A Computational Feasibility Study of Failure-Tolerant Path Planning

Transcription

1 A Computatonal Feasblty Study of Falure-Tolerant Path Plannng Rodrgo S. Jamsola, Jr., Anthony A. Macejewsk, & Rodney G. Roberts Colorado State Unversty, Ft. Collns, Colorado , USA Florda A&M-Florda State Unversty, Tallahassee, Florda , USA E-mals: & Abstract Ths work consders the computatonal costs assocated wth the mplementaton of a falure-tolerant path plannng algorthm proposed n [1]. The algorthm makes the followng assumptons: a manpulator s redundant relatve to ts task, only a sngle jont falure occurs at any gven tme, the manpulator s capable of detectng a jont falure and mmedately locks the faled jont, and the envronment s statc and known. The algorthm s evaluated on a three degree-of-freedom planar manpulator for a total of eleven thousand dfferent scenaros, randomly varyng the robot s start and goal postons and the number and locatons of obstacles n the envronment. Statstcal data are presented related to the computaton tme requred by the dfferent steps of the algorthm as a functon of the complexty of the envronment. I. INTRODUCTION Knematc falure tolerance gves a robot the ablty to gracefully recover from jont falures. Such a control strategy s most useful for robots performng tasks n hazardous or remote envronments, such as n space exploraton [2], [3], underwater exploraton [4], and nuclear waste dsposal [5], [6], [7]. It allows a robot to mmedately complete the task at hand wthout unnecessary delays due to robot repar and also avod the potentally sgnfcant danger assocated wth a robot falure durng task executon amongst hazardous materals. A worst-case falure-tolerance measure was frst ntroduced n [8] usng the mnmum sngular value of the robot Jacoban matrx. The nature of jont falures that have been studed nclude locked-jont [9], [10], [11] and freeswngng [12] jont falures. A real-tme mplementaton s gven n [11]. A number of falure-tolerant control strateges have been proposed, ncludng adaptve control [13], [14], [15], reflex control [16], and least squares approaches [17]. Generally, obstacles are not consdered n most knematc falure tolerance studes. One of the earlest works n knematc falure tolerance that does consder obstacles n the workspace s [18]. However, the method ntroduced extensvely checks every possblty of falure at every nstance n tme as the robot plans to move from a start to a goal workspace locaton. In more recent work [1], a method was ntroduced that guarantees task completon for any sngle locked-jont falures despte obstacles n the workspace wthout extensvely checkng every possblty of falure at every nstance n tme. The approach searches for a contnuous obstacle-free monotonc surface n the confguraton space that guarantees the etence of a soluton. The goal of ths work s to quantfy the computatonal feasblty of mplementng the knematc falure tolerance algorthm presented n [1]. We frst provde a revew of ths approach. Ths s followed by a set of smulaton experments consstng of eleven thousand scenaros, where a scenaro s defned as a workspace start locaton, a workspace goal locaton, and a set of obstacles. The resultng data s then analyzed to determne the types of scenaros where the mplementaton of a falure-tolerant path plannng strategy may be feasble. II. DEFINITION OF TERMS For an n degree-of-freedom (DOF) knematcally redundant robot operatng n an m DOF workspace, the degree-of-redundancy (DOR) s defned as r = n m. The set of confguratons n the confguraton space (Cspace) that result n the same end-effector workspace poston/orentaton x s called the pre-mage of x, denoted f 1 (x). The pre-mage can be wrtten as a unon of dsjont connected sets n m f 1 (x) = M (1) =1 where M s the -th r-dmensonal self-moton manfold n the nverse knematc pre-mage such that M M j = when j, and n m s the number of self-moton manfolds [19]. Fg. 1 shows a set of sngle dmensonal start and goal self-moton manfolds for a robot wth r =1. The dark portons of the self-moton manfolds denote confguratons of the robot that are n contact wth obstacles. A contnuous obstacle-free porton of the goal self-moton manfold s denoted as γ g whle an obstaclefree start confguraton s denoted as θ s. For a sngle locked-jont falure, the resultng C-space s an (n 1)-dmensonal hyperplane called a falure hyperplane. For an obstacle-free start confguraton θ s,

2 Fg. 2. A monotonc surface s defned as a surface where there are no closed contours along the ntersecton wth any plane parallel to ts falure planes. A monotonc surface has no local mnma or maxma. Fg. 1. The confguraton space for a sngle degree of redundancy robot showng a start and goal self-moton manfold. All the falure planes correspondng to an obstacle-free start confguraton θ s ntersect an obstacle-free porton of the goal self-moton manfold γ g. The falure cube contans an obstacle-free start confguraton θ s and an obstaclefree porton of the goal self-moton manfold γ g. The falure surface, shown as a web-lke network of paths, correspondng to an obstacle-free start confguraton θ s s dentfed by generatng monotonc paths wthn the falure cube and connectng the obstacle-free start confguraton θ s to ponts on the obstacle-free porton of the goal self-moton manfold γ g. Each node along the obstacle-free porton of the goal self-moton manfold γ g defnes an ntersecton wth ether a falure plane or a face of the falure cube. a falure hyperplane H assocated wth θ s s gven by H (θ s )={θ θ = θ s } (2) where θ denotes the -th component of θ n a falurenduced C-space, θ s s the fxed value of the locked -th component of start confguraton θ s, and =1,...,n. An obstacle-free start confguraton θ s s shown wth ts correspondng falure planes n Fg. 1. A falure hypercube s a hypervolume n C-space that contans an obstacle-free start confguraton θ s and an obstacle-free porton of the goal self-moton manfold γ g such that all the falure hyperplanes correspondng to the obstacle-free start confguraton θ s ntersect an obstaclefree porton of the goal self-moton manfold γ g. A falure hypercube assocated wth θ s and γ g has the form V = {θ θ H l θ θ H u for =1,...,n} (3) where H l s the lower boundng hyperplane, H l = {θ θ = θ H l }, (4) and H u s the upper boundng hyperplane, H u = {θ θ = θ H u }, (5) for = 1,...,n. The values of θ H l and θ H u defne the locaton of ther correspondng hyperplanes. Typcally, some of the faces of a falure hypercube wll le n the falure hyperplanes assocated wth θ s. The rest of the faces of the falure hypercube are defned by the extremal ponts of γ g. Note that a falure hypercube s not typcally obstacle free. A falure surface S s a contnuous obstacle-free monotonc surface wthn the falure hypercube that contans an obstacle-free start confguraton θ s and s bounded by three curves: an obstacle-free porton of the goal selfmoton manfold γ g, an obstacle-free monotonc curve lyng n the frst falure hyperplane that ntersects γ g, and an obstacle-free monotonc curve lyng n the last falure hyperplane that ntersects γ g. A monotonc surface has the property that ts ntersecton wth every plane parallel to a falure hyperplane s a non-closed curve (see Fg. 2). Thus a monotonc surface does not have any local nternal mnma or maxma. Fg. 1 shows a web-lke network of paths that represent a falure surface S wthn a falure cube. The outermost paths from θ s to γ g le on ther respectve falure planes. III. GUARANTEEING A FAILURE-TOLERANT PATH A. A Necessary Condton and a Suffcent Condton Two condtons are derved that determne the etence of a soluton. A necessary condton s used to dentfy feasble obstacle-free start confguratons θ s and a suffcent condton s used to determne the etence of a soluton from all the feasble obstacle-free start confguratons θ s. Necessary Condton. A gven obstacle-free confguraton θ s s called a feasble confguraton f all the correspondng falure hyperplanes assocated wth θ s ntersect a porton of the obstacle-free goal self-moton manfold γ g, that s, H (θ s ) γ g, for all =1,...,n (6) where H s the falure hyperplane at θ s correspondng to jont. Ths ensures a possblty of reachng the obstaclefree porton of the goal self-moton manfold γ g despte a jont falure at the start confguraton θ s. Note that ths s equvalent to the goal poston beng n the fault-tolerant workspace [10].

3 Suffcent Condton. Consder a gven falure hypercube V contanng an obstacle-free start confguraton θ s and an obstacle-free porton of the goal self-moton manfold γ g. If a falure surface S whch s a contnuous obstacle-free monotonc surface n the falure hypercube V ets such that the two condtons θ s S and γ g S (7) hold, then an obstacle-free path to the goal s guaranteed for any sngle locked-jont falure at any gven tme despte the presence of obstacles n the workspace. B. Algorthm The procedure for our proposed method s enumerated n the followng. 1. Determne the start self-moton manfold M s and goal self-moton manfold M g, from the gven start, x s, and goal,, workspace poston/orentaton, respectvely. 2. Identfy an obstacle-free start confguraton θ s and an obstacle-free porton of the goal self-moton manfold γ g. 3. Check for ntersectons of the falure hyperplanes H (θ s ) wth γ g for = 1,...,n. (Note that ths step uses the necessary condton n Secton III-A). 4. Check for the etence of a falure surface S. Ths s done by generatng monotonc paths from θ s to ponts n γ g and checkng for ntersectons wth obstacles. These paths are lmted to straght lnes or the set of quadratc curves that are monotonc. Straght lne connectons between paths are used to check for the contnuty of obstacle-free space between paths. (Ths step uses the suffcent condton n Secton III-A). Gven that a falure surface S ets, to move from the start poston/orentaton, x s, towards the goal poston/orentaton,, the manpulator confguraton traverses from the obstacle-free start confguraton θ s along the contnuous web of paths that represents the falure surface S toward the obstacle-free porton of the goal self-moton manfold γ g. Because S s known to be collson free, as long as the manpulator confguraton remans on the surface, the manpulator would be free from collson and at the same tme t can reach the goal for any sngle locked-jont falure at any tme. If no falure surface S ets, then t s not guaranteed that the robot can successfully complete ts task for any sngle lockedjont falure wth the gven obstacles n the envronment. The computatonal complexty of the proposed algorthm s hghly dependent on the method used for computng the start and goal self-moton manfolds, and the method used for collson detecton. For a sngle DOR, the computatonal complexty s O(mn 2 )+O(mnp) where p s the number of obstacles n the workspace. The frst term Fg. 3. The workspace of a 3-DOF planar manpulator used for the smulaton experments (all three lnk lengths are 100 unts long). The one thousand start locatons, x s, are randomly generated wthn the range [100, 200] unts along the x-a (shown as a thck bold lne). The one thousand goal workspace locatons,, (shown as dots) are randomly generated to be wthn the range [0, 200] unts away from ther correspondng start locaton (but nsde the reachable workspace). The center of the workspace s marked wth a bold cross, the workspace boundary wth a sold lne, and the boundary of the goal locatons wth a dashed lne. s the contrbuton for the computaton of the self-moton manfolds, whle the second term s the contrbuton due to collson detecton. IV. FEASIBILITY STUDY: 3-LINK PLANAR ROBOT A 3-DOF planar robot wth equal lnk lengths of 100 unts was used for ths feasblty study. The workspace contaned a number of crcular obstacles, each of a dameter of 40 unts, where the number of obstacles was vared from zero to twenty n two obstacle ncrements. For each workspace wth a gven number of obstacles, smulaton experments were performed for one thousand randomly generated scenaros, where a scenaro conssts of a gven start workspace locaton, a gven goal workspace locaton, and the specfed locatons of the obstacles. In each scenaro, the locatons for the correspondng number of obstacles are randomly selected from a unform dstrbuton throughout the entre robot workspace. The start workspace locaton x s s randomly selected from a unform dstrbuton of [100, 200] along the x-a, that s, at a dstance that corresponds to between one and two lnk lengths away from the base. The goal workspace locaton s randomly selected to be wthn a range of [0, 200] unts from the correspondng start workspace locaton (whle restrctng the goal to be wthn the reachable workspace of the manpulator). Fg. 3 presents an example of the start and goal locatons generated for one thousand scenaros. A total of eleven thousand scenaros

4 (f) (g) (h) () (j) (a) (b) (c) (d) (e) Fg. 4. Several examples selected from the eleven thousand scenaros where the fault-tolerant path plannng algorthm was appled to 3-DOF planar manpulator. The subfgures denoted (a) through (e) correspond to workspaces contanng from twenty to twelve obstacles, respectvely. In each case, the top scenaro represents one n whch the algorthm successfully dentfed a falure surface S, whereas the bottom scenaro represents a case where such a surface could not be found. In all cases the start confguraton shown for the manpulator s one that satsfes the necessary condton for a surface to et. were generated,.e., one thousand for each workspace contanng a gven number of obstacles. Ten examples from ths set of eleven thousand scenaros are presented n Fg. 4, where the number of obstacles s vared from twenty to twelve n two obstacle ncrements. The top row of ths fgure llustrates cases where the fault tolerant path plannng algorthm was able to dentfy a falure surface, wth the bottom row showng cases where t could not. The necessary condton for a surface to et was satsfed n all cases, wth the manpulator confguraton llustratng one such feasble startng confguraton. In the followng subsectons, data gathered from these eleven thousand smulaton experments wll be presented accordng to the order n whch these steps are performed n the algorthm descrbed n Sec. III-B. In all cases the algorthm was executed on a computer wth dual Intel Xeon processors runnng at 2.4 GHz. A. Computaton of Self-Moton Manfolds The frst step n the falure-tolerant path plannng algorthm s to compute the self-moton manfold(s) for both the start and goal workspace locatons. These are computed by dentfyng a sngle confguraton on each dsjont manfold and then steppng along the manfold (n two degree ncrements) by computng the null vector of the manpulator Jacoban matrx (whch corresponds to the tangent of the self-moton manfold). The average lengths for the sum of all manfolds correspondng to a workspace locaton are gven n Table I. The length for the start manfolds are larger than those for the goal manfold because the start workspace locatons were TABLE I COMPUTATION OF SELF-MOTION MANIFOLDS Ave. Std. Dev. Length of M s (deg) Length of M g (deg) Tme to compute M s, M g (ms) ntentonally restrcted to a regon of the workspace where these manfolds are larger, and thus, the locatons are more falure tolerant [10]. (A dstance of one lnk length from the base s optmally falure tolerant,.e., all jonts span ther entre range of moton, whle the locaton on the workspace boundary s most falure ntolerant,.e., ts selfmoton manfold conssts of a sngle pont.) The average computaton tme over all manfolds was 41.9 ms. B. Identfyng Obstacle-Free Portons of M s and M g After the start self-moton manfold M s and goal selfmoton manfold M g are computed, the next step s to determne the obstacle-free portons of the manfolds. Ths dentfes canddate feasble start confguratons θ s and contnuous obstacle-free porton of the goal selfmoton manfold γ g that can possbly satsfy the necessary condton. Fg. 5 shows the percentage of the start self-moton manfold M s and goal self-moton manfold M g that are obstacle free as a functon of the number of obstacles n the workspace. As expected, the percentage of the obstacle-free self-moton manfold decreases as the

5 Percent Fg. 5. Mllseconds Percent of a Manfold that s Obstacle Free Percent of the self-moton manfolds that are obstacle free. Tme to Compute Obstacle-Free Manfolds Fg. 6. Tme needed to determne the obstacle-free porton of the start self-moton manfold M s and the goal self-moton manfold M g as a functon of the number of workspace obstacles. number of workspace obstacles ncreases. Fg. 6 shows the tme, n mllseconds, requred to determne the obstaclefree porton of the self-moton manfolds. As expected, the tme requred to determne whch portons of a manfold are obstacle free s a lnear functon of the number of obstacles. In many cases, one could avod most of the computaton tme assocated wth checkng the entre manfold for collsons wth obstacles by performng the necessary condton check frst, and then verfyng that the start confguraton and correspondng porton of the goal selfmoton manfold are collson free. (Ths check s the same as determnng f the goal locaton s n the falure-tolerant workspace of the start locaton x s as frst consdered n [10].) C. Checkng the Necessary Condton The next step n the falure-tolerant path plannng algorthm s to apply the necessary condton to those portons of the start and goal manfolds that are collson free. Ths s done by selectng a start confguraton θ s from the obstacle-free porton of the start manfold and then checkng for ntersectons of all falure planes wth each γ g.ifaγ g ntersects all falure planes then the necessary condton s satsfed and ths step termnates. If no γ g satsfes the ntersecton tests then a new θ s s selected and the tests repeated untl ether the necessary condton s satsfed or all θ s have been tested. Thus f ths test fals t s guaranteed that no falure-tolerant obstacle-free path ets between the start and goal locatons. Table II presents the computatonal data assocated wth checkng the necessary condton. Note that even wthout any obstacles, the necessary condton can only be satsfed n 69% of the scenaros tested. Ths s essentally an ndcaton of the dffculty of guaranteeng that a goal locaton wll be n the falure-tolerant workspace of the start confguraton. Ths s clearly very dependent on both the startng locaton and the dstance between the start locaton and the goal locaton [10]. As the number of obstacles n the workspace ncreases, the percentage of cases that satsfy the necessary condton decreases, reachng a mnmum of 25% for the case wth twenty obstacles. For those cases where a θ s satsfed the necessary condton, we further processed the start manfold to see what percentage of the start manfold would be able to satsfy the necessary condton. (Ths s not requred by the algorthm and the tme requred to perform ths computaton was not ncluded n the overall executon tme data presented.) Table II shows that ths s also a monotoncally decreasng functon of the number of obstacles n the workspace. Thus, t becomes ncreasngly more tme consumng to dentfy a θ s that satsfes the necessary condton as the number of obstacles n the workspace ncreases. Ths s llustrated by the fact that the computaton tme s a monotoncally ncreasng functon of the number of obstacles. Ths s true despte the fact that there s less and less of the start manfold that s obstacle free (see Fg. 5) because an ncreasngly smaller percentage of the obstacle-free manfold s able to satsfy the necessary condton. In contrast, the tme to compute that the necessary condton s not satsfed s relatvely ndependent of the number of obstacles. Ths at frst appears anomalous because one would expect ths to be a monotoncally decreasng functon due to a smaller percentage of the start manfold needng to be checked (because less of t s obstacle free). However, ths s offset by the fact that larger and larger manfolds are now falng the necessary condton, thus keepng the computaton tme relatvely constant. Fg. 7 shows the average length of the γ g from a (θ s, γ g ) par that satsfes the necessary condton. Ths generally decreases as the number of obstacles ncreases due to the fact that t s more dffcult to have large γ g because they are requred to be obstacle free. The sze of a γ g that satsfes the necessary condton s correlated to the dstance between the start and goal workspace locatons so

6 TABLE II COMPUTATION TO CHECK THE NECESSARY CONDITION No. of Tme N.C. Tme N.C. % Cases % M s Obs. Sat. (ms) Not Sat. (ms) N.C. Sat. N.C. Sat Degrees Length of γ g Satsfes N.C. Satsfes S.C Fg. 7. Average length of a γ g that satsfes the necessary condton and that also satsfes the suffcent condton as a functon of the number of obstacles n the workspace. that, as expected, start and goal locatons must be closer together as more and more obstacles are added to the workspace. D. Computng a Falure Surface The fnal step n the algorthm s to check the suffcent condton by attemptng to compute a falure surface S that guarantees the etence of a soluton to the faluretolerant path plannng problem. Once a feasble start confguraton θ s s found from the prevous step, the search for a falure surface S begns. A falure surface S s dentfed by generatng monotonc paths that connect a feasble start confguraton θ s to ponts on ts correspondng γ g that satsfes the necessary condton. (The γ g curve s dscretzed at a resoluton of two degrees.) For each pont on γ g the algorthm frst attempts to use a straghtlne path. If ths path s not obstacle free, then t attempts to fnd a monotonc quadratc path that s obstacle free. If no such path can be found then the algorthm dscards ths (θ s, γ g ) par and uses the next (θ s, γ g ) par that satsfes the necessary condton. If all such pars are exhausted Percent TABLE III COMPUTATION TO CHECK THE SUFFICIENT CONDITION (ONCE THE NECESSARY CONDITION IS SATISFIED) Ave. Std. Dev % of tmes S found % lnear paths on S Tme to compute S(s) Tme to not fnd S(s) Overall Percentage N.C. Satsfed S.C. Satsfed Fg. 8. Percent of total cases where the necessary condton s satsfed and percent of total cases where the suffcent condton s satsfed,.e., where a falure surface S s found. wthout completng a falure surface then the algorthm termnates wth a message that s was unsuccessful. Table III presents data on the computaton of falure surfaces. It s nterestng to note that once the necessary condton s satsfed, t s hghly lkely that a falure surface wll be found,.e., ths occurs 84% of the tme. In addton, ths percentage s relatvely ndependent of the number of obstacles that are n the workspace as llustrated n Fg. 8 (except, of course, for the case of no obstacles). Ths s fortutous, because the constructon of falure surfaces s by far the most tme consumng porton of the algorthm. (The average tme to compute a falure surface s 1.16 s, and t takes 1.80 s, on average, to exhaust all possble canddates n cases where a surface cannot be found.) Thus the tme to evaluate surfaces s seldom wasted, wth the majorty of the cases where no soluton ets beng dentfed n a matter of mllseconds by the necessary condton test. It s also nterestng to note that nearly all (99.6%) of the paths n falure surfaces are lnear. However, t s mportant to pont out that even f only one path n a falure surface s quadratc, then that mples that that surface would not have satsfed the suffcent condton f only straght-lne paths were allowed. V. SUMMARY AND CONCLUSION Ths paper has presented and analyzed the behavor and computatonal cost of a falure-tolerant path plannng strategy orgnally proposed n [1]. Ths algorthm conssts

7 of four steps: (1) constructon of self-moton manfolds, (2) collson detecton for those manfolds, (3) testng of a necessary condton for a soluton to the problem, and (4) determnng f a suffcent condton was satsfed. The probablty of dentfyng a falure-tolerant path s strongly dependent on several factors,.e., the start workspace poston, the goal workspace poston, and the number and locaton of the obstacles. It was tested on 11,000 scenaros for a planar 3-DOF manpulator wth all of these factors randomly vared. Even wth no obstacles, faluretolerant paths only eted for 69% of the cases tested. For envronments wth 20 obstacles (the largest number of obstacles tested) ths number dropped to 15%. However, one advantage of the algorthm s that for most of the cases where a path does not et t s dentfed by the relatvely computatonally nexpensve test of the necessary condton. In all cases, the algorthm was computatonally feasble,.e., to determne that no possble path ets took an average of 0.2 s, to compute a falure-tolerant surface took an average of 1.2 s, and f the algorthm could not determne whether a surface ets took an average of 2.0 s. VI. REFERENCES [1] R. S. Jamsola, Jr., A. A. Macejewsk, and R. G. Roberts, A path plannng strategy for knematcally redundant manpulators antcpatng jont falures n the presence of obstacles, n Proc. Int. Conf. Intell. Robots Syst., Las Vegas, NV, Oct , pp [2] E. C. Wu, J. C. Hwang, and J. T. Chladek, Faulttolerant jont development for the space shuttle remote manpulator system: Analyss and experment, IEEE Trans. Robot. Automat., vol. 9, no. 5, pp , Oct [3] G. Vsentn and F. Ddot, Testng space robotcs on the Japanese ETS-VII satellte, ESA Bulletn- European Space Agency, pp , Sept [4] P. S. Babcock and J. J. Znchuk, Fault-tolerant desgn optmzaton: Applcaton to an autonomous underwater vehcle navgaton system, n Proc Symp. Autonom. Underwater Vehcle Technol., Washngton, D.C., June , pp [5] R. Colbaugh and M. Jamshd, Robot manpulator control for hazardous waste-handlng applcatons, J. Robot. Syst., vol. 9, no. 2, pp , [6] W. H. McCulloch, Safety analyss requrements for robotc systems n DOE nuclear facltes, n Proc. 2nd Specalty Conf. Robot. Challengng Envron., Albuquerque, NM, June , pp [7] M. L. Leuschen, I. D. Walker, and J. R. Cavallaro, Investgaton of relablty of hydraulc robots for hazardous envronment usng analytc redundancy, n Proc. Annual Rel. Mantan. Symp., Washngton, D.C., Jan , pp [8] A. A. Macejewsk, Fault tolerant propertes of knematcally redundant manpulators, n Proc. IEEE Int. Conf. Robot. Automat., Cncnnat, OH, May , pp [9] R. G. Roberts and A. A. Macejewsk, A local measure of fault tolerance for knematcally redundant manpulators, IEEE Trans. Robot. Automat., vol. 12, no. 4, pp , Aug [10] C. L. Lews and A. A. Macejewsk, Fault tolerant operaton of knematcally redundant manpulators for locked jont falures, IEEE Trans. Robot. Automat., vol. 13, no. 4, pp , Aug [11] K. N. Groom, A. A. Macejewsk, and V. Balakrshnan, Real-tme falure-tolerant control of knematcally redundant manpulators, IEEE Trans. Robot. Automat., vol. 15, no. 6, pp , Dec [12] J. D. Englsh and A. A. Macejewsk, Fault tolerance for knematcally redundant manpulators: Antcpatng free-swngng jont falures, IEEE Trans. Robot. Automat., vol. 14, no. 4, pp , Aug [13] Y. Tng, S. Tosunoglu, and D. Tesar, A control structure for fault-tolerant operaton of robotc manpulators, n Proc. IEEE Int. Conf. Robot. Automat., Atlanta, GA, May , pp [14] K. S. Tso, M. Hecht, and N. I. Marzwell, Faulttolerant robotc system for crtcal applcatons, n Proc. IEEE Int. Conf. Robot. Automat., Atlanta, GA, May , pp [15] Y. Tng, S. Tosunoglu, and R. Freeman, Actuator saturaton avodance for fault-tolerant robots, n Proc. 32nd Conf. Decson Contr., San Antono, TX, Dec. 1993, pp [16] T. S. Wkman, M. S. Brancky, and W. S. Newman, Reflexve collson avodance: A generalzed approach, n Proc. IEEE Int. Conf. Robot. Automat., Atlanta, GA, May , pp [17] J. Park, W.-K. Chung, and Y. Youm, Falure recovery by explotng knematc redundancy, n 5th Int. Workshop Robot Human Commun., Tsukuba, Japan, Nov , pp [18] C. J. J. Pareds and P. K. Khosla, Fault tolerant task executon through global trajectory plannng, Rel. Eng. Syst. Safety, vol. 53, pp , [19] J. W. Burdck, On the nverse knematcs of redundant manpulators: Characterzaton of the selfmoton manfolds, n Proc. IEEE Int. Conf. Robot. Automat., Scottsdale, AZ, May , pp