Kinematic Synthesis of Multi-fingered Robotic Hands for Finite and Infinitesimal Tasks

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1 Kiematic Sythesis of Multi-figered Robotic Hads for Fiite ad Ifiitesimal Tasks E. Simo-Serra, A. Perez-Gracia, H. Moo ad N. Robso Abstract I this paper we preset a ovel method of desigig multi-figered robotic hads usig tasks composed of both fiite ad ifiitesimal motio. The method is based o represetig the robotic hads as a kiematic chai with a tree topology. We represet fiite motio usig Clifford algebra ad ifiitesimal motio usig Lie algebra to perform fiite dimesioal kiematic sythesis of the multifigered mechaism. This allows tasks to be defied ot oly by displacemets, but also by the velocity ad acceleratio at differet positios for the desig of robotic hads. The additioal iformatio eables a icreased local approximatio of the task at critical positios, as well as cotact ad curvature specificatios. A example task is provided usig a experimetal motio capture system ad we preset the desig of a robotic had for the task usig a hybrid Geetic Algorithm/Leveberg- Marquadt solver. Key words: kiematic sythesis, multi-figered grippers, Clifford ad Lie algebra. 1 Itroductio The desig of ed-effector robotic tools has traditioally take place i a applicatioorieted fashio withi the framework of the mechaical desig theory [7]. Amog the rich variety of robotic ed-effectors, those geerally defied as robotic hads are cosidered suited ot oly for graspig, but also for dexterous maipulatio. We ca E. Simo-Serra Istitut de Robòtica i Iformàtica Idustrial (CSIC-UPC) esimo@iri.upc.edu A. Perez-Gracia Idaho State Uiversity, Pocatello, Idaho perealba@isu.edu H. Moo Texas A&M Uiversity, College Statio, TX hsmoo@eo.tamu.edu N. Robso Texas A&M Uiversity, College Statio, TX iarobso@tamu.edu 1

2 E. Simo-Serra, A. Perez-Gracia, H. Moo ad N. Robso defie a multi-figered robotic had as a ed-effector i which the base, or palm, spas several serial chais i a tree-like structure. There are a great variety of desigs for robotic hads. Some desigs mimic the huma had ad exhibit a high umber of degrees of freedom, [1] ad [15]; others are desiged for specific applicatios [4] ad may or ot be athropomorphic [5]. Most of the desigs have bee orieted either towards maximum athropomorphism or towards optimizig graspig, maipulability or workspace size. A good review o the efforts toward kiematic had desig ca be foud i [6]. As robotic hads become more commo i idustrial applicatios ad huma eviromets, it makes sese to thik that their desig will become more task-orieted. Soto Martell ad Gii [14] expose the eed for a task-based desig process for robotic hads. The use of kiematic sythesis for the desig of the multi-figered robotic had has bee applied to idividual figers, see []. We believe that the reaso why dimesioal sythesis has bee scarcely applied to robotic had desig is because of the lack of a method that takes a multi-figered task as the iput ad outputs a multi-figered desig. I this paper, we exted the work preseted i [13] by combiig it with the results o kiematic sythesis for ifiitesimal positios [8, 9] ad expressig the kiematics usig the Clifford algebra of dual quaterios [10]. Note that mechaical likages are traditioally sythesized by specifyig a task, cosistig of a umber of positios that the ed-effector has to move through, with the goal of determiig the desig parameters, i.e. fixed ad movig pivot locatios, as well as the size of the likage. The differece betwee the traditioal desig of mechaical likages ad the curret desig with cotact directio, used i this research, is basically i the task, which cosists ot oly of positios, but velocities ad acceleratios compatible with cotact ad curvature specificatios betwee the ed-effector/figers ad the object to be grasped. I compariso to the traditioal sythesis techiques, these velocities ad acceleratios yield to a more complicated system of positio, velocity ad acceleratio desig equatios, as well as more complicated trajectory plaig techiques. As a example, we apply this methodology to the desig of a multi-figered had for operatig a door kob. The motivatio for this desig arose from a idividual, who is cofied to a wheel chair after a accidet. He has limited movemet ad weakess i his hads, makig it difficult for him to grasp doorkobs at his workspace. The sythesis preseted here is the first step towards developig assistive maipulatio devices. Ifiitesimal Kiematics The geeric screw S for a twist ca be represeted as a elemet of the Lie algebra se(3) [1], S=λ(s;r s+ hs)=λ(s;s 0 + hs)=(ω;v) se(3) (1)

3 Kiematic Sythesis of Multi-figered Robotic Hads for Fiite ad Ifiitesimal Tasks 3 where s,r,ω,v R 3 with s s=1, s 0 = r s ad λ,h R. The relative velocities betwee a pair of rigid bodies form oe-dimesioal subalgebras of the Lie algebra se(3) [11]. The most geeric subalgebra is geerated by the screw or helical joit S which becomes a revolute joit S R =(s;s 0 ) with h=0 or a prismatic joit S P =(0;s) with the screw axis at ifiity. The biary operatio of the Lie algebra is the Lie bracket, which ca be expaded for screws as, [S 1,S ]=[(ω 1 ;v 1 ),(ω ;v )]=(ω 1 ω ;ω 1 v + v 1 ω ) () The velocity of the ed-effector for a serial articulated chai with joits i a give cofiguratio ca be writte as [1], dp dt =Ṗ= θ i S i (3) where Ṗ = (ω;v) with ω beig the agular velocities ad v beig the Cartesia velocities. The screwss i represet the ifiitesimal screws of each joit. The ifiitesimal screws ca be trasformed to a istataeous positio from a referece positio usig the Clifford algebra cojugatio actio, S k i = ( e ˆθ k i 1 S k i 1 )S i ( e ˆθ k i 1 S k i 1) = ( i 1 ) ˆθ k j e S j j=1 S i ( i 1 j=1 ˆθ k j e S j) (4) where ˆθ i k = ˆθ i k ˆθ i r with ˆθ i r is the joit parameter i the referece cofiguratio ads k i is the i-th screw i a serial chai at positio k. The velocity of a joit j i a chai is writte as the derivative of a fiite screw [3], ds j dt =Ṡ j = j 1 θ i [S i,s j ] (5) Cross terms ad the o-commutatio of the derivatio operator must be take ito accout as see by differetiatig each velocity compoet of (3) usig the chai rule. This ca be expaded to obtai the acceleratio of the ed-effector, d P dt = P= d dtṗ= ( θ i S i + θ i Ṡ i ) = 1 θ i S i + θ i θ j [S i,s j ] (6) j=i+1 where P=(α;a) with α beig the agular acceleratios ad a beig the Cartesia acceleratios. The approach is geeral i the sese that the chai rule ca be successively applied to obtai higher derivatives if ecessary.

4 4 E. Simo-Serra, A. Perez-Gracia, H. Moo ad N. Robso 3 Desig Equatios for Tree Topologies Tree topologies ca be see as may differet serial chais that share a umber of commo joits. The equatios ca be writte as for serial chais, but the task defiitio will vary with the topology. The fiite motio of a joit ca be expressed usig the expoetial map of a screw S. This ca be expressed usig the uit elemet of the Clifford eve subalgebra of the projective space Cl + (0,3,1) or dual quaterio, e ˆθ S =(cos θ d si θ ε)+(si θ + d cos θ ε)s=cos ˆθ ˆθ + si S. (7) where ε is the dual uit such that ε = 0. For a serial chai with joits, the forward kiematics of a serial chai ca be writte relative to a referece cofiguratio of the serial chai, ˆQ( ˆθ)= e ˆθ i S i = (cos ˆθ i + si ˆθ i S i) (8) where ˆθ i = ˆθ i ˆθ 0 with ˆθ 0 beig the joit parameters of the referece cofiguratio. For a task composed of fiite positios, the relative forward kiematics ca be compared to the relative motio from the referece cofiguratio to each positio ˆP 1k = ˆP k ˆP 1 1 [10], ˆP 1k = e ˆθ k i S i, k=,...,m p (9) where m p is the umber of positios cosidered ad ˆθ i k beig the referece cofiguratio. For a task with velocities, we ca use (3) to write, Ṗ k = = ˆθ k i θ 1 i, with k = 1 θ k i Sk i, k=1,...,m v (10) where Ṗ k is the absolute velocity iformatio for a give positio k i the form (ω;v). The istataeous joit screw axis S k i ca be calculated from (4). The same procedure ca be applied to acceleratio to obtai from (6), P k = θ i k S k i + 1 θ k i θ j[s k k i,s k j], k=1,...,m a (11) j=i+1 where P k is the absolute acceleratio for a give positio k i the form(α;a). The velocity ad acceleratio equatios ca be see as additioal pose iformatio that reduce the umber of poses eeded. Coutig the umber of idepedet ukows x ad idepedet equatios f we obtai,

5 Kiematic Sythesis of Multi-figered Robotic Hads for Fiite ad Ifiitesimal Tasks 5 x = s + j (m p + m v + m a 1) (1) f = c + d (m p + m v + m a 1) (13) where s is the umber of idepedet structural parameters, j the umber of joit degrees of freedom, c the umber of idepedet costraits ad d the degrees of freedom of the ed-effector motio. The umber of positios, velocities ad acceleratios are give by m p, m v ad m a respectively. If we cosider m=m p + m v + m a we obtai the familiar formula [10], m= s c d j + 1 (14) 4 Experimetal Set up ad Task Specificatio Sice the first step i our sythesis techique is related to choosig a specific task, the kiematic task selected for the desig is the operatio of a stadard door kob. I order to defie this kiematic task, the door kob graspig ad turig movemet was performed, from a start to ed spatial locatios. Durig the movemet, the subject emulates the opeig of the door motio with a apparatus show i Fig. 1a. The upper limb kiematics at specific poits of iterest are captured by a 3D Motio Capture System (Vico, OMG Plc., UK), available i our Huma Iteractive Robotics Lab at Texas A&M Uiversity. Three ifrared cameras track the positio of each marker relative to a predefied global coordiate frame, with a samplig rate of 100 Hz. Five movig frames are defied at the: elbow, wrist, ad tip of thumb, tip of idex ad tip of middle figers, respectively. Fig. 1b shows the marker attachmet. To oly sythesize the motio of the forearm, the positios chose were trasformed from absolute positios ˆP i to positios local to the elbow ˆP i j = ˆP 1 j ˆP i with ˆP J is the positio of the elbow. For velocities this ca be writte asṗ i j =Ṗ i Ṗ j. The obtaied positios ad velocities were the used as a task for the kiematic sythesis of the multi-figered robotic had. The kiematic specificatio cosists Fig. 1 1a The experimetal apparatus for emulatig the door kob task ad 1b the kiematic model cofigured i the 3D Motio Capture System. (a) (b)

6 6 E. Simo-Serra, A. Perez-Gracia, H. Moo ad N. Robso of set of three spatial displacemets defied by ˆP k = (ψ k,d k ),k = 1,,3, ad the associated agular ad liear velocitiesṗ k =(ω k ;v k ),k= 1, i two of the positios for each figertip. 5 Solvig Numerically The solver used is a updated versio of the kiematic sythesis solver for tree structures [13] updated to support the desig equatios (10) ad (11). The solver is composed of a Geetic Algorithm (GA) paired with a Leveberg-Marquadt (LM) local optimizer. For more details o the solvig approach see [13]. The full equatio system for a multi-figered robot with b figers formed by revolute joits ca be defied directly by maipulatio of the desig equatios, F(S, ˆθ, θ, θ)= ( c P c k θ k ˆP c 1k c Ṗ c k c c 1 i Sk i,c + e ˆθ k i S i,c, θ k i S k i,c, θ k i k=,...,m p c=1,...,b k=1,...,m v c=1,...,b c θ j k [Sk i,c,sk j,c ), ] k=1,...,m a c=1,...,b j=i+1 (15) where b is the umber of braches or figers, c is the umber of joits for a brach c, is the total umber of joits i the structure ad Si,c k are the istataeous axis of the joit i i the brach c for the frame k calculated by (4). A valid mechaism is said to be foud whe F(S, ˆθ, θ, θ)= 0. 6 Results The kiematic structure of the had cosists of a three degree of freedom palm+wrist complex (RRR), ad three figers, each of which is modeled as a two degree of freedom RR kiematic chai as see i Fig. a. This structure was chose for the kiematic sythesis of the task as it has fewer degrees of freedom tha the huma had while havig a o-fractioal umber of required samples. As this paper does ot deal with structural sythesis, a pre-determied topology is used. No additioal costraits were placed o the structure. For this kiematic structure with 9 revolute joits, a total of m=5 samples are eeded as obtaied from (14). The experimetal task cosists of may hudreds of frames of which m p = 3 were selected. For two of them velocity iformatio was also used providig m v =. Due to the ature of the door-kob opeig task, the acceleratios, related to curvature costraits (i.e. slidig motio of the figers alog the door-kob) are fairly small i compariso to the other task specificatios ad were ot take ito

7 Kiematic Sythesis of Multi-figered Robotic Hads for Fiite ad Ifiitesimal Tasks 7 (a) Topology (b) Full System (c) Idex Figer (d) Middle Figer (e) Thumb Fig. : The topology used ad overview of a solutio mechaism foud. accout. Therefore, our task cosists of positios, prescribed at the poit where the figers eed to grasp the door kob, ad velocities, describig the cotact of the figers with the door kob. The resultig equatio system has 90 ukows ad 10 equatios of which oly 7 are idepedet. A set of 50 solutios was obtaied takig a average of miutes per solutio ad eedig a average of geeratios per solutio with a populatio of 100 etities. A selected solutio mechaism of the doorkob task ca be see i Fig.. The thick joit axes are coected by thier lies at the itersectios of the commo ormals of the joits ad the joit axes. The origi is represeted by usig a square ad the ed-effectors are represeted by usig spheres. This solutio show is more compact tha the huma had as all the joits except oe are grouped together ad is able to perform the same task as the huma had. It was observed that geerally the solutios have a similarity betwee the idex ad middle figer, while the thumb has a differet shape, which is similar to how the huma had is desiged. 7 Coclusios This paper presets a ovel dimesioal sythesis methodology for articulated systems with a tree structure usig additioal costraits such as velocity ad acceleratio. It also presets a ew desig for umerically solvig kiematic systems usig these ew costraits, addig upo the previous work of umerically solvig tree structures with kiematic sythesis. The additio of velocity, acceleratio ad other derivatives at the positios allows a better local approximatio of the task motio, as well tasks with cotact ad curvature specificatios. For the selected kob-operatig task, a tree-like robot with three two-joited figers has bee desiged. Kiematic sythesis is just oe step i the desig process, oe that allows you to create iovative cadidates fitted for the kiematic tasks uder cosideratio. Eough solutios have bee foud to suggest that a method to aalyze ad rak those eeds to be a part of the desig process. These results show that the dimesioal sythesis of robotic multi-figered hads is possible. I additio, multi-figered hads appear ot to be redudat whe a task ivolvig several figers is to be performed.

8 8 E. Simo-Serra, A. Perez-Gracia, H. Moo ad N. Robso Ackowledgemets This work is partially supported by the Spaish Miistry of Sciece ad Iovatio uder project DPI Refereces [1] Borst, C., Fischer, M., Haidacher, S., Liu, H., Hirziger, G.: DLR Had II: Experimets ad experieces with a athropomorphic had. I: ICRA (003) [] Ceccarelli, M., Nava Rodriguez, N., Carboe, G.: Desig ad Tests of a Three Figer Had with 1-DOF Articulated Figers. Robotica 4, (006) [3] Cervates-Sáchez, J.J., Rico-Martíez, J.M., Gozález-Motiel, G., Gozález-Galvá, E.J.: The differetial calculus of screws: Theory, geometrical iterpretatio, ad applicatios. Proc. of the Istitutio of Mechaical Egieers, Part C: Joural of Mechaical Egieerig Sciece 3(6), (009) [4] Chalo, M.e.a.: Dexhad : a space qualified multi-figered robotic had. I: ICRA (011) [5] Dai, J.S., Wag, D., Cui, L.: Orietatio ad workspace aalysis of themultifigered metamorphic had-metahad. Tras. Rob. 5, (009) [6] Grebestei, M., Chalo, M., Hirziger, G., Siegwart, R.: A method for had kiematics desig. 7 billio perfect hads. I: ICABB (010) [7] Maso, M., Sriivasa, S., Vazquez, A., Rodriguez, A.: Geerality ad simple hads. Iteratioal Joural of Robotics Research (010) [8] Patarisky Robso, N., McCarthy, J.M.: Kiematic Sythesis With Cotact Directio ad Curvature Costraits o the Workpiece. I: IDETC (007) [9] Patarisky Robso, N., McCarthy, J.M.: Sythesis of a Spatial SS Serial Chai for a Prescribed Acceleratio Task. I: World Cogress i Mechaism ad Machie Sciece (IFToMM). Besaco, Frace (007) [10] Perez-Gracia, A., McCarthy, J.M.: Kiematic sythesis of spatial serial chais usig clifford algebra expoetials. Proc. of the Istitutio of Mechaical Egieers, Part C: Joural of Mechaical Egieerig Sciece 0(7), (006) [11] Rico, J.M., Gallardo, J., Ravai, B.: Lie algebra ad the mobility of kiematic chais. Joural of Robotic Systems 0(8), (003) [1] Selig, J.M.: Geometric Fudametals of Robotics (Moographs i Computer Sciece). SprigerVerlag (004) [13] Simo-Serra, E., Moreo-Noguer, F., Perez-Gracia, A.: Desig of oathropomorphic robotic hads for athropomorphic tasks. I: IDETC (011) [14] Soto Martell, J., Gii, G.: Robotic hads: Desig review ad proposal of ew desig process. World Academy of Sciece, Egieerig ad Techology 6 (007) [15] Ueda, J., Kodo, M., Ogasawara, T.: The multifigered aist had system for robot i-had maipulatio. Mechaism ad Machie Theory 45(), 4 38 (010)