Computational Issues in the Planning and Kinematics of Binary Robots Abstract 1. Introduction 2. Workspace Analysis and Optimization

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1 Computatonal Issues n the Plannng and Knematcs of Bnary Robots Matthew D. Lchter, Vvek A. Sujan, and Steven Dubowsky { lchter vasujan dubowsky@mt.edu } Department of Mechancal Engneerng Massachusetts Insttute of Technology Cambrdge, MA 019 USA Abstract To meet the objectves of many future mssons, robots wll need to be adaptable and reconfgurable. A concept for such a robotc system has been proposed prevously based on usng a large number of smple bnary actuators. Prevous researchers have addressed some of the ssues brought up by robots wth a few bnary actuators. Ths paper examnes the computatonal feasblty of controllng and plannng such bnary robotc systems wth a large number of actuators, ncludng computaton of ther workspace, forward knematcs, nverse knematcs and trajectory followng. Methods are proposed and evaluated by smulaton. Detaled error analyss and computatonal requrements are presented. An example of the plannng for a bnary walkng robot s presented. 1. Introducton Future robots wll be needed to perform complex tasks n dffcult envronments. For example, mssons to Mars wll requre robots to perform tasks such as scoutng, mnng, conductng scence experments, and constructng facltes for human explorers and settlers [9]. To accomplsh these objectves, robotc systems wll need to be lghtweght, relable and robust. Further, the elements of these systems need to be capable of large and fne moton, a large moton workspace, multple degrees of freedom, and have a small stowed volume. A new desgn concept has been proposed to meet these challenges [1, 15]. In ths concept, robotc systems use tens or hundreds of smple bnary actuators embedded n a flexble structures. Each of the bnary degrees of freedom contans a b-stable element so that the actuators smply flp the state of the jont. As the number of bnary degrees of freedom n the system ncreases, the capabltes of the devce approach that of a conventonal contnuous robotc system. Ths s analogous to the dgtal computer replacng the analog computer. The control of such devces s rather smple. To acheve a gven poston a set of jonts smply need to be flpped. No feedback s requred and theoretcally no servo errors wll exst. Control of such actuators has been classfed as sensor-less manpulaton [4, 6, 1]. The knematcs and control of such hyper-redundant manpulators, both wth and wthout bnary actuaton have been studed by a number of researchers [1,, 7, 8, 1]. However, many of the plannng and knematcs ssues of bnary robots are fundamentally dfferent and more dffcult than those of conventonal robotcs [, 1, 15]. For example, the nverse knematcs problem for a bnary robot nvolves searchng through a dscrete set of confguratons to fnd the one that best matches the desred state, rather than solvng geometrc equatons to determne jont angles or lnk lengths, as one would do for a contnuous systems. Most research has nvolved bnary robotc systems when the number of degrees of freedom s relatvely small, n the order of tens of DOF. Ths paper descrbes analyss and smulaton studes performed to examne the feasblty of controllng and plannng bnary-actuated robotc systems n real tme when the number of DOF s very large (hundreds or thousands). Such systems are currently under development [16]. The work also suggests plannng algorthms that can be used n future systems. The work outlnes some of the ssues and potental methods for the workspace analyss and optmzaton, the forward and nverse knematcs, and trajectory plannng of bnary robotc systems. These methods are then appled to bnary systems used for robot locomoton.. Workspace Analyss and Optmzaton The workspace of a robot generally refers to the locus of all ponts that a robot s end-effector can reach []. Wth a contnuous system, the workspace s usually a regon n contnuous space (see Fgure 1(a)). Many contnuous robots are also able to acheve a contnuous range of end-effector orentatons for a gven pont n the workspace. Understandng the sze of the workspace as well as the orentablty of the end-effector wthn ths workspace gves some measure of the ablty of the robot to perform dverse manpulaton tasks. (a) contnuous robot workspace (b) bnary-actuated robot workspace Fgure 1: The dstncton between bnary and contnuous robot workspace For bnary-actuated robots the noton of a workspace takes on some subtle dfferences [14]. For a bnary system, the workspace n not a contnuous volume but rather a fnte set of ponts n space (see Fgure 1(b)). For each pont there s an assocated orentaton of the endeffector, ndcated by the arrows orgnatng from each pont n Fgure 1(b). In such a workspace one can guarantee the exstence of at least one bnary confguraton of the robot that acheves a mnmum error of end-effector poston and orentaton. Thus for a bnary robot, the densty of the ponts wthn the workspace can be mportant, snce a dense set of ponts wll generally acheve small end-pont errors. The densty of ponts ncreases as the number of actuators n the system

2 ncreases. Each addtonal actuator doubles the number of workspace ponts. It s useful to vew a dscrete workspace cloud from the perspectve of a densty functon map. In desgnng a bnary robot, one mght want to optmze ts workspace. For example, t may be desrable for repeated pck and place tasks to have a workspace that has a great densty of ponts n the pck and place locatons. In other cases, t mght be desrable to have a unform dstrbuton of reachable ponts over the entre workspace. To deal wth such ssues the noton of workspace dstrbuton s proposed. For a planar robot, a densty map represents the densty of ponts (the z-axs) versus the Cartesan locaton n space (the x- and y-axes). Wth a dscrete cloud, the densty map appears as delta functons at each workspace pont, wth all other areas of the map havng a value of zero densty (see Fgure (a)). Applyng a lowpass flter (such as convoluton wth a Gaussan functon) to the densty map, the spkes blend together and provde a contnuous approxmaton of the densty of the workspace (see Fgure (b)). (a) dscrete pont cloud (b) contnuous representaton (c) optmzed unform workspace densty Fgure : Workspace of a 6 DOF seral bnary manpulator wth optmzaton Ths contnuous approxmaton can be a metrc for the unformty/dstrbuton of the workspace. Here t s defned based on the standard devaton of the workspace densty. A small standard devaton of the workspace densty ndcates a more unform dstrbuton. Ths method for quantfyng the dstrbuton of the workspace can be extended to three-dmensonal workspaces and nclude endpont orentaton nformaton. As an example of optmzng a bnary robot desgn to provde unform workspace densty, consder a seral planar manpulator, havng between four and ten bnary actuators. The lengths of each lnk, l, and the angles of devaton of each bnary rotary jont, ϕ I are to be optmzed. Ths robotc desgn results n a planar workspace composed of N ponts, where N s the number of bnary actuators. Usng an evolutonary algorthm the desgn varables of ths system can be optmzed. The algorthm generates a random set of canddate desgns and evaluated them based on ther unformty of workspace. The best canddates (those wth the most unform workspace denstes) are evolved n the classcal manner wth mutaton. A few hundred generatons result n good solutons to the problem. An example of one such optmzaton (for an un-optmzed system shown n fgure (a)) s shown n Fgure (c). Note that the densty map n ths fgure s much more unform than the one shown n Fgure (b).. Knematcs.1. Forward Knematcs For bnary robotc systems, t s convenent to formulate the forward knematcs usng four-by-four homogeneous transformaton matrces []. For example, the transformaton matrx A 0,M descrbng the poston and orentaton of the end-effector relatve to the base can be vewed as the product of the M ntermedate transformatons A -1, from module to module wthn the structure: M M 1, M = = 1 A0, = A0,1 A1, Κ A A (1) M 1, where M s the number of ntermedate modules comprsng the bnary robotc system [15]. Due to the dscrete nature of bnary devces, each term of the ntermedate transformaton A -1, can have only a fnte number of possble values. If each module has only a few bnary degrees of freedom, all the values that the terms of A -1, can be easly enumerated. For example, f a module has three bnary DOF, then the module has or 8 possble values for A -1, (notated by A - 1, (1), A -1, (),, A -1, (8) ). The soluton of the module knematcs may requre trgonometrc or more complex mathematcs, but these need only be solved once, and possbly offlne. Ths reduces onlne computatonal loads. Fgure : BRAID a seral chan of bnary-actuated parallel stages An example of such a robot s the Bnary Robotc Artculated Intellgent Devce (BRAID), developed at the Feld and Space Robotcs Laboratory, whch s a seral stack of dentcal parallel stages [15] (see Fgure ). Such

3 a desgn could be used for manpulatng nstruments, collectng sol samples, or matng two cooperatng robots, applcatons that requre only moderate precson (see Fgure 4). (a) matng two rovers (b) nstrument maneuverng Fgure 4: Potental BRAID applcatons In a sngle parallel lnk stage of the BRAID system, the three legs are postoned about the vertces of two equlateral trangles. Addtonally, based on the jont confguraton of each leg, the sngle stage has only three degrees of freedom ptch (θ x ) and yaw (θ y ) rotatons and a vertcal (z) translaton (couplng effects lead to nonndependent motons n the x and y drectons as well). The four by four transformaton matrx A -1,, of the th coordnate frame wth respect to the -1 th coordnate frame s defned based on these fve motons. The matrx A 0M defnes the forward knematcs from base to end-effector of the entre system. In ths formulaton the leg lengths are the control varables. The relatonshp between these leg lengths and the ptch, yaw, and translaton motons of the th coordnate frame wth respect to the -1 th coordnate frame s gven below. From Fgure 5 the deflecton parameters (δ, γ, ψ ) gve us the coupled x and y translaton of the th stage: δ x x = 1 sn( γ 1 ) r δ snγ r y = cosθ () x cosπ 6 δ j γ j l X' X 60 o Flexure to allow lateral moton of the lnks ψ j l x x -1 l z -1 (a) Sngle BRAID module C h 1 F D h E Z Z' 60 o l G h H (b) geometrcally equvalent representaton Fgure 5: Crtcal parameter representaton of BRAID system 60 o The followng relatons can be obtaned from fgure 5: B l 1 A y -1 z l 1 y Y' Y h1 snθ l1 snθ1 = r snθ () x h snα h snα1 = r snθ (4) y where θ 1 = DAB, θ = ADC, α 1 = EHG, and α = FEH, can be found from the leg lengths l 1, l, and l. Equatons and 4 gve two ndependent equatons n two unknowns. However, both are hghly non-lnear transcedental equatons and can only be solved numercally to gve the BRAID forward knematcs. A Newton-Raphson algorthm was mplemented to solve for the unknowns, θ x and θ y. Fnally, the vertcal translaton can be solved usng solutons for θ x, θ y and equaton z = h snα r snθ y = h1 snθ r snθ (5) x.. Inverse Knematcs The plannng and executon of practcal tasks generally requres the soluton to the nverse knematcs problem. The complexty of the trajectory plannng and nverse knematcs software of bnary devces are more complex than that of contnuous systems. The nverse knematcs problem cannot be expressed n a closed form soluton. Brute force or exhaustve search methods may prove appealng for systems wth few stages (less than 5), but become mpractcal for larger systems. As the number of degrees of freedom ncrease, the complexty of the workspace grows exponentally. For example, for every addtonal stage n the BRAID there s about an order of magntude ncrease n the number of states n the workspace. Hence, for large systems more effcent search methods are requred to fnd optmum solutons. In ths study two search methods for the nverse knematcs problem are studed. The frst s a combnatoral search algorthm and the second s a genetc search algorthm. The search metrc for both algorthms s to mnmze the error between the true endeffector and desred poston. Both algorthms are brefly descrbed below...1. Search Algorthms The combnatoral search algorthm was frst descrbed n [1]. To avod exponental growth of the search space as the number of actuators grow, the nverse knematcs are solved by changng the state of only a few actuators at a tme. Ths s perceved as a k-bt change to the gven system state, where any system state s defned by an m- bt word (m s the number of bnary actuators). At any state all possble changes (of up to k-bts, where k N) are evaluated to determne the one that optmzes the search metrc (.e. reduces the error between the endeffector poston and the desred poston). Ths optmal change forms the new state of the system and the search procedure repeats untl convergence. Ths reduces the computatonal complexty from O( N ) to O(N k ) or from exponental computatonal tme to polynomal computatonal tme [1]. The genetc algorthm used, s a classcal one, where each generaton conssts of N-bt bnary words descrbng

4 the manpulator state (where m s the number of bnary actuators). A full descrpton of genetc algorthms can be found n [5]. Comparng the genetc algorthm to the others dscussed, the sze of the search space explored by the genetc algorthm s gven by: search _ space _ sze = E G P (6) where E s the number of populatons separately evolved, G s the number of generatons for each populaton, and P s the number of ndvduals wthn the populaton. In studes done here, E, G, and P were kept constant relatve to the number of degrees of freedom, N. For more advanced algorthm development, these values could be made a functon of N. Wthn the algorthm, several computatons take place that are lnearly proportonal to N (such as forward knematcs computatons) and therefore computaton tme of the nverse knematcs usng a genetc algorthm grows approxmately lnearly wth the number of DOF of the system.... Algorthm Comparsons Performance of the two search methods s quantfed on a stochastc bass usng a Monte Carlo method. One thousand target ponts wth random orentatons are selected randomly wthn the volume of a bnary workspace cloud of a mult-staged BRAID system. The targets are chosen from wthn a sphercal regon, whose radus s roughly 90% of the radus of the actual pont cloud. The nverse knematcs for each target pose s then solved for and the soluton tmes and pose errors are computed and recorded. For comparson, results from exhaustve searchng are also presented. Fgure 6 shows the tmes for solvng the nverse knematcs problem for the two algorthms descrbed above. The tmes were computed from smulatons performed on a 600 MHz Pentum III processor. In these studes, the exhaustve search was observed to be the fastest for systems wth less than 1 DOF, the combnatoral algorthm was the fastest for systems havng between 1 and 40 DOF, and the genetc algorthm was the fastest for larger systems. Fgure 6: Inverse knematcs algorthms soluton tmes Errors n poston and orentaton for the algorthms are also quantfed. An example of the dstrbuton of the errors s shown n Fgure 7 for the case of a 0 DOF (a 10 stage) BRAID. The shapes of the error dstrbutons for any gven BRAID are very smlar for each of the algorthms and most closely resemble a gamma dstrbuton. The outlers are generally near the boundares of the workspace, and n practce tasks should be planned to avod these regons. Fgure 8 shows that the medan errors drop as a functon of the number of DOF for the combnatoral and genetc search algorthms. After about 0 DOF, there are only margnal mprovements as the number of DOF ncreases. For systems wth 0 DOF (a ten-stage BRAID), dsplacement errors are wthn a few percent of the characterstc manpulator length and angular errors are wthn ffteen degrees. Such a system s unsutable for precson work, but may be acceptable for such tasks as camera placement, crude nstrument manpulaton, and sample collecton. Fgure 7: Error dstrbuton for a 0-DOF BRAID: dsplacement error (1000 samples) Fgure 8: Medan errors vs. number of DOF for dfferent algorthms: dsplacement error (1000 samples per DOF) 4. Trajectory Followng The trajectory followng problem s also qute dfferent for a bnary devce than for contnuous ones. Instead of usng Jacoban matrces to compute actuator commands [], the problem s solved by a repeated search through the confguraton space to fnd the confguraton whose end-effector most closely matches a movng target [1]. Hence, ths problem s very close to the methods descrbed above and can be drectly appled. For low- DOF systems, the exhaustve search may prove to be the easest and most robust method for trajectory followng. For systems wth hgher DOF, genetc algorthms or combnatoral searches would be more effectve. However, t was found that the genetc algorthm used s not well suted for trajectory followng. A genetc algorthm, gven the same target and the same ntal condtons, wll produce dfferent solutons because of ts randomly selected ntal populaton and mutaton

5 component. Snce the hgh DOF system s hghly redundant, there can be a large number of confguratons that wll produce nearly the same end-effector poston yet wll have greatly dfferent confguratons. A relatvely smooth path planned n Cartesan space may have an erratc path n confguraton space (see Fgure 9). For power consumpton, relablty, and transent behavor, ths s very undesrable. Hence, the combnatoral search algorthm s found to be the most effectve method for trajectory followng. Ths algorthm searches only the subset of neghborng confguratons, and generates a path that s relatvely smooth n Cartesan and confguraton space. Between tme steps, only a few (specfcally defned) actuators wll be actuated at a tme. The combnatoral algorthm runs much faster n the trajectory plannng problem than for the nverse knematcs problem descrbed n Secton..1, snce t only makes one teraton per tme step (see Fgure 10). 5. Locomoton Plannng The trajectory followng problem can be extended to the locomoton plannng problem, where bnary devces are used as legs. Smulatons were done to explore the feasblty of plannng actuator sequences n real-tme for a robot havng sx bnary-actuated legs walkng n rough terran. Each of the sx legs s modeled as a BRAID and has 1 bnary DOF (see Fgure 1), yeldng a total of 16 DOF for the system. Desred ground contact ponts for the legs are chosen and the confguratons and actuator sequences are planned to acheve these contact ponts and body motons. Fgure 9: A smooth trajectory n Cartesan space s not necessarly smooth n confguraton space Fgure 1: Smulaton of a 6x1-DOF walkng robot composed of sx BRAIDs for legs, walkng n rough terran. Several ssues arse wth a bnary system for ths problem. Frst, the multple legs n contact wth the ground form a closed knematc chan that s overconstraned due to the dscrete nature of the leg motons. If the ground contact ponts are rgdly held, t wll be mpossble for the contactng legs to change confguratons due to the ncompatbltes between each leg s workspace (see Fgure 1(a)). Thus, t s mpossble to shft the body whle keepng the feet planted, as requred for walkng. Here a small amount of local complance n the ground contact s permtted and the lmted effects of the ncompatbltes between the workspaces of planted legs are gnored (see Fgure 1(b)). Fgure 10: Inverse knematcs soluton tmes for each trajectory followng step Smulatons showed that the errors mantaned durng trajectory followng were acceptable for a number of applcatons such as maneuverng a camera or nstrument, or manpulatng an object wth low precson (see Fgure 11). Typcal errors durng manpulaton were found to be of roughly the same sze as those dscussed n Secton... (a) rgd model Fgure 11: Smulaton of a camera maneuverng task desred trajectory: lghter path; actual trajectory: darker path. (b) sem-complant model Fgure 1: Knematc models n smulaton A second ssue arses n fndng the bnary confguraton of the legs that allows the body to move n a prescrbed manner. The system can be consdered as a sngle parallel 16 DOF system wth ground contact ponts modeled as contnuous revolute jonts wth

6 dsplacement complance. Wth the requrement that trajectores are smooth n both Cartesan and confguraton space, ths system becomes computatonally dffcult to solve quckly. To smplfy ths locomotonplannng problem, each leg was vewed as an ndependent trajectory plannng problem. Frst, the desred poston of the body for the next small tme step s selected. Then, the nverse knematcs for each leg s solved usng the one-pass combnatoral algorthm to make the leg move to the desred ground contact pont. Wth the confguraton of each leg beng solved ndependently, the actual body poston does not concde exactly wth the desred body poston. The actual body poston s obtaned from the equlbrum condton of the complant contact elements on the fxed confguraton robot. Ths problem s solved by mnmzng the potental energy stored n the complant elements as a functon of fnal body poston. The error requrng adjustments s small, roughly the sze of the errors n the legs themselves, generally a few percent of the characterstc sze (see Secton..). For most applcatons, these errors would be acceptable. Ths plannng method s found to be effectve and fast. Usng a Pentum III 9 MHz processor, the smulated robot plans and executes a strde at a rate of once per second. The robot s able to execute sdesteppng and turnng motons n rough terran. It mantans statc stablty whle walkng on slopes up to 0 degrees. Statc stablty s only lost occasonally ascendng, descendng, and traversng very steep slopes of around 45. Ths loss of stablty would need to be addressed by the hgh-level planner of the robot [11]. Ths requrement would be the same for robots wth contnuous degrees of freedom. 6. Conclusons Ths paper consders some of the computatonal challenges for the plannng of bnary robotc systems. The noton of a bnary workspace optmzaton was descrbed. The forward knematcs of bnary systems was dscussed, and the computatonal smplcty of ths operaton relatve to contnuous systems was shown. The methods to solve the nverse knematcs and trajectory plannng were addressed and compared to those of contnuous systems. The postonng errors of bnary systems were also quantfed n a probablstc manner. The methods were appled to a walkng system that mght be used for future space exploraton. Acknowledgements The authors would lke to acknowledge the NASA Insttute of Advance Concepts (NIAC) for ther support n ths work. References [1] Chrkjan, G.S.; Burdck, J.W. The knematcs of hyper-redundant robot locomoton. IEEE Transactons on Robotcs and Automaton. Volume: 11 Issue: 6, Dec Page(s): [] Crag J J. Introducton to Robotcs: Mechancs and Control. Second ed, Addson-Wesley, Readng, MA, [] Ebert-Uphoff, I.; Chrkjan, G.S. Inverse knematcs of dscretely actuated hyper-redundant manpulators usng workspace denstes. Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton, 1996, Volume: 1, Page(s): vol.1. [4] Erdmann, M.A. and Mason, M.T. Exploraton of sensor-less manpulaton. IEEE Journal of Robotcs and Automaton, Vol. 4, pp 69-79, August [5] Goldberg, D. Genetc Algorthms n Search, Optmzaton, and Machne Learnng. Addson-Wesley, Readng, MA [6] Goldberg, K. Orentng polygonal parts wthout sensors. Algorthmca, 199, Specal Robotcs Issue. [7] Huang, M.Z., Shou-Hung Lng. Knematcs of a class of hybrd robotc mechansms wth parallel and seres modules. Proceedngs of the 1994 IEEE Internatonal Conference on Robotcs and Automaton, Page(s): vol.. [8] Hughes, P.C. Trussarm a varable geometry truss manpulator. Journal of ntellgent materals, systems and structures, vol., pp , Aprl [9] Huntsberger, T.L., G. Rodrguez, and P. S. Schenker. Robotcs: challenges for robotc and human Mars exploraton. Proceedngs of ROBOTICS000, Albuquerque, NM, Mar 000. [10] Kwon, S., Youm, Y. General algorthm for automatc generaton of the workspace for n-lnk redundant manpulators. Proceedngs of the Internatonal Conference Advanced Robotcs, 'Robots n Unstructured Envronments', Page(s): vol.. [11] Latombe J. Robot Moton Plannng. Kluwar Academc Publshers, Boston, MA [1] Lees, D.S. and Chrkjan, G.S. A combnatoral approach to trajectory plannng for bnary manpulators. Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton, Mnneapols, Mnnesota, Aprl [1] Lchter, M.D., Sujan, V.A., Dubowsky, S. Expermental Demonstratons of a New Desgn Paradgm n Space Robotcs. Proceedngs of the Seventh Internatonal Symposum on Expermental Robotcs, ISER 00. Dec 10-1, 000, Honolulu, Hawa. [14] Sen D, Mruthyunjaya T S. A Dscrete State Perspectve of Manpulator Workspaces. Mech. Mach. Theory, Vol. 9, No.4, , [15] Sujan, V.A., Lchter, M.D., and Dubowsky, S. Lghtweght Hyper-redundant Bnary Elements for Planetary Exploraton Robots. Proceedngs of the 001 IEEE/ASME Internatonal Conference on Advanced Intellgent Mechatroncs (AIM '01) July 001, Como, Italy. [16] Hafez, M., Lchter, M.D., and Dubowsky, S. Optmzed Bnary Modular Reconfgurable Robotc Devces. Proceedngs of the 00 IEEE Internatonal Conference on Robotcs and Automaton. Washngton, D.C., May 11-15, 00.