Trajectory Control Method Based on Time-Varying Fuzzy Sliding Mode

Abstract

The present invention relates to a trajectory control method based on time-varying fuzzy sliding mode. The method includes: establishing a system dynamics model, designing a high-order observer to estimate the position, velocity, acceleration and lumped disturbance in real time, and avoiding the conservatism of fixed gains through adaptive gain adjustment; constructing a fuzzy sliding mode control law, combining the equivalent control term and the switching control term, and dynamically adjusting the error weight by using a fuzzy logic system; designing a dynamic limiter and a delay compensation mechanism to smoothly limit the control command to the physical limit of the actuator, and at the same time correcting the control command based on the observer feedback to offset the delay effect. The present invention effectively improves the trajectory tracking accuracy and system robustness through time-varying fuzzy sliding mode control, adaptive observer gain and anti-saturation compensation.

Description

Technical Field

[0001] The present invention belongs to the field of motion control, and particularly relates to a trajectory control method based on time-varying fuzzy sliding mode. Background Art

[0002] Traditional sliding mode control methods have the problem of high-frequency chattering in robot trajectory tracking, and fixed-gain observers rely on the upper bound of disturbances, which easily leads to conservative or insufficient gains. In addition, actuator saturation and response delay will reduce the control accuracy and even cause system instability. In the prior art, although the fuzzy logic system can partially suppress chattering, it is not combined with an adaptive observer gain and cannot dynamically respond to time-varying disturbances; anti-saturation compensation mostly uses hard limiting, resulting in non-differentiable control law boundaries and exacerbating chattering. Summary of the Invention

[0003] The present invention provides a trajectory control method based on time-varying fuzzy sliding mode, which includes the following steps:

[0004] Step 1: Perform system dynamics modeling on the robot, and its dynamic equation includes an inertia matrix, a Coriolis force matrix, a gravity term, and a lumped disturbance term;

[0005] Step 2: Design a high-order observer to estimate the position, velocity, acceleration, and disturbance in real time by expanding the state vector, and dynamically adjust the state estimation through a non-linear compensation term and an adaptive gain;

[0006] Step 3: Construct a time-varying fuzzy sliding mode control law, combine the equivalent control term and the switching control term, use the fuzzy logic system to dynamically adjust the tracking error weight, and use a non-linear switching term to compensate for unknown disturbances;

[0007] Step 4: Design a dynamic limiter to smoothly limit the control command to the physical limit of the actuator, obtain an anti-saturation control command, and correct the model through the observer feedback.

[0008] Further, the sliding mode surface is defined as a non-linear combination of the second derivative of the tracking error and the output of the fuzzy logic system.

[0009] Further, the observer gain satisfies:

[0010] , where η i is the adaptive rate parameter, and k i,max is the safety upper limit of the gain.

[0011] Further, the observer gain satisfies:

[0012] , where sat(⋅) is the saturation function, Δ is the dead zone threshold, and η i is the adaptive rate parameter.

[0013] Furthermore, the input variables of the fuzzy logic system are configured with Gaussian membership functions, the rule base is constructed based on the error dynamics, and the consequent parameters are adjusted online using the gradient descent method.

[0014] Furthermore, the switching control term of the sliding mode control is: , where K3 and K4 are adaptive gain matrices, ε is a smoothing factor, B is an input transformation matrix, B + is the pseudo-inverse of matrix B, -B + represents the inverse mapping of the control input through the pseudo-inverse matrix B + , M is the inertia matrix, d is the lumped disturbance, β is the smoothness parameter, and σ is the sliding mode surface.

[0015] Furthermore, the sliding mode surface of the sliding mode control includes a time-varying scaling term.

[0016] Furthermore, the dynamic limiter asymptotically saturates the control command through a smoothing function.

[0017] The present invention also provides a trajectory control system based on time-varying fuzzy sliding mode, which is used to execute the foregoing method.

[0018] Beneficial technical effects: The gain is adjusted in real time through an adaptive gain observer, the error weight is optimized by combining with a fuzzy logic system, and a dynamic limiter and a delay compensation mechanism are designed to smoothly transition the command. At the same time, the model is corrected by the observer feedback to improve the state estimation accuracy. This method can significantly reduce chattering, enhance the anti-interference ability, and suppress actuator saturation. Description of the Drawings

[0019] Att Figure 1 is the schematic diagram of the trajectory control based on time-varying fuzzy sliding mode according to the present invention.

[0020] Att Figure 2 is the tracking diagram of the robot joint angle error according to the present invention. Detailed Embodiments

[0021] As Figure 1 shown, this embodiment provides a trajectory control method based on time-varying fuzzy sliding mode, which includes the following steps:

[0022] Step 1, perform system dynamics modeling on the robot, and its dynamic equation includes an inertia matrix, a Coriolis force matrix, a gravity term, and a lumped disturbance term.

[0023] Step 2, design of a high-order observer:

[0024] Define an extended state vector x including the end position x1, velocity x2, and acceleration x3 of the robot, and construct a high-order observer:

[0025]

[0026] Among them, α1, α2, and α3 are homogeneous degree parameters, and Γ1 and Γ2 are nonlinear compensations used to offset model errors. The nonlinear compensation terms Γ1 and Γ2 improve the state estimation accuracy by correcting the inertia matrix, Coriolis force, and delayed compensation torque. u is the generalized control input, and k1, k2, and k3 are observer gains used to determine the estimation speed. The observer gains are adaptive gains, which can be adjusted in real time according to the observation error. For example, when the gain does not reach the safety upper limit, the gain is dynamically increased according to the power function of the error, or a saturation function is used to constrain the gain adjustment range. Specifically, it can be:

[0027] , where η i is the adaptive rate parameter used to control the adjustment speed of the gain. For example, it can be 0.5, and k i,max is the safety upper limit of the gain to prevent the gain from being too large due to noise. Alternatively, the observer gain k i can be adjusted through a saturation function, that is , where sat(⋅) is the saturation function, and Δ is the dead zone threshold, which can take 5% of the tracking error. When the observation error e i exceeds Δ, the gain k i grows linearly at a rate of η i .

[0028] In the formula is a nonlinear function used to accelerate convergence, where sign is the sign function.

[0029] Furthermore, a time-varying scaling term can be introduced in the observer design, specifically:

[0030]

[0031] Among them, θ is the scaling parameter that controls the attenuation rate of the time-varying scaling factor. The larger θ is, the faster the scaling factor decays with time, and the faster the dynamic component of the observer is suppressed. κ is the power decay parameter that controls the exponential decay rate of the nonlinear gain term. The larger κ is, the faster the gain term converges. In the initial stage, high gain quickly tracks the error, and in the steady state stage, low gain suppresses noise. The homogeneous degree parameter α i , determines the convergence characteristics of the nonlinear term. The gain is high when the error is large and low when the error is small.

[0032] By introducing a time-varying scaling term, in the initial stage, i.e., when t is small, the dynamic equation of the observer retains its original fast response characteristics, with a relatively high non-linear gain, accelerating the error convergence, thus ensuring that the observer can quickly track the true state and improving the initial tracking accuracy. In the steady state stage, i.e., when t is large, the high-frequency dynamic components of the observer (such as the oscillations caused by noise) are suppressed, the non-linear gain decreases, avoiding chattering caused by excessive gain, ensuring that the output of the observer tends to be smooth, and cooperating with the dynamic limiter to further reduce the risk of actuator saturation.

[0033] By adaptively adjusting the gain, there is no need to pre-determine the upper bound of the interference to determine the fixed gain, while the present invention can dynamically adjust the gain according to the real-time error, avoiding being overly conservative or insufficient.

[0034] Step 3, fuzzy sliding mode control:

[0035] The sliding mode surface is designed as a non-linear combination of the second derivative of the tracking error and the output of the fuzzy logic system, specifically: , where e is the tracking error, i.e., the difference between the actual position and the target position, γ1, γ2 are non-linear exponents used to control the convergence speed, and K1, K2 are diagonal matrices used to determine the weights of the error terms. Φ FLS (ξ) is the output of the fuzzy logic system, ξ is the input variable, and the specific calculation steps of the output are as follows:

[0036] (1) Define the input variables to include the tracking error, the derivative of the error, and the integral;

[0037] (2) Configuration of the membership functions: Three Gaussian membership functions are defined for each input variable;

[0038] (3) Construct the fuzzy rule base, and the rule base is constructed based on the error dynamics;

[0039] (4) Defuzzification: The center of gravity method is used to calculate the output Φ FLS (ξ), where the activation weight of a single rule can be taken as the minimum product of the input membership degrees, the initial values of the consequent parameters are set according to experience, and the gradient descent method is used for online adjustment.

[0040] The control law is: τ = τ eq +τ sw , τ eq is the equivalent control term, and τ sw is the switching control term used to compensate for unknown disturbances and unmodeled states, specifically

[0041] , where the settings of the B, M, and d parameters can be seen in the dynamic model. B is the input transformation matrix, B + is the pseudo-inverse of the matrix B, -B + represents the pseudo-inverse matrix B +Inverse map the control input, where M is the inertia matrix, d is the lumped disturbance, K3 and K4 are the adaptive gain matrices, β is the smoothness parameter, β ∈ (0, 1), which is used to control the smoothness of the switching term, and ε is the smoothing factor, which is used to determine the transition interval of the tanh function.

[0042] Alternatively, on the basis of retaining the output of the fuzzy logic system, a fractional stepped sliding mode surface and multi-scale gains can be constructed in the sliding mode control. The sliding mode surface can specifically be:

[0043]

[0044] where D t ζ is the fractional derivative, which is used to smooth the historical error and suppress high-frequency noise. ζ is the order, ζ ∈ (0, 1), and Λ k is the multi-scale gain matrix, which assigns weights to different levels of errors, such as position error, velocity error, and integral error. The multi-scale gain Λ k e |k / m| can be combined with the output of the fuzzy logic system to dynamically adjust the weights of each level to adapt to different working conditions, such as sudden load changes or friction changes, and enhance the adaptability to different dynamics.

[0045] By introducing a time-varying scaling term and a fractional stepped sliding mode surface, and combining the flexibility of the fuzzy logic system, the control scheme not only retains the advantages of the time-varying fuzzy sliding mode but also significantly improves the anti-disturbance ability, chattering suppression effect, and parameter robustness. The observer and the sliding mode surface are adaptively adjusted in different control stages to balance the fast response and steady-state smoothness. The multi-scale gain and fractional derivative achieve refined processing of the error and adapt to complex interference scenarios.

[0046] To match the dynamic characteristics of the fractional order sliding mode surface and avoid gain conflicts, the switching control term τ sw is correspondingly adjusted to:

[0047]

[0048] where the parameter definitions are the same as those described above.

[0049] The sliding mode control of the present invention can significantly reduce high-frequency chattering and at the same time enhance the anti-interference ability through feed-forward compensation.

[0050] Step 4, anti-saturation delay compensation, design a dynamic limiter, generate an anti-saturation control command, and perform delay compensation:

[0051] Limit the control instruction within the physical limit of the actuator to avoid actuator saturation or damage caused by excessive instructions. The observer may produce incorrect estimates due to unrealistic instructions. Therefore, a dynamic limiter can be designed to generate an anti-saturation control instruction. The dynamic limiter gradually saturates the control instruction through a smoothing function to avoid the non-differentiability problem at the boundary caused by hard limiting. The anti-saturation control instruction can be:

[0052] ,

[0053] Alternatively, the anti-saturation control instruction can be: , where τ max is the maximum value of the actuator torque, τ is the original control instruction, δ is the saturation factor used to adjust the curvature of the limiting transition region, and its value can be 1.

[0054] After implementing dynamic limiting, when τ is much smaller than τ max , the limited output τ cmd is approximately equal to τ, retaining the linear characteristics of the original instruction; when τ is much larger than τ max , the limited output τ cmd is approximately equal to τ max , achieving gradual saturation; thus avoiding the control law chattering caused by non-differentiability at the boundary of traditional hard limiting.

[0055] Delay compensation estimates the delay torque by comparing the original instruction and the limited instruction in real time and feeds it back to the observer to correct the model error. The delay compensation can be designed as:

[0056]

[0057] where τ max is the maximum value of the actuator torque, T m is the motor time constant reflecting the response speed of the actuator, k d is the convergence coefficient, k d >0, determining the convergence speed of the delay estimation. By converting the original control instruction τ from the sliding mode controller to the limited instruction τ cmd and sending it to the actuator, by comparing τ cmd with τ, the torque after delay compensation is estimated in real time and this value is fed back to the controller to enable it to adjust the instruction in advance to cancel the delay effect. At the same time, the torque after delay compensation is introduced into the observer, thereby further improving the state estimation accuracy of the observer, specifically: , where M, C, G, F are the inertia matrix, Coriolis force matrix, gravity term, and friction term respectively, and B is the input transformation matrix. By correcting the model, the accuracy of state estimation is improved.

[0058] The gain is adjusted in real time through an adaptive gain observer, the error weight is optimized by combining with a fuzzy logic system, and a dynamic limiter and a delay compensation mechanism are designed to smoothly transition the command. At the same time, the model is corrected through the observer feedback to improve the state estimation accuracy. This method can significantly reduce chattering, enhance the anti-interference ability and suppress actuator saturation. As Figure 2 shown, as the control progresses, the tracking error of the joint angle rapidly shrinks.

[0059] In several embodiments provided by the present invention, it should be understood that the disclosed method can be implemented in other ways. For example, the above-described invention embodiments are merely illustrative. 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. The components shown as modules may or may not be physical modules. They may be located in one place or distributed to multiple network modules. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0060] In addition, in each embodiment of the present invention, the functional modules and steps 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-mentioned integrated modules can be implemented in the form of hardware or in the form of hardware plus software functional modules.

[0061] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above-mentioned exemplary embodiments, and without departing from the basic characteristics of the present invention, the present invention can be implemented in other specific forms.

[0062] 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 time-varying fuzzy sliding mode, characterized in that Including the following steps: Step 1: Conduct system dynamics modeling on the robot, and its dynamics equation includes an inertia matrix, a Coriolis force matrix, a gravity term, and a lumped disturbance term; Step 2: Design a high-order observer to estimate the position, velocity, acceleration, and disturbance in real time by expanding the state vector, and dynamically adjust the state estimation through a nonlinear compensation term and an adaptive gain; Specifically, the observer gain satisfies or , where η i is the adaptive rate parameter, k i,max is the safety upper limit of the gain, x1 is the position of the robot end; sat(⋅) is the saturation function, Δ is the dead zone threshold, and e i is the observation error. Step 3: Construct a time-varying fuzzy sliding mode control law, combine the equivalent control term and the switching control term, use the fuzzy logic system to dynamically adjust the tracking error weight, and adopt a nonlinear switching term to compensate for unknown disturbances; Specifically, the synovial surface is or , where e is the tracking error, γ1, γ2 are non-linear exponents, K1, K2 are diagonal matrices, Φ FLS (ξ) is the output of the fuzzy logic system, ξ is the input variable, D t ζ is the fractional-order differential, ζ is the order, Λ k is the multi-scale gain matrix, Λ k e |k / m| is the multi-scale gain; The specific calculation steps for the output of the fuzzy logic system are as follows: (1) Define the input variables including the tracking error, the derivative of the error, and the integral; (2) Membership function configuration: Define three Gaussian membership functions for each input variable; (3) Construct a fuzzy rule base, and the rule base is constructed based on the error dynamics; (4) Defuzzification: Use the centroid method to calculate the output Φ FLS (ξ), and use the gradient descent method to adjust the consequent parameters online; Step 4: Design a dynamic limiter to smoothly limit the control command to the physical limit of the actuator, obtain an anti-saturation control command, and correct the model through the observer feedback.

2. The method according to claim 1, wherein When the synovial surface is , the switching control item of synovial control is: ； When the synovial surface is , the switching control item controlled by the synovium is: ； where $K_3$ and $K_4$ are adaptive gain matrices, $B$ is the input transformation matrix, $B$ + is the pseudo-inverse of matrix $B$, $-B$ + represents the inverse mapping of the control input through the pseudo-inverse matrix $B$ + $M$ is the inertia matrix, $d$ is the lumped disturbance, $\beta$ is the smoothness parameter, and $\varepsilon$ is the smoothing factor.

3. The method according to claim 1, wherein The dynamic limiter asymptotically saturates the control command through a smoothing function.

4. A trajectory control system based on time-varying fuzzy sliding mode, characterized in that, For implementing the method according to any one of claims 1-3.

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