Quaternions as a solution to determining the angular kinematics of human movement

Abstract The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. There are several options for describing rigid body orientation including Cardan angles, Euler angles, and quaternions; the utility of quaternions will be reviewed and elaborated. The orientation of a rigid body or a joint between rigid bodies can be described by a quaternion which consists of four variables compared with Cardan or Euler angles (which require three variables). A quaternion, q = (q0, q1, q2, q3), can be considered a rotation (Ω = 2 cos−1(q0)), about an axis defined by a unit direction vector $$ \left({q}_1/\sin \left(\frac{\Omega}{2}\right),{q}_2/\sin \left(\frac{\Omega}{2}\right),{q}_3/\sin \left(\frac{\Omega}{2}\right)\right) $$ q1/sinΩ2q2/sinΩ2q3/sinΩ2 . The quaternion, compared with Cardan and Euler angles, does not suffer from singularities or Codman’s paradox. Three-dimensional angular kinematics are defined on the surface of a unit hypersphere which means numerical procedures for orientation averaging and interpolation must take account of the shape of this surface rather than assuming that Euclidean geometry based procedures are appropriate. Numerical simulations demonstrate the utility of quaternions for averaging three-dimensional orientations. In addition the use of quaternions for the interpolation of three-dimensional orientations, and for determining three-dimensional orientation derivatives is reviewed. The unambiguous nature of defining rigid body orientation in three-dimensions using a quaternion, and its simple averaging and interpolation gives it great utility for the kinematic analysis of human movement.