Trajectory Planning: Continuous-Path Operations

Abstract

As a follow-up to Chapter 5, where we studied trajectory planning for pickand-place operations (PPO), we study in this chapter continuous-path operations. In PPO, the pose, twist, and twist-rate of the EE are specified only at the two ends of the trajectory, the purpose of trajectory planning then being to blend the two end poses with a smooth motion. When this blending is done in the joint-variable space, the problem is straightforward, as demonstrated in Chapter 5. There are instances in which the blending must be made in the Cartesian-variable space, in which advanced notions of interpolation in what is known as the image space of spatial displacements, as introduced by Ravani and Roth (1984), are needed. The image space of spatial displacements is a projective space with three dual dimensions, which means that a point of this space is specified by four coordinates similar to the homogeneous coordinates introduced in Section 2.5—of the form x _i + ε ζ_i, for i = 1, 2, 3, 4, where ε is the dual unity, which has the property that ε² = O. The foregoing coordinates are thus dual numbers, their purpose being to represent both rotation and translation in one single quantity. In following Ravani and Roth’s work, Ge and Kang (1995) proposed an interpolation scheme that produces curves in the image space with second-order geometric continuity, which are referred to as G ² curves.