Analysis and Computation of Two Body Impact in Three Dimensions

Abstract

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

The maximum dissipation principle in rigid-body dynamics with inelastic impacts

“…Using such an approach Stronge constructs an energetically consistent restitution hypothesis in [28], Mirtich solves rigidbody dynamics with impacts for virtual reality applications, where permanent contacts are treated as sequences of collisions [12,13]. And lately, Jia and Wang showed in [11] how contact reaction impulses can be computed from the limit of a contact model with linear normal stiffness and Coulomb friction for general collisions (central or eccentric, direct or oblique) in three dimensions. The authors established a condition that, if met, guarantees solution existence.…”

Computational modeling of impact and deformation

“…T , called the sliding velocity. The sliding velocity is governed by the differential equation Jia and Wang (2017), where B = (û,ŵ) T W (û,ŵ), d = (û,ŵ) T Wn, µ is the coefficient of friction. Here, the notation means differentiation with respect to the normal…”

“…Solve E(v − n ) = 0, which holds at the end of restitution for v − n . Below we will apply the analysis given in Jia and Wang (2017) for closed form computation of v − n .…”

“…Our goal is to tackle the batting task in full three dimension. A major obstacle for real-time planning is that the 3D impact problem generally does not have a closed-form solution and has to be solved through numerical integration, as analyzed in Jia and Wang (2017). The inverse problem of finding a motion of the bat to impart a desired change to the motion of the object could become prohibitively expensive.…”

“…apply the impact algorithm inJia and Wang (2017) to compute their generated impulses I 1 , I 2 , and I 3 3: for each I k do…”