Planning the Motion of a Sliding and Rolling Sphere

Transcription

1 2015 IEEE Internatinal Cnference n Rbtics and Autmatin (ICRA) Washingtn State Cnventin Center Seattle, Washingtn, May 26-30, 2015 Planning the Mtin f a Sliding and Rlling Sphere Yan-Bin Jia Department f Cmputer Science Iwa State University Ames, IA 50010, USA jia@iastate.edu Abstract This paper investigates the free mtin f a sphere with initial velcity and angular velcity n a plane under sliding and rlling frictin. The sphere will first slide alng a parablic trajectry (with a cnstant directin f its cntact velcity), and then rll alng a straight trajectry. Such a curved trajectry can be utilized fr bstacle avidance in path planning. A ne-tne crrespndence exists between trajectries cnnecting tw specified lcatins and pairs f sliding and rlling directins within sme 2-dimensinal regin, called the trajectry space, which is then used fr planning. A plane sweep algrithm perating in this space is presented t find all cllisin-free trajectries in the presence f cylindrical bstacles that are vertically riented. I. INTRODUCTION A sphere n a plane can achieve re-rientatin, that is, change the rientatin f its bdy frame, by simply rlling n the plane. Mtin planning fr the sphere, called the ball-plate prblem, explits the nnhlnmic rlling cnstraint t cnstruct a rlling path (desirably clsed) that results in sme prescribed change in the sphere s rientatin. A number f re-rientatin algrithms [5], [7], [8], [11] have been designed using techniues frm plane trignmetry r differential gemetry. Spherical rbts [1], [2] have als been designed t perfrm rlling and spinning peratins in a cntrlled manner. While rlling has received much attentin in path planning, mbile rbts, grasping, and dexterus manipulatin, sliding has largely been regarded as a negative phenmenn t avid, especially in the last tw areas abve. In this paper, we are ging t demnstrate that sliding ffers the benefit f curving the trajectry f a mving sphere, which can be utilized t avid bstacles in path planning. One mtivatin cmes frm billiard sprts. The cue ball after a massé sht will initially slide alng a curve, making slight nise frm scratching the felt, befre rlling straight ahead. Supprt fr this research was prvided in part by Iwa State University (ISU). II. MOVING SPHERE ON A PANE Figure 1 illustrates the general case f a sphere with radius ρ mving n a hrizntal plane P. The sphere P trajectry f ρ Fig. 1. Cntact velcity s f a sphere and cntact frictinal frce f. has n actuatin r cntrl during its mtin, which is cmpletely influenced by frictin. Des there exist initial velcity v 0 and angular velcity ω 0 that will take the sphere t sme lcatin while aviding bstacles? Withut lss f generality, we let the rigin be at the sphere s starting lcatin, the x-axis pint tward, and the xy-plane cincide with P. The fllwing tw assumptins are made: 1) There exists at least ne bstacle at distance ρ r less frm the line segment. 2) v 0 and ω 0 have n cmpnents in the z-directin. If the first assumptin des nt hld, the line segment wuld be a cllisin-free trajectry, making the prblem a trivial ne. The z-cmpnent f v 0 is assumed t be zer t ignre buncing f the sphere. The z-cmpnent f ω 0 plays n part in the initial velcity at the sphere s cntact pint with the plane, referred t as the sliding velcity: s 0 = v 0 + ρẑ ω 0 (1) due t the crss prduct with ẑ = (0, 0, 1) T. We shall see later in this sectin that s 0 and v 0 determine the sphere trajectry. The assumptin abut ω 0 will merely make ur presentatin clean. A. Sliding Sliding happens when s 0 is nt zer. et the sphere have mass m. Its angular inertia matrix is eual t the s ω v /15/$ IEEE 2396

2 2 5 mρ2 times the 3 3 identity matrix. Its velcity v and angular velcity ω yield the fllwing sliding velcity x r at t r sliding s 0 s = v + ρẑ ω. (2) The sphere is subject t a frictinal frce f = µ s mgŝ, where µ s is the cefficient f sliding frictin, g > 0 the magnitude f the gravitatinal acceleratin, and ŝ = s/ s, the sliding directin. Wrking ut the dynamics, we btain the sphere s acceleratin v = µ s gŝ and angular acceleratin ω = 5 2ρ µ sgẑ ŝ. We derive the cntact acceleratin by differentiating (2) and substituting the expressins f v and ω in: ṡ = µ s gŝ µ sgẑ (ẑ ŝ) = 7 2 µ sgŝ, (3) using ẑ ŝ = 0. The abve implies that ṡ is ppsite t s 0, which further implies that s will nt change its directin until it becmes t zer. We refer t ŝ = s 0 / s 0 as the sliding directin. With ŝ being cnstant, we can integrate ṡ, v, and ω t btain s, v, and ω, as derived earlier in [6, pp ]. The sliding trajectry is btained frm ne mre rund f integratin this time f v: x = (x, y) T = v 0 t 1 2 µ sgŝt 2, 0 t t r. (4) Sliding ends (i.e., s = 0) at time t r = 2 s 0 /(7µ s g) with velcity and lcatin given by v r = v 0 µ s g t r ŝ = v s 0, (5) x r = x(t r ) = ( v s 0) tr. (6) B. Pure Rlling At time t r, v r is fund rthgnal t the angular velcity ω r, after taking crss prduct f v with (2) under s = 0. The frictinal frce ppses the directin f v r. Neither ω nr v will change its directin frm t r n. The sphere starts pure rlling withut slip (s = 0). The rlling trajectry is a straight line segment. T derive the deceleratin f the sphere, we adpt Marlw s treatment [6, p. 12] based n the principle f energy dissipatin. Because the sphere rlls n a line, we can nw dente v and ω by their magnitudes v and ω with n ambiguity. Since v = ρω under pure rlling, the kinetic energy f the sphere is E = 1 2 mv (2 5 mρ2) ω 2 = 7 10 mv2. Its dissipatin rate Ė under rlling frictin euals µ r mgv. Frm this, we can shw that the sphere will cme t stp at time t e = t r +7 v r /(5µ r g), alng the fllwing line trajectry trajectry f rlling: x = x r + v r (t t r ) 5 14 µ rgˆr(t t r ) 2, (7) x e at t e rlling Fig. 2. Trajectry f a sphere with unit radius, initial velcities v 0 = (4, 3) T and ω 0 = (14, 25) T, which yield initial sliding velcity s 0 = (29, 17) T. The velcities are drawn as arrws f lengths prprtinal t their magnitudes. Rlling starts at x r = ( , ) T at time t r = and ends at x e = ( , ) T at time t e = The cefficients f frictin are µ s = 0.3 and µ r = 0.5. t r < t t e, where ˆr = v r / v r is the rlling directin. The sphere will be resting at the lcatin x e = x(t e ) = x r + (7 v r /(10µ r g))v r. (8) Euatins (4) and (7) tgether describe the sphere trajectry. An example is shwn in Fig III. TRAJECTORIES TO DESTINATION It is nt difficult t shw that v 0 ω 0 = 0 results in a straight sliding trajectry, using (1) and (4). Under Assumptin 1, a straight trajectry wuld result in cllisin with sme bstacle. Therefre, the case v 0 ω 0 = 0 will nt be cnsidered frm nw n. Therem 1: When v 0 ω 0 0, the sphere slides alng ( a parabla generated frm first rtating y = 1 2 µs g/ v 0 ŝ 2) x 2 abut the rigin thrugh φ+π/2, where ŝ = ( cs φ sin φ), and then translating it by x = (v 0 ŝ/(µ s g)) ( v (v 0 ŝ)ŝ ). Due t limited space, the prf f the therem is mitted, as will thse f ther prpsitin and therems in this paper. As an example, the sphere trajectry in Fig. 2 is a parabla y = x 2 underging a rtatin thrugh the angle and then a translatin by ( , ) T. We wuld like t describe the set f velcities (v 0, ω 0 ) that will take the sphere t rest at a given lcatin x e =. By rewriting (1) and (5) int a matrix euatin, the pair (s 0, v r ) f initial sliding and rlling velcities has a ne-t-ne crrespndence t (v 0, ρẑ ω 0 ), the secnd element f which in turn has a ne-t-ne crrespndence t ω 0 under Assumptin 2 and v 0 ω 0 0. Thus, we need nly cnsider (s 0, v r ). 1 Frm nw n, the sliding segment f a trajectry will be drawn in blue, and the rlling segment will be drawn in red. v 0 ω

3 Next, plug v 0 = 2 7 s 0 + v r, btained frm (5), int (6) and the result int (8) t rewrite = c s ŝ + c rˆr, where c s and c r are nn-negative expressins in terms f s 0 and v r. Prpsitin 2: There exist v 0 and ω 0 with v 0 ω 0 0 that will take the sphere t a stp at if and nly if the cne spanned by ŝ and ˆr cntains in its interir. π β π π Fig. 3. C-space f trajectries T (α, β) that take the sphere frm the rigin t a pint n the psitive x- axis. The space cnsists f tw pen triangular regins bunded by the tw axes and the lines α β = π and β α = π. π α The unit vectrs ŝ and ˆr are respectively determined by their plar angles α and β, referred t as the sliding angle and rlling angle. We can recnstruct the initial velcities v 0 and ω 0 (and thus the trajectry) frm α and β when the cne spanned by ŝ and ˆv cntains in the interir. It can be shwn that the crrespndence thus defined between (α, β) and (v 0, ω 0 ) is ne-t-ne. Nevertheless, the frms f v 0 and ω 0 in terms f α and β are t cmplicated with sines, csines, and suare rts t be up fr an analysis. Dente by T (α, β) the trajectry frm t determined by α and β. Since the trajectry is a parablic segment fllwed by a tangent line segment, it is cnvex and thus either abve the x-axis r belw it except at and. Under Prpsitin 2, we infer that t reach all feasible (α, β) frm the fllwing set: D = (0, π) (α π, 0) ( π, 0) (0, α + π). (9) Pltted in Fig. 3, D is the cnfiguratin space (r C- space) f trajectries, referred t as the trajectry space. Therem 3: Fr any (α, β) D, T (α, β) and T ( α, β) are reflectins f each ther in the x-axis. et R and R be the tw rays riginating respectively frm and in the directin ŝ, and R and R be the tw rays riginating respectively frm and in the directin ˆr. Therem 4: Fr a fixed sliding angle α (0, π), the fllwing statements hld: i) Fr any tw rlling angles β 1, β 2 (α π, 0) with β 2 < β 1, except at and, T (α, β 1 ) lies inside the regin bunded by T (α, β 2 ) and. ii) Every pint p in the interir f the regin bunded by, R, and R lies n exactly ne trajectry T (α, β), β (α π, 0). Therem 5: Fr a fixed rlling angle β ( π, 0), the fllwing statements hld: i) Fr any tw sliding angles α 1, α 2 (0, β + π) with α 1 < α 2, except at and, T (α 1, β) lies inside the regin bunded by T (α 2, β) and. ii) Every pint p in the interir f the regin bunded by, R, and R lies n exactly ne trajectry T (α, β), α (0, β + π). Fig. 4 plts tw grups f trajectries respectively with the same sliding and rlling angles. The values T (α10, π/6) T (π/6, β 1) (a) (b) T (α1, π/6) T (π/6, β 10) Fig. 4. Tw grups f 10 trajectries each frm the rigin t (3, 0) with (a) the same sliding angle α = π/6 but different rlling angles β i = i (π α), 1 i 10; and (b) the same rlling angle 11 β = π/6 but different sliding angles α i = i (β+π), 1 i µ s = 0.3 and µ r = 0.15 f the cefficients f frictin will als be used in the remaining examples in the paper. IV. MOTION PANNING Accrding t Therem 3, we need nly cnsider hw t find a cllisin-free path frm t that ges abve the x-axis. Such a path is represented by a pint in the lwer triangular regin D in Fig. 3. This implies that we need nly cnsider bstacles that are abve r intersected by the x-axis. T find the trajectry belw the x-axis, simply reflect all bstacles in the x-axis, invke the algrithm and then reflect the fund path in the x- axis again. T simplify, bstacles are vertical line segments standing n the xy-plane at the pints b 1,...,b n and having length at least the radius ρ f the sphere. In the C-space fr path planning, the sphere shrinks t a pint, and every bstacle induces a disk C i at b i f radius ρ. The disk is a C-space bstacle referred t as a C-circle

4 A. Cnstraint Curves in the C-Space f Trajectries Every C-circle C i defines sme regin in the trajectry space D that includes all the trajectries which wuld cllide with the bstacle b i. Such a regin is bunded by curves n which every pint (α, β) yields a trajectry T (α, β) tangent t C i. The bstacles nt belw the x-axis are either A) abve the x-axis; r intersecting the x-axis B) between and, C) t the left f, r D) t the right f. 1) Type A Obstacle: T simplify ntatin, we refer t b i as b and C i as C. There exist tw grups f tangent trajectries: thse that bund C frm abve and thse that bund it frm belw. T characterize them, we nte that there are tw tangent lines frm t C at p + frm the abve and at p frm the belw. Similarly, the tw tangent lines frm t C has tangencies p + and p. Every upper-bunding tangent trajectry has a tangency pint between p + and p+ n C. As illustrated in Fig. 5(a), the maximum β value, dented β max, + is the plar angle f the vectr p +. Frm Fig. 3, α β max + π α max. Meanwhile, the α value f the trajectry must be bunded frm belw by the plar angle f p +, dented α+ min. As α increases frm α+ min (exclusive), the tangent trajectry T (α, β) will have β increase frm α + min π (exclusive). This is the case with T 1 in Fig. 5(a). Eventually, α will increase t a value α + mid where the sphere starts rlling right at p+, as in the case f T 2 = T (α + mid, β+ mid ) in the figure. As α cntinues t increase, the β value will stay at β + mid, which is the case f T 3. All (α, β) values, where T (α, β) is an upperbunding tangent trajectry frm the curve γ + in D pltted in Fig. 5(c). The interir f the regin bunded by the curve and the line α β = π includes all trajectries that will enclse the bstacle b. The secnd grup f tangent trajectries bund the bstacle frm belw, as illustrated in Fig. 5(b). All such tangent trajectries are represented by the curve γ in Fig. 5(c). The interir f the regin in D bunded by the curve and the α- and β-axes represent all cllisinfree trajectries belw the bstacle. The shaded regin in Fig. 5(c) includes all trajectries that will cllide with b. 2) Type B Obstacle: The prtin f the C-circle abve the x-axis intrduces a cnstraint curve in D like γ + in Fig. 5(c). Cllisin-free trajectries are represented by the pen regin enclsed by the curve and the line α β = π. 3) Type C and D Obstacles: Cnstraint curves induced by bstacles f these tw types are very similar with differences in their lcatins in D. We discuss a type D bstacle but will present a type C ne later. b3 b1 b2 b5 (a) b4 γ 5 γ 3 (b) γ 2 γ 1 γ 4 α β = π Fig. 6. (a) Wrk space fr a unit sphere mving frm t = (8, 0) with five bstacles: b 1 = (3, 3), b 2 = (4, 1/2), b 3 = ( 12, 3/10), b 4 = (19/2, 1/2), and b 5 = (4, 3), which are shwn as the centers f C-space disks f radius 1. (b) Cnfiguratin space including five cnstraint curves induced by b k, 1 k 4. The blue curves γ 1 and γ 3 allw pints α, β (i.e., T (α, β) nt clliding their defining bstacles) belw them. The red curves γ 2, γ 4, and γ 3 allw pints abve them. Als shwn in (b) is a sweep line. Fig. 5(d) shws the same destinatin with a type D bstacle at ( 19 2, 1 2 ). Here, p is the pint f tangency frm n the C-circle that is abve the x-axis. It is easy t see that any tangent trajectry T (α, β) must have β β max, the plar angle f p. As α increases frm 0 +, the pint f tangency mves frm infinitely clse t p x, where the circle intersects the x-axis, t p (e.g., the trajectry T 1 ). The trajectry T 2 = T (α min, β min ) starts rlling at p. Frm that pint, β stps changing as fr the trajectry T 3, while α increases t β max + π. The cnstraint curve is shwn in Fig. 5(e). Therem 6: A cnstraint curve starts and ends n the line α β = π with α increasing mntnically and β nn-decreasing. In the trajectry space D, let us refer left, right, belw, and abve t the directins f the vectrs (1, 1), ( 1, 1), ( 1, 1), and (1, 1), respectively. B. Cnfiguratin Space Fig. 6(a) shws an example with five bstacles. The bstacles b 1 and b 4 are exactly the same as in Fig. 5(a) and (d). The bstacle b 5 is belw the rigin and eliminated when planning trajectries abve the x-axis. The ther fur bstacles are f types A, B, C, and D, respectively. Fig. 6(b) plts the cnstraint curves γ j, 1 j 5. The first tw f which are induced by b 1, and fr 2 k 4, γ k+1 is induced by b k. The cnstraint curves partitin D int pen regins. Every pint (α, β) frm the same regin vilate the same set f cnstraints (i.e., T (α, β) cllides with the same set f bstacles). The planning prblem has a slutin if and nly if there exists at least ne pen regin that vilates n cnstraint at all

5 T 3 T 1 T 2 p 1 p + p + b T 3 b C T 2 p p 1 p β β + max + π α + min (a) β + max β max + π T 1 α min (b) β max α T 3 D (α mid, β mid ) (α max, β max) (α + mid, β+ mid ) (α + max, β + max) γ γ + (α min, β min ) (α + min, β+ min ) α β = π T 1 T 2 p p 1 p x b (α mid, β mid) (α max, β max) γ β max (α min, β min) (c) (d) (e) Fig. 5. Tangent trajectries and C-space bstacles (ρ = 1). (a) Three tangent trajectries enclsing an bstacle disk f type A at (3, 3). (b) Three tangent trajectries belw the disk. (c) In the C-space, the curve γ + represents the set f all tangent and enclsing trajectries and the curve γ represents the set f all tangent and nn-enclsing trajectries. Nte that β + mid = β+ max and β mid = β max. (d) Tangent trajectries t an bstacle f Type D at (19/2, 1/2). (e) Cnstraint curve induced by the bstacle in (d) in the dmain D. The shaded regins in (c) and (e) represent all trajectries respectively clliding with the tw bstacles. C. Plane Sweeping We emply a plane sweep techniue frm cmputatinal gemetry [3]. A line initially aligned with α β = π will sweep acrss D, handling events at stps during the sweep. At an intermediate psitin, the line is partitined int segments by the cnstraint curves. Its status is a list f cnstraint curves rdered alng the sweep line, starting with the β-axis and ending with the α-axis. Each curve has a superscript r + t indicate that pints t the left r right f the curve yield trajectries nt clliding with the curve s defining bstacle. When intersected by, a cnstraint curve appears twice in the status with different superscripts. The sweep line status is stred in a data structure such as a red-black tree [4] r a splay tree [10]. Dente by s i the ith segment frm left t right n the sweep line. An array c[ ] is used such that the entry c[i] cunts the bstacles clliding with the trajectry defined by any pint in the regin immediately abve s i. D. Event Handling An event happens when the sweep line passes by either an intersectin pint f tw cnstraint curves r a pint n sme cnstraint curve that is the clsest t the rigin. A dynamically updated event ueue Q rders events in the increasing distance f the event pint t the line α β = π. Suppse the segment s k n at its current psitin is bunded by the curve γ a l n the left and the curve γ b r n the right. After the intersectin, the segment s k will be bunded by γ b r n the left and γ a l n the right. One f the fllwing updates is perfrmed based n (a, b): (, +): c[k] c[k] 2. (+, ): c[k] c[k] + 2. (, ): n change t c[k]. (+, +): n change t c[k]

6 e k e k e k γ 2 b2 γ 1 (a) (b) (c) e k (d) γ i ek Fig. 7. Fur types f intersectin pints: (a) (, +), (b) (+, ), (c) (, ), and (d) (+, +), and (e) an extremum pint. A slid arrw shws the sweeping directin, while a hllw arrw marks the side where pints define trajectries nt in cllisin with the bstacle assciated with the curve. See Fig. 7 (a) (d). Since γ a l and γ b r have switched their rder in the sweep line status, their neighbring curves in the status change. We need t check whether will intersect its new preceding curve and whether will intersect its new succeeding curve in the sweep line status. Whenever an intersectin exists, add its cmputed psitin (alng with the tw curves) t the event ueue Q. A segment s k n bunded by the same curve γ i may shrink t ne extreme pint n it. When this happens, we lcate the tw adjacent segments s k 1 and s k+1, and merge them tgether with s k int ne bunded by the left bunding curve f s k 1 and the right bunding curve f s k+1. N update n c[k 1] needs t be dne. Check if these tw curves will intersect. If s, add the intersectin t the ueue Q. If c[k] = 0, the regin immediately abve s k will be a cllisin-free ne. The entire regin can be traced ut as the sweep cntinues using a dubly-cnnected edge list [3, pp ]. E. Initializatin At the beginning, the sweep line cincides with the line α β = π. Fr each cnstraint curve, we cmpute its extremum pint and insert it int the event ueue Q. Every cnstraint curve yields tw entries in the sweep line status respectively assciated with its tw endpints. In the rdering, cmplicatins arise with multiple bstacles f type C r multiple bstacles f type D. The curves defined by multiple bstacles f type D all start at (0, π), as in Fig. 8 which shws an instance with tw bstacles defining the cnstraint curves γ 1 and γ 2. These curves can be rdered by their intersectins with a hrizntal line β = π +ǫ fr small enugh ǫ > (e) b1 Fig. 8. Tw cnstraint curves γ 1 and γ 2, due t bstacles b 1 and b 2, respectively, start at the lwest pint (0, π) f D. γ 5 e 6 e 5 γ + 3 e 9 γ 2 e 4 e 8 γ + γ e 7 e 1 γ + 4 e γ 3 e γ γ γ 4 1 Fig. 9. Shaded regins in the C-space represent all cllisin-free paths fr the envirnment in Fig. 6(a). 0. Apply Therem 5, we infer that the α value increase with the x-crdinates f the bstacles. Cnseuently, the curve γ i lies t the right f the curve γ j in D if b i has a larger x-crdinate than b j des. In Fig. 8, γ 2 lies t the right f γ 1. All the curves f type D are thus rdered. The cnstraint curves induced by type C bstacles all end at the pint (π, 0). Similarly, these curves appear in the status in the decreasing rder f the x-crdinates f the bstacles. Up t 2n, where n is the number f bstacles, cnstraint curves with superscripts, tgether with the α- and β-axes, are rdered. The array c[ ] can be initialized with a left-t-right scan alng the line α β = π as fllws. We let c[0] t be the number f bstacles f type B. These bstacles are the nly nes that define curves t exclude the pint (0, π). If the kth scanned curve has superscript, set c[k] c[k 1]+1. If the curve has superscript +, set c[k] c[k 1] 1. Table I details sweeping f the C-space in Fig. 9. Intervals n the sweep line are represented by their left bunding curves. The initial sweep line status is described in the secnd rw. Frm c[6] = 0 we see that the regin right abve the 6-th segment is cllisin free. Every rw starting at the third ne represents the updated status immediately after an event. T illustrate, the event e 1 is due t an intersectin between γ

7 i s i, c[i] y-axis, 1 γ 5, 2 γ+ 3, 1 γ 2, 2 γ+ 5,1 γ+ 1,0 γ 1, 1 γ 4,2 γ+ 2,1 γ 3,2 γ+ 4, 1 e 1 γ + 4,0 γ 3, 1 e 2 γ + 2,0 γ 4, 1 e 3 γ + 2,0 e 4 γ + 5,0 γ 2,1 e 5 γ + 3,0 γ 5,1 e 6 γ + 3, 0 e 7 γ 2,1 e 8 γ + 3,0 e 9 y-axis, 1 TABE I EXECUTION OVER THE C-SPACE IN FIG. 9. BANK ENTRIES ARE THE SAME AS IN THE ROWS IMMEDIATEY ABOVE. and γ + 4, which bund the tenth segment. After the intersectin, the tw curves switching their psitins. The bstacles b 3 and b 1 will nt cllide with paths in the regin bunded by γ + 2 and γ 4. The event e 3 happens at an extremum pint n the curve γ 4. In the previus rw labeled by e 2, γ 4 and γ + 4 are next t each ther. At e 3, the tw intervals leftbunded by them alng with the preceding interval leftbunded by γ + 2 are merged int ne labeled by γ+ 2. The regin immediately abve the segment is cllisin free. When the sweep finishes, the set f cllisin-free paths fr the sphere is the unin f the tw shaded pen regins shwn in Fig. 9. The plane sweep algrithm des nt explicitly cnstruct the cnstraint curves which d nt have algebraic frms. Efficient numerical subrutines have been designed t a) lcate a pint n a cnstraint curve given an α r β value; b) cmpute the starting, ending, and cnnectin pints n a cnstraint curve; c) find the extremum pint n a cnstraint curve; and d) intersect tw cnstraint curves. These subrutines emply bisectin r gld sectin search [9], by expliting the mntnicity f trajectries as stated in Therems 4 and 5. Under a specified accuracy, a call f any numerical subrutine abve takes cnstant time. Then, the running time f the sweep line algrithm fr n bstacles is easily analyzed t be O((n+n e )lg n), where n e is the number f events (which is O(n 2 ) in the wrst case). F. Discussin T have a gd clearance frm the bstacles, we may pick a pint (α, β) that lies clse t the center f a regin f cllisin-free trajectries in D. Cmpute the velcities (v 0, ω 0 ) frm α and β. Thugh presented fr vertical line bstacles, the algrithm trivially extends t vertical cylindrical bstacles. Extensin f the algrithm t plyhedral r ther generally shaped bstacles needs further investigatin. The presented wrk intrduces a new way f path planning which takes advantage f sliding under frictin. T apply this techniue t navigatin by spherical rbts, tw issues await investigatin. First, nn-unifrm mass distributin f such a rbt cmplicates the angular inertial matrix t change dynamic euatins significantly. Secnd, actuatin t generate a specified initial mtin can be uite challenging. REFERENCES [1] S. Bhattacharya and S. K. Agrawal. Spherical rlling rbt: a design and mtin planning studies, IEEE Trans. n Rbt. Autmat., vl. 16, pp , [2] A. Bicchi, A. Balluchi, D. Prattichizz, and A. Grelli. Intrducing the spherical: An experimental testbed fr research and teaching in nn-hlnmy, in Prc. IEEE Int. Cnf. Rbt. Autm., pp , [3] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkpf. Cmputatinal Gemetry: Algrithms and Applicatins, 3rd editin, Springer-Verlag, [4] T. H. Crmen, C. E. eisersn, R.. Rivest, and C. Stein. Intrductin t Algrithms, 3rd editin, The MIT Press, [5] Z. i and J. Canny. Mtin f tw rigid bdies with rlling cnstraint, IEEE Trans. n Rbt. Autmat., vl. 6, pp. 72, [6] W. C. Marlw. The Physics f Pcket Billiards, Marlw Advanced Systems Technlgies, [7] R. Mukherjee, M. A. Minr, and J. T. Pukrushpan. Mtin planning fr a spherical mbile rbt: Revising the classcial ballplate prblem, Trans. f ASME, vl. 124, pp , [8] A. Nakashima, K. Nagase, and Y. Hayakawa. Cntrl f a sphere rlling n a plane with cnstrained rlling mtin, in Prc. IEEE Cnf. Decisin and Cntrl, 2005, pp [9] W. H. Press, S. A. Teuklsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C++: The Art f Scientific Cmputing, 2nd editin, Cambridge University Press, [10] D. D. Sleatr and R. E. Tarjan. Self-adjusting binary search trees, J. ACM, vl. 32, pp , [11] M. Svinin and S. Hse. On mtin planning fr ballplate systems with limited cntact area, in Prc. IEEE Int. Cnf. Rbt. Autm., 2007, pp