An accurate wrist model is crucial to the understanding of human wrist mechanics and the development of forearm rehabilitation devices. This paper studied the nonlinear dynamics of the wrist through an ellipsoidal joint model. Compared to many studies where a universal joint is used to model the wrist, the proposed ellipsoidal model intends to better approximate the human wrist biomechanics with the use of kinematic constraints. The constraint on the original 3-dimensional rotation of the wrist is realized based on a quaternion formulation, reducing the wrist kinematics to the coupled 2-degree-of-freedom motions of flexion-extension and radial-ulnar deviation. The ellipsoidal joint also introduces additional coupling from the translational motion constraints. The multibody modeling of the wrist model is then established. The stability and control of the model are analyzed based on a constrained state-space model. Numerical simulations validate the analytical results and demonstrate the coupled dynamical behavior of the wrist. The simulations also show that the proposed model constraint is an ideal base regression function for wrist joint parameter identification. Finally, with the involvement of nonlinear stiffness and damping, chaotic-like behaviors and limit cycles are observed. The approach in this study is also generally applicable to a family of ellipsoidal joint systems.