In this article we examine the problem of designing a mechanism whose tool frame comes closest to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of distance metrics in SE(3), the Euclidean group of rigid-body motions. Using Lie theory, we show that no bi-invariant distance metric (i.e., one that is invariant under both left and right translations) exists in SE(3), and that because physical space does not have a natural length scale, any distance metric in SE(3) will ultimately depend on a choice of length scale. We show how to construct left- and right-invariant distance metrics in SE(3), and suggest a particular left-invariant distance metric parametrized by length scale that is useful for kinematic applications. Ways of including engineering considerations into the choice of length scale are suggested, and applications of this distance metric to the design and positioning of certain planar and spherical mechanisms are given.

In this article we generalize the concept of Be´zier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Be´zier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Be´zier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Be´zier sense) using this recursive algorithm. We apply this alogorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames.

The problem of globally stabilizing the attitude of a rigid body is considered. Topological and geometric properties of the space of rotations relevant to the stabilization problem, are discussed. Chevalley's exponential coordinates for a Lie group are used to represent points in this space. An appropriate attitude error is formulated and used for control design. A control Lyapunov function approach is used to design globally stabilizing feedback l a ws that have desirable optimality properties. Their performance is compared to the performance of previously developed proportional-derivative t ype control laws. The new control laws achieve the same or greater stabilization rate with less control e ort. Special issues in the Lyapunov stability proofs due to the topology of the space of rotations are identi ed and resolved.

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