The Ball Joint, also known as a spherical joint, has three rotational degrees of freedom between the two attached components. The two components are free to rotate relative to each other along all three axes.

Unlike other joints, the ball joint has no particular axis about which the relative degrees of freedom between the two components are possible. This joint has rotational symmetry, and hence, relative rotation is allowed about any axis. However, application of constraints, springs, dampers, and so forth needs the source (e1) and destination (ed1) axes. The local coordinate system of this joint is defined using the source axis.

You specify the initial source axis (e10) and initial destination axis (ed10) in terms of components along the axes of the selected coordinate system. The initial source axis must point toward the center of joint, whereas the initial destination axis must point away from the center of joint.

For a ball joint, the destination attachment is free to rotate relative to the source attachment about all three axes. The added degrees of freedom are contained in a relative quaternion (a, b, c, and d), which represents the rotation in 3D. The rest of the formulation is similar to that of the hinge joint.