Quadrotor System Identification Using the Multivariate Multiplex B-Spline

Abstract

A novel method for aircraft system identification is presented that is based on a new multivariate spline type; the multivariate multiplex B-spline. The multivariate multiplex B-spline is a generalization of the recently introduced tensor-simplex B-spline. Multivariate multiplex splines obtain similar or better approximation accuracy using less parameters (B-coefficients) than standard multivariate simplex B-splines which are currently used for aircraft system identification. The multiplex spline allows the user to incorporate a-priori knowledge of the modelled system in the definition of the model structure. In particular, while the standard simplex B-splines use a multi-dimensional triangulation in which all dimensions are coupled, the multiplex spline enables the user to decouple specific model dimensions based on expert knowledge of the system. The new method is used to approximate a 4-dimensional nonlinear quadrotor inflow dataset. The results show that the multiplex B-spline obtains a relative root mean square error of 0.672% using 1440 Bcoefficients. This compares favorably to results obtained with the standard 4-dimensional simplex B-spline on the same dataset, which resulted in an relative root mean square error of 1.608% using 1540 B-coefficients.

Distributed approach for Aerodynamic Model Identification of the ICE aircraft using multivariate splines

“…These studies have however focused on the two-and three-dimensional case only, or were limited to the simplex and the hypercube. Applications of the simplotope spline have been limited to a single study into quadrotor modeling (in which the basis is called a multiplex) [17].…”

“…As the dimension of a simplex increases, its corners become sharper and its content becomes more distributed (see [17,Fig. 5]).…”

“…To mitigate this problem we can choose to span the full-dimensional domain by taking the tensor-product of lower-dimensional simplex splines. The result is a B-spline defined on the product of simplices, a so called simplotope (or: multiplex in [17]). In this section we discuss the implications of using this basis for the definition of the geometric basis and tesselation, barycentric coordinates, basis polynomials, and continuity conditions.…”

“…Moreover, simplotopes are generally better shaped, easier to be filled with data, and often require less coefficients and less continuity constraints [17]. The tensor product between the layers will however result in high cross-coupling terms up to degree In Fig.…”

Simplex Solutions for Optimal Control Flight Paths in Urban Environments

“…Defining the coordinate system in 2  , let r 1 , r 2 , and r 3 be vertices of a simplex G. Any point, R, inside simplex G can be represented in terms of the vertices of G and the barycentric weights, used as a basis as follows [21][22][23]:…”