Abstract: A new invariant formulation of 3D eye-head kinematics improves on the computational advantages of quaternions. This includes a new formulation of Listing's Law parameterized by gaze direction leading to an additive rather than a multiplicative saccadic error correction with a gaze vector difference control variable. A completely general formulation of compensatory kinematics characterizes arbitrary rotational and translational motions, vergence computation, and smooth pursuit. The result is an invariant, quantitative formulation of the computational tasks that must be performed by the oculomotor system for accurate 3D gaze control. Some implications for neural network modeling are discussed.
Abstract: Invariant methods for formulating and analying the mechanics of the skeleto-muscular system with geometric algebra are further developed and applied to reaching kinematics. This work is set in the context of a neurogeometry research program to develop a coherent mathematical theory of neural sensory-motor control systems.
Abstract: Geometric algebra is used in an essential way to provide a coordinate-free approach to Euclidean geometry and rigid body mechanics that fully integrates rotational and translational dynamics. Euclidean points are given a homogeneous representation that avoids designating one of them as an origin of coordinates and enables direct computation of geometric relations. Finite displacements of rigid bodies are associated naturally with screw displacements generated by bivectors and represented by twistors that combine multiplicatively. Classical screw theory is incorporated in an invariant formulation that is less ambiguous, easier to interpret geometrically, and manifestly more efficient in symbolic computation. The potential energy of an arbitrary elastic coupling is given an invariant form that promises significant simplifications in practical applications.