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Adaptive Finite-Time Position-Force Control of Teleoperation Systems With Time-Varying Delays Using a Liquid State Machine Uncertainty Estimator

Shayan Akbari Haghighat, Mohammadali Ghaemifar, Armin Attarzadeh, Mohammadreza Piri Sangdeh

arXiv:2608.29544v1eess.SYeess.SP

TL;DR

The paper addresses uncertainty compensation for bilateral teleoperation when delays, nonlinear dynamics, and interaction histories complicate position–force tracking. It combines a finite-time adaptive hybrid position/force controller with an LSM uncertainty estimator, establishing finite-time convergence and reporting improved tracking and lower mean execution time than an RBFNN-based controller in simulations.

  • Problem

    History-dependent uncertainties, time-varying delays, and nonlinear bilateral dynamics challenge accurate position–force tracking, while feedforward approximators do not intrinsically retain temporal information.

  • Method

    The paper combines a finite-time hybrid position/force controller using velocity and force filters with an LSM estimator whose fixed reservoir preserves fading-memory information while only its linear readout adapts.

  • Results

    The method establishes bounded closed-loop signals and finite-time position/force error convergence through a Lyapunov–Krasovskii analysis and improves simulated tracking over an RBFNN-based controller.

  • Takeaways & Limitations

    LSM-based uncertainty estimation provides a teleoperation control structure that represents prior motion and interaction history without online recurrent-network training.

  • Takeaways & Limitations

    The controller and filter gains and reservoir hyperparameters were selected manually, and broader validation across hysteretic, contact-rich, and alternative reservoir settings remains future work.

Abstract

from arXiv · show

Teleoperation systems are increasingly used in medical, rehabilitation, and remote manipulation applications, where accurate position/force tracking and stable interaction are essential. In such applications, the remote environment may exhibit viscoelasticity, frictional memory, contact transitions, and other dynamic interaction effects, causing the system response to depend not only on the current state but also on its previous evolution. This history dependence, together with communication delays and uncertain nonlinear dynamics, makes accurate uncertainty compensation particularly challenging. Conventional feedforward neural approximators do not inherently retain temporal information, while fully recurrent architectures may introduce additional computational and online training complexity. To address this limitation, this article introduces the first application of a liquid state machine (LSM) to bilateral teleoperation control. A finite-time adaptive controller is developed using a hybrid position/force auxiliary error system with velocity and force filters, while the LSM is employed to estimate uncertain dynamics by exploiting its intrinsic temporal processing and fading-memory capabilities with a simple adaptation mechanism. Closed-loop stability and finite-time convergence are established through a Lyapunov--Krasovskii framework. Simulations in spring--damper and generalized Maxwell viscoelastic environments demonstrate improved position and force tracking and lower mean execution time compared with an RBFNN-based controller.

1. Introduction

The introduction identifies gaps in finite-time position–force teleoperation control and in uncertainty estimators that represent history-dependent dynamics. It proposes an LSM-based estimator within an acceleration-free adaptive controller, with Lyapunov–Krasovskii stability analysis.

  • Research gaps: Existing adaptive controllers based on prescribed regression models cannot directly reconstruct friction, contact-dependent effects, actuator nonlinearities, and other unmodeled terms.These limitations motivate broader uncertainty representations beyond selected parameterizations.
  • Research gaps: Conventional RBFNN and feedforward approximators require delayed or filtered regressors to supply temporal information for history-dependent uncertainties.Fixed-basis RBF designs also constrain the representation and may establish bounded weight errors rather than ideal-dynamics identification.
  • Proposed approach: The proposed LSM uncertainty approximator intrinsically preserves fading-memory information from motion, interaction, and delayed coupling conditions.Its liquid layer is intended to represent frictional memory, contact transitions, operator–environment variation, and delayed bilateral coupling.
  • Research gaps: Finite/fixed-time teleoperation research has emphasized position synchronization, while fewer controllers combine position–force tracking, acceleration avoidance, and asymmetric time-varying delays.The introduction presents this combination as an underdeveloped area.
  • Proposed approach: A finite-time hybrid position/force controller uses auxiliary error systems with velocity and force filters to avoid direct acceleration measurements and explicit time-delay derivatives.The controller targets position synchronization and force tracking under asymmetric time-varying delays.
  • Analysis: A Lyapunov–Krasovskii framework derives bounded closed-loop signals and finite-time convergence of position/force tracking errors under stated assumptions.The stability result accounts for delayed master–slave dynamics and adaptive compensation terms.

2. Preliminaries

The bilateral teleoperation model represents master and slave manipulators as serial revolute-joint chains with standard rigid-body dynamics. The stability analysis uses bounded inertia and skew-symmetry properties.

  • System notation: The indices j∈{m,s} denote the master and slave sides, while j′ denotes the opposite side.This indexing is used throughout the bilateral model.
  • Rigid-body dynamics: The manipulator dynamics include joint position, velocity, and acceleration, inertia, centripetal–Coriolis, gravity, friction, and commanded actuator torque terms.External partner torques act on each manipulator, with human torque at the master and environment torque at the slave.
  • Stability properties: The inertia matrix is symmetric positive definite and admits configuration-independent eigenvalue bounds.These bounds form one structural property used in the stability analysis.
  • Stability properties: The matrix Ṁ_j−2C_j is skew symmetric for each master and slave side.This is the second stated structural property used in the stability analysis.

2.2. Known and Unknown Parts of the Dynamics

The model separates available nominal dynamics from unknown residual components and aggregates the remaining uncertainty into a single term for estimation and compensation.

  • Dynamics decomposition: The inertial, Coriolis, and gravitational terms are decomposed into nominal components computable online and unknown residual components.Exact values of these dynamics are not assumed available in practice.
  • Dynamics decomposition: Friction is treated as entirely unknown, and unknown model terms together with interaction terms are lumped into Π_j.The aggregated uncertainty is assembled for subsequent adaptive estimation.

2.3. Operator and Environment Interaction

The operator and environment are modeled as passive spring–damper couplings. Their state-dependent stiffness and damping contributions are absorbed into the lumped uncertainty, so their parameters need not be known.

  • Interaction model: The operator and environment are modeled as passive spring–damper couplings.This interaction model supplies the external coupling structure for the bilateral system.
  • Interaction model: The damping and stiffness matrices are constant, diagonal, and positive definite for both operator and environment couplings.These properties render the couplings passive.
  • Uncertainty treatment: Measured-state-dependent stiffness and damping terms are absorbed into Π_j, eliminating the need to know the operator or environment coupling matrices.The controller therefore handles these interaction contributions through the lumped uncertainty representation.

2.4. Communication Delays

The teleoperation channel has separate forward and backward communication delays, and these delays are generally asymmetric and time-varying.

  • The forward delay T_m(t) runs from master to slave, while the backward delay T_s(t) runs from slave to master.
  • The channel is asymmetric because T_m(t) and T_s(t) are not generally equal.
  • Each delay is non-negative and has positive bounds on its magnitude and time-variation rate.

2.5. Standing Regularity Condition

The formulation assumes that lumped uncertainty is bounded and causally dependent on recent closed-loop history, including delayed signals and communication-delay variation.

  • On a compact admissible domain Ω_j, Π_j is a bounded causal fading-memory functional of closed-loop history and bounded delay rate.
  • Π_j depends on delayed position-side velocities and the varying-delay rate, rather than only on the instantaneous state.
  • The delay rate is unavailable pointwise but influences the time course of the received signal.

2.6. Problem Formulation and Control Objectives

The problem formulation defines locally available delayed position and force errors, then specifies finite-time stability, tracking, uncertainty reconstruction, and sparse computation objectives for asymmetric delays.

  • 2.6.1. Tracking errors: Position synchronization compares each local joint position with the delayed position received from the partner.
  • 2.6.1. Tracking errors: Force tracking errors are defined using interaction torques obtained from the force sensors.
  • 2.6.1. Tracking errors: Both position and force errors use locally available data by penalizing each side’s mismatch against the delayed partner signal.
  • 2.6.2. Control objectives: The controller must achieve finite-time reaching with bounded auxiliary variables and adaptive estimation errors under asymmetric time-varying delays and dynamic uncertainty.
  • 2.6.2. Control objectives: Position and force objectives separately assess transient and steady-state magnitudes of their respective delayed tracking errors.
  • 2.6.2. Control objectives: The estimator must reconstruct delay-dependent uncertainty while remaining linear in adaptive parameters and representing history without fixed hand-crafted delayed regressors.
  • 2.6.2. Control objectives: Sparse event-driven computation is required, with cost quantified by synaptic operations involving active units per control step.
  • 2.6.2. Control objectives: Position synchronization does not imply force tracking, and the relative position-force emphasis is controlled through χ_j1 and χ_j2.

2.7. Liquid State Machine as Uncertainty Estimator

The LSM represents history-dependent uncertainty with a fixed recurrent spiking reservoir and an online-adapted linear readout, combining temporal features with Lyapunov-compatible adaptation and sparse activity.

  • The estimator models Π_j as a functional of closed-loop history using an LSM with fixed internal connectivity and one adapted linear readout.
  • The four stages are input injection, a recurrent spiking liquid, spike filtering into continuous traces, and an online-adapted readout forming Π_j.
  • The liquid expands input history into a high-dimensional transient state, while the readout extracts task-relevant information from that state.
  • Input and recurrent weights remain fixed, so reservoir nonlinearity and memory are not adapted; only the linear readout is updated online.
  • Keeping the reservoir fixed preserves linearity in adaptive parameters, allowing the underlying adaptive law and Lyapunov certificate to carry over unchanged.
  • The bounded spike indicator and low-pass filtering produce continuous reservoir features bounded in [0, 1], yielding a bounded augmented readout vector.
  • The estimator assumes a sufficiently large reservoir and constant readout can approximate the uncertainty with bounded residual over the admissible domain.
  • The reservoir embeds input history through fading memory, allowing dynamic uncertainty recovery with a purely linear adaptive readout instead of fixed lagged inputs.

2.8. Preliminary Lemmas

The section introduces three lemmas supporting bounded filtered reservoir states, finite-time convergence, and practical finite-time convergence to a residual set.

  • Preliminary tools: Three lemmas provide the preliminary tools for the stability and finite-time analysis.They cover a power-sum inequality, finite-time convergence, and boundedness of a leaky integrator.
  • Power-sum inequality: A power-sum inequality supports combining multiple fractional-power terms into a single bound.The inequality applies to real-valued terms with exponent p in (0, 1).
  • Finite-time convergence: A Lyapunov function reaches the origin in finite time when its derivative satisfies the stated fractional-power inequality.The lemma also gives a finite-time upper bound for the settling time.
  • Practical finite-time convergence: When the derivative condition holds only outside a compact set, the trajectory reaches that set within the corresponding finite-time bound.The lemma does not assert behavior after the trajectory enters the set.
  • Bounded filtered states: A leaky integrator driven by a bounded signal remains in the unit interval when initialized there.Variation of constants shows the state is bounded by a convex combination of its initial value and the drive bound.

3. Control Design

The control design combines delayed position and force errors into filtered auxiliary variables and reconstructs lumped, history-dependent uncertainty with an LSM without acceleration or delay-rate regressors.

  • Hybrid error construction: The design combines position and force tracking errors into one hybrid error per side, then adds fractional-power and integral terms.The resulting auxiliary variable is the basis for the controller design.
  • Force filtering: A force-error filter admits force feedback without requiring operator or environment torque rates.The filter has unit dc gain, so a constant force error reaches the filtered state unattenuated.
  • Velocity filtering: A delay-consistent velocity error and velocity feedback filter avoid requiring the unavailable delay rate and acceleration during implementation.The filter is driven by delayed velocity received through the communication channel and coincides with the delay-consistent error in steady state.
  • Finite-time auxiliary variable: The fractional-power construction places the exponent below one under an integral, preventing the singularity associated with differentiating a negative power.The exponent above one remains continuous when differentiated.
  • Uncertainty reconstruction: The lumped uncertainty includes model, interaction, and delay-rate terms, with the unknown product involving the inertia mismatch reconstructed together with the remaining uncertainty.The delay rate is retained explicitly where its multiplier is known and absorbed into the lumped uncertainty where the multiplier is unknown.
  • LSM estimator: The LSM estimates history-dependent uncertainty from filtered and measured signals using fixed reservoir weights and an adapted linear readout.Its input uses filter states, measured positions and velocities, and delayed velocity; acceleration and delay rate are excluded.

4. Finite-Time Control Law and Stability Analysis

The controller cancels nominal dynamics, compensates lumped uncertainty, and uses adaptive laws selected from the Lyapunov derivative. Under the stated assumptions, closed-loop signals are bounded and the auxiliary errors reach a residual set in finite time.

  • Control and adaptation laws: The control torque cancels nominal feedforward, compensates lumped uncertainty through the reservoir estimate, and dominates the unmeasured delay-rate and residual terms.The adaptive laws are chosen to cancel sign-indefinite estimation-error terms in the Lyapunov derivative.
  • Lyapunov cancellation: Fixed reservoir weights and an adapted readout make the estimator linear in its adaptive parameters, preserving the Lyapunov cancellation structure.The cancellation requires the reservoir feature vector to be independent of the adapted readout parameters and does not require a bound on its norm at that step.
  • Stability guarantee: Theorem 4 establishes uniform ultimate boundedness of the closed-loop variables and finite-time reaching of the auxiliary residual set.The auxiliary variables reach the set within a finite time bounded by the theorem’s reaching-time expression.
  • Residual-set size: The residual set depends on plant and specification terms, leakage, and reconstruction residual, while increasing reservoir size reduces the reconstruction contribution.Increasing K_j1 contracts the set, whereas reducing leakage reduces the bound but slows parameter-error decay.
  • Scope of the guarantee: The finite-time guarantee does not depend on the reservoir’s internal spiking structure, recurrent connectivity, or fading memory; those properties affect only residual-set size.The reservoir enters the proof through parameter-independent features and the residual bound.
  • Design trade-off: The parameter θ trades residual-set size against reaching time, with values near one shrinking the set while lengthening the finite-time bound.The gains also shape the residual set and reaching time without requiring a lower bound beyond positive definiteness.

5. Simulation Results

Simulations evaluate the LSM-based finite-time controller under time-varying delays, uncertain robot dynamics, and spring–damper or history-dependent viscoelastic interactions. The results show bounded adaptive behavior, accurate tracking, and improved performance over an RBFNN-based method.

  • Simulation setup: The simulations use two-link planar master and slave manipulators with 0.001 s sampling and specified robot, operator, and environment parameters.The operator and environment are modeled with damping and spring matrices, while uncertain dynamics are introduced through time-varying perturbations of nominal model terms.
  • Simulation setup: The LSM estimators use identical 50-neuron liquid reservoirs for the master and slave, with fixed membrane, synaptic, threshold, and weight-initialization settings.The simulation also reports controller and LSM parameter sets in the corresponding tables.
  • Stability and tracking verification: Position synchronization and force-tracking errors decrease rapidly and remain near zero, while estimated uncertainty torques and adaptive signals remain bounded.The auxiliary variables also decrease toward zero, and the control torques remain bounded without sustained high-frequency oscillations.
  • Comparison with other control methods: Compared with the RBF-based controller, the LSM method reduces position and force errors, control torques, and computational requirements under identical simulation conditions.The comparison uses tracking errors, control effort, and computational efficiency, with RMSE values evaluated over multiple time intervals and repeated random initializations.
  • Temporal-memory evaluation: 31.09 × 10−4 Rad and 11.14 × 10−2 N.m. are the proposed method’s average spring–damper position and force RMSEs, versus 52.19 × 10−4 Rad and 11.35 × 10−2 N.m. for RBFNN.In the generalized Maxwell environment, the corresponding proposed-method RMSEs are 25.33×10−4 Rad and 14.12×10−2 N.m., versus 924.44×10−4 Rad and 94.29×10−2 N.m. for RBFNN.
  • Temporal-memory evaluation: 2.7205 s is the proposed method’s mean execution time, compared with 3.2803 s for the RBFNN-based method.The generalized Maxwell environment highlights the LSM’s ability to capture temporal interaction effects without modifying or retuning the controller structure.
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