  The kinematics of a robot manipulator are described in terms of the mapping
connecting its joint space and the 6-dimensional Euclidean group of motions
$SE(3)$. The associated Jacobian matrices map into its Lie algebra
$\mathfrak{se}(3)$, the space of twists describing infinitesimal motion of a
rigid body. Control methods generally require knowledge of an inverse for the
Jacobian. However for an arm with fewer or greater than six actuated joints or
at singularities of the kinematic mapping this breaks down. The Moore-Penrose
pseudoinverse has frequently been used as a surrogate but is not invariant
under change of coordinates. Since the Euclidean Lie algebra carries a pencil
of invariant bilinear forms that are indefinite, a family of alternative
hyperbolic pseudoinverses is available. Generalised Gram matrices and the
classification of screw systems are used to determine conditions for their
existence. The existence or otherwise of these pseudoinverses also relates to a
classical problem addressed by Sylvester concerning the conditions for a system
of lines to be in involution or, equivalently, the corresponding system of
generalised forces to be in equilibrium.
