A Unified Complementarity-based Approach for Rigid-Body Manipulation and Motion Prediction

Robotic manipulation in unstructured environments requires planners to reason jointly about free-space motion and sustained, frictional contact with the environment. Existing (local) planning and simulation frameworks typically separate these regimes or rely on simplified contact representations, particularly when modeling non-convex or distributed contact patches. Such approximations limit the fidelity of contact-mode transitions and hinder the robust execution of contact-rich behaviors in real time. This paper presents a unified discrete-time modeling framework for robotic manipulation that consistently captures both free motion and frictional contact within a single mathematical formalism (Unicomp). Building on complementarity-based rigid-body dynamics, we formulate free-space motion and contact interactions as coupled linear and nonlinear complementarity problems, enabling principled transitions between contact modes without enforcing fixed-contact assumptions. For planar patch contact, we derive a frictional contact model from the maximum power dissipation principle in which the set of admissible contact wrenches is represented by an ellipsoidal limit surface. This representation captures coupled force-moment effects, including torsional friction, while remaining agnostic to the underlying pressure distribution across the contact patch. The resulting formulation yields a discrete-time predictive model that relates generalized velocities and contact wrenches through quadratic constraints and is suitable for real-time optimization-based planning. Experimental results show that the proposed approach enables stable, physically consistent behavior at interactive speeds across tasks, from planar pushing to contact-rich whole-body maneuvers.

💡 Research Summary

The paper tackles a fundamental challenge in robotic manipulation: the need to plan and predict motions that involve both free‑space movement and sustained frictional contact with the environment, especially when contacts are distributed over non‑convex patches rather than idealized point contacts. Existing planners typically treat these regimes separately or simplify contact modeling (e.g., point contact, linearized friction cones), which leads to inaccurate contact‑mode transitions and limits real‑time execution of contact‑rich tasks.

To address this, the authors introduce a unified discrete‑time modeling framework called Unicomp that captures free motion and frictional contact within a single mathematical formalism based on complementarity theory. The core idea is to formulate the dynamics of a rigid body (or multiple bodies) as a Mixed Complementarity Problem (MCP) that simultaneously enforces Newton‑Euler dynamics, non‑penetration, unilateral contact, and friction constraints.

Key technical components

1. Equivalent Contact Point (ECP) – For a planar contact patch, the distributed pressure is replaced by an equivalent wrench acting at a single point (the ECP). This reduces an arbitrary set of contact points to a tractable single‑point representation while preserving the net force and moment.

2. Ellipsoidal Limit Surface – Frictional admissibility is modeled using the maximum power dissipation principle, yielding an ellipsoidal limit surface:
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