28 5 2010 9 JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering Vol. 28 No. 5 Sep. 2010 DOI: 10.3969/j.issn.0255-8297.2010.05.016 2T2R 1 1 1,2 1 1. 710048 2. 454150 4-UPU4-UPU/PPS. two-translational and two-rotational, 2T2R.. 20. TP242 0255-8297(2010)05-0546-05 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 710048, China 2. College of Mechanical and Electrical Engineering, Jiaozuo University, Jiaozuo 454150 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. 1999 Pierrot Company 1 [2]. [3-6] 2010-05-07 2010-07-19 (No.102-210911) E-mail: jiye_xaut@yeah.net E-mail: liu-hongzhao@163.com
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 3 1 4-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 2.. 2 / 3 3 6 6 2 5. 1 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. 4. 4 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, 1. 3.1. P
548 28 (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 0.5 0.2 w min = 0.2 + k k max (7) w min = 1 1 0.8 k k max (8) PSO 1. 24 (6) (8)k max 3 000 k 1 D 4 c 1 c 2 1.7 (9) 2 24 x t i, p t xt i, p t gd. 3 (4) (5) 4 p t,. p t gd
5 2T2R 549 p t gd pt gd,. 5 k 1 p t gd 1 10 6, 3... 5 (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 6 6.1 100 mm a = 50 mm; 200 mm b =100 mm. 20.. 5 1. x z [x 10, x + 10] [z 20, z + 20] β γ[ π/4, π/4]. 1 2 f(x i )/mm 10 2 10 0 10 2 10 4 10 6 0 1 000 2 000 3 000 2 1 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 5 320 10π/180 10π/180 7 326 15π/180 13π/180 13 331 17π/180 14π/180 19 321 10π/180 14π/180 22 317 16π/180 18π/180 4.999999533022 320.000000001687 0.174532926677 0.174532925284 6.999999918550 326.000000000189 0.261799388053 0.226892802801 13.000001151567 330.999999965481 0.296705969291 0.244346094840 18.999996642011 321.000000207434 0.174532935360 0.244346095454 22.000000741434 316.999999958306 0.279252677798 0.314159264900
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