Optimization method for the rotation minimum Euler-Rodrigues frames to Pythagorean-hodograph curve

The rotation-minimizing frames (RMF) optimization method based on Euler-Rodrigues frames (ERF) of quintic spatial PH curves is proposed for the rational minimum rotation frame problem of spatial Pythagorean-hodograph (PH) curves. The PH curve is constructed by Bézier method, and then expressed by Bernstein polynomial and quaternion, and the ER frames of the curve is obtained simply. When the ER frames of PH curve has the smallest rotation angle along the curve arc, then the minimum rotation ER frames is also called RMF of the curve. In the process of calculating the RMF of the curve, the key problem is to solve the rotation angle function. Because there are many rational polynomial integrals, and it is generally difficult to find the concrete function form of angle function. Using the least square approximation algorithm, a polynomial is constructed to approximate the rotation angle function, and the error of different degree polynomials and angle function is compared to obtain the appropriate degree polynomial approximate angle function. The time required to solve the minimum rotation ER frames of a curve is compared between the polynomial approximation method of angle function and the direct calculation method of angle function, and the calculation amount of each method is analyzed. The results show that the method of optimal square approximation can greatly reduce the amount of computation and achieve the purpose of small error.