Piecewise Linear Motion Research Articles

A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

Stochastic methods for the neutron transport equation I: Linear semigroup asymptotics

The neutron transport equation (NTE) describes the flux of neutrons through an inhomogeneous fissile medium. In this paper, we reconnect the NTE to the physical model of the spatial Markov branching process which describes the process of nuclear fission, transport, scattering, and absorption. By reformulating the NTE in its mild form and identifying its solution as an expectation semigroup, we use modern techniques to develop a Perron–Fröbenius (PF) type decomposition, showing that growth is dominated by a leading eigenfunction and its associated left and right eigenfunctions. In the spirit of results for spatial branching and fragmentation processes, we use our PF decomposition to show the existence of an intrinsic martingale and associated spine decomposition. Moreover, we show how criticality in the PF decomposition dictates the convergence of the intrinsic martingale. The mathematical difficulties in this context come about through unusual piecewise linear motion of particles coupled with an infinite type-space which is taken as neutron velocity. The fundamental nature of our PF decomposition also plays out in accompanying work (Harris, Horton and Kyprianou (2020), Cox et al. (2020)).

A full 3D numerical model of the industrial electrostatic coating process for moving targets

Abstract In this paper, a full three dimensional FLUENT numerical model of the electrostatic coating process as used in the automotive industry is presented and selected parametric studies on the numerical model have been conducted. The model incorporates a moving mesh capability in order to simulate movements of the target. The target follows piecewise linear motion and the entire area of the target has been coated in multiple passes. All the dominant mechanical and electrical phenomena were taken into account and cases with both neutral and charged particles were examined. It was confirmed that the transfer efficiency increases when the electrical forces are present in the model and that the coating uniformity improves with the movement of the target. The deposition pattern corresponds to the motion of the target and gradually builds up in a cumulative manner. Due to the aggregation effect associated with the target movement, the deposition uniformity is dramatically improved.

Continuous billiards and the weakly dissipative Kolmogorov – Arnold – Moser theory

Within the framework of the simplest model of the weakly dissipative Kolmogorov–Arnold–Moser theory, known as the Bogdanov-map, we compute two families of phase portraits of vector fields. These phase curves describe the moving boundary in the variational problem governing a piecewise linear motion of a test particle in an inhomogeneous continuous medium. Bibliography: 6 titles. Illustrations: 1 figure.

Trajectory planning for two manipulators to deform flexible beams

Coordinating two manipulators to deform a flexible and uniform beam is studied in this paper. Two coordination methods are identified for the manipulators. The first method allows the relative position to be altered, but no relative orientation changes. The second method allows both the position and the orientation changes. By studying the mechanics of elastic deformation, the performance of the methods, in terms of the interaction forces and moments between the object and the manipulators, is analyzed. The analysis shows that the second method may result in less interaction forces and moments than the first method. Optimal motion trajectories for generating zero interaction moments in the second method are then derived. To reduce the computation load for specifying the optimal trajectory, a piece-wise linear motion is proposed to replace the optimal trajectory. The performance of the piece-wise linear motion is then analyzed. Finally, experimental results are presented to verify the effectiveness of the proposed methods.

Optimal piecewise linear motion of an object among obstacles

We present an algorithm for determining the shortest restricted path motion of a polygonal object amidst polygonal obstacles. The class of motions which are allowed can be described as follows: a designated vertex,P, of the polygonal object traverses a piecewise linear path, whose breakpoints are restricted to the vertices of the obstacles. The distance measure being minimized is the length of the path traversed byP. Our algorithm runs in timeO(n 4kogn). We also discuss a variation of this algorithm which minimizes any positive linear combination of length traversed byP and angular rotation of the ladder aboutP. This variation requiresO(n 5) time.