Composite Adaptive Bilateral Control

• Composite adaptive bilateral control is a method that integrates tracking and prediction errors to update parameters without relying on persistent excitation.
• It employs Lyapunov–Krasovskii functional analysis to ensure closed-loop stability and robust master–slave synchronization in the presence of time-varying delays.
• Simulation studies show improved tracking accuracy and exponential parameter convergence compared to classical adaptive schemes.

Composite adaptive bilateral control is a methodology for bilateral teleoperation systems, in which both the master and slave manipulators are controlled adaptively in the presence of dynamic uncertainties and time-varying communication delays. The central feature of composite adaptive control is updating parameter estimates using both tracking errors and prediction errors, enabling parameter convergence without relying on the persistent excitation (PE) condition. This approach, exemplified in the work of Li et al., addresses stability, synchronization, and parameter adaptation challenges inherent to nonlinear teleoperation with uncertain dynamics and delayed bidirectional communication (Li et al., 2018).

1. Mathematical Model of the Bilateral Teleoperation System

The bilateral teleoperation system consists of a single master and single slave robotic manipulator, each described by standard joint-space dynamics: $$M_m(q_m)\,\ddot q_m + C_m(q_m,\dot q_m)\,\dot q_m + G_m(q_m) = \tau_m + F_m,$$

$$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$

where $q_i$ encapsulates joint positions ($i=m,s$), $M_i(q_i)$ is the symmetric positive definite inertia matrix, $C_i(q_i,\dot q_i)$ models Coriolis and centrifugal effects, $G_i(q_i)$ represents gravity, $\tau_i$ is the control input torque, and $F_m$, $F_s$ are the interaction forces from the human operator and environment, respectively.

These dynamics admit a linear parametric form: $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$0 with known regressor $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$1 and unknown constant parameter vector $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$2.

2. Composite Adaptive Control Architecture

The composite adaptive bilateral control law is structured around the following elements:

Tracking Error Design: The round-trip position errors under time-varying delays $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$3, $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$4 are defined by

$$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$5

with filtered error variables

$$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$6

where $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$7 are positive definite diagonal gain matrices.

Control Input Law: By leveraging the robots' structural properties,

$$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$8

where $$M_s(q_s)\,\ddot q_s + C_s(q_s,\dot q_s)\,\dot q_s + G_s(q_s) = \tau_s + F_s,$$9 are positive definite diagonal feedback gain matrices, and $q_i$0 is the adapted regressor corresponding to delayed interactions.

Composite Adaptive Parameter Update: For parameter estimation, the law is formulated as

$q_i$1

$q_i$2

using the auxiliary prediction error variable $q_i$3 (with $q_i$4), adaptation gain $q_i$5, and “covariance-like” matrix $q_i$6, lower bounded by $q_i$7. The terms $q_i$8, $q_i$9, and $i=m,s$0 encode a forgetting factor, σ-modification (for parameter drift prevention), and error scaling, respectively.

This composite mechanism simultaneously exploits tracking error ($i=m,s$1) and prediction error feedback ($i=m,s$2), removing the requirement for a PE condition in driving $i=m,s$3.

3. Stability and Convergence Guarantees

Stability Analysis: Closed-loop stability is established via a Lyapunov–Krasovskii functional designed to accommodate time-varying delays: $i=m,s$4 with positive-definite $i=m,s$5, scalar $i=m,s$6.

The time-derivative of $i=m,s$7 admits an upper bound characterized by a matrix inequality: $i=m,s$8 with explicit expressions for block matrix $i=m,s$9, ensuring that the linear matrix inequality (LMI) $M_i(q_i)$0 yields $M_i(q_i)$1 under free motion ($M_i(q_i)$2).

Parameter Convergence: The inclusion of composite terms $M_i(q_i)$3 in the parameter update guarantees a non-vanishing negative-definite term $M_i(q_i)$4, ensuring exponential parameter convergence $M_i(q_i)$5 without PE, supported by Barbalat’s lemma.

4. Performance Quantification

To systematically assess synchronization between the master and slave manipulators, dimensionless performance measures are defined:

• Position Tracking Ratio (free motion):

$M_i(q_i)$6

• Force Tracking Ratio (contact scenario):

$M_i(q_i)$7

Corresponding integral indices over time,

$M_i(q_i)$8

allow for quantitative comparison of tracking accuracy between schemes.

5. Simulation-Based Evaluation

A planar two-degree-of-freedom teleoperation system was considered with varying physical and control parameters:

• Dynamics: Masses, lengths as specified for master and slave.
• Delays: Non-constant, time-varying; $M_i(q_i)$9, $C_i(q_i,\dot q_i)$0.
• Control Parameters: $C_i(q_i,\dot q_i)$1, $C_i(q_i,\dot q_i)$2, $C_i(q_i,\dot q_i)$3, $C_i(q_i,\dot q_i)$4, $C_i(q_i,\dot q_i)$5, $C_i(q_i,\dot q_i)$6.
• Scenarios: Free motion (no environmental contact) and interaction with a stiff wall.

Key findings:

Feature	Composite Adaptive Law	Classical Adaptive Law (Sarras et al. 2014)
Parameter Convergence	Exponential; no PE needed	Fails to converge if PE is absent
Synchronization	Accurate (by $C_i(q_i,\dot q_i)$7s)	Inaccurate under same conditions
Performance Indices	Lower $C_i(q_i,\dot q_i)$8	Higher (poorer tracking)
Sensor Requirements	Position/velocity only	Position/velocity only

This suggests that the composite adaptive approach confers substantial performance improvements over classical adaptive schemes, particularly in environments with significant communication delays and insufficient excitation.

6. Notable Implications, Limitations, and Distinctions

Composite adaptive bilateral control ensures (i) delay-dependent closed-loop stability via LMI constraints, (ii) master–slave asymptotic output synchronization, and (iii) precise convergence of uncertain dynamic parameters without the PE condition. The design obviates the need for acceleration or delay-derivative measurement, reducing sensor complexity. A plausible implication is increased robustness in teleoperation applications encountered in haptic feedback, remote surgery, and space robotics with nonstationary delays and model uncertainties.

The approach is distinct from earlier adaptive methods which depend on persistent excitation for parameter convergence and are thus less reliable when system excitation is insufficient. No factual information is present regarding experimental hardware validation or limitations beyond the considered parametric uncertainties and delay assumptions (Li et al., 2018).