Bilateral Control Law Overview

• Bilateral control law is a dual feedback mechanism that applies actuation at both ends of a system to ensure synchronized command and force reflection.
• It employs techniques such as backstepping, passivity-based synthesis, and adaptive control to maintain stability under uncertainties and dynamic interactions.
• Applications include teleoperation, PDE boundary control, and model-free regulation, offering enhanced performance and fault tolerance over unilateral designs.

A bilateral control law is a systematic feedback mechanism in which control signals or actuation channels are applied simultaneously at multiple, usually opposing, points of a physical system—typically at both ends of a spatial domain or both sides of a coupled system—enabling enhanced performance in synchronization, force/motion reflection, stability, and adaptability under dynamic interactions or uncertainties. Bilateral control is foundational in teleoperation, distributed sensing/actuation in PDEs, and cooperative human–robot interaction, characteristically employing techniques such as backstepping, passivity-based synthesis, adaptive control, and signal extraction for imitation learning.

1. Bilateral Control Law in Robot Teleoperation

Bilateral control architecture in robot teleoperation aims to couple a human operator (master) and a remote/executive device (slave) such that the master’s position and the slave’s environmental interaction forces are bidirectionally reflected (Sasagawa et al., 2019, Takanashi et al., 2023). The canonical instantiation is the four-channel bilateral control law, composed of:

• Master–slave system: Two robots (3-DoF haptic manipulators) with measurable joint angles $(\theta_m, \theta_s)$, velocities $(\dot\theta_m, \dot\theta_s)$, reaction torques $(\tau_m, \tau_s)$, and actuator commands $(\tau_m^{\text{ref}}, \tau_s^{\text{ref}})$;
• Control objectives:
  • Position transparency: $\theta_m^i - \theta_s^i = 0$ for all joints $i$,
  • Force reflection: $\tau_m^i + \tau_s^i = 0$ for all joints $i$.

The feedback structure is: $$\tau_m^{\text{ref}} = -C_p(\theta_m^{\text{res}} - \theta_s^{\text{res}}) - C_f(\tau_m^{\text{res}} + \tau_s^{\text{res}})$$

$$\tau_s^{\text{ref}} = +C_p(\theta_m^{\text{res}} - \theta_s^{\text{res}}) - C_f(\tau_m^{\text{res}} + \tau_s^{\text{res}})$$

where $C_p$ and $C_f$ are position and force feedback operators. In the Laplace domain, $C_p(s) = (J/2)(K_p+K_ds)$ and $C_f(s) = (1/2)K_f$ with specified inertia $J$, gain $K_p$, derivative $K_d$, and force feedback $K_f$.

This law allows for independent extraction of command (human-intended) and response (robot-felt) signals, enabling high-fidelity skill transmission and subsequent end-to-end imitation learning via neural networks (Sasagawa et al., 2019). Stability and passivity criteria are met provided $C_p$ and $C_f$ are passive, making the closed loop behave as a lossless transmission line. The architecture extends to encrypted cyber-secure implementations with homomorphic encryption, inheriting performance and stability (Takanashi et al., 2023).

2. Bilateral Control Laws for Distributed Parameter Systems (PDEs)

Bilateral control extends naturally to boundary control of PDEs—reaction-diffusion, wave, and hyperbolic systems—where actuators are positioned at both spatial boundaries. The bilateral backstepping methodology generalizes the one-sided transformation by introducing integral kernels defined on hourglass-shaped (Goursat) domains allowing the simultaneous design of multiple boundary control laws (Vazquez et al., 2016, Sun et al., 3 Sep 2024).

For a 1-D reaction–diffusion system: $$u_t = \epsilon u_{xx} + \lambda(x)u, \quad u(-L,t) = U_2(t), \quad u(L,t) = U_1(t)$$ the backstepping map

$w(t,x) = u(t,x) - \int_{-x}^x K(x,\xi)u(t,\xi)d\xi$

yields the bilateral law

$$U_1(t) = \int_{-L}^L K(L,\xi)u(t,\xi)d\xi, \quad U_2(t) = -\int_{-L}^L K(-L,\xi)u(t,\xi)d\xi$$

The kernel $K$ solves a hyperbolic PDE on the symmetric domain, with closed-form Bessel function solutions in constant-coefficient cases.

For $2 \times 2$ hyperbolic systems (e.g., traffic or wave equations), bilateral controllers stabilize both states in finite time, even for spatially varying coefficients, using four kernel functions $L^{ij}(z,w)$ determined from coupled first-order PDEs (Sun et al., 3 Sep 2024), with explicit feedback laws: $$U_1(t) = \int_{-1}^1 \left( L^{11}(z,-1)u(z,t) + L^{12}(z,-1)v(z,t) \right)dz$$ and analogously for $U_2(t)$.

3. Adaptive and Robust Bilateral Control Schemes

Composite adaptive bilateral control laws provide robustness against dynamic uncertainties and time-varying communication delays, crucial for teleoperation and distributed coordination under practical conditions (Li et al., 2018, Wang et al., 2021). The use of auxiliary prediction error states $z_i$ and parameter covariance matrices $P_i$ removes the need for persistent excitation:

$$\dot{\hat{\theta}}_i = \Gamma_i \left( Y_i^T \eta_i + (\xi_i + \delta_i)z_i \right)$$

$$\dot{z}_i = -\mu_i z_i + Y_{io}^T e_{io} - P_i \dot{\hat{\theta}}_i$$

$\dot{P}_i = -\mu_iP_i + Y_{io}^T Y_{io}$

Performance measures $\Delta_{J_p}^i, \Delta_{J_f}^i$ quantify position/force tracking tightness. These schemes guarantee synchronized tracking and parameter convergence even under heterogeneous, time-varying delays by satisfying appropriate matrix inequalities (LMI conditions).

Controllers for closed architecture robots (with unknown PD/PID inner loops) utilize dynamic feedback and input–output analysis to ensure stability without assuming exponential convergence of inner-loop adaptation (Wang et al., 2021). The combination of outer velocity commands derived from adaptive observers and appropriate Lyapunov-Krasovskii functionals ensures closed-loop stability and robustness.

4. Bilateral Output-Feedback and Observer Design

When full-state measurement is unavailable or fault-tolerance is necessary, bilateral output-feedback and observer-based control methodologies are implemented (Chen et al., 2019, Bekiaris-Liberis et al., 2018). Observer design via folded diffusion domains and Volterra transformations enables rapid and robust estimation across both boundaries, and output feedback controllers retain stability and convergence properties analogous to full-state feedback.

• For coupled diffusion–ODE systems, bilateral boundary controllers are designed via two-tier Volterra transformations, yielding closed-loop exponential stabilization in $L^2 \times \mathbb{R}^n$ (Chen et al., 2019).
• For nonlinear PDEs (Hamilton–Jacobi classes), feedback linearization interlaced with bilateral backstepping and boundary observers achieves regional stability under explicit integral boundary law constructions (Bekiaris-Liberis et al., 2018).

5. Applications, Performance, and Comparison with Unilateral Designs

Bilateral controllers are deployed in domains requiring high transmission fidelity, disturbance rejection, and force/motion reflection—robotic teleoperation (serving food, manipulation under uncertainty (Sasagawa et al., 2019)), cyber-secure manipulation (Takanashi et al., 2023), distributed parameter system stabilization (chemical reactors, traffic networks (Yu et al., 2019, Vazquez et al., 2016)), and advanced PDE control with delay compensation (Guan et al., 2023).

Empirically, bilateral designs outperform unilateral counterparts when system coefficients (reaction rates, coupling intensities) are large, reducing per-actuator effort and offering intrinsic fault tolerance if one actuator fails (Vazquez et al., 2016, Sun et al., 3 Sep 2024).

6. Stability, Passivity, and Design Constraints

All major bilateral control laws cited are structured to ensure passivity—energy dissipation—and stability under arbitrary environment or delay conditions. Lyapunov-based analysis, invertible Volterra transformations, and passivity-based criteria are standard tools (Sasagawa et al., 2019, Takanashi et al., 2023, Li et al., 2018). Key assumptions for well-posedness include boundedness of kernel functions, regularity of system coefficients, actuator placement at controllable boundaries, and compatibility of initial/boundary data with domain geometry.

7. Data-Driven and Model-Free Bilateral Control

Recent advancements include direct data-driven bilateral regulation in discrete-time bilinear systems, where the optimal bilateral control law is learned from sample autocorrelation without explicit system identification, using matrix equality recursions derived from Pontryagin’s principle (Clarke et al., 2022). The resulting law,

$u_k = K^*(x_k)x_k$

is computed via iterative updates on data-derived Riccati equations and stationarity conditions, enabling model-free synthesis of optimal bilateral feedback policies.

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Bilateral control laws, encompassing real-time multichannel feedback, observer-augmented architectures, and robust/adaptive schemes, are fundamental to modern high-performance control applications, especially in complex, distributed, or uncertain environments requiring dynamic coordination at multiple actuated boundaries. These laws are mathematically characterized by symmetry, passivity, invertibility, and adaptability, offering rigorous guarantees for synchronization, reflection, estimation, and learning capacities.