Dynamics of Human Head and Eye Rotations Under Donders' Constraint

On geodesics of the rotation group SO(3)

“…Interestingly, the normal vector in Listing's constraint is the normal vector to V rather than L. Listing's constraint (25) is integrable (holonomic) and restricts the rotation tensor R to lie in a submanifold of SO(3). As discussed in [7,31], this submanifold is the real projective plane RP 2 .…”

“…6 The plane V is denoted by Pω in [2,7] and is known as the displacement plane in the works by Tweed, Villis et al [40,41,42] who credit the discovery of the plane to Helmholtz [12]. Tweed et al's papers contain experimental evidence for the existence of V. The basis set w 1 , w 2 , w 3 that we are using is novel.…”

“…Tweed et al's papers contain experimental evidence for the existence of V. The basis set w 1 , w 2 , w 3 that we are using is novel. 7 In the classical literature, the vector n is parallel to what Helmholtz [12, p. 78] between V and Listing's plane L can be verified by comparing the angle between the respective normal vectors n and E 1 to these planes. …”

“…The geodesics correspond to the motions whereω = 0 and we are able to use [29] [7,8,31] on geodesics in the context of motions of the eye. When the eye is modeled as an axisymmetric or asymmetric rigid body, then some of the geodesics for the symmetric case are still present but new classes of geodesics also arise.…”

“…These models started in the 1970s with the work of Robinson [25,35] and has lead to models where the muscles are modeled with improved fidelity, such as [33,43], models for a robotic eye by Cannata and Maggiali [2] and a rigid body model with generalized control torques by Ghosh et al [7,8,31]. The resulting models can be combined with control strategies to gain a deeper understanding of how the brain controls the ocular system and enables vision.…”