Advanced B-spline interpolation with the use of quaternions for spatial rotations: formulas for derivatives and non-uniform parametric distribution

Mechanics abounds in machines involving one- or two-degree-of-freedom subsystems with various purposes, such as vehicle suspension. Such systems are usually addressed according to the multibody approach, with the kinematics possibly involving a large number of constraint equations repeatedly solved around near configurations. As an alternative, this paper proposes solving the kinematics for a few key configurations (knots) and interpolating them in terms of parameters whose number is equal to the number of degrees of freedom. This work explores the application of cumulative B-splines for a consistent interpolation of either translations or rotations. Owing to the C 2 continuity requirements, 4th-order B-splines are retained. The interpolation extends the computation of control points (de Boor points) to the case of non-uniformly distributed parameter values. On the other hand, expressions are provided to express the first and second derivatives necessary for velocities and accelerations. The method is illustrated on the example of a double wishbone suspension, and the influence of the knot density is analyzed according to kinematic and dynamic simulations. Moreover, a method is proposed to minimize the number of knots while controlling the interpolation error. An example of a spatial slider crank mechanism illustrates the ability of the method to manage large motions.