A new development of homotopy continuation method, applied in solving nonlinear kinematic system of equations of parallel mechanisms

Lafmejani, Amir Salimi; Kalhor, Ahmad; Masouleh, Mehdi Tale

doi:10.1109/icrom.2015.7367874

Cited by 8 publications

(5 citation statements)

References 13 publications

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“…Homotopy methods, also referred to as continuation methods, may be used for finding solutions to polynomial equations and systems of polynomial equations [209,210,211]. Polynomial equations often arise in kinematics and robotics related problems [14,12]; papers [6,7] apply homotopy methods to solve systems of power flow equations; large-scale integrated circuit designs and protein-protein interaction equation are solved using homotopy method in [8], [27] applies homotopy methods for solving SNEs arising in geodesy; [22] applies homotopy for finding string vacua, [4] applies homotopy methods in chemical engineering. Homotopy (or deformation) of a system of equations F n (x) = Θ n ≡ (0, 0, .…”

Section: Homotopy / Continuation Methodsmentioning

confidence: 99%

“…As per [213], advantages of homotopy methods are 1) handling of singular solutions, 2) possibility to obtain multiple solutions using one homotopy path, and 3) preserving Morse indices for gradient systems. Paper [12] introduces a collision-based homotopy continuation technique. Article [214] discusses the problem of divergent homotopy paths and proposes an algorithm which performs projective path tracking.…”

Section: Homotopy / Continuation Methodsmentioning

confidence: 99%

“…designs [8], climate modeling [9], materials engineering [10], robotics [11,12,13,14], nuclear engineering [15], image restoration [16], protein interaction networks [8], neurophysiology [17], economics [18], finance [19], applied mathematics [20], physics [21], finding string vacua [22], machine learning [23,24], geometric constraint solving (used in computer aided design) [25], and geodesy [26,27] among others. The problem of solving even a system of polynomial equations has been proven to be NP-hard [28].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

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This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications.To accomplish these objectives, we present a multi-part survey. In part one, we focus on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we will introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.

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Section: Homotopy / Continuation Methodsmentioning

confidence: 99%

Section: Homotopy / Continuation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Finding one or more solutions to a SNE is a challenging and ubiquitous task faced in many fields including chemistry [1,2,3], chemical engineering [4], automotive steering [5], power flow [6,7], large-scale integrated circuit designs [8], climate modeling [9], materials engineering [10], robotics [11,12,13,14], nuclear engineering [15], image restoration [16], protein interaction networks [8], neurophysiology [17], economics [18], finance [19], applied mathematics [20], physics [21], finding string vacua [22], machine learning [23,24], and geodesy [25,26] among others. The problem of solving even a system of polynomial equations has been proven to be NP-hard [27].…”

Section: Introductionmentioning

confidence: 99%

Survey of Methods for Solving Systems of Nonlinear Equations, Part II: Optimization Based Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

View full text Add to dashboard Cite

This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications.To accomplish these objectives, we present a multi-part survey. In part one, we focused on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. We emphasize the important characteristics of each method, and we discuss promising directions for future research. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.

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“…In [9], according to a motion-tracking system, a vision-based control has been proposed for a 6-DoF parallel robot by measuring posture of movable platform. In [10], based on Homotopy Continuation approach, the nonlinear equations in FKP of a parallel robot were solved. Their method has been experienced in solving problem with high precision.…”

Section: Introductionmentioning

confidence: 99%

Research on Foot Trajectory Tracking of Parallel Wheel-legged Robot based on Dynamic Model Predictive Control

Liu

Wang²,

Chen

et al. 2020

Preprint

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In this paper, the foot trajectory tracking control for parallel structure of the sixwheel-legged robot is investigated. The accuracy of trajectory tracking and dynamic responsewith heavy load are the main challenges of parallel mechanism. To guarantee the tracking performance and improve dynamic response frequency to posture input, a method based on dynamic model predictive control is proposed under the establishment of dynamic model of single leg. Newton-Eulerian equation is derived and converted into a discrete state space expression for velocity loop control, appropriate parameters including prediction time domain, control time domain and proportional gain are determined by co-simulation. Desired sinusoidal trajectories with different frequencies are tracked with satisfactory performance in terms of accuracy and response frequency. Finally, comparative experimental results using BIT-NAZA robot derived from the proposed control strategy indicate that the delay error and amplitude error are better than PI controller under the same conditions. This research can provide theoretical and engineering guidance for accurate planning of intelligent robot, and facilitate the control performance of wheel-legged robot in practical system.

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A new development of homotopy continuation method, applied in solving nonlinear kinematic system of equations of parallel mechanisms

Cited by 8 publications

References 13 publications

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Survey of Methods for Solving Systems of Nonlinear Equations, Part II: Optimization Based Approaches

Research on Foot Trajectory Tracking of Parallel Wheel-legged Robot based on Dynamic Model Predictive Control

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