The Space Redundant Robotic Manipulator Chaotic Motion Dynamics Control Algorithm

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1 Sesors & rasducers, Vol. 75, Issue 7, July 24, pp Sesors & rasducers 24 by IFSA Publishig, S. L. he Space Redudat Robotic Maipulator Chaotic Motio Dyamics Cotrol Algorithm Yu G, Xifeg G College of Mechaical & lectrical gieerig, Xuchag Uiversity, No. 88 of Bayi Road Xuchag Hea, 46, Chia el.: , fax: mail: xifeg.ge@gmail.com Received: 23 April 24 /Accepted: 3 Jue 24 /Published: 3 July 24 Abstract: A free-floatig space redudat robotic maipulator dyamics model is established ad the dyamics equatio is derived, the delayed feedback cotrol time ad the self-adaptive gai coefficiet are obtaied by substitutig ito delayed feedback cotrol equatios. he simulated dyamics cotrol chaotic motio of the space redudat robotic maipulator, the simulatio results show that the chaotic motio of the space redudat robotic maipulator becomes the periodic motio by dyamics cotrol, therefore it is effective that dyamics cotrol as the space redudat robotic maipulator chaotic motio cotrol. Copyright 24 IFSA Publishig, S. L. Keywords: Space, Free-floatig, Redudat robotic maipulator, Dyamic cotrol, Chaotic cotrol.. Itroductio With the developmet of space sciece ad techology, the huma desire to explore space is more ad more ehacig; space robotic maipulator ca help humas to accomplish tasks such as air rescue, air repair, air refuelig, air maiteace ad other air tasks, so the space robotic maipulator becomes a hot research []. he robotic maipulator are ofte required to complete some tasks, such as trackig the target i ay posture, i a flexible way i replacig huma work, that s eed to icrease DOF (the degree of freedom) of the robotic maipulator to improve flexibility, ad the umber of DOF are more tha the dimesios eeded to complete the task, this additioal DOF called redudat DOF, robotic maipulator with redudat DOF called redudat robotic maipulator [2-5]. Redudat robotic maipulator has greater flexibility ad more accurate target trackig capability, better obstacle ad sigularity avoidace ability, therefore ofte used as itelliget robotic maipulator actuators, ad become a hot research field of the mechaical egieerig ad automatic cotrol [6-7]. Redudat robotic maipulator has ull space or redudat space, ad its ull space self-motio ca geerate chaotic motio; the redudat robotic maipulator chaotic motio cotrol is oe of a difficulty i robotic maipulator cotrol [8]. Matthew [9] studied the redudat robotic maipulator chaotic motio, a plaar 3R redudat robotic maipulator chaotic motio i self-motio was foud by the system Poicare figure ad power spectrum, but did t study the coditios geeratig chaotic motio ad cotrol algorithms. Li [] studied the dual pedulum mechaism chaotic pheomea by umerical method, ad obtaied parameter coditios geerated chaotic motio. Zhag [] studied the plaar 3R redudat robotic maipulator chaotic motio, the parameters were 27

2 Sesors & rasducers, Vol. 75, Issue 7, July 24, pp selected appropriately ad the chaotic motio was chaged ito regular periodic motio successfully by delayed feedback cotrol method at the expese flexibility. hese studies are based o groud redudat robotic maipulator, ad the study that the chaotic motio cotrol based o the free-float space redudat robotic maipulator is still relatively little, a cotrol algorithm is proposed that ca cotrol the chaotic motio i the space free-float redudat robotic maipulator self-motio i this paper. I this paper, the robotic maipulator dyamic equatios are derived firstly uder the coditio that the mometum ad agular mometum coservatio by usig of the rigid body dyamics modelig methods [2-4], the dyamic cotrol algorithm is proposed combied with self-adaptive delayed feedback cotrol, ad fially simulated the chaotic motio cotrol i the space free-float redudat robotic maipulator by dyamic cotrol algorithm, the simulatio results show that the dyamic cotrol algorithm proposed i this paper is a effective i cotrollig the chaotic motio cotrol i the space free-float redudat robotic maipulator. 2. he Space Redudat Robotic Maipulator Dyamics 2.. Descriptio of the Space Redudat Robotic Maipulator he model of the space free-float redudat robotic maipulator with -DOF is show i Fig.. robotic maipulator, q = [q,q 2,q 3 q ] is the joit variables i joit space, r i is the vector from the coordiate system origi to the cetroid of the i th lik i the iertial coordiate system, m is the mass of B, m i is the mass of the i th lik, l i is the positio vector form the i th joit to the (i+) th joit, ω is the agular velocity of B, k i is the uit vector of the i th lik axis directio i the iertial coordiate system, p i is the positio vector of the i th joit i the iertial coordiate system, I i is the momet of iertia that the i th lik relative to the ceter of mass i the iertial coordiate system, 3 is the uit matrix of he Space Redudat Robotic Maipulator Kiematics quatios he space redudat robotic maipulator edeffector positio ca be expressed as: r=rb+r+ li, () Differetiate the formula () o time; the space redudat robotic maipulator ed-effector velocity ca be obtaied: v = r = r + ω ( r r ) + [ k ( r p )] q, (2) b i i he space redudat robotic maipulator edeffector agular velocity ca be expressed as: ω = ω +kq, (3) i Combied the formula (2) ad (3), the space redudat robotic maipulator kiematics equatios ca be obtaied: x = J x + J q, (4) O O O2 Fig.. he model of the space free-float redudat robotic maipulator with -DOF. Coordiate system ad the symbols are defied as follows: O -X Y Z is the based coordiate system; O 2 -X 2 Y 2 Z 2 is the iertial coordiate system; O CM -X CM Y CM Z CM is the cetroids coordiate system; B is the base of the space free-float redudat where x = [ v ω ], = [ ] I ( r r ) 3 O2 J O =, I3 x v ω, J O2 [ k ( r p )] i i = ki is a operator, which meas that the atisymmetric matrix of a vector he Space Redudat Robotic Maipulator Kiematics quatios he free-floatig space redudat robotic maipulator is i microgravity eviromet, igorig other iterferece, ad the the free-floatig space redudat robotic maipulator is mometum coservatio, assumig the iitial mometum of the, 28

3 Sesors & rasducers, Vol. 75, Issue 7, July 24, pp robotic maipulator system is, the free-floatig space redudat robotic maipulator liear mometum ad agular mometum are as follows respectively: m r =, (5) i i ( Iω + mr r ) =, (6) i i i i i Differetiate the formula (5) ad (6) o the time, the substituted ito formula (4), the follows is available: X = ( J J + J ) q= J q, (7) O B O2 where J = JO J B + J O is the free-floatig space 2 redudat robotic maipulator geeralized Jacobia matrix, the geeralized Jacobia matrix describes the relatioship betwee the robotic maipulator ed-effector liear ad agular velocities ad the joits velocities. he free-floatig space redudat robotic maipulator total kietic eergy is [5]: = ( ω Iω + mr r = q Mq, (8) 2 2 i i i i i i) he free-floatig space redudat robotic maipulator system dyamics equatio ca be obtaied by the Lagrage equatio [5] i the microgravity eviromet of space, igorig the freefloatig space redudat robotic maipulator system potetial eergy is: ( ) (, ) M q q+ C q q = τ, (9) where τ is the drivig torque of each joit, M is the free-floatig space redudat robotic maipulator symmetric positive iertial matrix, C is the freefloatig space redudat robotic maipulator Coriolis force ad cetripetal force vector. 3. Dyamics Cotrol Algorithm For the free-floatig space redudat robotic maipulator iverse problem, Liegeois [7] proposed the geeral solutio of the robotic joit velocity based o geeralized iverse matrix of the geeralized Jacobia matrix: ( ) q = J X + I J φ, () + + e e e + where J e is the geeralized iverse matrix of the geeralized Jacobia matrix, X e is the desired trajectory of the ed-effector, φ is a arbitrary vector. Differetiate the formula (), the formula () ca be available, that is: + + q = J ( X J q ) + ( I J J ) φ, () e e e e e Itroductio the follow cotrol: τ = C( qq, ) + M( q ) μ, (2) Compare the dyamic equatios of the freefloatig space redudat robotic maipulator, ad have: ( ) M( qq ) = M q μ, (3) M (q) is a ivertible matrix, the formula (3) equivalet to a liear time-ivariat system of decouplig. Itroductio PD cotrols: q = μ, (4) μ = X + k ( X q ) + k ( X q ), (5) e v e p e where k v ad k p are the feedback gai matrixes. As log as ( X e q ) ad ( X q e ), the chaotic motio cotrol is realized, the + τ = M( q){ Je[ Xe + kv( Xe q ) + k ( X q) Jq + φ]}+ C( qq, ), (6) p e e 4. Simulatio Aalysis he three joits free floatig space redudat robotic maipulator operated plae motio which is take as a example ad computer umerical simulatio is made i this paper. Simulatio requires the ed-effector movig by desired trajectory. System parameters are as follows: the body mass is m =2 kg, each lik mass are m = m 2 = m 3 = kg, the body rotary iertia is I =2 kg/m 2, each lik rotary iertia are I =.3 kg/m 2 I 2 =.2 kg/m 2 I 3 =. kg/m 2, the legth of liks are l = l 2 = l 3 = m, the distace from the body ceter of mass to the first joit is l = m, positio vector of the body ceter of mass i iertial coordiate system is r C =(,.2), the desired trajectory is: X e =[+.5cos(2πt), 2+.5cos(2πt)], there exist the chaotic motio i the self-motio of the free-floatig space redudat robotic maipulator [8]. For the chaotic motio existig i the self-motio of the free floatig space redudat robotic maipulator, addig the cotrol ad simulatig, the phase diagrams of the joits (Fig. 2) ad the Poicare sectio diagrams of the joits (Fig. 3) are obtaied. 29

he Poicare sectio diagrams of the robotic maipulator joit after cotrollig. Refereces (c) he phase diagram of q3 Fig. 2. he phase diagrams of the robotic maipulator joit after cotrollig.

periodic accordig to the theory of chaotic motio [8], that is, the chaotic motio ca be cotrolled by the dyamics cotrol algorithm. []. W. H. Zhag, X. P. Ye, X. M.

4 Sesors & rasducers, Vol. 75, Issue 7, July 24, pp (a) he Poicare sectio diagram of q (a) he phase diagram of q (b) he Poicare sectio diagram q2 (b) he phase diagram of q2 (c) he Poicare sectio diagram of q3 Fig. 3. he Poicare sectio diagrams of the robotic maipulator joit after cotrollig. Refereces (c) he phase diagram of q3 Fig. 2. he phase diagrams of the robotic maipulator joit after cotrollig. Whe the free floatig space redudat robotic maipulator joits phase diagram is a closed curve, ad Poicare sectio diagram is a poit, the free floatig space redudat robotic maipulator joits motio is periodic accordig to the theory of chaotic motio [8], that is, the chaotic motio ca be cotrolled by the dyamics cotrol algorithm. []. W. H. Zhag, X. P. Ye, X. M. Ji, Developmet summarizig of space robot techology atioal ad outside, Flight Mechaics, Vol. 3, Issue 3, 23, pp [2]. Z. Lu, Priciple ad applicatio of redudat robot, Chia Machie Press, 27. [3]. X. C. Liu, Suppressio of chaos i kiematic ad dyamic cotrol for free-floatig redudat space robot, Najig Uiversity of Aeroautics ad Astroautics, 2. [4]. L. G. Wag ad B. L. Ma, Self-motio cotrol of redudat maipulators, Cotrol ad Decisio, Vol. 8, Issue 2, 23, pp [5]. Q. H. Zhag, H. X. Su ad S. M. Wei, A ew method for improvig flexibility of redudat robot, Joural of Beijig Uiversity of Posts ad elecommuicatios, Vol. 3, Issue 7, 25, pp [6]. C. ag, C. Yu ad S. Lua, Aalysis of kiematics ad dexterity for ew surgery robot, Joural of Beijig Uiversity of Aeroautics ad Astroautics, Vol. 27, Issue 4, 24, pp

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X. F. Ge ad Z. F. Yi, A spatial redudat robotic maipulator s chaotic self-motio, Chiese Joural of Space Sciece, Vol. 32, Issue 6, 22, pp. 897-92. [9]. M. Varghese, A.

Li, Numerical aalysis for chaotic pheomea of a double-pedulum, Mechaical Sciece ad echology, Vol. 2, Issue 4, 22, pp. 596-599. []. D. C. Zhag ad L.

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5 Sesors & rasducers, Vol. 75, Issue 7, July 24, pp [7]. Q. X. Jia, M. Chu, H. X. Su, Research o the optimal algorithm for trajectory plaig of a 9-DOF hyper-redudat robot, Joural of Beijig Uiversity of Posts ad elecommuicatios, Vol. 3, Issue 2, 28, pp [8]. X. F. Ge ad Z. F. Yi, A spatial redudat robotic maipulator s chaotic self-motio, Chiese Joural of Space Sciece, Vol. 32, Issue 6, 22, pp [9]. M. Varghese, A. Fuchs, Mukuda Ragaswamy, Chaotic zero dyamics i kiematically redudat robots, I rasactios o Aerospace ad lectroic Systems, Vol. 27, Issue 5, 99, pp []. K. F. Li ad L. Li, Numerical aalysis for chaotic pheomea of a double-pedulum, Mechaical Sciece ad echology, Vol. 2, Issue 4, 22, pp []. D. C. Zhag ad L. Li, Chaotic motio ad its cotrol i self-motio of redudat robot, Robot, Vol. 26, Issue 2, 24, pp [2]. Z. H. Che ad L. Che, Dyamic ad cotrol for dual-arm space robot with closed chai, Chiese Quarterly of Mechaics, Vol. 33, Issue 4, 22, pp [3]. W. F. Wag, J. J. Luo ad W. H. Ma, Study o dyamic modelig ad simulatio of free-floatig space robot, Sciece echology ad gieerig, Vol., Issue 3, 2, pp [4]. S. Liu, Studies o itelligece cotrol algorithm of trajectory trackig of free-floatig space robot, Xi'a Istitute of Optics ad Precisio Mechaics Chiese Academy of Scieces, 2. [5]. Z. X. ia, Research o mutibody dyamics ad target capturig for free-floatig space robot, Najig Uiversity of Aeroautics ad Astroautics, 2. [6]. X. Li, J. H. Wag ad X. K. Fag, Dyamic modelig ad simulatio of 5-DOF upper-limb rehabilitatio robot, Cotrol gieerig of Chia, Vol. 9, Issue 5, 22, pp [7]. A. Liëgeois, Automatics supervisory cotrol of cofiguratio ad behavior of multibody mechaisms, I rasactios o System, Ma ad Cyberetics, Vol. 7, Issue 2, 997, pp [8]. W. L. Jiag, S. Q. Zhag ad Y. Q. Wag, Numerical experimetal aalysis for chaotic motio characteristics, Chiese Joural of Mechaical gieerig, Vol. 36, Issue, 2, pp Copyright, Iteratioal Frequecy Sesor Associatio (IFSA) Publishig, S. L. All rights reserved. ( 3

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