Related papers: A detailed analysis of the brachistochrone problem
An analysis is presented of a Brownian particle moving on the half-line, subject to a restoring force proportional to its displacement and an absorbing boundary at the origin. When the initial displacement is large, the central moments of…
Preliminary spacecraft trajectory optimization is a parameter dependent global search problem that aims to provide a set of solutions that are of high quality and diverse. In the case of numerical solution, it is dependent on the original…
We solve the problem concerning a time optimal return of a particle with a prescribed velocity to the origin by applying a magnitude-bounded force. The equations of controlled motion are derived and explicitly integrated, and the optimal…
Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a non-zero drift.…
We explore the gravitational dynamics of falling through planetary interiors. Two trajectory classes are considered: a straight cord between two surface points, and the brachistochrone path that minimizes the falling time between two…
Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points.…
We study the navigation problem for a robot moving amidst static and dynamic obstacles and rely on a hierarchical approach to solve it. First, the reference trajectory is planned by the safe interval path planning algorithm that is capable…
The variational problem of the least uncomfortable journey between two locations on a straight line is simplified by a choice of the dependent variable. It is shown that taking the position, instead of the velocity, as the optimal function…
We analyse a mechanical system in two-dimensional relative motion with friction. Although the system is simple, the peculiar interplay between two kinetic friction forces and gravity leads to the wide range of admissible solutions exceeding…
We discuss a fluid dynamic variant of the classical Bernoulli's brachistochrone problem. The classical brachistochrone for a non-dissipative particle is governed by maximization of the particle's kinetic energy resulting in a cycloid. We…
The steady development of motor vehicle technology will enable cars of the near future to assume an ever increasing role in the decision making and control of the vehicle itself. In the foreseeable future, cars will have the ability to…
Motion planning for multi-jointed robots is challenging. Due to the inherent complexity of the problem, most existing works decompose motion planning as easier subproblems. However, because of the inconsistent performance metrics, only…
Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the…
The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example,…
In this study a new approach is proposed for the generation of aircraft trajectories. The relative guidance of an aircraft, which is aimed to join in minimum time the track of a leader aircraft, is particularly considered. In a first place,…
When a large collection of objects (e.g., robots, sensors, etc.) has to be deployed in a given environment, it is often required to plan a coordinated motion of the objects from their initial position to a final configuration enjoying some…
We consider the problem of trajectory planning in an environment comprised of a set of obstacles with uncertain locations. While previous approaches model the uncertainties with a prescribed Gaussian distribution, we consider the realistic…
The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural…
This paper focuses on spatial time-optimal motion planning, a generalization of the exact time-optimal path following problem that allows the system to plan within a predefined space. In contrast to state-of-the-art methods, we drop the…
We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct…