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Adaptive Neuro-Fuzzy Control of a Spherical Rolling Robot Using Sliding-Mode-Control-Theory-Based Online Learning Algorithm

Erkan Kayacan, Erdal Kayacan, Herman Ramon, Wouter Saeys

arXiv:2104.07160v1cs.ROeess.SY

TL;DR

Conventional controllers for spherical rolling robots can be degraded by unmodeled dynamics, parameter variations, uncertainties, and disturbances. The paper combines an adaptive neuro-fuzzy controller with an SMC-theory-based online learning algorithm, yielding lower steady-state error and improved transient performance in simulations.

  • Problem

    Unmodeled dynamics, parameter variations, uncertainties, and disturbances can make conventional model-based controllers poorly tuned for spherical rolling robots.

  • Method

    The paper combines a neuro-fuzzy network with a conventional controller and updates the neuro-fuzzy parameters using an SMC-theory-based online learning algorithm.

  • Results

    The proposed scheme eliminates steady-state error with a PD controller and reduces settling time below 1 second versus approximately 2.5 seconds for stand-alone PID control under viscous-friction variation.

  • Takeaways & Limitations

    The adaptive neuro-fuzzy scheme provides better performance and higher robustness than the conventional stand-alone controller in the reported simulations.

Abstract

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As a model is only an abstraction of the real system, unmodeled dynamics, parameter variations, and disturbances can result in poor performance of a conventional controller based on this model. In such cases, a conventional controller cannot remain well tuned. This paper presents the control of a spherical rolling robot by using an adaptive neuro-fuzzy controller in combination with a sliding-mode control (SMC)-theory-based learning algorithm. The proposed control structure consists of a neuro-fuzzy network and a conventional controller which is used to guarantee the asymptotic stability of the system in a compact space. The parameter updating rules of the neuro-fuzzy system using SMC theory are derived, and the stability of the learning is proven using a Lyapunov function. The simulation results show that the control scheme with the proposed SMC-theory-based learning algorithm is able to not only eliminate the steady-state error but also improve the transient response performance of the spherical rolling robot without knowing its dynamic equations.

I. INTRODUCTION

Spherical rolling robots offer mechanical and mobility advantages but are difficult to control because their dynamics are highly nonlinear and affected by uncertainties. The paper therefore proposes an adaptive FNN controller with an SMC-theory-based online learning algorithm.

  • I. INTRODUCTION: Spherical mechanisms provide greater instantaneous mobility than wheels, easier direction changes, and reduced risk of falling over.These advantages accompany highly complex nonlinear dynamics.
  • I. INTRODUCTION: Existing spherical robots use diverse actuation designs, including internal wheeled vehicles, flywheels, pendulums, and mass-center displacement.These designs support different applications and motion capabilities.
  • I. INTRODUCTION: Highly nonlinear dynamics make simplified linearized and kinematic models insufficient for controlling spherical rolling robots.The cited approaches omit system properties or are impractical for the complex nonlinear equations.
  • I. INTRODUCTION: Unmodeled dynamics, parameter variations, friction, uncertainties, and disturbances can invalidate conventional time-invariant controllers.The paper motivates advanced intelligent control techniques for these effects.
  • I. INTRODUCTION: The proposed controller combines a fuzzy neural network with a conventional feedback controller and uses SMC theory for online adaptation.The SMC-based algorithm is intended to address parameter variations, uncertainties, and disturbances while improving online convergence.

A. Kinematic Model

The model describes a pendulum-actuated spherical robot using reference frames, rolling and pendulum coordinates, energy-based dynamics, and viscous friction. The resulting equations support velocity control for translation along O −y.

  • A. Kinematic Model: The spherical rolling mechanism is modeled with a sphere and pendulum under explicit geometric and no-slip assumptions.The sphere rolls on a perfectly horizontal surface, its shell center coincides with the system center of mass, and the pendulum hangs vertically at equilibrium.
  • A. Kinematic Model: The model uses frames attached to the ground and sphere center, with relative angular positions described by standard angle parameterizations.Rf0 is inertial, while Rf1 and Rf2 describe translation and rotation associated with the sphere.
  • B. Dynamic Model: The dynamic formulation combines kinetic and potential energy terms for the sphere and pendulum in a Lagrangian function.Masses, moments of inertia, linear and angular velocities, and pendulum height contribute to the formulation.
  • B. Dynamic Model: For translation along O −y, Euler–Lagrange equations are written in matrix form using sphere and pendulum generalized coordinates.The coordinates are q1 = θ and q2 = α, with the pendulum input torque producing an opposing reaction torque.
  • B. Dynamic Model: Viscous friction is included as an energy-dissipation term, distinguishing this model from the cited earlier approach.The equations are used for velocity control of the spherical rolling robot.

A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System

The proposed scheme combines a conventional PD or PID controller with a feedback neuro-fuzzy controller to address uncertainty and surface-dependent disturbances in spherical rolling robots. The FNN maps tracking-error signals through fuzzy inference to generate a compensating torque.

  • A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System: The controller is designed for unmodeled dynamics, parameter variations, uncertainties, disturbances, and changing surface properties that challenge conventional control.The paper specifically notes that friction and surface conditions affect spherical rolling-robot behavior.
  • A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System: The conventional PD or PID controller operates in parallel with the neuro-fuzzy controller to guarantee global asymptotic stability in compact space.The overall torque combines the conventional-controller torque and the neuro-fuzzy torque.
  • A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System: The FNN uses the tracking error and its derivative as inputs and produces a feedback-controller output.The inputs are x1(t) = e(t) and x2(t) = ė(t).
  • A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System: The FNN implements a zeroth-order Takagi-Sugeno-Kang rule system whose fuzzy rules use two inputs and constant consequents fij = dij.Its layers fuzzify inputs, compute rule firing strengths, normalize them, adapt node outputs, and sum the resulting signals into τn.
  • A. The Control Scheme and the Adaptive Neuro-Fuzzy Inference System: The neuro-fuzzy output τn is summed with the conventional-controller torque τc to form the torque input τ applied to the system.The passage identifies τc and τn as the PD-controller and neuro-fuzzy feedback-controller torques, respectively.

B. The Sliding Mode Learning Algorithm

The learning algorithm applies sliding-mode conditions to update the FNN parameters online. Its stability analysis uses Lyapunov functions, with the adaptation strategy ensuring finite-time learning convergence and negative Lyapunov-function derivative.

  • B. The Sliding Mode Learning Algorithm: The analysis assumes bounded input signals, their time derivatives, the system input torque, and its time derivative.These boundedness conditions support the stability and convergence analysis.
  • B. The Sliding Mode Learning Algorithm: The method defines the conventional-controller output τc(t) as a time-varying sliding surface and also defines a tracking-error sliding surface Sp(e, ė).The parameter λ determines the slope of the tracking-error sliding surface.
  • B. The Sliding Mode Learning Algorithm: The proposed FNN parameter-update laws are derived by applying the sliding-mode condition to the online learning mechanism.The update laws are stated as the adaptation laws for the adjustable FNN parameters.
  • B. The Sliding Mode Learning Algorithm: The adaptation strategy ensures that the learning error τc(t) converges to zero within finite time th for any arbitrary initial condition τc(0).This result follows when the learning rate α satisfies the specified design inequality.
  • B. The Sliding Mode Learning Algorithm: The adaptation strategy ensures negative definiteness of the Lyapunov-function time derivative, establishing stability of the learning.The stability proof is formulated around the Lyapunov function in equation (35).

IV. SIMULATION RESULTS AND DISCUSSION

The simulations use fixed robot and controller settings while replacing the discontinuous sign function to reduce sliding-mode chattering.

  • The simulations use Ms = 3 kg, mp = 2 kg, R = 0.2 m, l = 0.075 m, g = 9.81 m/s2, ζ = 0.2, and a 0.001 s sampling period.
  • The FNN uses three membership functions for each of its two inputs in all simulations.
  • The paper replaces the SMC sign function with a modified equation to decrease high-frequency chattering.

A. Case 1: PD controller

Case 1 compares standalone PD control with PD operating in parallel with the FNN for a time-varying reference signal. The FNN-assisted controller eliminates the steady-state error observed with standalone PD control.

  • The reference velocity is 1 rad/s for 0 < t ≤5, 2 rad/s for 5 < t ≤10, and 1.5 rad/s for 10 < t ≤15.
  • The PD controller cannot eliminate steady-state error for the time-varying step input, whereas PD+FNN produces no steady-state error.
  • Figures 5 and 6 report the velocity response and tracking error for standalone PD and PD operating in parallel with the FNN.

B. Case 2: PID controller

Case 2 evaluates PID and PID+FNN control under transient, noisy, and abruptly varying damping conditions. The FNN-assisted controller improves transient response and robustness to parameter variations.

  • The FNN learns system dynamics after a finite duration, producing smaller rise time, overshoot, and settling time than PID alone.
  • Under damping coefficient 0.5 and noise level SNR = 20dB, the adaptive neuro-fuzzy approach is reported as more robust and gives smaller rise and settling times than PID alone.
  • When damping changes among 0.2, 0.5, and 0.8 rad/s, PID settling time is approximately 2.5 seconds versus less than 1 second for PID+FNN.
  • Figure 13 compares the PID-only and parallel FNN/PID control signals during the changing-damping experiment.

V. CONCLUSION

The study concludes that SMC-theory-based online learning improves velocity control of a spherical rolling robot under uncertainties and parameter variations. The parallel neuro-fuzzy scheme removes PD steady-state error, improves PID transients, and adapts controller parameters.

  • The adaptive neuro-fuzzy scheme achieves better performance and higher robustness than the conventional stand-alone controller in simulations.
  • The algorithm eliminates steady-state error with standalone PD and improves transient response with standalone PID.
  • SMC-theory-based online learning automatically adapts controller parameters to parameter variations and dynamic uncertainties.
  • The proposed learning algorithm is described as computationally simple compared with gradient-descent and evolutionary algorithms.

APPENDIX A PROOF OF THEOREM 1

The proof develops time derivatives and a Lyapunov function to establish learning stability. It requires the Lyapunov derivative to remain negative, including when τ̇ reaches its maximal value.

  • APPENDIX A PROOF OF THEOREM 1: The proposed learning stability is assessed with a Lyapunov function.The proof explicitly introduces a Lyapunov function for the stability analysis.
  • APPENDIX A PROOF OF THEOREM 1: The stability condition requires the Lyapunov derivative V̇ to be smaller than zero.This negativity condition is stated as necessary for learning stability.
  • APPENDIX A PROOF OF THEOREM 1: The proof evaluates the resulting condition when τ̇ reaches its maximal value Bτ̇.Equation (49) is rewritten under this maximal-value condition.

APPENDIX B PROOF OF THEOREM 2

The second theorem’s proof proceeds by differentiating the Lyapunov function introduced in equation (35).

  • APPENDIX B PROOF OF THEOREM 2: The proof begins by writing the time derivative of the Lyapunov function in (35).This derivative supplies the starting expression for the theorem’s stability analysis.
  • APPENDIX B PROOF OF THEOREM 2: Equation (35)'s Lyapunov derivative is the central analytical quantity in this proof.The supplied passage identifies the derivative as the next proof step.
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