Kinematic analysis of a 3-UPU parallel Robot using the Ostrowski-Homotopy Continuation

📝 Abstract

The direct kinematics analysis is the foundation of implementation of real world application of parallel manipulators. For most parallel manipulators the direct kinematics is challenging. In this paper, for the first time a fast and efficient Homotopy Continuation Method, called the Ostrowski Homotopy continuation method has been implemented to solve the direct and inverse kinematics problem of the parallel manipulators. This method has advantage over conventional numerical iteration methods, which is not rely on the initial values and is more efficient than other continuation method and it can find all solutions of equations without divergence just by changing auxiliary Homotopy function. Numerical example and simulation was done to solve the direct kinematic problem of the 3-UPU parallel manipulator that leads to 16 real solutions. Results obviously reveal the fastness and effectiveness of this method than the conventional Homotopy continuation methods such as Newton Homotopy. The results shows that the Ostrowski-Homotopy reduces computation time up to 80-97 % with more accuracy in solutions in comparison with the Newton Homotopy.

💡 Analysis

The direct kinematics analysis is the foundation of implementation of real world application of parallel manipulators. For most parallel manipulators the direct kinematics is challenging. In this paper, for the first time a fast and efficient Homotopy Continuation Method, called the Ostrowski Homotopy continuation method has been implemented to solve the direct and inverse kinematics problem of the parallel manipulators. This method has advantage over conventional numerical iteration methods, which is not rely on the initial values and is more efficient than other continuation method and it can find all solutions of equations without divergence just by changing auxiliary Homotopy function. Numerical example and simulation was done to solve the direct kinematic problem of the 3-UPU parallel manipulator that leads to 16 real solutions. Results obviously reveal the fastness and effectiveness of this method than the conventional Homotopy continuation methods such as Newton Homotopy. The results shows that the Ostrowski-Homotopy reduces computation time up to 80-97 % with more accuracy in solutions in comparison with the Newton Homotopy.

📄 Content

Kinematic analysis of a 3-UPU parallel Robot using the Ostrowski-Homotopy Continuation Milad Shafiee-Ashtiani, Aghil Yousefi-Koma, Sahba Iravanimanesh, Amir Siavosh Bashardoust Center of Advanced Systems and Technologies (CAST) School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. shafiee.a@ut.ac.ir, aykoma@ut.ac.ir, siravanimanesh@ut.ac.ir, as.bashardoust@ut.ac.ir

Abstract— The direct kinematics analysis is the foundation of implementation of real world application of parallel manipulators. For most parallel manipulators the direct kinematics is challenging. In this paper, for the first time a fast and efficient Homotopy Continuation Method, called the Ostrowski Homotopy continuation method has been implemented to solve the direct and inverse kinematics problem of the parallel manipulators. This method has advantage over conventional numerical iteration methods, which is not rely on the initial values and is more efficient than other continuation method and it can find all solutions of equations without divergence just by changing auxiliary Homotopy function. Numerical example and simulation was done to solve the direct kinematic problem of the 3-UPU parallel manipulator that leads to 16 real solutions. Results obviously reveal the fastness and effectiveness of this method than the conventional Homotopy continuation methods such as Newton Homotopy. The results shows that the Ostrowski-Homotopy reduces computation time up to 80-97 % with more accuracy in solutions in comparison with the Newton Homotopy.
Keywords-Direct Kinematics; Parallel manipulators; Homotopy continuation method; Ostrowski-Homotopy;

I. INTRODUCTION The parallel robotic manipulator has attracted the attention of many researchers and it also has growing applications in robotics, machine tools, positioning systems, measurement devices, and so on and even the other mechanism such as biped robots in double support phase behave like a parallel mechanism [5-6]. This popularity is a result of the fact that the parallel manipulators have more advantages in comparison to serial manipulators in many aspects, such as stiffness in mechanical structure, high position accuracy and high load carrying capacity [1]. However, they have some disadvantages such as limited workspace and complex forward position kinematics problems. In contrast to serial manipulators, the inverse position kinematics of parallel manipulators is relatively straightforward and the direct position kinematics is quite complicated [2]. It involves the solution of a system of coupled nonlinear algebraic equations that its variables describing platform posture and has many solutions [3].
Except in a limited number of parallel manipulators, there is no exact closed form solution [4]. So these nonlinear equations should be solved using numerical methods. Up to now, so many numerical methods has been developed to solve system of coupled nonlinear algebraic equations, such as the Newton– Raphson method which is efficient in the convergence speed [7]. Unfortunately, there always needs to guess the initial value in the iteration process. Good initial guess value may converge to solutions but bad initial guess value usually leads to divergence. Homotopy continuation method (HCM) is a type of perturbation and Homotopy method [8-9]. It can guarantee the answer by a certain path, if the auxiliary Homotopy function is chosen well. It does not have the drawbacks of conventional numerical algorithms, in particular the requirement of good initial guess values and the problem of convergence [10]. The recent development of the HCM was conducted by Morgan [11], Allgower [12]. Wu [8-10] introduced some techniques by combining Newton’s and Homotopy methods to avoid divergence in solving system of coupled nonlinear algebraic equations. Also Wu [8], Varedi [13] and Abbasnezhad [14] have employed Newton-HCM in kinematics analysis of manipulators. Ostrowski’s method was introduced by Alexander Markowich which has fourth order convergence and is an extension of Newton’s method. [15, 16].
The latest advancement on Ostrowski’s method was done by Grau [17], Chun [18] . Nor [19] presented some techniques by combining Ostrowski’s and Homotopy Method to solve Polynomial Equations. As it is clear, the numerical methods have the problem of expensive computational cost and for real-time application of robotics are not sufficient.
In this paper, for the first time the Ostrowski-HCM is employed to solve the direct and inverse kinematic of 3-UPU parallel manipulator with a very small computation time that allows for implementing kinematic analysis of parallel robots in real time applications. At the end the comparison was made between the Newton-HCM and Ostrowski-HCM results.
II. THE HOMOTOPY CONTINUATION METHOD Numerical iterative methods such as Newton-Raphson have two drawb

This content is AI-processed based on ArXiv data.