RNEA Computed-Acceleration Controller

• The paper introduces an efficient computed-acceleration controller using RNEA to perform O(n) inverse dynamics, making it ideal for high-DoF robots.
• It integrates differentiable RNEA with explicit impedance control and friction compensation to achieve precise acceleration tracking under physical constraints.
• Experimental validations on multi-DoF platforms demonstrate robust performance with low error rates and rapid online adaptation in dynamic environments.

An RNEA-based computed-acceleration controller is a model-based control system that computes joint-level or task-space accelerations using the Recursive Newton-Euler Algorithm (RNEA) for efficient evaluation of robot dynamics. This architecture enables precise tracking and compliance shaping in robots—most notably in multi-DoF manipulators and legged locomotion modules—while accommodating physical constraints, friction compensation, and real-time adaptation. Recent works demonstrate differentiable RNEA, explicit impedance shaping, and integration with high-bandwidth sensor feedback, offering high-fidelity control and robust performance across simulated and physical platforms.

1. Principles of RNEA-Based Computed-Acceleration Controllers

Controllers of this class leverage the RNEA to compute inverse dynamics efficiently, mapping desired accelerations to actuator torques. Given the state $(q, \dot{q}, \ddot{q})$, the RNEA recursively evaluates the equations of motion for robots with arbitrary tree topologies. Unlike general matrix inversion, the RNEA allows $O(n)$ computation, making it suitable for high-DoF robots.

A typical control objective is to impose a desired acceleration profile in task or joint space: $M(q)\ddot{q} + C(q, \dot{q})\dot{q} + g(q) = \tau$ where $M, C, g$ represent the inertia matrix, Coriolis/centrifugal, and gravity terms, respectively.

Feedback linearization, fundamental to computed-acceleration control (Ros, 2023), cancels nonlinearities by constructing a torque command: $\tau = M(q)v + C(q, \dot{q})\dot{q} + g(q)$ with $$v = \ddot{q}_d - k_p(q-q_d) - k_v(\dot{q}-\dot{q}_d)$$ governing error convergence.

The RNEA may be directly parameterized for differentiable learning (Sutanto et al., 2020), or it may underlie an impedance controller that renders virtual springs and dampers in task space (Wang, 2 Oct 2025).

2. Differentiable Implementation and Physical Constraints

A key advance is the construction of differentiable RNEA, in which the algorithm forms a computational graph for gradient-based optimization (Sutanto et al., 2020). In practical deployments (e.g., PyTorch), the full forward and backward passes enable learning of inertial parameters via gradient descent: $$\text{ID} = \sum_{t=1}^T \|\tau_t - f(q_t, \dot{q}_t, \ddot{q}_t)\|^2$$ with $f(\cdot)$ performed by RNEA.

To preserve physical plausibility, structured parameterizations are imposed:

• Mass: $m = (\theta_{\sqrt{m}})^2 + b$, ensuring $m > 0$.
• Inertia: Cholesky ($I = LL^T + bI_3$), symmetric matrices, and covariance/triangular inequalities assure symmetry and positive-definiteness.

Structured approaches—DiffNEA SPD, Tri, Cov—constrain learning to physically consistent domains, yielding not only faster convergence but superior generalization when compared to unstructured models. These properties are evaluated on a 7-DoF KUKA IIWA, with structured methods requiring 2–3 training epochs and achieving normalized mean squared error (NMSE) commensurate with ground-truth models.

3. Friction Compensation and Impedance Objectives

Rendering precise compliance necessitates friction modeling. The controller may augment its torque commands with a smooth approximation to Stribeck friction: $$\tau_{\text{friction}} = [\tau_c + (\tau_s-\tau_c)e^{-(|\dot{q}|/|\dot{q}_s|)^a}]\tanh(\beta \dot{q}) + b\dot{q}$$ where $\tau_c$ is Coulomb friction, $\tau_s$ is static friction, $\dot{q}_s$ is the Stribeck velocity, and $(a,\beta)$ shape parameters select smoothness (Wang, 2 Oct 2025).

For impedance control, desired task-space behavior is expressed as: $$\Lambda(x)\ddot{\tilde{x}} + D\dot{\tilde{x}} + K\tilde{x} = F_{\text{ext}}$$ where $\tilde{x} = x_d - x$ is the error, $K$ and $D$ are commanded stiffness/damping, and $\Lambda(x)$ is the inertia matrix (not explicitly shaped in some implementations).

The computed-acceleration control law in joint coordinates can be written: $$\tau_{\text{imp}} = M(q)J^\dag(q)[\ddot{x}_d - \dot{J}(q, \dot{q})\dot{q}] + J^T(q)[D\dot{\tilde{x}} + K\tilde{x}] + C(q, \dot{q})\dot{q} + g(q)$$ where $J^\dag$ is a damped least-squares pseudoinverse of the Jacobian, conferring robustness near singularities.

4. Sensor Integration and Disturbance Observer Design

High-bandwidth sensor feedback—particularly acceleration measurement—enhances disturbance rejection in digital control loops (Sariyildiz, 2021). Conventional velocity-based disturbance observers (DOb) are subject to the waterbed effect, manifesting as unwanted sensitivity peaks at mid-to-high frequencies upon increasing observer gain $g_{\text{DOB}}$:

$$a\,g_{\text{DOB}}\,T_s \leq 2(1 - T_{su}), \quad a\,g_{\text{DOB}}\,T_s \leq 2(1 + T_{Tv})$$

where $a$ reflects plant parameterization and $T_s$ is sampling time.

In contrast, acceleration-measurement-based DObs remove the waterbed effect, and the sensitivity function

$S_A(z) = z^{-1}(1 + a\,g_{\text{DOB}}\,T_s)^{-1}$

improves monotonicity. Only accelerometer noise limits the attainable bandwidth, permitting aggressive disturbance rejection in RNEA-based controllers without compromising robust stability. This is formally elucidated via the Bode Integral Theorem for discrete systems.

5. Experimental Validation and Performance Metrics

Bench experiments demonstrate the capabilities of RNEA-based computed-acceleration controllers in physical platforms (Wang, 2 Oct 2025):

• Push–pull tests on SPARC’s spine module reveal force-displacement curves with commanded horizontal stiffness spanning 300–700 N/m, achieving ≤ 1.5% relative error and $R^2 \geq 0.992$ (with narrow 95% confidence intervals).
• Dynamic displace–release trials confirm mass–spring–damper behavior across damping settings ($b$ = 0, 2, 20, 40 N·s/m), with interpretable phase deviations arising from configuration-dependent inertia and frictional effects.
• A basic task-space PD controller produces roughly linear stiffness but exhibits increased variability and sensitivity to friction, underscoring the value of the RNEA-based approach.

Differentiable RNEA algorithms (Sutanto et al., 2020) in tracking and operational space control show fast convergence—2–3 epochs—and generalization to unseen trajectories both in simulation and on a real 7-DoF arm.

6. Structures and Implications for Robotics

A spectrum of RNEA-based controllers exists, from unstructured models (DiffNEA No Str) to highly structured parameterizations (Symm, SPD, Tri, Cov) (Sutanto et al., 2020). Increased structural embedding correlates with improved training and generalization, guiding learning toward physically consistent models and enabling rapid online adaptation when robot payloads or configuration change.

Applied to legged robots, such controllers allow programmable compliance shaping, support impact attenuation, and facilitate task-adaptive behaviors. Embedding physical constraints directly in the learning and control law streamlines online reparameterization and adaptation, reducing the need for repeated high-gain feedback interventions.

7. Applications and Broader Context

RNEA-based computed-acceleration controllers have emerged as principal tools in:

• Multi-DoF robot arms and manipulators (real-time inverse dynamics evaluation, adaptive compliance)
• Compliant spine modules in quadrupeds—SPARC’s architecture exemplifies closed-loop stiffness and damping shaping for sagittal-plane compliance (Wang, 2 Oct 2025)
• Hybrid control strategies in autonomous vehicles and traffic coordination, where tunable gains govern smooth acceleration profiles across heterogeneous conditions (Tzortzoglou et al., 2023)
• High-performance racing platforms, enabling accurate and robust acceleration-based pursuit control under dynamic tire and friction effects (Becker et al., 2022)

A plausible implication is that such controllers, equipped with structured learning and advanced observer design, will continue to accelerate closed-loop adaptation and physical interaction in increasingly complex robotic environments. This suggests a trajectory towards modular, open-source compliance platforms and integrable high-bandwidth control—supported by efficient real-time dynamics computation via RNEA—across the spectrum of modern robotics.