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Kinematic Analysis of a 3-UPU Parallel Robots Using the Ostrowski

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Kinematic Analysis of a 3-UPU Parallel Robots Using the Ostrowski 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. [email protected], [email protected], [email protected], [email protected] Abstract— The direct kinematics analysis is the foundation of coupled nonlinear algebraic equations, such as the Newton– implementation of real world application of parallel manipulators. Raphson method which is efficient in the convergence speed [7]. For most parallel manipulators the direct kinematics is Unfortunately, there always needs to guess the initial value in challenging. In this paper, for the first time a fast and efficient Homotopy Continuation Method, called the Ostrowski Homotopy the iteration process. Good initial guess value may converge to continuation method has been implemented to solve the direct and solutions but bad initial guess value usually leads to divergence. inverse kinematics problem of the parallel manipulators. This Homotopy continuation method (HCM) is a type of perturbation method has advantage over conventional numerical iteration and Homotopy method [8-9]. It can guarantee the answer by a methods, which is not rely on the initial values and is more efficient certain path, if the auxiliary Homotopy function is chosen well. than other continuation method and it can find all solutions of It does not have the drawbacks of conventional numerical equations without divergence just by changing auxiliary Homotopy function. Numerical example and simulation was done algorithms, in particular the requirement of good initial guess to solve the direct kinematic problem of the 3-UPU parallel values and the problem of convergence [10]. manipulator that leads to 16 real solutions. Results obviously The recent development of the HCM was conducted by Morgan reveal the fastness and effectiveness of this method than the [11], Allgower [12]. Wu [8-10] introduced some techniques by conventional Homotopy continuation methods such as Newton combining Newton’s and Homotopy methods to avoid Homotopy. The results shows that the Ostrowski-Homotopy divergence in solving system of coupled nonlinear algebraic reduces computation time up to 80-97 % with more accuracy in solutions in comparison with the Newton Homotopy. equations. Also Wu [8], Varedi [13] and Abbasnezhad [14] have employed Newton-HCM in kinematics analysis of manipulators. Keywords-Direct Kinematics; Parallel manipulators; Homotopy Ostrowski’s method was introduced by Alexander Markowich continuation method; Ostrowski-Homotopy; 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 I. INTRODUCTION Grau [17], Chun [18] . Nor [19] presented some techniques by The parallel robotic manipulator has attracted the attention of combining Ostrowski’s and Homotopy Method to solve many researchers and it also has growing applications in Polynomial Equations. robotics, machine tools, positioning systems, measurement As it is clear, the numerical methods have the problem of devices, and so on and even the other mechanism such as biped expensive computational cost and for real-time application of robots in double support phase behave like a parallel mechanism robotics are not sufficient. [5-6]. This popularity is a result of the fact that the parallel In this paper, for the first time the Ostrowski-HCM is employed manipulators have more advantages in comparison to serial to solve the direct and inverse kinematic of 3-UPU parallel manipulators in many aspects, such as stiffness in mechanical manipulator with a very small computation time that allows for structure, high position accuracy and high load carrying capacity implementing kinematic analysis of parallel robots in real time [1]. However, they have some disadvantages such as limited applications. At the end the comparison was made between the workspace and complex forward position kinematics problems. Newton-HCM and Ostrowski-HCM results. In contrast to serial manipulators, the inverse position kinematics of parallel manipulators is relatively straightforward II. THE HOMOTOPY CONTINUATION METHOD and the direct position kinematics is quite complicated [2]. It Numerical iterative methods such as Newton-Raphson have two involves the solution of a system of coupled nonlinear algebraic drawbacks. One is that bad initial guesses may leads to equations that its variables describing platform posture and has divergence and another is related to whether the iterative process many solutions [3]. will converge to desirable solutions. The HCM can rectify these Except in a limited number of parallel manipulators, there is no deficiency [10]. In this algorithm we first write the Homotopy exact closed form solution [4]. So these nonlinear equations continuation equations with auxiliary Homotopy function and should be solved using numerical methods. Up to now, so many then solve this system of nonlinear equations by Newton- numerical methods has been developed to solve system of Raphson method and also with Ostrowski method. Let us consider the following system of nonlinear equations: 푓(푥 ) (7) 푦 = 푥 − 푖 푖 푖 푓´(푥푖) 푓(푥푖) 푓(푦푖) (8) 푓(푥, 푦, … , 푧) = 0, 푥푖+1 = 푦푖 − 푓(푥푖) − 2푓(푦푖) 푓´(푥푖) 푔(푥, 푦, … , 푧) = 0, 퐹(푥) = 0 . 푒. { (1) ⋮ Where (7) is the conventional Newton’s iterative formula ℎ(푥, 푦, … , 푧) = 0, while (8) is the Ostrowski’s formula. Ostrowski’s method which has fourth order convergence is an extension of Given a system of equation in n variables 푥,푦,…,z we Newton’s method which has second order of convergence. modify the equations by omitting some of the terms and adding new ones until we have a new system of equations. The solution of this new system of equation may be easy to detect. We then B. Ostrowski Homotopy Continuation Method modify the coefficients of the new system into the coefficient As we saw in previous section, there have been several studies of the original system by a series of small increments and in last that combine local methods with HCM such as Newton-HCM, step we will solve the original system. This is called Homotopy Secant-HCM. Nor [19] combined local Ostrowski methods continuation technique. In order to find the solution of with HCM. Equations (1), we choose a auxiliary simple Homotopy function Ostrowski-HCM is as follows: [8-10], as: 퐻(푥푖) 퐺(푋) = 0, (3) 푦푖 = 푥푖 − , 퐻(́푥 ) 푖 퐻(푥 ) 퐻(푦 ) The roots of G(X) must be easy to detect. Then, we can write 푥 = 푦 − 푖 푖 (9) 푖+1 푖 ́ the Homotopy continuation function as follows: 퐻(푥푖) − 2퐻(푦푖) 퐻(푥푖) = 0,1,2, … , 푘 − 1 퐻(푋, 푡) ≡ 푡퐹(푋) + (1 − 푡)퐺(푋) = 0, (4) Where = 0,1,2, … , 푘 − 1 and t ∈ [0, 1]. To solve a system 푡 is an arbitrary parameter and varies from 0 to1, i.e., 푡 ∈ [0,1]. of system of coupled nonlinear algebraic equations, we have: Therefore, we have the following 2 boundary condition [8-10]: −1 푦푖 = 푥푖 − [퐷푥푯(푥, 푡)] 푯(푥, 푡) 퐻(푋, 0) = 퐺(푋) , 퐻(푋, 1) = 푓(푋) (5) ( ) ( ) −1 퐻 푥푖 퐻 푦푖 푥푖 = 푦푖 − [퐷푥푯(푥, 푡)] (10) Our goal is to solve the 퐻(푋, 푡) = 0 instead of 퐹(푥) = 0 by 퐻(푥푖) − 2퐻(푦푖) varying parameter 푡 by a series of small increments from 0 to 1. = 0,1,2, … , 푘 − 1 Hence numerical iteration formula of Homotopy Newton’s method for solving these equations is as follows: as [10]: Where 푥푖+1 , 푥푖 and 푯(푥푖, 푡) are vectors with dimensions 푇 푇 n×1, 푦푖 = (푦1 푦2 ⋯ 푦푛) , 푥푖 = (푥1 푥2 ⋯ 푥푛) , and the 퐷푥푯(푥, 푡) 푠 Jacobian matrix of size n×n and the 퐻(푥푖) and 휕퐻 (푥 ,푦 ,… ) 휕퐻 (푥 ,푦 ,… ) 1 푛 푛 1 푛 푛 … 퐻(푦푖) are scalars. 휕푥 휕푦 푥푛+1 − 푥푛 휕퐻2(푥푛,푦푛,… ) 휕퐻2(푥푛,푦푛,… ) [푦 − 푦 ] = … 푛+1 푛 휕푥 휕푦 ⋮ III. KINEMATICS MODEL OF AN OFFSET 3-UPU [ ] ⋮ ⋮ ⋱ (6) MANIPULATOR −퐻1(푥푛, 푦푛, … ) Here we study kinematic analysis of offset 3-UPU [−퐻2(푥푛, 푦푛, … )] ⋮ (Universal–Prismatic– Universal) parallel manipulator with an equal offset in its six universal joints, as shown in Fig.1 [20]. 3- To rectify divergence problem, Wu [10] suggested some UPU manipulator has three prismatic limbs (legs) that link the useful choice of auxiliary Homotopy function. They are base to the mobile platform by universal joints, and it can polynomial, harmonic, exponential or any combinations of construct a 3-DOF pure translational motion [20]. Fig.3 shows them. a schema of the ith limb = (1,2,3), which links point B on the mobile platform and point A on the base by a passive universal A. Ostrowski’s Method (U) joint, an active prismatic (P) joint, and another passive Ostrowski’s method was introduced by Alexander universal (U) joint. So, it is called a 3-UPU parallel manipulator Markowich Ostrowski to find the roots of a single-variable [20]. For the facility of kinematics analysis, three local frames, nonlinear function. The method composed of two-step namely: ∑3: 푂푋3푌3푍3 , ∑2: 푂푋2푌2푍2 푎푛푑 ∑1: 푂푋1푌1푍1 are iterations using the following equations [15, 16]: considered and attached to the base in order to describe the position and the first axis of a universal joint. Actually, frame ∑ ( = 1,2,3) is rotated along the X axis by angle 훽푖 from global frame∑. 훽1 = 0 since ∑1 is chosen in a way to be parallel to ∑. Thus coordinates {푋푖, 푌푖 , 푍푖}( = 1,2,3) of a point in a local frames ∑푖 and its coordinates {푋, 푌, 푍} in the global frame have the following relation (Figs.2 and 4) [20]: 푋푖 1 0 0 푋 {푌푖 } = [0 cos (훽푖) sin (훽푖)] {푌} = 푍푖 0 −sin (훽푖) cos (훽푖) 푍 푋 { 푌 cos(훽푖) + 푍 sin (훽푖) } (11) −푌 sin(훽 ) + 푍 cos(훽 ) 푖 푖 Figure 2.
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