[Paper Review] Dynamic Model of Planar Sliding

Jiayin Xie, Nilanjan Chakraborty|arXiv (Cornell University)|Sep 14, 2018

Dynamics and Control of Mechanical Systems18 references3 citations

TL;DR

This paper presents a principled discrete-time dynamic model for planar sliding with distributed contact patches, using an equivalent contact point (ECP) to represent the net effect of distributed friction and normal forces. By decoupling contact impulse computation (via a system of four quadratic equations) from state and ECP estimation (via linear equations), the method enables accurate, closed-form solutions for quasi-static and pure translation motion without ad hoc point approximations, significantly improving simulation fidelity over traditional three-point contact models.

ABSTRACT

In this paper, we present a principled method to model general planar sliding motion with distributed convex contact patch. The effect of contact patch with indeterminate pressure distribution can be equivalently modeled as the contact wrench at one point contact. We call this point equivalent contact point. Our dynamic model embeds ECP within the equations of slider's motion and friction model which approximates the distributed contact patch, and eventually brings us a system of quadratic equations. This discrete-time dynamic model allows us to solve for the two components of tangential friction impulses, the friction moment and the slip speed. The state of the slider as well as the ECP can be computed by solving a system of linear equations once the contact impulses are computed. In addition, we derive the closed form solutions for the state of slider for quasi-static motion. Furthermore, in pure translation case, based on the discrete-time model, we present the closed form expressions for the friction impulses the slider suffers and the state of it at each time step. Simulation examples are shown to demonstrate the validity of our approach.

Motivation & Objective

• To address the lack of accurate dynamic models for planar sliding with distributed contact patches in robotic manipulation.
• To eliminate ad hoc approximations like three-point contact models that can lead to inaccurate motion prediction when the center of friction lies outside the convex hull.
• To develop a principled method that models the entire contact patch as a single equivalent contact point (ECP) while preserving dynamic fidelity.
• To derive closed-form solutions for quasi-static and pure translational motion, enabling efficient motion prediction.
• To validate the model against a nonlinear complementarity problem (NCP)-based simulation, demonstrating accuracy and computational efficiency.

Proposed method

• Introduces the Equivalent Contact Point (ECP) as the unique point in the contact patch where the net moment due to normal forces is zero, equivalent to the center of friction.
• Derives a discrete-time dynamic model composed of four quadratic equations in four unknowns: two tangential friction impulses, frictional moment, and slip speed.
• Decouples the computation of contact impulses (solved via quadratic system) from the state and ECP estimation (solved via linear system) after impulse computation.
• Uses a generalized Coulomb friction model assuming contact force and moment lie within a convex cone and friction is independent of contact area.
• Applies Newton-Euler equations of motion with the ECP-based contact wrench to model slider dynamics under external forces and gravity.
• Validates the model through numerical simulations with convex and non-convex contact patches (e.g., square and ring-shaped), comparing results with an NCP-based model.

Experimental results

Research questions

• RQ1 Can a dynamic model be developed for planar sliding that accurately represents distributed contact patches without ad hoc point approximations?
• RQ2 How can the equivalent contact point (ECP) be computed dynamically during sliding motion, and what is its role in force and moment balance?
• RQ3 What closed-form solutions exist for the state of the slider under quasi-static and pure translational motion conditions?
• RQ4 How does the proposed model compare in accuracy and computational cost to existing NCP-based simulation methods?
• RQ5 Can the ECP shift relative to the center of mass during dynamic sliding, and does this affect motion prediction accuracy?

Key findings

• The ECP dynamically shifts relative to the center of mass during acceleration, confirming that inertial effects cause the center of friction to deviate from the center of mass in non-quasi-static motion.
• The model achieves numerical agreement with the NCP-based model within a tolerance of 1e-6 for angular and linear velocities, validating its accuracy.
• For pure translation, closed-form expressions are derived for contact impulses, ECP, and slider state at each time step, enabling efficient computation.
• The system of equations is significantly smaller than the NCP-based model (24 nonlinear equations vs. 24 unknowns), reducing computational complexity.
• Simulations with non-convex ring-shaped contact patches confirm the model’s validity where three-point approximations would fail due to ECP lying outside the convex hull.
• The method is independent of contact geometry and pressure distribution, relying only on rigid body dynamics and a generalized Coulomb friction model.