Hyperspherical path tracking methodology as correction step in homotopic continuation methods

Oliveros-Muñoz, Juan Manuel; Jiménez‐Islas, H.

doi:10.1016/j.ces.2013.03.053

Cited by 16 publications

(30 citation statements)

References 33 publications

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“…The successful collision-free curve in an environment with the presence of obstacles can be formed by combining two or more external curves, this depending on the parameters of repulsion and proximity of obstacles with the direct path [1]. Due to the existence of multiple homotopic curves a tracking technique to prevent jumps between the successful path and the other homotopic curves is necessary [2], [4], [5]. IV.…”

Section: Homotopy Path Planing Methodsmentioning

confidence: 99%

“…The solution of this system is found using the Newton-Raphson (N-R) method [6]. Combining predictor-corrector scheme and the SA [2], [3], we obtained an appropriate methodology for spherical tracking. …”

Section: Homotopy Path Planing Methodsmentioning

confidence: 99%

“…where (x p , y p , λ p ) is the predictor point, (c x , c y , c λ ) is the center of the sphere, r is the radius of the sphere and v p is the tangent vector [2], [3]. The N-R corrector is described by the expression:…”

Section: Predictor-corrector Algorithmmentioning

confidence: 99%

See 2 more Smart Citations

Path optimization for terrestrial robots using Homotopy Path Planning Method

Diaz-Arango

Sarmiento-Reyes

Hernandez-Martinez

et al. 2015

2015 IEEE International Symposium on Circuits and Systems (ISCAS)

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The path planning method is an important element in the architecture of all mobile robots with certain degree of autonomy. An appropriate path planning method provides the robot the ability to move in a previously known environment while avoiding collisions with obstacles along the way. Additionally, this method should generate optimal trajectories from one point to another. In this work the optimal path is defined as the shortest one. Path planning is not an easy task, some algorithms and methods have been developed for solving this problem, however, its success is not guaranteed. On the other hand, Homotopy Path Planning Method (HPPM) is a new tool used to find collision-free paths, this takes the properties of the Homotopy Continuation Methods (HCM) to find a successful path. However, the path found by this method is not the optimal. In this work, a path planning method for a terrestrial mobile robot based on HPPM and Spherical Algorithm (SA) is presented. Furthermore, a new strategy able to obtain the optimal path is proposed. Finally, simulation results for several environment maps with hundreds of circular obstacles are shown.

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Section: Homotopy Path Planing Methodsmentioning

confidence: 99%

Section: Homotopy Path Planing Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Path optimization for terrestrial robots using Homotopy Path Planning Method

Diaz-Arango

Sarmiento-Reyes

Hernandez-Martinez

et al. 2015

2015 IEEE International Symposium on Circuits and Systems (ISCAS)

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“…The homotopy formulation contains n equations and ( n + 1) variables, where x i ( i = 1,…, n ) represent the variables of the system and x n +1 is the homotopy parameter λ . Nevertheless, if we add the equation that describes a sphere [ 2 , 3 , 13 , 37 , 66 ] with center at c (initial point of the trajectory) and radius r expressed by

then, it is possible to apply a regular NRM to solve the homotopy formulation.…”

Section: The Proposed Homotopy Schemementioning

confidence: 99%

Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices

Vázquez-Leal

Jimenez-Fernandez

Benhammouda

et al. 2014

The Scientific World Journal

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We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation.

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“…As an alternative, the homotopy continuation method (HCM) (Jimenez-Islas et al. 2013 ; Oliveros-Munoz and Jimenez-Islas 2013 ; Jimenez-Islas 1996 ; Bates et al. 2008 ; Ushida et al.…”

Section: Introductionmentioning

confidence: 99%

Speed-up hyperspheres homotopic path tracking algorithm for PWL circuits simulations

et al. 2016

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In the present work, we introduce an improved version of the hyperspheres path tracking method adapted for piecewise linear (PWL) circuits. This enhanced version takes advantage of the PWL characteristics from the homotopic curve, achieving faster path tracking and improving the performance of the homotopy continuation method (HCM). Faster computing time allows the study of complex circuits with higher complexity; the proposed method also decrease, significantly, the probability of having a diverging problem when using the Newton–Raphson method because it is applied just twice per linear region on the homotopic path. Equilibrium equations of the studied circuits are obtained applying the modified nodal analysis; this method allows to propose an algorithm for nonlinear circuit analysis. Besides, a starting point criteria is proposed to obtain better performance of the HCM and a technique for avoiding the reversion phenomenon is also proposed. To prove the efficiency of the path tracking method, several cases study with bipolar (BJT) and CMOS transistors are provided. Simulation results show that the proposed approach can be up to twelve times faster than the original path tracking method and also helps to avoid several reversion cases that appears when original hyperspheres path tracking scheme was employed.

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Hyperspherical path tracking methodology as correction step in homotopic continuation methods

Cited by 16 publications

References 33 publications

Path optimization for terrestrial robots using Homotopy Path Planning Method

Path optimization for terrestrial robots using Homotopy Path Planning Method

Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices

Speed-up hyperspheres homotopic path tracking algorithm for PWL circuits simulations

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