4-Channel Bilateral Control System

• 4-channel bilateral control system is a teleoperation architecture that synchronizes both position and force between master and slave devices for enhanced transparency.
• The system employs four logical channels, combining delay compensation and advanced state estimation methods like fuzzy modeling and moving horizon estimation.
• Recent developments extend the approach with encrypted teleoperation, Cartesian scaling for multi-DOF robots, and multilateral control to improve performance and stability.

The four-channel bilateral control system is a canonical architecture in teleoperation and distributed control for achieving bidirectional synchronization of both position and force between two physically separated systems. Originally developed for haptic robotics, the four-channel formulation generalizes two-channel (position-error-based) and three-channel (force-reflecting) schemes, offering superior transparency and stability, especially in the presence of time-varying delays and model uncertainties. Modern developments span fuzzy-model-based state estimation, encrypted teleoperation for cybersecurity, Cartesian-decoupled scaling for arbitrary manipulator designs, and discrete-time implementations for digital networks.

1. System Architecture and Channel Definitions

The four-channel bilateral teleoperation architecture comprises two symmetric subsystems—"Master" and "Slave"—each typically a multi-degree-of-freedom (DOF) manipulator with joint states $q_i, \dot{q}_i$, control torques $T_i$, and external torques $T_{h}$ (human) and $T_{e}$ (environment). Four logical channels structure the controller and communication interfaces:

Channel	Direction	Signal(s)
Command	Master → Slave	Positions $q_m \to q_s$
Reaction	Slave → Master	Positions $q_s \to q_m$
Human-force	Master → Slave	Estimated $\hat{T}_h$
Environment-force	Slave → Master	Estimated $\hat{T}_e$

Each channel may incur transmission delays $T_1(t)$ and $T_2(t)$, which can be time-varying but are bounded.

The block-diagram realization typically cascades observer blocks (velocity, disturbance/force) with controller blocks (PD position, P force) on each side, with delayed channels interconnecting the master and slave controllers (Liao et al., 2018, Takanashi et al., 2023, Ghavifekr et al., 2020).

2. Dynamic Modeling and State Estimation

Dynamic modeling in four-channel control is critical for both feedback accuracy and system transparency. Conventional models utilize the $n$-DOF manipulator equation:

$$M(q) \ddot{q} + C(q,\dot{q})\dot{q} + g(q) + f_v(\dot{q}) + f_c(q) + F_d = T + T_e$$

Recent advances favor data-driven, uncertainty-aware models. The interval Type-2 Takagi–Sugeno (T-S) fuzzy model is notable for capturing dynamic uncertainties without requiring exact physics, using rules of the form:

$$\text{IF } x(k) \text{ is } \tilde{A}^l \text{ THEN } y_i(k) = \sum_j a_{ij}^l x_j(k) + f_i^l$$

with intervals for membership functions and consequent offsets, and blending via

$$y(k) = \sum_{l=1}^L \mu^l(x) [A^l x(k) + f^l(k)], \quad f^l(k) = \bar{f}^l + \frac{\Delta f^l}{2} \vartheta(k)$$

Identification leverages Gustafson–Kessel clustering and weighted least squares (Liao et al., 2018).

For observer design, moving horizon estimation (MHE) solves for joint states and uncertainties over a sliding window, robustly filtering sensor noise and dynamic variations by minimizing a quadratic cost:

$$J_k = \Vert X(k-N|k) - X(k-N) \Vert^2_{P_x} + \sum_{i=k-N}^{k-1} \Vert \vartheta(i|k) \Vert^2_{P_\vartheta} + \sum_{i=k-N}^{k} \Vert y(i) - C_i X(i|k) \Vert^2_{P_y}$$

The solution yields input-to-state stable state and force estimation under mild observability and weighting assumptions (Liao et al., 2018).

Alternative approaches employ disturbance observers (DOB) and reaction-force observers (RFOB), e.g., for sensorless force estimation in low-cost manipulators (Yamane et al., 8 Jul 2025), or encrypted DOB/RFOB chains under homomorphic cryptography (Takanashi et al., 2023).

3. Four-Channel Control Laws and Implementation

Canonical four-channel laws combine position and force information from both systems using structured gains and explicit delay compensation. Representative control laws:

Master control:

$$T_m(t) = K_m [\hat{\alpha}_s(t-T_2) - \hat{\alpha}_m(t)] - B_m \hat{T}_h(t) + K_h [\hat{T}_h(t) - \hat{T}_e(t-T_2)]$$

Slave control:

$$T_s(t) = K_s [\hat{\alpha}_m(t-T_1) - \hat{\alpha}_s(t)] - B_s \hat{T}_e(t) + K_e [\hat{T}_e(t) - \hat{T}_h(t-T_1)]$$

where $\hat{\alpha}$ are estimated joint accelerations, $K_m,K_s,B_m,B_s,K_h,K_e$ are diagonal gain matrices, and time-delayed signals compensate for channel lags. Each term directly maps to one of the four logical channels.

Discrete-time implementations recast controllers and observers as difference equations under sampling period $T$, with passivity preserved via wave-variable (scattering) transforms and frequency-domain bounds (Ghavifekr et al., 2020):

$$u_m[k] = \frac{\tau_m[k] + b_m v_m[k]}{\sqrt{2 b_m}}, \quad v_m[k] = \frac{\tau_m[k] - b_m v_m[k]}{\sqrt{2 b_m}}$$

Selection of gain and sampling parameters must negotiate a trade-off between transparency (operator feel) and stability (through frequency-domain passivity inequalities).

4. Extensions: Cartesian Coordination, Encryption, and Multilateral Control

Recent research extends four-channel bilateral control to more complex domains:

• Decoupled Cartesian Scaling: For disparate manipulator structures, control is performed in Cartesian space, decoupling each axis and allowing user-tunable scaling via rotation matrices and block-Jacobians (Yamane et al., 12 Oct 2025). The error dynamics for each axis become:

$$\ddot{x}_{e,j} + K_{d,j} \dot{x}_{e,j} + K_{p,j} x_{e,j} = 0, \quad \dot{w}_{e,j} + K_{w,j} w_{e,j}=0$$

Independent gains $K_p, K_d, K_w$ per dimension facilitate arbitrarily scaled synchronization and force-feedback.

• Encrypted Teleoperation: End-to-end encryption using homomorphic ElGamal cryptosystems ensures cyber-secure transmission of all control signals and parameters. Controller state $\xi_k$ and parameters $\Phi$ are quantized, encrypted, and multiplied homomorphically to produce encrypted commands, preserving closed-loop performance with minimal computational overhead (Takanashi et al., 2023).
• Multilateral and Motion Retouch: Expanding beyond bilateral, systems like Motion ReTouch introduce a third editing agent, enabling post hoc correction of both position and force trajectories within the same four-channel (and multichannel) architecture. This is achieved through blending and force-share laws, improving high-speed, complex task execution and success rates in benchmarks (Inami et al., 28 Feb 2025).

5. Stability, Passivity, and Performance Analysis

Stability and passivity remain paramount, particularly under time-varying delay, quantization, and model uncertainties. Composite Lyapunov functions demonstrate exponential convergence of velocity observers (Liao et al., 2018), input-to-state stability of MHE (Liao et al., 2018), and passivity of the closed-loop system so long as controller gains satisfy:

$$B_m \ge (T_{\max}^1 + T_{\max}^2) I, \quad B_s \ge (T_{\max}^1 + T_{\max}^2) I$$

and appropriate sign choices for the reflection gains.

Discrete systems follow frequency-domain passivity conditions; wave impedance parameters set bounds on allowable controller gains relative to the sampling rate (Ghavifekr et al., 2020). Delay and packet loss under digital networks are addressed using wave-variable transforms, ensuring preserved passivity even with asynchronous or lossy communication.

Transparency is assessed via force reflection accuracy, position RMS error, bandwidth, and operator impedance match. Proper gain scheduling and model compensation (nonlinear, inertial) yield sub-degree positional MAE, sub-0.05 Nm force bias, and millinewton-level tracking, outperforming unilateral or non-model-based implementations (Yamane et al., 8 Jul 2025, Liao et al., 2018).

6. Applications and Empirical Validation

The four-channel bilateral framework supports:

• Haptic teleoperation with high-fidelity force feedback, e.g., Phantom-like haptic devices under soft/hard contact and delay disturbances (Liao et al., 2018)
• Secure distributed manipulation over encrypted networks, using real-time homomorphic encryption (Takanashi et al., 2023)
• Cartesian-synchronized control for complex 6-DOF robotic arms, facilitating operator-intuitive scaling and task-space coordination (Yamane et al., 12 Oct 2025)
• High-resolution sensorless force estimation and teleoperation with low-cost manipulators for contact-intensive and high-speed tasks (Yamane et al., 8 Jul 2025)
• Motion retouch and multilateral correction for post-editing of robotic trajectories in both position and force—empowering efficient, reliable imitation learning (Inami et al., 28 Feb 2025)
• Fault-tolerant stabilization for hyperbolic PDE systems with bilateral boundary control, leveraging unique four-actuator backstepping transformations (Sun et al., 3 Sep 2024)

Experimental platforms range from dual-CRANE-X7 arm setups at 1 kHz, dual SCARA-style arms under encrypted control, to PDE boundary-controlled plants using 4-channel actuation for finite-time nullification.

7. Practical Design Rules and Future Directions

Design guidelines for four-channel bilateral teleoperation systems include:

1. Model-based observer chain: Employ interval Type-2 fuzzy models and MHE for robust force/velocity estimation without exact dynamics.
2. Explicit four-channel delays: Buffer and compensate all channels to mitigate time-varying network or computational delays.
3. Gain selection: Diagonalize and tune passive gain matrices for both position and force, respecting delay bounds and sampling constraints.
4. Passivity and transparency validation: Quantify operator/environment impedance match via RMS error, bias, bandwidth, and transparency indices.
5. Fault-tolerance: Utilize redundant actuator channels as in PDE boundary control to ensure stabilization under hardware faults (Sun et al., 3 Sep 2024).
6. Security: Where required, integrate homomorphic encryption on all controller states and signals with negligible quantization error (Takanashi et al., 2023).
7. Multilateral extension: For complex tasks, incorporate motion retouch and multi-agent blending for post-hoc correction and adaptive learning (Inami et al., 28 Feb 2025).

Ongoing research addresses robustness to sensor and model uncertainty, null-space motions in redundant manipulators, effects of sampling/discretization, and hardware validation with increased DOF and non-ideal communication.

In sum, the four-channel bilateral control system constitutes a mature and versatile methodology underpinning advanced teleoperation, distributed control, and cyber-physical robotic systems in both continuous and discrete-time domains.