Computational modeling of impact and deformation
This thesis tackles several problems arising in robotics and mechanics: analysis and computation of two- and muti-body impacts, planning a contact velocity for robotic batting, impact of an elastic rod onto a fixed foundation, robotic pickup of soft three-dimensional objects, and recovery of their gravity-free shapes.
Impact is an event that lasts a very short period of time but generates a very large interaction force. Assuming Stronge’s energy-based restitution, a formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb friction in three dimensions (3D). Based on this analysis, we describe a complete algorithm to take advantage of fast numerical integration and closed-form evaluation. For a simultaneous collision involving more than two bodies, we describe a general computational model for predicting its outcome.
Based on the impact model, we then look into the task of planning an initial contact velocity between a bat and an in-flight object to send the latter to a target. In certain situations, a closed-form solution can be found, while in others, a bounding triangle algorithm of iterative nature can be employed.
An alternative way of modeling impact is to consider the engaged objects to be elastic rather than rigid. A damped one-dimensional wave equation can model an elastic rod bouncing off the ground at a given initial velocity, under the influence of gravity. We derive an explicit solution based on the Method of Descent and D’Alembert’s formula. We also obtain formulas for the time of contact and analyze the dependence of the energetic coefficient of restitution on the physical constants.
I conclude the thesis with two pieces of work involving deformable objects. First, an algorithm for picking up a 3D object is introduced. Homotopy continuation method is applied to solve a non-linear system for slips between objects and fingers. Some simulation and experimental results are compared. Second, I discuss an iterative fixed-point method for recovering the gravity-free shape of an object. An experiment shows that the resulting stiffness matrix gives better predictions on deformations than the conventional stiffness matrix influenced by gravity.