Abstract. This work presents a methodological framework, based on an indirect approach, for the automatic generation and numerical solution of Optimal Control Problems (OCP) for mechatronic systems, described by a system of Differential Algebraic Equation (DAEs). The equations of the necessary condition for optimality were derived exploiting the DAEs structure, according to the Calculus of Variation Theory. A collection of symbolic procedures was developed within a general-purpose Computer Algebra Software. Those procedures are general and make it possible to generate both OCP equations and their jacobians, once any DAE mathematical model, objective function, boundary conditions and constraints are given. Particular attention has been given to the correct definition of the boundary conditions especially for models described with set of dependent coordinates. The non-linear symbolic equations, their jacobians with the sparsity patterns, generated by the procedures above mentioned, are translated into a C++ source code. A numerical code, based on a Newton Affine Invariant scheme, was also developed to solve the BVPs as generated by such procedures. The software and methodological framework here presented were successfully applied to the solution of the minimum-lap time problem of a racing motorcycle. 1

Abstract — This paper develops an indirect optimal control methodology to achieve green driving optimisation for series hybrid electric vehicles. Starting from a given vehicle mission, specified in terms of a road journey that has to be completed in a given amount of time, the power sharing among the powertrain sources and the vehicle speed profile along the journey are optimised and found. The scheme combines para-metric modelling of the vehicle and powertrain together with computationally efficient optimal control software to provide an optimization strategy that works in real-time. Simulation results that demonstrate the success of the method and provide further insight into efficient driving, are presented. I.

A new effective solution to the problem of Hermite G1 interpolation with a clothoid curve is here proposed, that is a clothoid that interpolates two given points in a plane with assigned unit tangent vectors. The interpolation problem is a system of three nonlinear equations with multiple solutions which is difficult to solve also numerically. Here the solution of this system is reduced to the computation of the zeros of one single function in one variable. The location of the zero associated to the relevant solution is studied analytically: the interval containing the zero where the solution is proved to exists and to be unique is provided. A simple guess function allows to find that zero with very few iterations in all possible configurations. The computation of the clothoid curves and the solution algorithm call for the evaluation of Fresnel related integrals. Such integrals need asymptotic expansions near critical values to avoid loss of precision. This is necessary when, for example, the solution of the interpolation problem is close to a straight line or an arc of circle. A simple algorithm is presented for efficient computation of the asymptotic expansion. The reduction of the problem to a single nonlinear function in one variable and the use of asymptotic expansions make the present solution algorithm fast and robust. In particular a comparison with algorithms present in literature shows that the present algorithm requires less iterations. Moreover accuracy is maintained in all possible configurations while other algorithms have a loss of accuracy near the transition zones.

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Investigation of the influences of tyre–road friction and engine power on motorcycle racing performance