Projection Metrics for Rigid-body Displacements

An open research question is how to define a useful metric on SE(n) with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group SO(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements (hence the term projection metric. A bi-invariant metric on SO(n+1) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection metrics on SE(n). These metrics have applications in motion synthesis, robot calibration, motion interpolation, and hybrid robot control. INTRODUCTION Simply stated a metric measures the distance between two points in a set. There exist numerous useful metrics for defining the distance between two points in Euclidean space, however, defining similar metrics for determining the distance between two locations of a finite rigid body is still an area of ongoing research, see [22], [14], [2], [26], [23], [21], [8], [11], [30], [4], [7], and [1]. In the cases of two locations of a finite rigid body in ddress all correspondence to this author. 1 either SE(3) (spatial locations) or SE(2) (planar locations) any metric used to measure the distance between the locations yields a result which depends upon the chosen reference frames, see [2] and [23]. However, a metric that is independent of these choices, referred to as being bi-invariant, is desirable. It is well known that for the specific case of orienting a finite rigid body in SO(n) bi-invariant metrics do exist. For example, Ravani and Roth [27] defined the distance between two orientations of a rigid body in space as the magnitude of the difference between the associated quaternions and a proof that this metric is bi-invariant may be found in [21]. In [21] and [18] Larochelle and McCarthy presented an algorithm for approximating displacements in SE(2) with orientations in SO(3). By utilizing the metric of Ravani and Roth [27] they arrived at an approximate bi-invariant metric for planar locations in which the error induced by the spherical approximation is of the order 1 R2 , where R is the radius of the approximating sphere. Their algorithm for a projection metric is based upon an algebraic formulation which utilizes Taylor series expansions of sine() and cosine() terms in homogeneous transforms, see [24]. Etzel and McCarthy [8] extended this work to spatial displacements by using orientations in SO(4) to approximate locations in SE(3). Their algorithm is also based upon Taylor series expansions of sine() and cosine() terms, see [10], and here too the error is of the order 1 R2 . This paper presents an efficient alternative approach for defining projection metrics on SE(n) to those presented by Larochelle and McCarthy [21] and Etzel and McCarthy [8]. Copyright c © 2005 by ASME Here, the underlying geometrical motivations are the sameto approximate displacements with hyperspherical rotations. However, an alternative approach for reaching the same goal is presented. We utilize the singular value and polar decompositions to yield projections of planar and spatial finite displacements onto hyperspherical orientations. PROJECTING SE(n) ONTO SO(n+1) First, we review how spherical displacements may be used to approximate planar displacements with some finite error associated with the radius R of the sphere, see [15] and [21]. This approach is based upon the work of McCarthy [24] in which he examined spherical and 3-spherical motions with instantaneous invariants approaching zero and showed that these motions may be identified with planar and spatial motions, respectively. Recall that a general planar displacement (a,b,α) in the z = R plane (an element of SE(2)) may be expressed as a homogeneous coordinate transformation,