A Simple Mono-Dimensional Approach for Lap Time Optimisation

Lap time minimisation methods have great relevance in the analysis of race tracks, and in the design and optimisation of race vehicles. Several lap time minimisation approaches have been proposed in the literature, which are computationally demanding because they need to either solve differential equations or to implement a forward−backward integration based on an apex-finding method. This paper proposes an alternative method, based on a mono-dimensional quasi-steady-state numerical approach. The proposed approach uses a simplified vehicle model accounting for combined tyre−road interactions, aerodynamic effects, and power limitations. The method exploits the knowledge of the curvature of the trajectory, which is worked out through a rigorous approach that allows for the use trajectories defined with respect to ageneric curve parameter and not necessarily the arc length. An iterative routine is implemented that exploits the vehicle dynamics, without solving differential equations or performing forward−backward integrations from the trajectory apexes. Simulations are carried out on three different tracks and are shown to be computationally efficient. Despite being intentionally simple, the proposed method allows to grasp key aspects of the problem, such as the effect of the combined tyre−road interactions on the acceleration profiles, and the effect of aerodynamic drag and downforce on the position of the braking point on the track and on the speed profile

Impressions of convexity - An illustration for commutator bounds

We determine the sharpest constant $C_{p,q,r}$ such that for all complex matrices $X$ and $Y$, and for Schatten $p$-, $q$- and $r$-norms the inequality $$\|XY-YX\|_p\leq C_{p,q,r}\|X\|_q\|Y\|_r$$ is valid. The main theoretical tool in our investigations is complex interpolation theory.Comment: 32 pages, 88 picture

Calibration Using Matrix Completion with Application to Ultrasound Tomography

We study the calibration process in circular ultrasound tomography devices where the sensor positions deviate from the circumference of a perfect circle. This problem arises in a variety of applications in signal processing ranging from breast imaging to sensor network localization. We introduce a novel method of calibration/localization based on the time-of-flight (ToF) measurements between sensors when the enclosed medium is homogeneous. In the presence of all the pairwise ToFs, one can easily estimate the sensor positions using multi-dimensional scaling (MDS) method. In practice however, due to the transitional behaviour of the sensors and the beam form of the transducers, the ToF measurements for close-by sensors are unavailable. Further, random malfunctioning of the sensors leads to random missing ToF measurements. On top of the missing entries, in practice an unknown time delay is also added to the measurements. In this work, we incorporate the fact that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the missing ToFs, we apply a state-of-the-art low-rank matrix completion algorithm, OPTSPACE . To find the correct positions of the sensors (our ultimate goal) we then apply MDS. We show analytic bounds on the overall error of the whole process in the presence of noise and hence deduce its robustness. Finally, we confirm the functionality of our method in practice by simulations mimicking the measurements of a circular ultrasound tomography device.Comment: submitted to IEEE Transaction on Signal Processin