Averages of Rotations and Orientations in 3-space

Finding the mean (average) of n arbitrary 3D rotations or orientations is conceptually challenging both in its algebra and its geometric interpretation. To my knowledge, no solutions have been found for the general case, although a number of approximate solutions have been proposed [3, 4, 5] to accurately and e ciently handle those limiting cases typical of engineering interest. This report proposes an exact, closed form solution which is extended from the 2D complex number domain where the averaging methods are widely accepted. Representing rotations as rotation (unit) quaternions [1] is convenient in that a rotation composed of a sequence of successive rotations can be simply represented as the (non-commutative) product of the successive rotation quaternions. But this very property is problematical for averaging. A mean average is conventionally taken as a sum of averagands divided by their number. If applied to rotation quaternions however, the result is not in general a rotation quaternion; moreover, there is no sensible geometric interpretation of such a procedure. The multiplicative analogue of this procedure, taking the n root of the product of rotation quaternions does return a rotation quaternion. However the result is dependent on the order of multiplication, while a meaningful average should be order independent. This report explores the extension of rotations by unit complex numbers in the complex plane to rotations by unit quaternions in 3-space. Unit complex numbers u = e form a proper subspace of the rotation quaternions. Multiplication of any complex number z by u will yield a number z with identical magnitude, rotated by an angle about the origin in the complex plane. Analogous to quaternion rotations, a complex rotation may be composed of a product (commutative in this case) of successive rotations. In averaging, the commutative property of complex multiplication provides a unique result, the principal n root of which does indeed geometrically represent the average rotation. Mapped into logarithmic space, the averaging procedure becomes the conventional sum of averagands divided by their number. We'll show how this logarithmic mapping can be extended from complex to quaternion space, where the summation remains commutative, thus satisfying the order independence requirement for averaging. Distinct from rotations are orientations which pose an additional set of problems, which we shall discuss in the second half of this report. We shall use notational representations of, e.g., r for real or complex numbers, v for vectors, v̂ for unit vectors and Q for quaternions. This notation notwithstanding, it should be understood that all these quantities should be regarded as equivalent to their quaternion form.