This paper proposes an analytical method of solving the inverse kinematic problem for a humanoid manipulator with five degrees-of-freedom (DOF) under the condition that the target orientation of the manipulator's end-effector is not constrained around an axis fixed with respect to the environment. Since the number of the joints is less than six, the inverse kinematic problem cannot be solved for arbitrarily specified position and orientation of the end-effector. To cope with the problem, a generalized unconstrained orientation is introduced in this paper. In addition, this paper conducts the singularity analysis to identify all singular conditions.

For redundant robot kinematics with a degree of redundancy 1 a self-motion vector field is examined whose equilibrium points lie at singular configurations of the kinematics, and whose orbits determine the self-motion manifolds. It is proved that the self-motion vector field is divergence-free. Locally, around singular configurations of corank 1, the self-motion vector field defines a 2-dimensional Hamiltonian dynamical system. An analysis of the phase portrait of this system in a neighbourhood of a singular configuration solves completely the question of avoidability or unavoidability of this configuration. Complementarily, sufficient conditions for avoidability and unavoidability are proposed in an analytic form involving the self-motion Hamilton function. The approach is illustrated with examples. A connection with normal forms of kinematics is established.