Kinematic analysis of a 3-UPU parallel 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]. 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. , 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 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(푥푖) (7) 푦푖 = 푥푖 − 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 𝑖. 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) 𝑖 = 0,1,2, … , 푘 − 1 퐻(푋, 푡) ≡ 푡퐹(푋) + (1 − 푡)퐺(푋) = 0, (4) Where 𝑖 = 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. 𝑖 = 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) 𝑖푠 Jacobian matrix of size n×n and the 퐻(푥푖) and 휕퐻1(푥푛,푦푛,… ) 휕퐻1(푥푛,푦푛,… ) 퐻(푦 ) are scalars. … 푖 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 𝑖 = (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

∑𝑖 (𝑖 = 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}(𝑖 = 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. The base and global frame. [20]

Figure 1. The 3-UPU parallel manipulator. [20]

A. Inverse kinematics

Let us consider the coordinates of 푂푝 be (푋, 푌, 푍) in the global frame∑. For the inverse kinematics problem, the origin point 푂푝(푋, 푌, 푍) of the mobile platform in the global Figure 3. The ith limb of the manipulator [20] frame ∑ is known and the link lengths 퐿푖(𝑖 = 1,2,3) are need to be found. Since each branch limb can be regarded as a 3-DOF serial mechanism with its joint variables

{휃 1푖 , 휃2푖 , 퐿푖} . Point 푂푝 in a local frame ∑𝑖 (𝑖 = 1,2,3) can be represented as:

푋푖 0 0 {푌푖 } = {퐴푖퐵푖} + {0} − {0 } (12) 푍푖 푟 푟푝

Where {푋푖, 푌푖 , 푍푖} is the coordinates of 푂푝 in the local frame ∑푖 (𝑖 = 1,2,3) r is the circumradius of triangle 훥퐴1퐴2퐴3 in the base (Figs. 2 and 3) and 푟 is the circumradius of triangle 푝 훥퐵1퐵2퐵3 with the circumcenter 푂푝 in the mobile platform Figure 4. The mobile platform and its frame. [20] (Figs. 3 and 4) [20]. From Fig.3, the vector from 퐴푖 to 퐵푖 in a local frame ∑푖 (𝑖 = 1,2,3) is: (퐿푖 cos(휃2푖) + 2푒) cos(휃1푖) {퐴푖퐵푖} = {(퐿푖 cos(휃2푖) + 2푒) sin (휃1푖) } (13) 퐿푖 sin (휃2푖)

Where 퐿푖the length of ith prismatic joint, e is is the offset in a universal joint and 휃 1푖 휃2푖 are the rotation angles around its

corresponding axis (Fig. 2) [20]. Substituting the value of 퐴푖퐵푖 The results shows Ostrowski-HCM can reach the Newton- from Eq. (13) into Eq. (12), and some manipulation, yields [20]: HCM accuracy with a quite less iteration and more accuracy. Based on results it’s obvious that Ostrowski-HCM reduces Computation time for solving direct kinematic problem of the 푋 (퐿푖 cos(휃2푖) + 2푒) cos(휃1푖) 푖 proposed manipulator up to 97 % in comparison with Newton- {푌 } = {(퐿 cos(휃 ) + 2푒) sin (휃 ) } (𝑖 = 1,2,3) (14) 푖 푖 2푖 1푖 HCM. 푍푖 퐿푖 sin(휃2푖) + 푑푟 B. Forward Kinematics Where 푑푟 = 푟 − 푟푝 When three link lengths 퐿푖(𝑖 = 1,2,3) are known, the For finding the roots of this system of nonlinear equations we forward kinematics is to find 푂푝(푋, 푌, 푍). By eliminating can use the Homotopy continuation method. Considering the 휃 1푖 휃2푖 Eq. (10) becomes an equation in terms of e, 푑푟 and 퐿푖: structure parameters of the manipulator as: 푑푟 = 0.382 푒 = 0.182 푋푖 = 0.5 , 푌푖 = −1.5 푎푛푑 푍푖 = 1 we can rewrite 2 2 2 2 2 2 2 2 [푋푖 + 푌푖 + (푍푖 − dr) − 4푒 − 퐿푖 ] − 16푒 [(퐿푖 − equations (15) in the following simultaneous non-linear forms: 2 (푍푖 − 푑푟) ] = 0 (17) 푓 = (퐿푖 cos(휃2푖) + 2 × 0.182) cos(휃1푖) − 0.5 = 0 (15-a) From equations (11) and (17), the following three equations 푔 = (퐿 cos(휃 ) + 2 × 0.182) sin(휃 ) + 1.5 = 0 (15-b) 푖 2푖 1푖 can be exploited: 2 2 2 2 2 2 2 2 ℎ = 퐿푖 sin(휃2푖) + 0.382 = 0 (15-c) 푋 + 푌 + (푍 − dr) − 4푒 − 퐿1] − 16푒 [(퐿푖 − (푍푖 − 2 Therefore, we can write the Homotopy continuation function 푑푟) ] = 0 (18-a) as follows: 2 2 푋 + (푌 푐표푠(훽2) + 푍 sin (훽2)) + (−푌 sin (훽2) + 퐻 = [(퐿 cos(휃 ) + 2 × 0.182) cos(휃 ) − 0.5] × 푡 + (1 − 2 2 2 2 2 2 1 푖 2푖 1푖 푍 푐표푠(훽2) − 푑푟) − 4푒 − 퐿2] − 16푒 [(퐿2 − 푡) × 퐺1 = 0 (16-a) 2 (−푌 푠𝑖푛(훽2) + 푍 푐표푠(훽2) − 푑푟) ] = 0 (18-b) 퐻2 = [(퐿푖 cos(휃2푖) + 2 × 0.182) sin(휃1푖) + 1.5] × 푡 + (1 − 2 2 푡) × 퐺2 = 0 (16-b) 푋 + (푌 푐표푠(훽3) + 푍 sin (훽3)) + (−푌 sin (훽3) + 2 2 2 2 2 2 푍 푐표푠(훽3) − 푑푟) − 4푒 − 퐿3] − 16푒 [(퐿3 − 퐻3 = [퐿푖 sin(휃2푖) + 0.382] × 푡 + (1 − 푡) × 퐺3 = 0 (16-c) 2 (−푌 푠𝑖푛(훽3) + 푍 푐표푠(훽3) − 푑푟) ] = 0 (18-c) We solve equations (16a)–(16c) by the Newton–Raphson and also by Ostrowski method and change the Homotopy parameter Each of equations. (18a)–(18c) represents one equation in t from 0 to 1. We choose the initial guesses of unknown terms of X, Y, and Z. In order to solve the above equation, let us parameters as: (휃 1푖0 , 휃2푖0 , 퐿푖0) = (−1 , −0.5, 1.5) define:

By appropriately regulating the auxiliary Homotopy function, 2 2 2 we will obtain in total two sets of the roots of this problem 푃 = 푋 + 푌 + 푍 (19) 2휋 without any divergence. The auxiliary Homotopy functions of 퐿 = 1.486 , 퐿 = 1.386 , 퐿 = 1.576 , 훽 = 0 , 훽 = , these results are given in Table 1. The criterion used for 1 2 3 1 2 3 4휋 measuring the iteration number of Ostrowski method in 훽 = , 푒 = 0.182 , 푑푟 = 0.382 (20) 3 3 simulation is based on the both following terms simultanesly: Substituting the value of P from Eq. (19) into equations −6 (18a)–(18c), upon some simplifications and Substituting the |(푠표푙푢푡𝑖표푛(푛푒푤푡표푛−퐻퐶푀) − (푠표푙푢푡𝑖표푛(표푠푡푟표푤푠푘푖−퐻퐶푀))| < 10 퐻 (푠표푙푢푡𝑖표푛 ) − 퐻 (푠표푙푢푡𝑖표푛 ) > 0 values of the geometric parameters from (20), then we can write 푖 (푛푒푤푡표푛−퐻퐶푀) 푖 (표푠푡푟표푤푠푘푖−퐻퐶푀) the Homotopy continuation function as follows

TABLE I. THE RESULTS OF HOMOTOPY CONTINUATION METHOD FOR INVERSE KINEMATICS

Calculation Runtime No. G1 , G2 , G3 Method ° ° Iteration Time (휃1 ) , (휃2 ) , (L) (seconds) Reduction% Ostrowski (288.435) , (26.91928) , (1.36504) 100 0.062951 (cos(휃 )) , (sin(휃 )) , -HCM 97.735 1 1i 2푖 (퐿2) Newton- 푖 (288.434) , (26.91909) , (1.36504) 100000 2.778823 HCM (-(cos(휃 ))*2) , Ostrowski 0.069221 1푖 (108.43492) ,(162.37417) , (2.04095) 290 (sin(휃 )+2*cos(휃 )+1) , -HCM 97.5467 2 2푖 2푖 (퐿2) Newton- 2.821601 푖 (108.43494) , (162.37416) , (2.04095) 100000 HCM

[(푃 − 0.764 × 푍 − 2.194768)2 − 1.170308549 + result. The auxiliary Homotopy functions and their results are 2 0.529984 × (푍 − 0.382) ] × 푡 + (1 − 푡) × 퐺1 = 0, (21-a) given in Table 2. The simulation implemented in a hardware with the Intel® Core™ 2 Duo CPU and 4GB RAM in 64-bit [(푃 + 0.382 × 푍 + 0.661624 × 푌 − 1.907568)2 − Operating system by MATLAB program. we have found all of 1.018097144 + 0.529984 × (−0.866 × 푌 − 0.5 × 푍 − possible real solutions of direct kinematic of proposed parallel 2 0.382) ] × 푡 + (1 − 푡) × 퐺2 = 0, (21-b) robot by this method. Ostrowski-HCM can reach to better accuracy with a quite less iteration than the Newton-HCM. [(푃 + 0.382 × 푍 + 0.661624 × 푌 − 2.470348)2 − Based on results it’s clear that Ostrowski-HCM, with better 1.31636154 + 0.529984 × (0.866 × 푌 − 0.5 × 푍 − accuracy, reduces computation time for solving direct 2 kinematic problem of the proposed manipulator up to 80-97 % 0.382) ] × 푡 + (1 − 푡) × 퐺3 = 0 (21-c) in comparison with Newton-HCM. Final 16 solutions by using The Homotopy parameter 푡 changes from 0 to 1 and the initial Eq. (19) and result of Table.2 are listed in Table.3. The comparison of these results and the results reported by Wu [20] guesses of the unknown parameters are: (푃0, 푌0, 푍0) = (1,1,1) We solve Equations (21a)–(21c) by the Ostrowski method and are in excellent agreement. change the Homotopy functions 퐺푖(𝑖 = 1,2,3) to obtain the

TABLE II. THE RESULTS OF HOMOTOPY CONTINUATION METHOD FOR DIRECT KINEMATICS( INITIAL GIESS:(푃0, 푌0, 푍0) = (1,1,1)) Calculation Result Runtime G , G , G Method (P) , (Y) , (Z) Iteration Time No. 1 2 3 (seconds) Reduction% 0.022847 Ostrowski-HCM (1.57197) , (0.71419) , (0.58723) 68

1 P,Y,Z 92.2063 0.293147 Newton-HCM (1.57197) , (0.71419) , (0.58723) 10000

0.0238 Ostrowski-HCM (2.37060) , (0.93539) , (-0.71988) 143

2 (-P),Y,Z 91.97367 0.296524 Newton-HCM (2.37060) , (0.93539) , (-0.71988) 10000

0.031543 Ostrowski-HCM (1.17091) , (-0.43286) , (0.03721) 253

3 P,(-Y),Z 89.54744 0.301773 Newton-HCM (1.17091) , (-0.43286) , (0.03721) 10000

0.019755 Ostrowski-HCM (1.54796) , (-0.28820) , (-1.08312) 58

4 (-P),(-Y),Z 93.18873 0.290034 Newton-HCM (1.54796) , (-0.28820) , (-1.08312) 10000

0.019237 (-P+5), Ostrowski-HCM (2.40438) , (-0.28933) , (1.34928) 81

5 (-Y-1), 93.4556 0.293946 (-Z+3) Newton-HCM (2.40438) , (-0.28933) , (1.34928) 10000

0.026975 (-P+5), Ostrowski-HCM (3.21157) , (-0.42875) , (-0.02965) 181

6 (-Y-1), 90.71652 0.29057 (Z+3) Newton-HCM (3.21157) , (-0.42875) , (-0.02965) 10000

0.032144 (-P-1), Ostrowski-HCM (2.48980) , (-1.38318) , (-0.64092) 323

7 (-Y+5), 88.89841 0.289544 (Z+3) Newton-HCM (2.48980) , (-1.38318) , (-0.64092) 10000

0.078746 (-P-1), Ostrowski-HCM (1.47615) , (-1.10076) , (0.47274) 791

8 (Y+5), 80.34387 0.400618 (-Z-3) Newton-HCM (1.47615) , (-1.10076) , (0.47274) 10000

TABLE III. THE FINAL 16 SOLUTIONS OF THE DIRECT KINEMATICS [5] Khadiv, Majid, et al. "Optimal gait planning for PROBLEM humanoids with 3D structure walking on slippery Solution surfaces." Robotica (2015): 1-19. P X Y Z Number [6] Khadiv, Majid, S. Ali A. Moosavian, and Majid Sadedel. 1 2.37060 ±0.98863 0.93539 -0.71988 "Dynamics modeling of fully-actuated humanoids with 2 1.47615 ±0.20244 -1.10076 0.47274 general robot-environment interaction."Robotics and 3 2.48980 ±0.40720 -1.38318 -0.64092 Mechatronics (ICRoM), 2014 Second RSI/ISM 4 3.21157 ±1.73978 -0.42875 -0.02966 International Conference on. IEEE, 2014.. 5 1.54719 ±0.54013 -0.28820 -1.08312 [7] Ben-Israel, Adi. "A Newton-Raphson method for the solution of systems of equations." Journal of Mathematical 6 1.57197 ±0.84678 0.71419 0.58723 analysis and applications 15, no. 2(1966):243-252 7 1.17091 ±0.99103 0.43286 0.03721 [8] Wu, Tzong-Mou. "The inverse kinematics problem of 8 2.40416 -0.28929 1.34969 ±0.70625 spatial 4P3R robot manipulator by the Homotopy IV. CONCLUSION continuation method with an adjustable auxiliary Homotopy function." Nonlinear Analysis: Theory, In this paper the Ostrowski-HCM is conducted to solve the Methods & Applications 64, no. 10 (2006): 2373-2380 forward kinematics of parallel manipulators. Analysis showed [9] Wu, Tzong-Mou. "A study of convergence on the Newton- that that there are 16 real solutions for non-linear direct Homotopy continuation method." Applied Mathematics kinematic equations of 3-UPU manipulator. In result it has been and Computation 168.2 (2005): 1169-1174. proved that Ostrowski-HCM method reduces Computation time [10] Wu, Tzong-Mou. "Solving the nonlinear equations by the for solving direct kinematic problem and it is more effective Newton-Homotopy continuation method with adjustable and faster than conventional HCM methods. Also as shown in auxiliary Homotopy function." Applied Mathematics and Table.4 solutions doesn’t depend on initial guess. This method Computation 173, no. 1 (2006): 383-388 is so suitable for kinematic analysis of parallel robots and in [11] Morgan, Alexander P. "A method for computing all particular in applications that fastness and accuracy of the solutions to systems of polynomials equations. " ACM method is important such as obstacle and singularity avoidance. Transactions on Mathematical Software (TOMS)9, no. 1 (1983): 1-17 TABLE IV. SOME INITIAL GUESSES WHICH LEAD TO THE SAME ANSWER WITH COMMON HOMOTOPY FUNCTIONS IN TABLE 1 (NO.1) [12] Allgower, Eugene L., and Kurt Georg. Numerical continuation methods: an introduction. Vol. 13. Springer ° ° ° ° Number (휃1,0 , 휃2,0 , 퐿0) (휃1 , 휃2 ,퐿) Science & Business Media, 2012 (288.435) , [13] Varedi, S. M., H. M. Daniali, and D. D. Ganji. "Kinematics 1 (1,10,1.5) (26.91928) , of an offset 3-UPU translational parallel manipulator by the (1.36504) Homotopy continuation method."Nonlinear Analysis: Real (288.435) , World Applications 10, no. 3 (2009): 1767-1774 2 (5729,85.94,0.5) (26.91928) , [14] Abbasnejad, Ghasem, Soheil Zarkandi, and Misagh Imani. (1.36504) "Forward kinematics analysis of a 3-PRS parallel (288.435) , manipulator." World Academy of Science, Engineering and 4 (-150, -81, 10) (26.91928) , Technology 37 (2010): 329-335 (1.36504) [15] Ostrowski, M. Solutions of Equations and System of (288.435) , Equations. New York: Academic Press, 1960. 5 (-0.05, 0.45, 100) (26.91928) , [16] Ostrowski, Alexander M. Solution of equations in (1.36504) Euclidean and Banach spaces. Elsevier, 1973. [17] Grau, Miquel, and José Luis Díaz-Barrero. "An improvement to Ostrowski root-finding method." Applied REFERENCES Mathematics and Computation 173, no. 1 (2006): 450-456. [1] Taghirad, Hamid D. “Parallel robots”: mechanics [18] Chun, Changbum, and YoonMee Ham. "Some sixth-order and control. CRC press, 2013. variants of Ostrowski root-finding methods." Applied [2] Merlet, Jean-Pierre. Parallel robots. Vol. 74. Springer Mathematics and Computation 193, no. 2 (2007): 389-394 Science & Business Media, 2012. [19] Nor, Hafizudin Mohamad, Amirah Rahman, Ahmad Izani [3] Abbasnejad, G., H. M. Daniali, and A. Fathi. "Closed form Md Ismail, and Ahmad Abdul Majid. "Numerical Solution solution for direct kinematics of a 4PUS+ 1PS parallel of Polynomial Equations using Ostrowski Homotopy manipulator." Scientia Iranica 19.2 (2012): 320-326. Continuation Method." MATEMATIKA 30 (2014): 47-57. [4] Fu, Jianxun, et al. "Forward kinematics solutions of a [20] Ji, Ping, and Hongtao Wu. "Kinematics analysis of an offset 3-UPU translational parallel robotic special six-degree-of-freedom parallel manipulator with manipulator." Robotics and Autonomous Systems42, no. 2 three limbs." Advances in Mechanical Engineering 7.5 (2003): 117-123 (2015): 1687814015582118.I. S.