Marco Frego

G1 fitting with clothoids

An effective solution to the problem of Hermite

Semi-analytical minimum time solutions for a vehicle following clothoid-based trajectory subject to velocity constraints

G^1

Semi-analytical minimum time solutions with velocity constraints for trajectory following of vehicles

interpolation with a clothoid curve is provided. At the beginning the problem is naturally formulated as a system of nonlinear equations with multiple solutions that is generally difficult to solve numerically. All the solutions of this nonlinear system are reduced to the computation of the zeros of a single nonlinear equation. A simple strategy, together with the use of a good and simple guess function, permits to solve the single nonlinear equation with a few iterations of the Newton--Raphson method. The computation of the clothoid curve requires the computation of Fresnel and 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 interpolation problem is close to a straight line or an arc of circle. Moreover, some special recurrences are deduced for the efficient computation of asymptotic expansion. The reduction of the problem to a single nonlinear function in one variable and the use of asymptotic expansions make the solution algorithm fast and robust.A new effective solution to the problem of Hermite

Trajectory planning for car-like vehicles: A modular approach

G^1

Preconditioning Complex Symmetric Linear Systems

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.

Fast and accurate clothoid fitting

A possible strategy for finding the optimal path that connects two different configurations of a car-like vehicle can be addressed by solving two different sub-problems: 1. identifying a curve that connects the two points respecting the geometric constraints, 2. finding an optimal control strategy that drives the vehicle along the trajectory accounting for its dynamic constraints. In this paper, we focus on the second problem. Assuming that the vehicle moves along a specified clothoid, we find a semi-analytical solution for the optimal profile of the longitudinal acceleration (the control variable). Our technique explicitly considers non-linear dynamics, aerodynamic drag effect and bounds on the longitudinal and on the lateral acceleration.

Fast and accurate

Abstract We consider the problem of finding an optimal manoeuvre that moves a car-like vehicle between two configurations in minimum time. We propose a two phase algorithm in which a path that joins the two points is first found by solving a geometric optimisation problem, and then the optimal manoeuvre is identified considering the system dynamics and its constraints. We make the assumption that the path is composed of a sequence of clothoids. This choice is justified by theoretical arguments, practical examples and by the existence of very efficient geometric algorithms for the computation of a path of this kind. The focus of the paper is on the computation of the optimal manoeuvre, for which we show a semi-analytical solution that can be produced in a few milliseconds on an embedded platform for a path made of one hundred segments. Our method is considerably faster than approaches based on pure numerical solutions, it is capable to detect when the optimal solution exists and, in this case, compute the optimal control. Finally, the method explicitly considers nonlinear dynamics, aerodynamic drag effect and bounds on the longitudinal and on the lateral acceleration.

G^1

We consider the problem of trajectory planning with geometric constraints for a car-like vehicle. The vehicle is described by its dynamic model, considering such effects as lateral slipping and aerodynamic drag. We propose a modular solution, where three different problems are identified and solved by specific modules. The execution of the three modules can be orchestrated in different ways in order to produce efficient solutions to a variety of trajectory planning problems (e.g., obstacle avoidance, or overtake). As a specific example, we show how to generate the optimal lap on a specified racing track. The numeric examples provided in the paper are good witnesses of the effectiveness of our strategy.

fitting of clothoid curves

A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.

On the G2 Hermite Interpolation Problem with clothoids

An effective solution to the problem of Hermite