Gray-Box Computed Torque Controller

• Gray-box computed torque controller is a hybrid nonlinear control approach that fuses physical modeling with data-driven estimation to compensate for model inaccuracies.
• It employs adaptive learning methods like Gaussian Processes and deep kernel learning to enhance robustness and ensure stable trajectory tracking in robotic systems.
• The approach offers improved sample efficiency, precise tracking, and analytically interpretable controller parameters across various applications.

A Gray-Box Computed Torque Controller (CTC) is a nonlinear control approach that leverages partial model knowledge and learning-based components to compensate for complex, nonlinear, and uncertain robot dynamics. Originating from the classical computed torque method, the gray-box CTC fuses available physical modeling (e.g., rigid-body or manipulator kinematics/dynamics) with data-driven estimation and online adaptation mechanisms. This hybridization enables stable trajectory tracking with higher robustness to model inaccuracies, improved sample efficiency in learning-based adaptation, and preservation of physically interpretable controller parameters.

1. Computed Torque Control: Fundamentals and Limitations

The computed torque method employs feedback linearization to decouple and linearize the nonlinear equations of motion in fully actuated robotic systems. Given a dynamics model,

$M(q)\ddot{q} + C(q, \dot{q})\dot{q} + g(q) = \tau$

the input torque is synthesized as

$\tau = M(q) v + C(q, \dot{q})\dot{q} + g(q) - u_f$

yielding closed-loop dynamics $\ddot{q} = v$, where $v$ is a virtual input, and $u_f$ is the frictional force. Standard design replaces $v$ with a PD or PID term based on tracking errors.

Advantages include global linearization and applicability of linear control design. However, the method critically relies on a precise dynamic model. Model errors due to unknown friction, compliance, or unmodeled dynamics degrade tracking performance and can induce instability or actuator saturation for aggressively-tuned gains (Ros, 2023).

Gray-box enhancements address these limitations by incorporating learning-based components to estimate and compensate for modeling inaccuracies while retaining analytic model structure.

2. Gray-Box CTC Structure and Model Integration

Gray-box CTCs embed partial knowledge of the dynamic system—such as structure derived from the Euler–Lagrange or Newton–Euler formulation—with data-driven estimators or adaptive parameterization.

Hybrid Model Formulation

A typical gray-box CTC for manipulator systems is

$$u_c = \hat{H}(q)\ddot{q}_d + \hat{C}(q, \dot{q})\dot{q}_d + \hat{g}(q) + f_{\mathrm{GP}}(q_c) - K_d(\dot{q}-\dot{q}_d) - K_p(q-q_d)$$

where $\hat{H}, \hat{C}, \hat{g}$ are estimated inertia, Coriolis, and gravity terms, and $f_{\mathrm{GP}}(q_c)$ is a machine learning-based term (e.g., Gaussian Process regression) trained to approximate the residual dynamics $\tilde{\tau} = \tau - \hat{\tau}$ (Beckers et al., 2018).

In mobile robot or exoskeleton settings, moderately simplified models—sometimes omitting Coriolis/centrifugal terms for computational efficiency—are combined with PID inner loops and learning/adaptive gain mechanisms (Hasan, 24 Sep 2024).

Parameter Constrained Learning

Controllers may allow learning or adaptation only on a subset of parameters (e.g., friction coefficients, gains), keeping the overall law analytically interpretable. Learning algorithms (such as deep RL) then optimize parameters within physically plausible value ranges enforced via transformation functions, e.g., $V_i = O_{c_i} + O_{r_i}\tanh(z_i)$ (Pishkhani, 30 Aug 2025).

3. Learning-Based Compensation and Adaptive Control Elements

To address unmodeled or unknown dynamics, gray-box frameworks rely on data-driven estimators:

Gaussian Process Regression

A GP is used to model residual dynamics, providing the correction term $f_{\mathrm{GP}}(q_c)$. Each output component is predicted as

$$f_{\mathrm{GP}, i}(q_c) \sim \mathcal{N}(\mu_i(q_c), \mathrm{var}_i(q_c))$$

with mean $\mu_i$ computed from a kernelized function of joint states and measurement data. As $N \rightarrow \infty$ training points, consistency guarantees that the mean of the GP cancels the model error (Beckers et al., 2018). The GP also quantifies uncertainty, enabling principled adaptation of control gains (Jorge et al., 2022).

Deep Kernel Learning

Combining neural network-based feature extraction $g(x, w)$ with kernel learning,

$k(g(x_i, w), g(x_j, w) \mid \theta, w)$

yields a model for the feedforward torque mapping incorporating both learned non-linearities and explicit model structure. This enhances scalability and uncertainty quantification, essential for real-time adaptation in robot control (Jorge et al., 2022).

Model-Free and Gray-Box Feedback Optimization

Model-based sensitivities (derived from the estimated model) and model-free zeroth-order gradient estimates are adaptively fused. The update direction is

$$\tilde{\phi}_k = \alpha_k\phi_{k,1} + (1-\alpha_k)\phi_{k,2}$$

where $\phi_{k,1}$ uses the model’s sensitivity and $\phi_{k,2}$ is an exploration-driven gradient. The weighting $\alpha_k$ adapts according to the reliability of the model and the cumulative model error (He et al., 5 Apr 2024).

4. Feedback Gain Scheduling and Compliance

Gray-box CTC frameworks employ uncertainty-aware gain scheduling to combine trajectory tracking accuracy and compliant interaction, particularly for human-robot collaboration:

• Gains $K_P, K_D$ in the control law are adapted based on the prediction variance $\sigma(\tau)_t$ returned by the learning model:

$$K_{P, t} = K_{\min} + (1-z)(K_{\max}-K_{\min}), \quad z = \exp\left(-C\frac{\sigma(\tau)_t-\sigma_n}{\sigma_f-\sigma_n}\right)$$

where $C$ is a tunable constant, and $\sigma_f, \sigma_n$ represent signal and noise variances (Jorge et al., 2022).

• In RL-tuned controllers, reward design encourages critically damped closed-loop behavior and penalizes excessive overshoot or control effort (Pishkhani, 30 Aug 2025).

A reduced gain produces a more compliant controller (lower impedance), crucial for safety and comfort in wearable robotics (Hasan, 24 Sep 2024).

5. Stability, Robustness, and Theoretical Guarantees

Several gray-box CTC strategies provide stochastic and asymptotic stability guarantees even under model uncertainties:

• Stochastic Boundedness: For finite data, the tracking error $e = q - q_d$ satisfies

$$\mathbb{P}\left( \sup_t \|e(t)\| \leq \delta \right) > 1 - \epsilon$$

for any $\epsilon>0$, ensuring closed-loop safety despite residual uncertainties (Beckers et al., 2018).

• Asymptotic Stability: As data and learning converge, tracking error approaches zero almost surely (Beckers et al., 2018).
• Convex Combination Adaptivity: Closing the loop between model-based and model-free correction ensures long-term convergence to local optima even if the analytic model is deficient, given sufficiently decaying model error and adaptive mixing (He et al., 5 Apr 2024).

6. Application Examples and Empirical Evidence

Gray-box CTCs have demonstrated superior performance in a range of robotic contexts:

System	Learning Mechanism	Empirical Result Highlights
SCARA manipulator	GP-based correction	CT-GP achieves lower RMSE than classical CT and low-gain controllers; competitive with high-gain PD (Beckers et al., 2018)
KUKA LBR iiwa manipulator	Deep kernel learning	Variable-gain control enables increased compliance and low RMSE across multiple task and payload scenarios (Jorge et al., 2022)
Differential-drive mobile	RL-tuned gray-box	Achieves larger domain of attraction and improved tracking vs. kinematic and standard CTM controllers (Pishkhani, 30 Aug 2025)
Lower-limb exoskeleton	Model reference, PID	Trajectory errors remain sub-degree under weight/height variability; PID loop robustly corrects parametric deviations (Hasan, 24 Sep 2024)

These results consistently demonstrate that gray-box CTCs can reduce training time or required data (as few as 11 short RL learning episodes (Pishkhani, 30 Aug 2025)), outperform black-box learning and traditional controllers in low-data and uncertain settings, and maintain low computational cost suitable for embedded applications.

7. Extensions, Generalization, and Future Directions

Gray-box CTC approaches generalize to diverse robot morphologies and operational environments:

• Generalization: Deep kernel models successfully adapt from simulated to real data, different payloads, and gripper configurations, with uncertainty-based gain adaptation ensuring uniform performance (Jorge et al., 2022).
• Time-Varying/Constrained Problems: Projections and adaptive gradient mixing extend gray-box feedback optimization to handle constraints, time-varying objectives, and steady-state maps (He et al., 5 Apr 2024).
• Architectural Variations: Exclusion of low-magnitude dynamic terms (e.g., Coriolis, centrifugal) to enhance computational efficiency, with fast feedback loops compensating for omitted dynamics (Hasan, 24 Sep 2024).

A plausible implication is that further integration of hybrid modeling, RL-based adaptation, and principled uncertainty quantification will enable resource-efficient, robust control for increasingly complex, dynamic, or partially known robotic systems—without sacrificing stability guarantees or real-time responsiveness.

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In summary, the Gray-Box Computed Torque Controller unifies analytic model structure, data-driven learning, and adaptive gain scheduling within a single, dynamically robust framework. It provides a coherent pathway to stable, high-performance control under both parametric and nonparametric uncertainty, with applications ranging from dexterous manipulation to safe, compliant exoskeleton assistance, and mobile robot tracking in uncertain environments (Beckers et al., 2018, Jorge et al., 2022, Ros, 2023, He et al., 5 Apr 2024, Hasan, 24 Sep 2024, Pishkhani, 30 Aug 2025).