  This paper presents a novel contact-implicit trajectory optimization method
using an analytically solvable contact model to enable planning of interactions
with hard, soft, and slippery environments. Specifically, we propose a novel
contact model that can be computed in closed-form, satisfies friction cone
constraints and can be embedded into direct trajectory optimization frameworks
without complementarity constraints. The closed-form solution decouples the
computation of the contact forces from other actuation forces and this property
is used to formulate a minimal direct optimization problem expressed with
configuration variables only. Our simulation study demonstrates the advantages
over the rigid contact model and a trajectory optimization approach based on
complementarity constraints. The proposed model enables physics-based
optimization for a wide range of interactions with hard, slippery, and soft
grounds in a unified manner expressed by two parameters only. By computing
trotting and jumping motions for a quadruped robot, the proposed optimization
demonstrates the versatility for multi-contact motion planning on surfaces with
different physical properties.
