Passivity-Based Whole-Body Control (PB-WBC)

• PB-WBC is a passivity-based control paradigm that leverages energy functions and Lyapunov proofs to guarantee robust stability in multi-DOF robotic systems.
• It employs quadratic programming to compute compliant contact forces under strict physical constraints, enabling reliable task execution across diverse platforms.
• The modular design of PB-WBC facilitates consistent parameter tuning and resilience against disturbances, including sensor noise and actuator uncertainties.

Passivity-Based Whole-Body Control (PB-WBC) is a rigorous and physically grounded control paradigm for complex robotic and biomechanical systems, wherein stability and robustness are systematically enforced through the passivity property of all control, task, and projection modules. PB-WBC has been developed for a wide array of platforms—humanoids, quadrupeds, exoskeletons, collaborative mobile manipulators, and even virtual avatars. Unlike acceleration-based approaches, PB-WBC is formulated primarily in task-force or generalized-impedance spaces, with passivity proofs derived from explicit Lyapunov (storage) energy functions at the system- or task-level. Its operational advantages include inherent robustness to unmodeled or dissipative effects, physically meaningful and uniform parameter tuning, and formal stability guarantees even in the presence of hard contacts, transmission delays, or noisy sensing.

1. Fundamental Principles and Mathematical Formulation

PB-WBC is universally constructed atop the full floating-base rigid-body dynamics, typically expressed in Center-of-Mass (CoM) coordinates for maximum physical transparency. For a system with $n$ internal DOFs, generalized configuration $q\in\mathbb{R}^{n+6}$, and velocity $\nu \in \mathbb{R}^{n+6}$, the dynamics are written as: $$M_c(q)\,\dot \nu + C_c(q,\nu)\,\nu + \tau_g = \begin{bmatrix}0_6\\tau\end{bmatrix} + J^T \begin{bmatrix} f_{\text{grf}} \ f_{\text{imp}}\end{bmatrix} - \tau_{\text{dist}}$$ Where $M_c$ and $C_c$ are the centroidal inertia and Coriolis matrices, $\tau$ are joint torques, $J$ stacks the relevant task and contact Jacobians, $f_{\text{grf}}$ are ground reaction/contact wrenches (with friction- and unilateral-constraints), $f_{\text{imp}}$ models desired impedance forces, and $q\in\mathbb{R}^{n+6}$0 collects all unmodeled disturbance terms (e.g., joint friction, unknown payloads, actuator dynamics) (Sovukluk et al., 24 Jul 2025, Fahmi et al., 2018).

The PB-WBC paradigm enforces a virtual impedance or “PD+” control law at the task level:

• For the main (e.g., CoM, torso) task, a desired wrench $q\in\mathbb{R}^{n+6}$1.
• For secondary tasks (arms, swing legs), $q\in\mathbb{R}^{n+6}$2.

The controller solves for feasible $q\in\mathbb{R}^{n+6}$3 via a quadratic program (QP), imposing physical constraints (friction cones, unilateral ground contact, actuation bounds) and force-tracking objectives. Once $q\in\mathbb{R}^{n+6}$4 is computed, joint torques follow from the dynamics' nullspace (Sovukluk et al., 24 Jul 2025, Fahmi et al., 2018).

2. Passivity and Lyapunov-Based Stability

Passivity in PB-WBC requires the closed-loop system to never inject more energy than it stores, even under external perturbations, delays, or dissipation. The canonical storage function for PB-WBC is

$q\in\mathbb{R}^{n+6}$5

Time-derivative yields

$q\in\mathbb{R}^{n+6}$6

if $q\in\mathbb{R}^{n+6}$7 is dissipative, thus establishing Lyapunov (energy) stability (Sovukluk et al., 24 Jul 2025, Moro et al., 2017, Fahmi et al., 2018).

PB-WBC retains passivity even with additional modules (e.g., task-space hierarchical projections, contact solvers, virtual guides), provided each is designed to be passive at its port. If projections are used, they must be implemented via physically realizable, energy-storing/dissipating operators (e.g., virtual springs and dampers), as purely algebraic projections can break passivity and introduce instability (0707.2275).

3. Optimization and Task Prioritization

The PB-WBC QP is typically formulated with decision variables corresponding to contact forces (and optionally, accelerations) and subject to strict task/physical constraints. For humanoids and quadrupeds,

$q\in\mathbb{R}^{n+6}$8

with constraints:

• $q\in\mathbb{R}^{n+6}$9 encodes CoM/task-space wrench errors.
• $\nu \in \mathbb{R}^{n+6}$0 penalizes deviation from desired contact ground forces.
• Physical feasibility: friction cones, unilateral contact, actuation, and joint limits (Sovukluk et al., 24 Jul 2025, Fahmi et al., 2018).

In exoskeletons and virtual avatars, prioritized projections into the null-space of higher-priority tasks are realized using passive mechanical analogues (not purely algebraic projections), leveraging secondary tasks for redundancy exploitation and internal posture shaping without violating passivity (Moro et al., 2017, 0707.2275).

In collaborative and kinematic (velocity-controlled) settings, the QP includes energy-tank constraints and convex Control Barrier Function (CBF) inequalities to arbitrate between passivity, safety, and secondary objectives (Benzi et al., 2022, Risiglione et al., 2021).

4. Robustness and Real-World Implementation

PB-WBC demonstrates marked invariance to unknown joint friction, actuator dynamics, and sensor noise, as passivity ensures that any such dissipative effects only accelerate convergence to the equilibrium without destabilization (Sovukluk et al., 24 Jul 2025). For robust contact handling (e.g., during non-ideal landings or impacts), additional dissipative terms (e.g., small contact damping proportional to contact point velocity) are injected to account for instantaneous rigid-contact model violation.

Parameter tuning in PB-WBC is physically transparent: all task-space gains (stiffness, damping) have consistent interpretations, regardless of the controlled task, and there is no inertia-dependent gain adjustment. This contrasts with acceleration-based WBCs, which require inertia-adaptive tuning due to the explicit appearance of mass in steady-state disturbance rejection (Sovukluk et al., 24 Jul 2025).

In the case of actuation or friction uncertainty, PB-WBC can reject constant disturbances up to bounded steady-state errors, the size of which scales as $\nu \in \mathbb{R}^{n+6}$1 and is inertia-invariant.

The modular passivity-based architecture generalizes naturally to collaborative human-robot interaction (HRI), haptic teleoperation under delays (using energy-tank/port-Hamiltonian frameworks), and virtual simulation where real-time passivity guarantees are essential (0707.2275, Benzi et al., 2022, Risiglione et al., 2021).

5. Comparative Evaluations and Application Domains

Experimental benchmarks on humanoids, quadrupeds, exoskeletons, and mobile manipulators have demonstrated that PB-WBC achieves:

• Comparable or superior task accuracy to acceleration-based (ID-WBC) schemes in swing foot tracking, squatting, and jumping (Sovukluk et al., 24 Jul 2025, Fahmi et al., 2018).
• Higher robustness to unmodeled payloads and environmental uncertainty (e.g., rough terrain, incorrect friction estimation) (Fahmi et al., 2018).
• Uniform ease in gain tuning, regardless of affected inertia or task context (Sovukluk et al., 24 Jul 2025).
• Stable physical human-robot interaction in challenging collaborative scenarios, enabling online parameter adaptation and constraint enforcement while ensuring passivity at all times (Benzi et al., 2022, Moro et al., 2017).
• Seamless integration in hybrid teleoperation architectures, tolerating arbitrary communication delays due to passivity-enforcing energy tanks at both ends (Risiglione et al., 2021).

For example, on a 90kg quadruped over ramps, stepping stones, and slippery floors, PB-WBC reduced CoM and orientation errors by ~50% compared to quasi-static controllers, bounded foot slip under 150% friction-parameter error, and redistributed load under torque saturation with no loss of task execution (Fahmi et al., 2018). In humanoid jumping experiments, PB-WBC error was consistently smaller (by several mm) and showed no instabilities (Sovukluk et al., 24 Jul 2025).

6. Limitations and Extensions

A primary tradeoff for PB-WBC is the requirement of a full-rank task mapping Jacobian—over-determined or conflicting task sets must be handled offline or by nullspace decompositions. Task switching at contact transitions (e.g., heel-strike, jump landing) may require explicit dissipative damping to augment the passivity margin when the rigid-contact assumption is transiently invalid (Sovukluk et al., 24 Jul 2025).

In velocity-controlled or collaborative mobile platforms, the number and type of CBF and energy-tank constraints can become a computational bottleneck as task dimensionality increases, demanding fast solvers or architecture adaptation (Benzi et al., 2022).

Applications to virtual humans and exoskeletons demonstrate that passivity-based analogues of nullspace projections (via mechanical models, not algebraic projections) preserve stability even as the number of active constraints and degrees of freedom scales up, but complexity in choosing appropriate potentials and constraints may grow (0707.2275, Moro et al., 2017).

7. Broader Impact and Outlook

PB-WBC constitutes the core of several state-of-the-art robust WBC architectures for legged robotics, human-robot physical interaction, and simulation of virtual avatars. Its principles—modularity via passivity, physically meaningful parameterization, and strong robustness to real-world disturbances—support both deeper theoretical guarantees and practical deployability.

For future research, the integration of PB-WBC with data-driven adaptation, stochastic safety guarantees via CBFs, and scalable distributed implementations for multi-agent or large-DOF virtual systems is an active domain, building on the energy-based modularity of the passivity paradigm (Sovukluk et al., 24 Jul 2025, Benzi et al., 2022, Risiglione et al., 2021, Fahmi et al., 2018, Moro et al., 2017, 0707.2275).