Smooth Invariant Interpolation on Lie groups with Prescribed Terminal Conditions for Robot Motion Planning and Modeling of Soft Robots

Interpolation of rigid body motions, or a general frame motion in Euclidean space, is a recurring topic in robotics. It boils down to generating trajectories in a Lie group, either SE (3) or SO (3) × ℝ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup>, with given initial and/or terminal values. To this end, spline interpolation schemes were developed where canonical coordinates are represented by cubic splines. They allow for prescribing initial velocity and acceleration only. In many robotic applications, terminal conditions are prescribed, however. In this paper, a novel interpolation scheme is presented that admits prescribing the terminal pose, velocity and acceleration, or the initial condition. As example, the scheme is applied to a rendezvous task of a UAV and describing the deformation of a Cosserat beam as relevant for soft robotics. The presented interpolation scheme can be directly applied to the motion parameterization in terms of (dual) quaternions.