4-Channel Bilateral Control in Teleoperation

• 4-Channel Bilateral Control is a teleoperation framework where a master and a slave exchange both motion (position/velocity) and force/torque signals to achieve synchronized movement and replicate interaction forces.
• It employs sensorless force estimation, dynamic compensation, and observer designs to accurately model system dynamics in both joint-space and Cartesian-space implementations.
• Recent studies validate the approach in high-speed, hard-contact, and secure teleoperation settings, enhancing motion transparency and enabling robust data acquisition for imitation learning.

Searching arXiv for papers on four-channel bilateral control, sensorless force control, and related teleoperation architectures. 4-channel bilateral control is a bilateral teleoperation architecture in which a leader or master manipulator and a follower or slave manipulator exchange both motion and force information so as to synchronize their motion while reproducing interaction forces. In the cited literature, the standard four-channel architecture is described as exchanging both position/velocity and force/torque signals between the master and slave and using dynamic compensation on each side to maximize transparency (Noohian et al., 1 Jul 2026). Recent work places the same architecture in several distinct settings: force-sensorless low-cost manipulators with accurate dynamics models (Yamane et al., 8 Jul 2025), human-scale teleoperation without external force/torque sensors (Noohian et al., 1 Jul 2026), Transformer-based imitation learning from position and torque information (Kobayashi et al., 2024), motion modification via multilateral control (Inami et al., 28 Feb 2025), Cartesian-space control with independent scaling of rotation, translation, and force (Yamane et al., 12 Oct 2025), cyber-secure teleoperation with homomorphic encryption (Takanashi et al., 2023), and arbitrary-scale force-sensor-less teleoperation on dissimilar master-slave systems (Lampinen et al., 2020).

1. Core objectives and transparency criteria

The formal objectives of 4-channel bilateral control are typically expressed as motion synchronization and action-reaction consistency. In joint space, representative formulations impose

$$\theta_l(t)-\theta_f(t)=0, \qquad \tau_l(t)+\tau_f(t)=0,$$

and, in steady state, also require velocity synchronization, so that the differential mode satisfies $x_l-x_F \to 0$ and $v_l-v_F \to 0$ while the common mode satisfies $\tau_l+\tau_F \to 0$ (Kobayashi et al., 2024, Inami et al., 28 Feb 2025). These relations summarize the two principal goals of the architecture: the follower should reproduce the leader’s motion, and the leader should feel the environment reaction encountered by the follower.

A frequency-domain formulation for a linearized 1-DOF two-port model writes the leader and follower dynamics as

$$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$

with a hybrid matrix $H(s)$ relating $\{X_l,F_e\}$ to $\{F_h,X_f\}$. Perfect transparency requires

$H=\begin{pmatrix}0&1\1&0\end{pmatrix},$

equivalently $h_{11}=0$, $x_l-x_F \to 0$0, $x_l-x_F \to 0$1, and $x_l-x_F \to 0$2 (Noohian et al., 1 Jul 2026). In human-scale experiments, two measurable transparency metrics are emphasized: hard-contact force tracking

$x_l-x_F \to 0$3

and maximum transmittable impedance

$x_l-x_F \to 0$4

These metrics are used to compare two-channel and four-channel schemes under both free motion and hard contact (Noohian et al., 1 Jul 2026).

The literature therefore treats 4-channel bilateral control less as a single controller than as a structural template for teleoperation. The same template can be instantiated in joint space or Cartesian space, with direct sensing or sensorless estimation, and with local dynamic compensation ranging from gravity cancellation to inverse-dynamics feedforward. What remains invariant is the four-way exchange of motion and force information and the associated transparency objective.

2. Signal architecture and channel decompositions

A canonical implementation consists of symmetric leader and follower controllers, each containing local control and observation blocks, with two motion-related signals and two force-related signals traversing the communication link. In one explicit formulation, the four logical channels are: position command, position feedback, torque command, and torque feedback (Kobayashi et al., 2024). In another, the architecture is described as two position channels and two force channels, with position and velocity feedforward combined with force feedback (Inami et al., 28 Feb 2025).

The low-cost sensorless architecture reported for fast teleoperation makes the channel structure more explicit at the controller level. At each control cycle of 1 kHz, the two sides exchange joint positions $x_l-x_F \to 0$5, estimated velocities $x_l-x_F \to 0$6, and estimated external torques $x_l-x_F \to 0$7 (Yamane et al., 8 Jul 2025). The control law is then decomposed through an oblique $x_l-x_F \to 0$8 coordinate transform into a position-position channel, a velocity-velocity channel, a force-force channel, and a zero-dynamics channel in which the average coordinate

$x_l-x_F \to 0$9

has pure inertia (Yamane et al., 8 Jul 2025). This decomposition separates synchronization dynamics from the average motion mode and is directly tied to gain selection in the difference and average coordinates.

In joint-space PD/P realizations, the torque references can be written as a sum of a position-coupling term and a force-reflection term. A representative form is

$v_l-v_F \to 0$0

$v_l-v_F \to 0$1

with the total reference adding a disturbance estimate (Kobayashi et al., 2024). This structure is mirrored in encrypted implementations, where each side contains a disturbance observer, a reaction-force observer, a position-PD branch, and a force-P branch, and the four network signals are the leader and follower joint angles together with the corresponding estimated reaction torques (Takanashi et al., 2023).

A recurring point across these variants is that “four channels” does not mean a single fixed list of transmitted variables. Some implementations transmit measured torques, others transmit estimated external torques; some make velocity an explicit exchanged signal, whereas others absorb velocity into local differentiation or PD action. The common feature is the simultaneous closure of motion and force loops across the leader-follower pair.

3. Dynamic compensation and sensorless force estimation

The effectiveness of 4-channel bilateral control depends strongly on how accurately each side compensates its own dynamics. For rigid serial-link manipulators, one cited formulation uses

$v_l-v_F \to 0$2

where $v_l-v_F \to 0$3 is the inertia matrix, $v_l-v_F \to 0$4 collects centrifugal and Coriolis terms, $v_l-v_F \to 0$5 is viscous friction, and $v_l-v_F \to 0$6 is gravity torque (Yamane et al., 8 Jul 2025). After feedforward compensation with identified models and insertion of observer estimates, the controller is posed in an acceleration form, and the reference torque is chosen as

$v_l-v_F \to 0$7

This realizes approximate feedback linearization and supports acceleration-based bilateral control on affordable hardware (Yamane et al., 8 Jul 2025).

A parallel sensorless formulation for human-scale teleoperation uses the Euler-Lagrange model

$v_l-v_F \to 0$8

with $v_l-v_F \to 0$9 denoting viscous friction (Noohian et al., 1 Jul 2026). The model is written in linear-in-parameters form,

$\tau_l+\tau_F \to 0$0

and rich excitation trajectories together with least squares yield an estimate $\tau_l+\tau_F \to 0$1 (Noohian et al., 1 Jul 2026). In contact, external torque is estimated by inverse dynamics:

$\tau_l+\tau_F \to 0$2

and the Cartesian wrench maps through the Jacobian as $\tau_l+\tau_F \to 0$3 (Noohian et al., 1 Jul 2026). This eliminates the need for end-effector force/torque sensors and, in the human-scale study, avoids the limitation that such sensors measure only tip forces and are blind to body contacts (Noohian et al., 1 Jul 2026).

Observer design is another major branch of the literature. In the fast sensorless low-cost system, a minimal-order observer estimates velocity and external torque under the assumption that the disturbance $\tau_l+\tau_F \to 0$4 is slowly varying, and the resulting Laplace-domain expressions reduce to second-order filter forms parameterized by damping $\tau_l+\tau_F \to 0$5 and cut-off $\tau_l+\tau_F \to 0$6 (Yamane et al., 8 Jul 2025). In the encrypted two-axis implementation, a disturbance observer estimates lumped disturbance torque and a reaction-force observer subtracts the known nonlinear term $\tau_l+\tau_F \to 0$7 to recover external torque, with both DOB and RFOB cut-off frequencies set to $\tau_l+\tau_F \to 0$8 (Takanashi et al., 2023). In arbitrary-scale force-sensor-less teleoperation, reaction-force estimation is obtained from inverse dynamics and filtered by a first-order filter, while exogenous hand force is modeled as $\tau_l+\tau_F \to 0$9 and estimated adaptively (Lampinen et al., 2020).

Taken together, these formulations show that sensorless 4-channel control is not a single estimation algorithm but a family of model-based strategies. Accurate inverse dynamics, DOB/RFOB pipelines, and minimal-order observers all serve the same teleoperation objective: replacing dedicated force sensing with sufficiently accurate joint-side estimation of reaction torque or external wrench.

4. Passivity, Lyapunov stability, and time delay

Passivity is the dominant analytical framework for 4-channel bilateral control. In one formulation, the position-PD and force-coupling mappings

$$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$0

are positive for $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$1, and the master-slave interconnection becomes lossless when communication delay is negligible (Kobayashi et al., 2024). In the low-cost acceleration-based formulation, full dynamic compensation together with passive desired impedances in the $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$2 directions renders each side a lossless or strictly passive map from $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$3 to $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$4, and a theorem attributed there to Lawrence states that a 4-channel network of two passive one-ports interconnected in $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$5 coordinates or via wave variables remains passive and hence Lyapunov-stable under arbitrary constant communication delay (Yamane et al., 8 Jul 2025).

The standard passivity-preserving transformation under delay is the scattering or wave-variable representation

$$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$6

with wave impedance $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$7 (Kobayashi et al., 2024, Yamane et al., 8 Jul 2025). In a delayed channel, sending $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$8 and $$Z_l(s)X_l(s)=F_h(s)+F_l(s), \qquad Z_f(s)X_f(s)=-F_e(s)+F_f(s),$$9 preserves passivity even under constant delay (Yamane et al., 8 Jul 2025). This same device appears in discussions of encrypted and LAN-based implementations as the standard remedy when non-negligible constant delay is present (Kobayashi et al., 2024).

Lyapunov analysis appears in both joint-space and Cartesian-space variants. In the decoupled-scaling Cartesian method, the closed-loop energy

$H(s)$0

is positive definite if $H(s)$1, and along the desired dynamics its derivative satisfies $H(s)$2 (Yamane et al., 12 Oct 2025). In the arbitrary-scale dissimilar master-slave system, local Lyapunov-like functions are built for master and slave subsystems and then combined with a small-gain analysis showing that arbitrary constant delay does not induce instability when the algebraic gain conditions are satisfied (Lampinen et al., 2020).

A notable practical controversy concerns how much model compensation should be retained inside a passive design. The human-scale WAM study explicitly reports that only viscous friction was retained to avoid energy injection, that static/Coulomb friction degraded passivity, and that the force-feedback gain was constrained to $H(s)$3 and chosen as $H(s)$4 to trade off maximal $H(s)$5 versus passivity (Noohian et al., 1 Jul 2026). This indicates that “better compensation” is not automatically preferable if it compromises passivity margins.

5. Scaling, Cartesian formulations, and secure/networked variants

Several recent studies generalize 4-channel bilateral control beyond symmetric joint-space teleoperation. One direction is arbitrary scaling. In the dissimilar force-sensor-less system, the communication layer incorporates both position scaling $H(s)$6 and force scaling $H(s)$7, and experiments report a force scaling factor of up to $H(s)$8 and a position scaling factor of up to $H(s)$9 (Lampinen et al., 2020). The cited experiments use a commercial haptic master and a 2-DOF hydraulic slave, with rigorous stability guarantees under arbitrary time delays (Lampinen et al., 2020).

A second direction is Cartesian-space 4-channel control. The decoupled-scaling method on a 6-DoF manipulator represents pose as

$\{X_l,F_e\}$0

and defines scaled orientation, translation, and wrench errors by

$\{X_l,F_e\}$1

(Yamane et al., 12 Oct 2025). The control law injects desired second-order error dynamics through an inverse of a stacked matrix $\{X_l,F_e\}$2 built from Cartesian error Jacobians and inverse inertias, with $\{X_l,F_e\}$3 and $\{X_l,F_e\}$4 acting as virtual stiffness and damping and $\{X_l,F_e\}$5 acting as a virtual mass felt by the operator (Yamane et al., 12 Oct 2025). The stated goal is to achieve desired dynamics by decoupling each dimension in Cartesian coordinates regardless of the scaling factor (Yamane et al., 12 Oct 2025).

A third direction is cyber-secure implementation. The encrypted system uses ElGamal homomorphic encryption with key length $\{X_l,F_e\}$6 bits, quantizes controller matrices and signals with scale $\{X_l,F_e\}$7, and keeps both controller parameters and all four network signals encrypted in transmission and in the controller’s memory (Takanashi et al., 2023). The real-time implementation runs on leader and follower PCs with 20 ms sampling, and the reported average compute times are approximately $\{X_l,F_e\}$8 ms for the unencrypted controller and approximately $\{X_l,F_e\}$9 ms for the encrypted controller, remaining below the deadline (Takanashi et al., 2023). Quantization error in motor current is reported to be on the order of $\{F_h,X_f\}$0 A, compared with D/A resolution of approximately $\{F_h,X_f\}$1 A (Takanashi et al., 2023).

These variants expand the scope of 4-channel bilateral control without changing its defining bilateral structure. A plausible implication is that the framework is better viewed as a teleoperation topology onto which scaling laws, Cartesian error definitions, and security mechanisms can be layered.

6. Data collection, motion editing, and imitation learning

Recent work uses 4-channel bilateral control not only for teleoperation but also as a data-generation and motion-editing substrate. In ILBiT, bilateral teleoperation data are logged at 100 Hz as master and follower state vectors

$\{F_h,X_f\}$2

which are stacked over five motors into 15-dimensional vectors $\{F_h,X_f\}$3 (Kobayashi et al., 2024). A 4-layer Transformer encoder receives the follower state $\{F_h,X_f\}$4 and predicts the next master state $\{F_h,X_f\}$5, which then replaces the human leader during closed-loop execution (Kobayashi et al., 2024). The bilateral controller therefore serves both as the source of demonstration data and as the low-level stabilization mechanism during deployment.

The 2025 low-cost sensorless study states directly that using data collected by 4-channel bilateral control and incorporating force information into both the input and output of learned policies improves performance in imitation learning (Yamane et al., 8 Jul 2025). It further reports that task demonstrations such as nut turning, cucumber peeling, and dual-arm pick-and-place succeeded only with 4-channel force feedback, and that imitation-learning policies trained on 4-channel data with force in input and output achieved 100% success where unilateral data failed (Yamane et al., 8 Jul 2025). This places force reflection not merely in the haptics loop but inside the learning representation itself.

Motion ReTouch extends the same logic to post hoc correction of demonstrations. The method records follower command trajectories $\{F_h,X_f\}$6 at 500 Hz during four-channel teaching and replays them through a three-node multilateral system composed of leader, follower, and editor (Inami et al., 28 Feb 2025). The multilateral constraints are

$\{F_h,X_f\}$7

and the follower torque reference blends leader replay and editor intervention through a parameter $\{F_h,X_f\}$8, set to $\{F_h,X_f\}$9 in the experiments (Inami et al., 28 Feb 2025). The authors state that the method retroactively modifies not only position but also force information and that no explicit batch optimization is performed (Inami et al., 28 Feb 2025).

These developments shift the role of 4-channel bilateral control from an operator-facing interface to a data-centric control primitive. This suggests that its importance in current robotics extends beyond transparency alone to the acquisition, editing, and deployment of contact-rich demonstrations.

7. Reported operating regimes and empirical results

The empirical record across the cited studies spans low-cost manipulators, human-scale WAM systems, dissimilar haptic-hydraulic platforms, encrypted two-axis robot arms, and multilateral editing systems.

Study	Setting	Reported outcomes
(Yamane et al., 8 Jul 2025)	High-speed shake test, base joint swung $H=\begin{pmatrix}0&1\1&0\end{pmatrix},$0 at $H=\begin{pmatrix}0&1\1&0\end{pmatrix},$1 Hz	Angle MAE: unilateral 2.39; symmetric-position 1.58; force-feedback 1.69; 4-ch fixed-inertia 2.29; 4-ch w/o C & C 0.71; 4-ch pseudo-diff 0.81; 4-ch (proposed) 0.61. Velocity MAE: proposed 9.2 vs. 13.1–48.1 for others. Torque MAE: proposed 0.52. Stability with zero added damping; crisp transparency; tolerated contact impacts.
(Noohian et al., 1 Jul 2026)	Customized WAM bilateral teleoperation; free motion, hard contact, door opening	4c-DC outperforms all baselines in free-motion NRMSE, leader impedance, and maximum transmittable impedance ($H=\begin{pmatrix}0&1\1&0\end{pmatrix},$2). Force-tracking NRMSE similar across controllers ($H=\begin{pmatrix}0&1\1&0\end{pmatrix},$3 on leader joints). External torque estimation for 4c-DC: leader joint 2 NRMSE 3.23% ($H=\begin{pmatrix}0&1\1&0\end{pmatrix},$4), follower joint 2 11.72% ($H=\begin{pmatrix}0&1\1&0\end{pmatrix},$5).
(Inami et al., 28 Feb 2025)	Test-tube transfer with 3× speed-up	Original-speed motion copying: 10/10. 3× speed open-loop copying: 0/10. 3× speed with Motion ReTouch ($H=\begin{pmatrix}0&1\1&0\end{pmatrix},$6): 10/10. Joint 4 reaction torque fluctuations during search/insert are stabilized around zero.
(Takanashi et al., 2023)	Encrypted two-axis teleoperation, free space and contact	Free-space unencrypted errors: yaw $H=\begin{pmatrix}0&1\1&0\end{pmatrix},$7, pitch $H=\begin{pmatrix}0&1\1&0\end{pmatrix},$8; encrypted errors: yaw $H=\begin{pmatrix}0&1\1&0\end{pmatrix},$9, pitch $h_{11}=0$0. Estimated reaction torque remains within $h_{11}=0$1 Nm. Encrypted and unencrypted systems are reported as indistinguishable in posture synchronization and force feedback.
(Lampinen et al., 2020)	Dissimilar master-slave system with arbitrary scaling; planar contact-slide-release	Position tracking error stayed below a few mm; 2 DOF RMS error $h_{11}=0$2 mm even at 4× scaling. Force-tracking error remained bounded and small. Stable tracking under 80 ms one-way delay, with modest overshoot in x-y force curves.

The combined evidence shows that 4-channel bilateral control is routinely evaluated in regimes where unilateral position transmission is insufficient: high-speed motion, hard contact, sustained body contact, scaled teleoperation, encrypted communication, and learning-oriented data acquisition. A plausible implication is that current performance bottlenecks lie primarily in model fidelity, friction handling, observer bandwidth, and delay/security integration rather than in the bilateral architecture itself.