Objective and Frame-Invariant Kinematic Metric Functions for Rigid Bodies

Kinematic metric functions, which allow us to make metric measurements such as “distance,” “length,” and “angle,” are often needed in robotics. However, some commonly used metric functions are not well-defined because of their dependence on choice of reference frame. This paper presents formal and practical conditions for kinematic metric functions to be well-defined. To properly analyze this issue, we introduce an intrinsic formulation of the configuration space of a rigid body that does not depend on frames. Based on this intrinsic definition, the principle of objectivity, or observer indifference, is introduced to derive a formal condition for well-definedness of kinematic metric functions. The principle of objectivity is further used to gain physical insight into left, right, and bi-invariances on the Lie group SE(3). The abstract notion of objectivity is then related to the more practically useful notion of frame invariance, which is shown to be necessary and sufficient for objectivity. Thus, frame invariance can be used as a practical condition for determining objective functions, and we give an explicit formula to determine frame invariance of a given metric function. Finally, by use of examples, we present several frame-invariant norms of rigid body velocities and wrenches. These examples demonstrate that norms with interesting physical interpretations need not be derived from inner products of velocities or wrenches.