JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering. two-translational and two-rotational, 2T2R TP (2010)

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1 JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering Vol. 28 No. 5 Sep DOI: /j.issn T2R 1 1 1, UPU4-UPU/PPS. two-translational and two-rotational, 2T2R TP (2010) A Novel Parallel Manipulator of 2T2R and Positional Analysis JI Ye 1, LIU Hong-zhao 1, FAN Cai-xia 1,2, QIAO Zhan-hong 1 1. Faculty of Mechanical and Precision Instrument Engineering, Xi an University of Technology, Xi an , China 2. College of Mechanical and Electrical Engineering, Jiaozuo University, Jiaozuo Henan Province, China Abstract: Based on five degree-of-freedom (5-DOF) of parallel manipulator of 4-UPU, a novel spatial parallel manipulator of 4-UPU/PPS is deduced. From the theory of screw, we show that this manipulator possess 2T2R (two-translational and two-rotational) DOFs, and establish an equation of position. Since forward positional analysis of parallel manipulator is difficult, based on the constrained length of limps, we build an unconstrained nonlinear optimization fitness function of this manipulator. The problem of forward positional analysis is thus converted to a multi-objective optimization problem that can be solved with the dynamic variable-weighting multi-objective particle swarm optimization (PSO) algorithm which seeks the global minimum of the nonlinear optimization function. The result of 20 groups of random forward positional analysis is consistent with the actual position and orientation. proposed parallel manipulator. Therefore the problem of forward positional analysis is solved using the Keywords: parallel manipulator, particle swarm optimization, forward positional analysis, four degree-offreedom 2T2R 2T2R 2T2R [1] 2T2R Pierrot Company 1 [2]. [3-6] (No ) jiye_xaut@yeah.net liu-hongzhao@163.com

2 5 2T2R 547 [7] SP [8] 2T2R. PSO particle swarm optimization 1995 Kennedy Eberhart [9]. PSO. PSO. [10] PSO [11] PSO (time variant multi-objective particle swarm optimization) [12]PSO [13] PSO PSO 4-UPU 4-UPU/PPS2T2R 4 UPU PPS PSO. 1 4-UPU. 1 5 z. 1 PPS. B 1 A 1 A 4 P(A 5 ) B 4 z z A 2 y y O B 5 B 2 A 3 x x B UPU/PPS Figure 1 Schematic diagram of 4-UPU/PPS parallel manipulator 4 UPU 1 PPS. UA i (i = 1, 2,, 4); B i (i = 1, 2,, 4). A 1 A 1 A 2 A 1 A 4. S A 5 P B 5 B 5 O B 1 B / UPU UPU 5 1 $ τ 1 = (0, 0, 0; 0, 0, 1) z. 1 PPS 5 y $ τ 2 = (0, 1, 0; 0, 0, 0) y UPU P. 3. Oxyz. B 1 B 2 B 3 B 4 2b x B 1 B 2, y B 2 B 3, 1. P xyz, A 1 A 2 A 3 A 4 2a x A 1 A 2, y A 2 A 3, P

3 (x p, z p ) (β, γ). X Op = c = (x p ; 0; z p ; β; γ) T ϕ x p z p c = (x p ; 0; z p ) T ; β γ ϕ = (0; β; γ) T. Y-X Euler C β S β S γ S β C γ R(β γ, γ X ) = 0 C γ S γ (1) S β C β S γ C β C γ S β = sin β, C β = cos β,. (A ix, A iy, A iz ), (A ix, A iy, A iz ); (B ix, B iy, B iz )(i = 1, 2,, 4), A i = R(β y, γ x )A i + c, i = 1, 2,, 4 (2) A i l i, i = 1, 2,, 4 (3) l i = (A ix B ix ) 2 + (A iy B iy ) 2 + (A iz B iz ) 2 = f i (x p, z p, β, γ), i = 1, 2,, 4 (3) P (3) 3.2. PSO 4 PSO PSO.. PSOi t x t i = (x t i1, x t i2,, x t ) T x t [L d, U d ], L d U d v i t = (v i1, v i2,, v ) T, v t [v min,d, v max,d ], v min v max p t i = [p t i1, pt i2,, pt il ]T ; p t g = [p t g1, p t i2,, pt id ]T ; t + 1 (4) v t+1 = wv t + c 1 r 1 (p t x t ) + c 2 r 2 (p t gd x t ) (4) x t+1 = x t + v t+1 (5) r 1 r 2 (0,1) c 1 c 2 w. PSO. [14] w w [15]. w w = w min + (k max k)(w max w min )/k max (6) w min [0.2,0.5] w max [0.8,1] k max k. (6) w min w max w min = k k max (7) w min = k k max (8) PSO (6) (8)k max k 1 D 4 c 1 c (9) 2 24 x t i, p t xt i, p t gd. 3 (4) (5) 4 p t,. p t gd

4 5 2T2R 549 p t gd pt gd,. 5 k 1 p t gd , (3) f(x i ) = 4 (A ix B ix ) 2 + (A iy B iy ) 2 + (A iz B iz ) 2 l i i=1 (9). f i (X). npso min f{(x i )} X Op = c = (x p ; 0; z p ; 0; β; γ) T (10) ϕ X Op mm a = 50 mm; 200 mm b =100 mm x z [x 10, x + 10] [z 20, z + 20] β γ[ π/4, π/4]. 1 2 f(x i )/mm Figure 2 Fitness function optimization of the 6.2 first position and orientation [16] 6-CPS [17] 200 s[18] Stewart 10 µm 0.03 [19] Levenberg- Marquardt, (LM)6-PRRS k 1 5 Table 1 Results of five random choice positions and orientations x p/mm z p/mm β/rad γ/rad PSO π/180 10π/ π/180 13π/ π/180 14π/ π/180 14π/ π/180 18π/

5 PSO mm rad 1 µm rad CPU AMD2.7 GHz 2G PC 1 s 7 4-UPU 4-UPU/ PPS 4 UPU. PSO : [1]. [M] [2] Pierrotf, Company O. H4: a new family of 4- DOF parallel robot[c]//proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM 99, September 19-23, 1999, Atlanta, Georgia, USA. New York: IEEE, 1999: [3] Grigore G. Structural synthesis of fully-isotropic parallel robots with Schnflies motions via theory of linear transformations and evolutionary morphology[j]. European Journal of Mechanics: A/Sols, 2007, 26(2): [4] Altuzarra O, Loizaga M, Pinto C, Petuya V. Synthesis of partially decoupled multi-level manipulators with lower mobility[j]. Mechanism and Machine Theory, 2010, 45(1): [5] Motevalli B, Zohoor H, Sohrabpour S. Structural synthesis of 5 DOFs 3T2R parallel manipulators with prismatic actuators on the base[j]. Robotics and Autonomous Systems, 2010, 58(3): [6] Zong Guanghua, Pei Xu, Yu Jingjun, Bi Shusheng. Classification and type synthesis of 1-DOF remote center of motion mechanisms[j]. Mechanism and Machine Theory, 2008, 43(12): [7] Kong Xianwen, Gosselin C M. Type synthesis of 4- DOF SP-equivalent parallel manipulators: a virtual chain approach[j]. Mechanism and Machine Theory, 2006, 41(11) [8] 2T2R[J] 2010, 21(9): Fan Caixia, Liu Hongzhao, Zhang Yanbin. Type synthesis of 2T2R 4-DOF parallel mechanism based on configuration evolution and Lie group theory[j]. China Mechanical Engineering, 2010, 21(9): (in Chinese) [9] Kennedy J, Eberhart R C. Particle swarm optimization[c]//proceedings IEEE International Conference on Neural Net works, IV. Piscataway, NJ: IEEE Service Center, 1995: [10] Wang Yu, Li Bin, Thomas W, Wang Jianyu. Selfadaptive learning based particle swarm optimization[j]. Information Sciences, 2010, 180(14): [11] Praveen K T, Sanghamitra B, Sankar K P. Multi-objective particle swarm optimization with time variant inertia and acceleration coefficients[j]. Information Sciences, 2007, 177(22): [12] Leandro dos S C. Gaussian quantum-behaved particle swarm optimization approaches for constrained engineering design problems[j]. Expert Systems with Applications, 2010, 37(2): [13] Serkan K, Turker I, Alper Y. Evolutionary artificial neural networks by multi-dimensional particle swarm optimization[j]. Neural Networks, 2009, 22(10): [14] Shi Y H, Eberhart R C. A modified particle swarm optimizer[c]//proceedings of the IEEE Congress on Evolutionary Computation. Piscataway, USA: IEEE Service Center, 1998: [15] [J] (5): Wang Lin, Cao Jianfu, Han Chongzhao. A robot multi-sensor self-calibration method based on particle swarm optimization[j]. Robot, 2009, 31 5 : (in Chinese) [16] 6-CPS [J] 2009, 40(11): Che Linxian. Forward positional analysis of 6-cps orthogonal parallel manipulators[j]. Transactions of the Chinese Society for Agricultural Machinery, 2009, : (in Chinese) [17] 5SPS-1CCS [J] 2007, 43(5): Huang Xiguang, Liao Qizheng, Li Duanling, Wei Shimin. Forward displacement analysis of generalized 5sps-1ccs parallel robot mechanism based on quaternion[j]. Chinese Journal of Mechanical Engineering, 2007, 43(5): (in Chinese) [18] Stewart [J]2003, 30(2): Zheng Chunhong, Jiao Licheng. Forward kinematics of a general Stewart parallel manipulator using the genetic algorithm[j]. Journal of Xian University, 2003, 30(2): (in Chinese) [19] 6-PRRS [J]2008, 38(3): Yang Yonggang, Zhao Jie, Liu Yubin, Zhu Yanhe. Solving forward kinematics of 6-PRRS parallel robot[j]. Journal of Jilin University: Engineering and Technology Edition, 2008, 38(3): (in Chinese) ( : )

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