The need for using multivariate simplex B-splines on problematic identification data arose from replacing data tables in a tiltrotor aircraft quasiLinear Parameter Varying (qLPV) flight dynamics model. Tiltrotors are characterized by their ability to transition between helicopter and fixed-wing aircraft mode. Therefore, no single linear state-space model is capable of adequately covering the entire flight envelope. The qLPV technique relies on interpolation between stability and control derivatives, as well as trim points. For this purpose, a set of scheduling parameters, which have the largest influence on aircraft behaviour, is identified. Typical examples for a tiltrotor are air speed, nacelle incidence and wing flap angle. Point models are obtained, which cover a certain scheduling parameter range. The result is preprocessed to fit the rigid, rectangular shape of a multidimensional lookup table and interpolated during runtime. This procedure has been successfully applied to a number of aerial vehicles. However, data tables scale badly when dimension or accuracy are increased. Furthermore, local updates using new data are inherently difficult. It was therefore recommended to replace lookup tables by multivariate simplex B-splines, which do not suffer from these drawbacks. Their high approximation power enables the fitting of scattered, globally or locally nonlinear data sets characteristic of aerospace systems. Also, computations are efficient and yield transparent results. There was, however, a data deficiency whose impact only became apparent late in the project. Large portions are coplanar due to an imbalance in scheduling parameter resolution. This so-called data collinearity caused ill-conditioned regression matrices and associated numerical problems with coefficient estimates. Consequently, the project focus shifted from converting the qLPV model to answering the following research question: How can multivariate simplex B-splines be used effectively and reliably to fit collinear aerodynamic data sets?

Collinearity and its negative effects have been acknowledged for decades and a substantial amount of work and time dedicated to solutions. A very popular remedy, due to its power and simplicity, is the Ridge Regression Estimator (RRE). Being part of the Tikhonov regularization family, it augments the basic least-square estimator with a penalty term and a tuning parameter. On the one hand this introduces bias to the estimation, however on the other variance is reduced. No regularization scheme could be found for multivariate simplex B-splines, despite a thorough literature study. This led to the following research subquestions:

How can regularization be integrated in the existing B-coefficient estimation framework?
How well does it work against regression matrix ill-conditioning?
How can good tuning parameter candidates be determined?

Regularization of other, more commonly employed spline types is well-known. Eilers and Marx in 1996 developed what they called the P-spline by adding a coefficient difference penalty to an ordinary B-spline. Although it was invented and popularized as a tool for smoothing noisy data, the P-spline can cope with collinearity just like the ridge regression estimator. The initial article presented the concept for univariate B-splines, which later was extended by the same authors to multivariate tensor product B-splines. The P-spline inherited the tensor product spline’s major limitations being an inherent difficulty to fit scattered data sets and rather cumbersome calculations in higher dimensions. Simplex B-splines, by contrast, do not suffer from these. Additional research subquestions were therefore:

What is the simplex counterpart to the multivariate tensor product P-spline?
How does it compare to ridge regression in terms of ill-conditioning reduction, goodness-of-fit and smoothing properties?

This master thesis report is intended to provide answers and act as entry point for further research.

The Unmanned Aerial Vehicle (UAV) market is growing, as the field of application becomes increasingly diverse. Various missions performed by UAVs such as transport, surveillance and monitoring missions can be less challenging and more efficient compared to missions carried out by manned aircraft. The new design principle of combining a helicopter and a conventional aircraft has been proven to be prominent, as it allows a UAV to be versatile and have capability of being retrofitted to different mission profiles; these Hybrid UAVs are able to operate at broader conditions, as they can take-off and land without a runway and have a horizontal flight performance of a conventional aircraft.