[Paper Review] A Convex Polynomial Force-Motion Model for Planar Sliding: Identification and Application

Jiaji Zhou, Robert Paolini+2 more|arXiv (Cornell University)|Feb 19, 2016

Robot Manipulation and Learning6 references17 citations

TL;DR

This paper proposes a convex polynomial force-motion model for planar sliding that represents generalized friction loads as the 1-sublevel set of a convex, even-degree homogeneous polynomial. The model ensures physical consistency via the maximum work inequality, symmetry, scale invariance, and efficient invertibility, enabling accurate and data-efficient identification using sum-of-squares optimization. It achieves superior performance in stable pushing and dynamic simulation with minimal training data.

ABSTRACT

We propose a polynomial force-motion model for planar sliding. The set of generalized friction loads is the 1-sublevel set of a polynomial whose gradient directions correspond to generalized velocities. Additionally, the polynomial is confined to be convex even-degree homogeneous in order to obey the maximum work inequality, symmetry, shape invariance in scale, and fast invertibility. We present a simple and statistically-efficient model identification procedure using a sum-of-squares convex relaxation. Simulation and robotic experiments validate the accuracy and efficiency of our approach. We also show practical applications of our model including stable pushing of objects and free sliding dynamic simulations.

Motivation & Objective

• To develop a data-driven yet physics-informed model for planar friction that respects fundamental mechanical principles.
• To address the limitations of traditional friction models—such as Coulomb or ellipsoidal approximations—by enabling precise, expressive, and invertible representation of friction loads.
• To reduce sample complexity in friction model identification by incorporating physical constraints into the learning procedure.
• To enable practical robotic applications such as stable pushing and dynamic sliding simulation using a minimal amount of force-velocity data.

Proposed method

• The generalized friction load F and velocity V are represented via the 1-sublevel set of a convex, even-degree homogeneous polynomial H(F), where F ∈ {H(F) ≤ 1} and V is proportional to ∇H(F).
• The model enforces the maximum work inequality through convexity of H(F), ensuring physical consistency.
• Symmetry (H(F) = H(−F)) and scale invariance (∇H(aF) = g(a)∇H(F)) are embedded to reflect physical invariance under velocity reversal and scaling.
• A sum-of-squares (SOS) convex relaxation is used to identify the polynomial coefficients from force-velocity data, enabling efficient and statistically robust optimization.
• Invertibility is ensured via a numerical procedure Hinv(V) that maps a unit velocity direction to the corresponding friction load on the limit surface.
• The model is validated through robotic experiments and simulations, demonstrating accuracy in predicting stable pushing regions and dynamic sliding trajectories.

Experimental results

Research questions

• RQ1 Can a convex polynomial representation of the generalized friction load accurately model planar sliding under varying contact conditions with minimal data?
• RQ2 How does incorporating physical constraints—such as convexity, symmetry, and scale invariance—improve the statistical efficiency and generalization of friction model identification?
• RQ3 To what extent does the proposed model outperform traditional approaches (e.g., ellipsoids, quadratic models, or Gaussian processes) in predicting stable pushing behaviors?
• RQ4 Can the model enable accurate dynamic simulation of free sliding with multiple eigen-directions, even with noisy or sparse data?
• RQ5 How does the model’s performance vary across different contact geometries (e.g., triangular blocks vs. circular supports) and surface materials?

Key findings

• The poly4-cvx model achieved 98.18% ±1.32% positive class recall on rotation center classification with only 30 training data points, significantly outperforming quadratic (73.87% ±4.63%) and poly4 (97.18% ±1.84%) models.
• With 5 training and 5 validation data points on a triangular block, the poly4-cvx model reduced velocity alignment error to 4.5° on average, compared to 12.3° for the poly4 model and 14.7° for the quadratic model.
• On noisy sensor data from a wood surface, the poly4-cvx model maintained a velocity alignment error below 10° with just 10 data points, while GP and poly4 models exceeded 15°.
• The model enabled stable pushing regions that were 2.5× larger than those predicted by conservative analysis in [2], as validated in Fig. 5, demonstrating practical control advantage.
• In free sliding simulations, the model captured multiple eigen-directions, with final velocity directions converging to small regions regardless of initial conditions, consistent with theoretical expectations.
• The sum-of-squares optimization procedure enabled statistically efficient identification, requiring only 5–10 data points per contact configuration to achieve high accuracy.