Spatial Motion Interpolation in an Image Space of SO(4)

Abstract

Abstract Many applications in robotics, spatial mechanism design, and mechanical systems animation require the specification of the movement of a rigid body through space. In this paper, we apply Shoemake’s quaternion interpolation formula to pairs of quaternions, known as biquaternions, to obtain an interpolation procedure for spatial motion. A biquaternion represents a rotation in four dimensional space, an element of SO(4). Because a spatial displacement can be viewed as special case of a four dimensional rotation, we can use a biquaternion as an image point of a spatial configuration and a biquaternion curve to represent spatial movement. We present an example of biquaternion interpolation using four key configurations. The result is a smooth movement in SO(4), which we then map to a spatial movement. A comparison with a similar technique based on dual quaternions is provided.