9 11 1 11 : 1 815(1)11 1433 7 Control Theory & Applications Vol. 9 No. 11 Nov. 1 / 1, 1,, 1 (1., 119;., 119) : /,, / /.,,.,. : ; ; ; ; : TP73 : A Rebound model between spinning table tennis ball and table/racket REN Yan-qing 1, XU De 1,, TAN Min 1 (1. State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 119, China;. Research Center of Precision Sensing and Control, Institute of Automation, Chinese Academy of Sciences, Beijing 119, China) Abstract: Based on the stress, impulse, and impulse moment received by a spinning table tennis ball from the table/racket in the rebound process, a concept of critical friction angle is introduced and a criterion is put forward to distinguish the types of the friction effects, thus a physical rebound model is built. In addition, a linear rebound model is obtained through the learning algorithm and multiple linear regressions. Experiments and the error analysis validate the effectiveness of the two rebound models. Key words: table tennis robot; rebound model; critical friction angle; multiple linear regressions; parameter estimation 1 (Introduction). [1]., [].,, : 1) ( ),, ; ),, ; 3),,,., /. /,,. /. Zhang [3],,. [4], ( ).,. Andersson [5],,,,. Andersson,. [6 7] : 1 3 ; : 1 5 1. : (617535, 6173337).
1434 9,,. (Problem analysis) : 1) :,,,, ; ) ; 3),... Garwin [8], ;. Brody [9],,,. Cross [1 11],.,.,, ;,.,, [1],. [6 7],, /, /.,,.,. 3 (Critical friction angle) 3.1 (Definition of critical friction angle), α, α ϕ,, α ϕ,, ϕ.,,. 3. (Derivation of critical friction angle) ( 1 )., X, Y, Z, V i ω i, V e ω e, X, Y, Z x, y, z, Z V iz. i e. 1 Fig. 1 Rebound phenomenon between the table tennis ball and the table,, Z, M, Z V iz =.,.,. r,,, O M OM = (,, r) T, M V M = V + OM ω = ω = [ω x ω y ω z ]. v x rω y v y + rω x, M., M v xy = (v x rω y ) + (v y + rω x ), v x rω y sin θ = (vx rω y ) +(v y +rω x ) = v x rω y v xy, v y +rω x cos θ = (vx rω y ) + (v y +rω x ) = v y+rω x v xy. [6 7],, M,
11 : / 1435,,,. P xy M v xy, f v xy,., X Y P x = P xy sin θ = µ P z sin θ, P y = P xy cos θ = µ P z cos θ, P = P x P y P z µ P z sin θ = µ P z cos θ. P z (1) () M Fig. The velocity in the horizontal direction of the table at the point M 3.3 (Rebound model of spinning ball and table while sliding), M. N f, µ, f f = µn, M. m, P, I = 3 mr, 1, mv e mv i = P, (1) Iω e Iω i = OM P. (),. N, e, V ez = ev iz., S mv ez mv iz = P z = Ndt = (1 + e)mv iz, S. P xy, f = µn, P xy S mv exy mv ixy = P xy = fdt = S µ Ndt = µp z = (1 + e)mµv iz. V e = V i + P m = V ix (1 + e) V iz µ V ix rω iy V iy (1 + e) V iz µ V iy + rω ix, ev iz OM P ω e = ω i + I 3(1 + e) ω ix r 3(1 + e) ω iy + r = V iz µ V iy + rω ix V iz µ V ix rω iy ω iz. α, α, ctg α= V iz,, V e = V i + P m = V ix (1 + e)µctgα(v ix rω iy ) V iy (1 + e)µctgα(v iy + rω ix ), ev iz OM P (3) ω e = ω i + = I 3(1 + e) ω ix µctgα(v iy + rω ix ) r 3(1 + e) ω iy + µctgα(v ix rω iy ). r ω iz 3.4 (Determination of critical friction angle) 3.1, M v ex rω ey V em = v ey + rω ex =
1436 9 V ix rω iy ctg αµ(v ix rω iy ) r V iy + rω ix ctg αµ(v iy + rω ix ) = r V ix rω iy (1 µctg α) = V iy + rω ix (1 µctgα)v im. (4) V em, V im, 1 µctg α, tg α µ, α arctan( µ). α arctan( µ), ϕ = arctan( µ). [1]. 3.5 (Rebound model between spinning ball and table while rolling),,, (3).,,,, P xy = P x = P y =.,,,,., V ix V e = V iy, ev iz ω e = ω i. (5) 4 (Learning rebound model between spinning ball and table),.,,,, : { V ez = ev iz, (6) ω ez = ω iz., 3 : X X Y, X Y., Y X Y, X Y., V ex = k vx1 V ix + k vx ω iy + b vx, V ey = k vy1 V iy + k vy ω ix + b vy, ω ex = k wx1 V iy + k wx ω ix + b wx, ω ey = k wy1 V ix + k wy ω iy + b wy. (7),. 5 (Rebound model between spinning ball and racket) :,. = +.. : V = T eq + T ev, (8) : V, T, T T, e Q, e V., T, T. T ( q x, q y, q z ) T, V ph = (v phx, v phy, v phz ) T, q e x q x T eq q y e q y = q z e q z = 1 v e phx q x v phx v phy e q y v phz e q z = v phy v phz = V ph. (9) 1 (9) (8) V = V ph + T e V. (1) V i ω i, V e ω e (1), : e V i = T 1 (V i V ph ), (11)
11 : / 1437 e V e = T 1 (V e V ph ). (1),, ω = T e ω, e ω = T 1 ω. e V i e V e (3) (5) (6) (7),. 6 (Parameter estimation) (1 + e)µ = η, (3) η e V ix ctgα(v ix rω iy ) V e V iy = η ctgα(v iy + rω ix ) e, V iz 3 r ctgα(v iy + rω ix ) ω e = ω i + η 3 r ctgα(v ix rω iy ), V ex V ix V ey V iy V ez = ω ex ω ix ω ey ω iy ω ez ω iz η ctgα(v ix rω iy ) ctgα(v iy + rω ix ) 3ctgα(V iy + rω ix ) r 3ctgα(V ix rω iy ) r + e V iz. (13) [13] 1, [14], 1. 1 Table 1 The measured values of velocity and angular velocity V i / (m s 1 ) ω i / (rad s 1 ) V e / (m s 1 ) ω e / (rad s 1 ) 1 (.9,.89,.46) T ( 35.44, 4.13, 7.9) T (.54, 1.97,.) T (11.35,.85,.) T (., 3.3,.57) T ( 45.95, 43.76, 41.16) T (.15,.8,.8) T (1.4, 13.56, 9.98) T 3 (.45, 3.1,.6) T (3.5, 3.41, 15.86) T (.6,.14,.31) T (17.45, 7.45, 6.44) T 4 (.7,.96,.61) T (47.7,.3, 1.83) T (.41, 1.9,.31) T (.3, 8.3, 5.7) T 5 (.41, 3.14,.64) T ( 1.55, 77., 85.94) T (.4,.14,.3) T (14.4, 11.74, 1.66) T 6 (.9,.7,.5) T ( 33.5, 38.83, 43.38) T (.16, 1.8,.5) T (14.3, 5.38, 1.3) T 7 (.7,.71,.59) T ( 37.3, 55.61, 5.33) T (.19, 1.8,.3) T (16.57, 15.1, 6.47) T 8 (.41, 3.4,.86) T ( 47.1, 3., 17.1) T (.9,.9,.45) T (14.34, 4.63, 18.79) T 9 (.46,.64,.98) T ( 54.17, 59.9, 49.5) T (.3, 1.78,.51) T (16.53, 16.76, 8.) T 1 (.4, 3.7,.6) T ( 45., 6.57,.9) T (.18,.11,.31) T (1.95,.98, 16.85) T 11 (.5, 3.18,.68) T ( 7.7, 51.55, 43.15) T (.35,.17,.35) T (16.11,.68, 16.5) T 1 (.35, 3.17,.64) T ( 53.6, 1.51, 19.13) T (.5,.,.3) T (11.59,.3, 18.5) T, (13), η e, η e, µ = η/(1 + e) µ, ϕ = arctan( µ), : e =.8788, µ =.149, ϕ = 6.318. 7 (Experiment) (3) (5) (physical model), (6) (7) (learning model). µ e, [3, 6 7],, 3 4. 5 6,. 3 6,,. [3, 6 7] 4, : 1), 4,,,
控 制 理 论 与 应 用 1438 第 9 卷 预测准确性的影响不大, 因此不影响该模型的有 效性. 图 5 几种反弹模型的速度估计结果与测量值的误差 Fig. 5 Errors between the estimated velocities of several rebound models and measured values 图 3 几种反弹模型的速度估计结果与测量值的比较 Fig. 3 Comparison of the estimated velocities of several rebound models and measured values 图 6 几种反弹模型的角速度估计结果与测量值的误差 Fig. 6 Errors between the estimated angular velocities of several rebound models and measured values 图 4 几种反弹模型的角速度估计结果与测量值的比较 Fig. 4 Comparison of the estimated angular velocities of several rebound models and measured values ) 在平行于球台方向上, 本文学习模型获得 最佳估计效果, 尤其是角速度的估计误差明显小 于本文物理模型和文献 [6 7]中的模型. 文献 [3]中
11 : / 1439,.,,. 3) 3 :,, [6 7],,. ( 3 6, 3 4 ), [6 7],. Table Average errors, maximum errors, and mean square errors between estimated values of the several models and the measured values [6 7] [3] V e/ (m s 1 ) (.4,.5344,.3) T (,,.3) T (.951,.385,.3) T (,,.3) T (.54, 1.7,.58) T (.73,.81,.58) T (.54,.549,.58) T (.119,.866,.58) T (.374,.631,.) T (,,.) T (.4,.195,.) T (,.1,.) T (9.19, 4.33, 4.11) T (,, 4.11) T (.36, 36.35, 4.11) T ω e/ (7.46, 67.6, 98.59) T (.31,.43, 98.59) T (7.46, 67.6, 98.59) T (rad s 1 ) (8.9, 558.39, 976.8) T (.49, 3.87, 976.8) T (1734.3, 81.8, 976.8) T 8 (Conclusiona) /,, /,,,, ;,, /.,,,.. (References): [1] ZHANG Z, XU D, YU J. Research and latest development of pingpong robot player [C] //Proceedings of the 7th World Congress on Intelligent Control and Automation. Chongqing: IEEE, 8: 4881 4886. []. [D]. :, 1. (ZHANG Zhengtao. Visual measurement and control for table tennis robot [D]. Beijing: Chinese Academy of Sciences, 1.) [3] ZHANG Z T, XU D, YANG P. Rebound model of table tennis ball for trajectory prediction [C] //Proceedings of the 1 IEEE International Conference on Robotics and Biomimetics. Tianjin: IEEE, 1: 376 38. [4],,,. [J]. ( ), 7, 6(4): 433 437. (PENG Bo, HONG Yongchao, DU Sensen, et al. An approach to hit point prediction for ping pong robot [J]. Journal of Jiangnan University (Natural Science Edition), 7, 6(4): 433 437.) [5] ANDERSSON R L. Dynamic sensing in a ping-pong playing robot [J]. IEEE Transactions on Robotics and Automation, 1989, 5(6): 78 739. [6] CHEN X P, TIAN Y, HUANG Q, et al. Dynamic model based ball trajectory prediction for a robot ping-pong player [C] //Proceedings of the 1 IEEE International Conference on Robotics and Biomimetics. Tianjin: IEEE, 1: 63 68. [7] AKIRA N, YOSUKE K, YUKI O, et al. Modeling of rebound phenomenon between ball and racket rubber [C] //ICROS-SICE International Joint Conference 9. Japan: IEEE, 9: 95 3. [8] GARWIN R. Kinematics of an ultraelastic rough ball [J]. American Journal of Physics, 1969, 37(1): 88 9. [9] BRODY H. That s how the ball bounces [J]. The Physics Teacher, 1984, (8): 494 497. [1] CROSS R. Effects of friction between the ball and strings in tennis [J]. Sport England,, 3: 85 97. [11] CROSS R. Measurements of the horizontal coefficient of restitution for a superball and a tennis ball [J]. American Journal of Physics, 1, 7(5): 48 489. [1]. [J]., 1985, 19(): 83 88. [13]. [D]. :, 11. (YANG Ping. Research on motion planning and control strategy for cartesian robots [D]. Beijing: Chinese Academy of Sciences, 1.) [14] HUANG Y L, XU D, TAN M, et al. Trajectory prediction of spinning ball for ping-pong player robot [C] //11 IEEE/RSJ International Conference on Intelligent Robots and Systems. San Francisco: IEEE, 11: 3434 3439. : (1983 ),,,, E-mail: yanqing.ren@ia.ac.cn; (1965 ),,,,, E-mail: de.xu@ia.ac.cn; (196 ),,,,, E-mail: min.tan@ ia.ac.cn.