Cubic Interpolation of Four Poses
This page briefly explains the implemented algorithm
MotionInterpolation.interpolate_cubic() introduced by Hegedüs et al.[1]
for cubic interpolation of 4 poses on the Study quadric. The example is shown
in Motion Interpolation.
The interpolation is done for the poses \(\mathbf{p}_0, \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3\), where the \(\mathbf{p}_0\) is the identity, i.e. it is the dual quaternion:
Taking one of the poses as the identity simplifies the problem, and on the other hand, the result can be always transformed by a static transformation to the desired initial pose.
The 4 poses span a projective 3-space, which is intersected with Study quadric. This intersection gives another quadric \(\mathcal{L}\) containing all 4 poses, and it also contains cubic curves if it contains lines. The algorithm later searches for one of the cubic curves that interpolates the 4 poses.
The test of the existence of lines is done by computing the straight lines on \(\mathcal{L}\) through the identity (\(\mathbf{p}_0\)). This can be done py computing two dual quaternions \(k_1\) and \(k_2\). These two dual quaternions have to be vectorial (they represent half-turns), which are in the Study quadric and 3-space spanned by the 4 poses. The interpolation works only if such \(k_1\) or \(k_2\) exist.
The algorithm takes one of the dual quaternions \(k_1\) or \(k_2\) and computes the parameter \(t_i\) for the cubic curve, where it corresponds to the pose \(\mathbf{p}_i\). It automatically chooses the dual quaternion \(k_1\), and defines a new pose \(\mathbf{p}_4\)
where \(\lambda\) is a parameter in \(\mathbb{R}\) that defines a family of cubic curves. The existence of the pose \(\mathbf{p}_4\) and its inclusion to the set of given poses \(\mathbf{p}_{0..3}\) guarantees, that the final cubic curve will lie on the Study quadric, i.e. it will be a proper rigid body motion polynomial that is also suitable for factorization, since it is a degree 3 polynomial.
The next step is to calculate the \(t_{1..3}\) parameters, since the \(t_0 = \infty\), which assures that the result will be a monic polynomial suitable for factorization, and \(t_4 = \lambda\).
Now, with the parameters \(t_{0..4}\) an interpolation for poses \(\mathbf{p}_{0..4}\) can be performed. The Lagrange interpolation is used for this purpose.
By its construction, the Lagrange polynomial interpolates \(\lambda_i \mathbf{p}_i\) for \(i = 0..3\). The algorithm solves for the \(\lambda_i\) parameters, such that the polynomial also interpolates \(\lambda_4 \mathbf{p}_4\). If the interpolation was done directly for all 5 poses, the result would be a polynomial of degree 4, which is not suitable for the construction of a 6R linkage.