on the high-order finite-time observer according to the present invention;

[0019] Att Figure 2 is the schematic diagram of the tracking error of the joint angle according to the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0020] As Figure 1 shown, this embodiment provides a trajectory control method based on a high-order finite-time observer, which includes the following steps:

[0021] Step 1, system dynamics modeling, defining the extended state vector x = [x1 T x2 T x3 T T , where x1, x2, and x3 are position, velocity, and acceleration respectively, and a third-order chain differential equation is established:

[0022] where M, C, and G are the inertia matrix, Coriolis force matrix, and gravity term respectively, and τ d is the lumped disturbance, including friction, external load, etc., τ is the control input, u is the generalized control input, specifically the first derivative of τ.

[0023] Step 2, Design of high-order finite-time observer:

[0024] Construct an r-order finite-time homogeneous observer:

[0025]

[0026] where Φ is the nonlinear dynamic term, and k i , α i are gain coefficients and can be designed as:

[0027] , where λ i , L are observer parameters, and λ i is determined by the root configuration of the Hurwitz polynomial. Specifically: construct the observer error dynamic equation, linearize it to obtain the characteristic polynomial, and according to Hurwitz stability, select λ i such that the roots of this polynomial are all in the left half-plane of the complex plane (i.e., the real part is negative), and L is the upper bound of the Lipschitz constant of the unmodeled dynamics and disturbance terms in the system dynamics model, which can be determined through experiments or theoretical analysis.

[0028] Among them, the lumped disturbance estimation can use the projection operator to constrain the parameter range. The projection operator dynamically truncates or adjusts the parameter update direction according to the preset upper limit of the disturbance amplitude to prevent parameter drift. The lumped disturbance estimation error can be designed as , where σ is the integration variable, representing the historical time points during the integration process, t is the continuous time variable, , γ is the disturbance control coefficient, and proj( ) is the projection operator. In the disturbance estimation, the true lumped disturbance τ d is usually bounded, but the integral term in the adaptive law may cause the adaptive parameters to exceed the physically allowed range. Therefore, the projection operator proj( ) is introduced to limit the value range of the adaptive parameters by forcing truncation or adjusting the parameter update direction, preventing it from growing infinitely due to integral accumulation, that is, preventing parameter drift. The calculation of the projection operator proj( ) is specifically:

[0029] ​

[0030] where τ max is set according to the characteristics of the robot system, such as the maximum torque of the motor and the load limit.

[0031] Alternatively, the fixed-gain disturbance compensation may not be able to cope with sudden disturbances. Therefore, a fractional-order time-varying gain adaptive law can be designed. The disturbance compensation term is updated through the fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding mode surface error to balance the intensity of sudden disturbance compensation and steady-state noise suppression. The improved adaptive disturbance compensation law is:

[0032]

[0033] D is a fractional-order differential operator used to balance the dynamic response and suppress high-frequency noise, and it can be: , Γ(ν) is the Gamma function, f (1) is the first derivative of f. ν is the fractional-order differential order, ν ∈ (0, 1), which reflects the influence weight of historical errors on the current compensation and is used to adjust the historical error weight. By adjusting ν, the adaptive compensator can cope with both slow-varying (such as temperature drift) and sudden disturbances (such as impact loads) simultaneously. γ(t) is the time-varying gain, μ is the gain adjustment exponent, μ ∈ (0, 1), which is used to determine the balance relationship between gain increase and decay; κ is the gain growth rate, κ > 0, which reflects the adjustment intensity according to the sliding mode surface error, and θ is the gain decay rate, θ > 0, which prevents unbounded growth of the gain. When the sliding mode surface error is large, γ(t) increases rapidly to enhance the disturbance compensation intensity. When the sliding mode surface error approaches the steady state, γ(t) converges to the equilibrium point to avoid overcompensation.

[0034] The improved control system can resist sudden disturbances, enhance the steady-state accuracy, and enhance the robustness of parameter adjustment. When the system suddenly adds a load (such as a robotic arm grasping an object of unknown mass), the sliding mode surface error increases rapidly, triggering the adjustment mechanism, causing γ(t) to jump within milliseconds, and the compensation speed is several times faster than that of the fixed gain. When in the steady state, it can avoid high-frequency chattering caused by excessive gain.

[0035] In the formula, where sign is the sign function.

[0036] Alternatively, to solve the problem that the convergence speed may decrease when there is a unilateral disturbance (such as one-way friction) in the design of the symmetric gain of the observer, an asymmetric barrier function can be introduced to reconstruct the observation error, that is, the observer introduces an asymmetric barrier function, and the convergence exponent is independently set in the positive and negative error directions to optimize the observation accuracy and convergence speed under unilateral disturbances. The asymmetric barrier function is specifically:

[0037]

[0038] Thus, the observer is changed to an asymmetric r-order observer:

[0039]

[0040] where α i , β i ∈(0, 1) are the positive / negative error direction convergence exponents respectively, controlling the convergence speeds of the positive / negative error directions respectively, and λ max (M), λ min (M) are the maximum / minimum eigenvalues of the inertia matrix respectively.

[0041] In the presence of unilateral disturbances, the observation accuracy can be significantly improved through asymmetric design, making the observer adaptive to asymmetric dynamics.

[0042] Furthermore, to improve the convergence, a time-varying scaling fixed-time convergence mechanism can be introduced into the observer. A time-varying scaling term is injected into the observer equation. In the initial stage, the fast response characteristic is retained, and as time goes by, the high-frequency noise components are gradually suppressed, achieving a balance between the dynamic convergence speed and the steady-state smoothness. Specifically:

[0043] . where β is the scaling parameter and ε is the power parameter, used to balance the convergence speed and the smoothness. The larger β and ε are, the faster the decay is.

[0044] By injecting a time-varying scaling term into the observer equation, in the initial stage of control, the scaling factor is 1, retaining the fast response characteristic of the original observer. As the control progresses and time t increases, the scaling factor gradually decreases and tends to 0, suppressing the high-frequency dynamic components of the observer. In the fast convergence stage, the time-varying scaling term hardly plays a role, and the exponential decay term maintains a high power, enabling the observer to quickly track the true state. While in the steady state stage, the scaling term suppresses high-frequency noise, and the exponential decay term reduces the gain, achieving smooth control.

[0045] Step 3, backstepping sliding mode differential control:

[0046] To reduce the high-frequency chattering in traditional sliding mode control, the sliding surface can be designed as a fractional-order stepped sliding surface, where the sliding surface dynamically adjusts the error weighting through a fractional-order differential operator. The control law includes an integral term and a time-varying smooth nonlinear damping term, used to suppress high-frequency chattering and improve the tracking accuracy. The sliding surface and the control law are specifically as follows:

[0047]

[0048] Where D is the fractional differential, through which the sliding surface characteristics are dynamically adjusted to reduce the sensitivity of the controller to high-frequency noise while maintaining a fast response to low-frequency errors. ζ is the fractional differential order, and m is the order of the sliding surface, that is, the number of backstepping recursions. When m is an odd number, the suppression of high-frequency error components is enhanced, and when m is an even number, the integrity of the low-frequency tracking information is retained. k is the gain matrix, Λ k is a diagonal matrix; the multi-scale weighting of the error is realized by the index k / m, that is, the gain of the higher level k is more sensitive to the high-frequency error; M is the inertia matrix, which can be directly quoted from the inertia matrix in the dynamic model, and its estimated value can be used in actual control; K ω is the integral gain matrix, which is used to adjust the weight of the integral term (the subscript ω is only a mark and has nothing to do with the integral term power parameter); ω is the integral term power parameter, which can be ω=ζ / (ζ-1). ω controls the cumulative effect of the integral term. When ω tends to 1, the integral degenerates into pure accumulation. When ω tends to infinity, it is approximate proportional control. Through the coupling design of ω and ζ, the steady-state phase lag in the traditional sliding mode control is eliminated; H(s) is a nonlinear continuous operator, specifically: , when |s| is large, it is approximate proportional control to enhance robustness, and when |s| is small, it is approximate linear damping control to eliminate high-frequency chattering, where δ is the smoothing factor, the smoothing factor δ is a time-varying parameter, δ>0, δ0 is the initial value of the smoothing factor, δ ∞ is the steady-state value; η is the power parameter. ρ is the robust gain, which can be adjusted by the adaptive regulation law , ρ0 is the initial value of the robust gain, Κρ is the robust parameter, Κρ>0, the gain is automatically increased in the initial stage to accelerate convergence, and the gain is reduced in the steady state to reduce energy consumption.

[0049] The sliding film control of this embodiment can suppress high-frequency chattering, maintain convergence within a limited time, and dynamically design the control parameters so that the controller can automatically adjust under different working conditions. Figure 2 As shown, as the control proceeds, the tracking error of the joint angle decreases rapidly.

[0050] In the several embodiments provided by the present invention, it should be understood that the disclosed methods can be implemented in other ways. For example, the above-described embodiments of the invention are merely schematic. For example, the division of method modules and steps is only a logical function division, and there may be other division methods in actual implementation. The modules and steps described as separate components or separate steps may or may not be physically separated, and the components displayed as modules may or may not be physical modules, which may be located in one place or distributed on multiple network modules. Some or all of the modules may be selected according to actual needs to achieve the purpose of the scheme of this embodiment.

[0051] In addition, each functional module and step in various embodiments of the present invention can be integrated into a processing module or method, or each module and step can exist physically alone, or two or more modules and steps can be integrated into one module or method. The above integrated modules can be implemented in the form of hardware, or in the form of a combination of hardware and software functional modules.

[0052] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and without departing from the basic features of the present invention, the present invention can be implemented in other specific forms.

[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention.

Claims

1. A trajectory control method based on a high-order finite-time observer, characterized in that: The following steps are involved: Step 1, establish a multi-degree-of-freedom robot dynamics model and define an extended state vector to describe position, velocity, acceleration and lumped disturbance; Step 2, design a high-order finite-time homogeneous observer, which consists of a recursive form of the observation error equation, limits the drift of adaptive parameters through a projection operator, or uses a fractional-order time-varying gain adaptive law to compensate for disturbances in real time; Step 3, design a fractional step sliding surface and control law, where the sliding surface dynamically adjusts the error weighting through a fractional order differential operator, and the control law includes an integral term and a time-varying smooth nonlinear damping term; Specifically, the sliding film surface and the control law are: , where D is the fractional differential, ζ is the fractional differential order, m is the order of the sliding surface, Λ k is the gain matrix, M is the inertia matrix, K ω is the integral gain matrix, ω is the integral term power parameter, H(s) is the nonlinear continuous operator, σ is the integral variable, and ρ is the robust gain; the nonlinear continuous operator is: , δ is the smoothing factor, δ0 is the initial value of the smoothing factor, η is the power parameter, δ ∞ is the steady-state value.

2. The method according to claim 1, characterized in that The observer gain coefficients of the high-order finite-time homogeneous observer are configured through Hurwitz polynomials, and the gain parameters are designed by combining the upper bound of the Lipschitz constant of the unmodeled dynamics of the system.

3. The method according to claim 1, characterized in that The lumped disturbance estimation uses a projection operator to constrain the parameter range, and the projection operator dynamically cuts off or adjusts the parameter update direction according to a preset disturbance amplitude upper limit.

4. The method according to claim 1, characterized in that: The disturbance compensation term is updated through a fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding surface error.

5. The method according to claim 1, characterized in that The observer introduces an asymmetric barrier function and sets the convergence index independently through the positive and negative error directions.

6. The method according to claim 1, characterized in that A time-varying scaling term is injected into the observer equation, which retains the fast response characteristics in the initial stage and gradually suppresses the high-frequency noise components over time.

7. A trajectory control system based on a high-order finite-time observer, used to implement the method described in any one of claims 1 to 6.

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