Nonlinearity Index of the Cayley Form

The nonlinearity index is a measure of the nonlinearity of dynamical systems based on computing the initial-condition sensitivity of the state-transition matrix. The Cayley form is a representation for dynamical systems that relates their motion to N-dimensional rotations. The generalized coordinates of the system are used to define an N-dimensional orientation, and a set of quasi velocities is defined equal to the corresponding angular velocity. The nonlinearity index of the Cayley-form representation is computed for an elastic spherical pendulum and a planar satellite example. These results are compared to values for alternative dynamical representations. Additionally, the nonlinearity is evaluated by analyzing how well the linearized equations of each representation capture certain properties of the motion. These results show that the Cayley form can have lower nonlinearity than traditional representations, in particular those representations that suffer from kinematic singularities.