Averaging Rotations

Suppose A1, . . . , AN are rotation matrices on n-dimensional Euclidean space R, i.e., Aj ∈ SO(n). We want to consider some element of SO(n) that represents an “average” of these elements Aj . There are a number of possible ways to define the notion of an average in this context. One approach has been to write Aj = ej with Zj a real, skew-symmetric n × n matrix (i.e., Zj ∈ skew(n)), and define an average as e , with Z = (Z1 + · · ·+ZN )/N . This has ambiguities, arising from the fact that the exponential map exp : skew(n) → SO(n) is not one-to-one; one then needs to tackle the problem of finding the “best” candidates Zj (logarithms of Aj) to produce the desired e ; cf. [V]. Another drawback to this approach is mentioned in §4. We discuss a different approach, considering the following minimization problem. Given A1, . . . , AN ∈ SO(n), define