1 Models of vehicles with differential constraints. 2 Traveling salesperson problems. 3 The heavy load case. 4 The light load case

Transcription

1 Dynamic Vehice Routing for Robotic Networks Lecture #7: Vehice Modes Francesco Buo 1 Emiio Frazzoi 2 Marco Pavone 2 Ketan Sava 2 Stephen L. Smith 2 Outine of the ecture 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 1 CCDC University of Caifornia, Santa Barbara buo@engineering.ucsb.edu 2 LIDS and CSAIL Massachusetts Institute of Technoogy {frazzoi,pavone,ksava,ssmith}@mit.edu Workshop at the 2010 American Contro Conference Batimore, Maryand, USA, June 29, 2010, 8:30am to 5:00pm 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 1 / 36 Vehice routing with differentia constraints FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 2 / 36 Modes of vehices with differentia constraints What happens if the vehices are subject to non-integrabe differentia constraints on their motion? Minimum turn radius, constant speed UAVs, Dubins cars Minimum turn radius, abe to reverse Reeds-Shepps cars Differentia drive robots e.g., tanks. Bounded acceeration vehices e.g., heicopters, spacecraft. Fundamentay different probems, combining combinatoria task specifications with differentia geometry and optima contro. Decompose the probem, study the asymptotic cases: Heavy oad: Traveing saesperson probems. Light oad: optima oitering stations. Dubins vehice ẋ = cos θ ẏ = sin θ θ = ω ω 1/ρ Differentia drive ẋ = 1 2 ω + ω r cos θ ẏ = 1 2 ω + ω r sin θ θ = 1 ρ ω r ω ω 1; ω r 1 Reeds-Shepp car ẋ = v cos θ ẏ = v sin θ θ = ω v { 1, 1} ω 1/ρ Doube integrator ẍ = u ẋ 1 u 1 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 3 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 4 / 36

2 DTRP formuation DTRP formuation Probem setup m identica vehices in Q Spatio-tempora Poisson process: rate λ and uniform spatia density On-site service time s = 0 Q Probem setup m identica vehices in Q Spatio-tempora Poisson process: rate λ and uniform spatia density On-site service time s = 0 Q Objective Contro poicy π = {task assignment, scheduing, oitering} T π := im sup i E[wait time of task i]; T = inf π T π Design π for which T π is equa to or within a constant factor of T Objective Contro poicy π = {task assignment, scheduing, oitering} T π := im sup i E[wait time of task i]; T = inf π T π Design π for which T π is equa to or within a constant factor of T FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 5 / 36 Stabiizabiity FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 5 / 36 Stabiizabiity λ }{{} task generation rate n: # outstanding tasks n m TSPengthn }{{} task service rate = task growth rate λ }{{} task generation rate n: # outstanding tasks n m TSPengthn }{{} task service rate = task growth rate TSPengthn stricty sub-inear = stabiity λ, m TSPengthn stricty sub-inear = stabiity λ, m Eucidean TSPengthn = Θn 1/2 Beardwood et. a. 59 Eucidean TSPengthn = Θn 1/2 Beardwood et. a. 59 Eucidean TSP based path panning heuristic = On Eucidean TSP based path panning heuristic = On Traveing saesperson probems for differentia vehices. Traveing saesperson probems for differentia vehices. FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 6 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 6 / 36

3 Outine of the ecture Traveing Saesperson Probem 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad Probem Statement Find the shortest cosed curve feasibe for the vehice through a given finite set of points in the pane NP-hardness a consequence of the NP-hardness of the Eucidean TSP. Does the cost of this TSP increase SUBLINEARLY with n? Is there a poynomia-time agorithm that returns a tour of ength on?? What is the quaity of the soution? FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 7 / 36 Traveing Saesperson Probem FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 8 / 36 Literature review Probem Statement Find the shortest cosed curve feasibe for the vehice through a given finite set of points in the pane NP-hardness a consequence of the NP-hardness of the Eucidean TSP. Does the cost of this TSP increase SUBLINEARLY with n? Is there a poynomia-time agorithm that returns a tour of ength on?? 1 K. Sava, E. Frazzoi, and F. Buo. Traveing Saesperson Probems for the Dubins vehice. IEEE Transactions on Automatic Contro, 536: , K. Sava, F. Buo, and E. Frazzoi. Traveing Saesperson Probems for a doube integrator. IEEE Transactions on Automatic Contro, 544: , J. J. Enright, K. Sava, E. Frazzoi, and F. Buo. Stochastic and dynamic routing probems for mutipe UAVs. AIAA Journa of Guidance, Contro, and Dynamics, 344: , J. J. Enright and E. Frazzoi. The stochastic Traveing Saesman Probem for the Reeds-Shepp car and the differentia drive robot. In IEEE Conf. on Decision and Contro, pages , San Diego, CA, December K. Sava and E. Frazzoi. On endogenous reconfiguration for mobie robotic networks. In Workshop on Agorithmic Foundations of Robotics, Guanajuato, Mexico, December M. Pavone, K. Sava, and E. Frazzoi. Sharing the oad. IEEE Robotics and Automation Magazine, 162:52 61, F. Buo, E. Frazzoi, M. Pavone, K. Sava, and S. L. Smith. Dynamic vehice routing for robotic systems. Proceedings of the IEEE, May Submitted 8 S. Rathinam, R. Sengupta, and S. Darbha. A resource aocation agorithm for muti-vehice systems with non hoonomic constraints. IEEE Transactions on Automation Sciences and Engineering, 41:98 104, J. Le Ny, E. Feron, and E. Frazzoi. On the curvature-constrained traveing saesman probem. IEEE Transactions on Automatic Contro, to appear 10 S. Itani. Dynamic Systems and Subadditive Functionas. PhD thesis, Massachusetts Institute of Technoogy, 2009 What is the quaity of the soution? FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 8 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 9 / 36

4 Stochastic TSP: A nearest-neighbor ower bound Outine of the cacuations Cacuate an upper bound on expected distance from an arbitrary vehice configuration to cosest point, δ Cacuate an upper bound on the area of the set reachabe with a path of ength δ, R δ. Prδ δ max{0, 1 n R δ /} Expected ength of the tour cannot be ess than n times E[δ ] Stochastic TSP: A nearest-neighbor ower bound Outine of the cacuations Cacuate an upper bound on expected distance from an arbitrary vehice configuration to cosest point, δ Cacuate an upper bound on the area of the set reachabe with a path of ength δ, R δ. Prδ δ max{0, 1 n R δ /} Expected ength of the tour cannot be ess than n times E[δ ] Exampe: Dubins vehice!"!"$!" Exampe: Dubins vehice!"!"$!" R δ = δ3 3ρ E[δ ] = 3 3ρ 1/3. 4 n E[TSPn] im n 3 n 2/3 4 3ρ1/3.!"#!"'!!!"'!!"#!!"!!"$!!"!!"#!"$!"%!"& ' R δ = δ3 3ρ E[δ ] = 3 3ρ 1/3. 4 n E[TSPn] im n 3 n 2/3 4 3ρ1/3.!"#!"'!!!"'!!"#!!"!!"$!!"!!"#!"$!"%!"& ' FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 10 / 36 Towards an upper bound: tiing based agorithms The way the ETSP tours are constructed reies on the scaing properties of tours: the ength of the tour scaes as the coordinates of the points. FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 10 / 36 Towards an upper bound: tiing based agorithms The way the ETSP tours are constructed reies on the scaing properties of tours: the ength of the tour scaes as the coordinates of the points. No such scaing exists for the TSP for vehices with differentia constraints, e.g., the bound on the curvature for the Dubins vehice does not scae with the coordinates of the points! Any tiing-based agorithm must account for a preferentia direction, e.g., by penaizing turning for Dubins vehices FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 11 / 36 No such scaing exists for the TSP for vehices with differentia constraints, e.g., the bound on the curvature for the Dubins vehice does not scae with the coordinates of the points! Any tiing-based agorithm must account for a preferentia direction, e.g., by penaizing turning for Dubins vehices FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 11 / 36

5 Bead construction Bead construction ρ ρ B B Bead properties Lengthp, q, p + + o 2 for a q B Width: w = 2 8ρ + o3 The beads tie the pane Usefu for Dubins vehice, Reeds-Shepp car and doube integrator Bead properties Lengthp, q, p + + o 2 for a q B Width: w = 2 8ρ + o3 The beads tie the pane Usefu for Dubins vehice, Reeds-Shepp car and doube integrator Diamond-ike ce for differentia drive p B p + Diamond-ike ce for differentia drive p B p + FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 12 / 36 The singe-sweep tiing agorithm FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 12 / 36 The singe-sweep tiing agorithm Ketan Sava LIDS, MIT Short tite Dat B B + B B + Tie the region with beads Sweep the bead rows, whie servicing a the targets in every bead as foows: Service every task q in B using the p q p protoco Ketan Sava LIDS, MIT Short tite Date 2 / 2 Tie the region with beads Sweep the bead rows, whie servicing a the targets in every bead as foows: Service every task q in B using the p q p protoco Ketan Sava LIDS, MIT Short tite Dat Move from p to p + Service every task q in B + using the p + q p + protoco Move from p to p + Service every task q in B + using the p + q p + protoco FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 13 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 13 / 36

6 Anaysis of the singe-sweep tiing agorithm Path ength cacuations TSPn = bead row ength + move to next bead row # bead rows + move to service each task # tasks + tour cosure ength Anaysis of the singe-sweep tiing agorithm Path ength cacuations TSPn = bead row ength + move to next bead row # bead rows + move to service each task # tasks + tour cosure ength B B + B B + For a Reeds-Shepp car, as 0: TSPn + /2 w/2 + n ρπ Pick = 32ρ n 16ρ + 8ρ 2 1/3 i.e., B = 2 n + n ρπ w 2 8ρ = TSPn = On 2/3. For a Reeds-Shepp car, as 0: TSPn + /2 w/2 + n ρπ Pick = 32ρ n 16ρ + 8ρ 2 1/3 i.e., B = 2 n + n ρπ w 2 8ρ = TSPn = On 2/3. FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 14 / 36 Anaysis of the singe-sweep tiing agorithm Path ength cacuations TSPn = bead row ength + move to next bead row # bead rows + move to service each task # tasks + tour cosure ength FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 14 / 36 Anaysis of the singe-sweep tiing agorithm Path ength cacuations TSPn = bead row ength + move to next bead row # bead rows + move to service each task # tasks + tour cosure ength B B + B B + For a Reeds-Shepp car, as 0: TSPn + /2 w/2 + n ρπ Pick = 32ρ n 16ρ + 8ρ 2 1/3 i.e., B = 2 n + n ρπ w 2 8ρ = TSPn = On 2/3. For a Dubins vehice, as 0: TSPn + w 2 + κ w/2 + + κn κ 16ρ κ 2 + n + κn κ The κn term grows ineary in n for a = TSPn = On FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 14 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 15 / 36

7 Anaysis of the singe-sweep tiing agorithm The recursive sweep tiing agorithm Path ength cacuations TSPn = bead row ength + move to next bead row # bead rows + move to service each task # tasks + tour cosure ength B B + B Tie Q with beads such that: = 1 2n i.e., n 1/3 Sweep the bead rows, visiting one target per non-empty bead. Iterate, using at the i-th phase a meta-bead composed of 2 i 1 beads. After og n phases, visit the outstanding targets in any arbitrary order, e.g., with a greedy strategy. For a Dubins vehice, as 0: TSPn + w 2 + κ w/2 + + κn κ 16ρ κ 2 + n + κn κ The κn term grows ineary in n for a = TSPn = On Phase 1 Phase 2 Phase 3 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 15 / 36 The recursive sweep tiing agorithm FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 16 / 36 The recursive sweep tiing agorithm B Tie Q with beads such that: = 1 2n i.e., n 1/3 Sweep the bead rows, visiting one target per non-empty bead. Iterate, using at the i-th phase a meta-bead composed of 2 i 1 beads. After og n phases, visit the outstanding targets in any arbitrary order, e.g., with a greedy strategy. B Tie Q with beads such that: = 1 2n i.e., n 1/3 Sweep the bead rows, visiting one target per non-empty bead. Iterate, using at the i-th phase a meta-bead composed of 2 i 1 beads. After og n phases, visit the outstanding targets in any arbitrary order, e.g., with a greedy strategy. Phase 1 Phase 2 Phase 3 Phase 1 Phase 2 Phase 3 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 16 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 16 / 36

8 Anaysis of the recursive agorithm Anaysis of the recursive agorithm Theorem: For a Dubins vehice, with probabiity one, TSPn im sup n n 2/ ρ π ρ Theorem: For a Dubins vehice, with probabiity one, TSPn im sup n n 2/ ρ π ρ Outine of the proof Prim n # tasks remaining after phase i > 24 og n = 0 Path ength cacuations: Phase 1 path ength O 1 = O n 2/3 n 1/3 2 Subsequent phase path engths are decreasing geometric series; path ength for a i phases is O n 2/3 Path ength by greedy heuristic is Oog n Outine of the proof Prim n # tasks remaining after phase i > 24 og n = 0 Path ength cacuations: Phase 1 path ength O 1 = O n 2/3 n 1/3 2 Subsequent phase path engths are decreasing geometric series; path ength for a i phases is O n 2/3 Path ength by greedy heuristic is Oog n FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 17 / 36 Summary of TSPs FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 17 / 36 Summary of TSPs Lower bound: E [TSPn] Ωn 2/3 Upper bound: E [TSPn] On 2/3 TSPn is of order n 2/3 ; constant factor approximation agorithms Computationa compexity of the agorithms is of order n Lower bound: E [TSPn] Ωn 2/3 Upper bound: E [TSPn] On 2/3 TSPn is of order n 2/3 ; constant factor approximation agorithms Computationa compexity of the agorithms is of order n Stabiizabiity of the DTRP n }{{} λ m TSPn task generation rate }{{} n: # outstanding tasks task service rate = task growth rate Stabiizabiity of the DTRP n }{{} λ m TSPn task generation rate }{{} n: # outstanding tasks task service rate = task growth rate E[TSPn] Θn 2/3 = trivia receding horizon TSP-based poicies are stabe for the DTRP for a λ and m E[TSPn] Θn 2/3 = trivia receding horizon TSP-based poicies are stabe for the DTRP for a λ and m FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 18 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 18 / 36

9 Outine of the ecture 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad The heavy oad case: nearest neighbor ower bound Outine of the cacuations Let n π be the number of outstanding tasks at steady-state under stabe poicy π Cacuate an upper bound on expected distance from an arbitrary vehice configuration to cosest among n π points, δ n π At steady-state: λ m = 1 E[δ n π ] Litte s formua: λt π = n π Exampe: Dubins vehice 1/3 E[δ n π ] = 3 3ρ 4 n π Steady state+ Litte s formua: λ m = 4 3 im inf λ m + T m 3 λ ρ 1/3 λtπ 3ρ FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 19 / 36 The heavy oad case: nearest neighbor ower bound Outine of the cacuations Let n π be the number of outstanding tasks at steady-state under stabe poicy π Cacuate an upper bound on expected distance from an arbitrary vehice configuration to cosest among n π points, δ n π At steady-state: λ m = 1 E[δ n π ] Litte s formua: λt π = n π Exampe: Dubins vehice 1/3 E[δ n π ] = 3 3ρ 4 n π Steady state+ Litte s formua: λ m = 4 3 im inf λ m + T m 3 λ ρ 1/3 λtπ 3ρ FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 20 / 36 The mutipe sweep tiing agorithm The singe vehice version 1 Tie Q with beads of ength = c/λ 2 Update outstanding task ist 3 Execute singe sweep tiing agorithm 4 Goto 2. The muti-vehice version Divide Q into m equa strips Assign one vehice to every strip Each vehice executes the mutipe sweep agorithm in its own strip FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 20 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 21 / 36

10 The mutipe sweep tiing agorithm The singe vehice version 1 Tie Q with beads of ength = c/λ 2 Update outstanding task ist 3 Execute singe sweep tiing agorithm 4 Goto 2. The muti-vehice version Divide Q into m equa strips Assign one vehice to every strip Each vehice executes the mutipe sweep agorithm in its own strip Anaysis of the mutipe sweep agorithm Genera protoco Each bead can be treated as a separate queue, with Poisson arriva process with intensity λ B = λ B The vehice visits each bead with at a rate no smaer than µ B singe sweep path ength 1 The system time is no greater than the system time for the corresponding M/D/1 queue: T 1 µ B λ B 2 µ B λ B Optimize over Exampe: Dubins vehice λ B = 3 λ 16ρ ; µ B im sup λ m + T m 3 λ 2 2 m 16ρ 71ρ π ρ π ρ FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 21 / 36 Anaysis of the mutipe sweep agorithm Genera protoco Each bead can be treated as a separate queue, with Poisson arriva process with intensity λ B = λ B The vehice visits each bead with at a rate no smaer than µ B singe sweep path ength 1 The system time is no greater than the system time for the corresponding M/D/1 queue: T 1 µ B λ B 2 µ B λ B Optimize over Exampe: Dubins vehice λ B = 3 λ 16ρ ; µ B im sup λ m + T m 3 λ 2 2 m 16ρ 71ρ π ρ π ρ FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 22 / 36 Outine of the ecture 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 22 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 23 / 36

11 The ight oad case The ight oad case The target generation rate is very sma: λ/m 0 + In such case: Amost surey a vehices wi have enough time to return to some oitering station between task competion/generation times The probem is reduced to the choice of the oitering stations that minimizes the system time The target generation rate is very sma: λ/m 0 + In such case: Amost surey a vehices wi have enough time to return to some oitering station between task competion/generation times The probem is reduced to the choice of the oitering stations that minimizes the system time Introducing differentia constraints Nove chaenges: Vehices possiby cannot stop e.g., Dubins vehice, Reeds-Shepp car Strategies are more compex than defining a oitering point How many of the resuts from the Eucidean case carry over to this case? Introducing differentia constraints Nove chaenges: Vehices possiby cannot stop e.g., Dubins vehice, Reeds-Shepp car Strategies are more compex than defining a oitering point How many of the resuts from the Eucidean case carry over to this case? FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 24 / 36 A simpe ower bound FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 24 / 36 The Median Circing MC Poicy The ength of shortest feasibe path from a vehice positioned at p R 2 to an arbitrary point q Q is ower bounded by q p A simpe ower bound on T is obtained by reaxing differentia constraints T H mq Assign virtua generators to each agent. A agents do the foowing, in parae possiby asynchronousy: Update the generator position according to a gradient descent aw. Service targets in own region, returning to a oitering circe of radius 2.91ρ centered on their generator position when done We have im T λ/m 0 + MC HmQ ρ HmQ = Θ 1 m Furthermore, T MC im Hm +,λ/m 0 + T = 1. Q FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 25 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 26 / 36

12 The Median Circing MC Poicy IIustration of the MC poicy Assign virtua generators to each agent. A agents do the foowing, in parae possiby asynchronousy: Update the generator position according to a gradient descent aw. Service targets in own region, returning to a oitering circe of radius 2.91ρ centered on their generator position when done We have im T λ/m 0 + MC HmQ ρ Furthermore, T MC im Hm +,λ/m 0 + T = 1. Q FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 26 / 36 Tighter ower bound using differentia constraints Genera protoco Consider a frozen moment in time Consider the modified Voronoi diagram of the vehices. Reaxation: approximate vehice Voronoi region by their reachabe sets Optimize over the vehice configurations Exampe: Dubins vehice FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 27 / 36 Tighter ower bound using differentia constraints Genera protoco Consider a frozen moment in time Consider the modified Voronoi diagram of the vehices. Reaxation: approximate vehice Voronoi region by their reachabe sets Optimize over the vehice configurations Exampe: Dubins vehice!"!"!"$!"$!"!"!"#!"#!"'!"'!!!!"'!!"'!!"#!!"#!!"!!"!!"$!!"$!!"!!"#!"$!"%!"& '!!"!!"#!"$!"%!"& ' For m m crit, T k 1,ρ m 1/3 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 28 / 36 For m m crit, T k 1,ρ m 1/3 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 28 / 36

13 The Strip Loitering SL poicy { } Divide the environment Q into strips of width min k2 Q,ρ, 2ρ m 2/3 Design a cosed oitering path that bisects the strips. A vehices move aong this path, equay spaced, with dynamic regions of responsibiity. Each vehice services targets in own region, returning to the nomina position on the oitering path. The Strip Loitering SL poicy { } Divide the environment Q into strips of width min k2 Q,ρ, 2ρ m 2/3 Design a cosed oitering path that bisects the strips. A vehices move aong this path, equay spaced, with dynamic regions of responsibiity. Each vehice services targets in own region, returning to the nomina position on the oitering path. ρ ρ Q d 1 point of departure d 2 target δ Q d 1 point of departure d 2 target δ im m + T SL m 1/3 k 3 Q, ρ, and im m + T SL T k 4Q, ρ. im m + T SL m 1/3 k 3 Q, ρ, and im m + T SL T k 4Q, ρ. FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 29 / 36 Iustration of the SL poicy FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 29 / 36 Outine of the ecture 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 30 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 31 / 36

14 Phase transition in the ight oad Phase transition in the ight oad We have two poicies: Median Circing MC, and Strip Loitering SL. Which is better? We have two poicies: Median Circing MC, and Strip Loitering SL. Which is better? Define the non-hoonomic density d ρ = ρ2 m. MC is optima when d ρ 0, SL is within a constant factor of the optima as d ρ +. Define the non-hoonomic density d ρ = ρ2 m. MC is optima when d ρ 0, SL is within a constant factor of the optima as d ρ +. phase transition: the optima organization changes from territoria MC to gregarious SL depending on the non-hoonomic density of the agents. phase transition: the optima organization changes from territoria MC to gregarious SL depending on the non-hoonomic density of the agents. FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 32 / 36 Estimate of the critica density L w FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 32 / 36 Estimate of the critica density L w Ignoring boundary conditions e.g., consider the unbounded pane, we can compare the coverage cost for the two poicies anayticay: T SL < T MC d ρ > i.e., transition occurs when the area of the dominance region is about 4-5 times the area of the minimum turning radius circe. Ignoring boundary conditions e.g., consider the unbounded pane, we can compare the coverage cost for the two poicies anayticay: T SL < T MC d ρ > i.e., transition occurs when the area of the dominance region is about 4-5 times the area of the minimum turning radius circe Simuation resuts yied dρ crit within a factor 1.3 of the anaytica resut. TURNING RADIUS SQUARED MC poicy SL poicy Simuation resuts yied dρ crit within a factor 1.3 of the anaytica resut. TURNING RADIUS SQUARED MC poicy SL poicy AREA PER VEHICLE AREA PER VEHICLE FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 33 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 33 / 36

15 Dynamic Vehice Routing Summary Lecture outine Eucidean vehice Dubins vehice, Reeds-Shepp car Doube integrator, Differentia drive E[TSP Length] Θn 1 2 Θn 2 3 n T Θ λ Θ λ2 m 2 m 3 λ m T Θm 1 2 Θm 1 2 λ m 0, m 0 T Θm 1 2 Θm 1 3 λ m 0, m 1 Modes of vehices with differentia constraints 2 Traveing saesperson probems 3 The heavy oad case 4 The ight oad case 5 Phase transition in the ight oad FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 34 / 36 FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 35 / 36 Workshop Structure and Schedue 8:00-8:30am Coffee Break 8:30-9:00am Lecture #1: Intro to dynamic vehice routing 9:05-9:50am Lecture #2: Preims: graphs, TSPs and queues 9:55-10:40am Lecture #3: The singe-vehice DVR probem 10:40-11:00am Break 11:00-11:45pm Lecture #4: The muti-vehice DVR probem 11:45-1:10pm Lunch Break 1:10-2:10pm Lecture #5: Extensions to vehice networks 2:15-3:00pm Lecture #6: Extensions to different demand modes 3:00-3:20pm Coffee Break 3:20-4:20pm Lecture #7: Extensions to different vehice modes 4:25-4:40pm Lecture #8: Extensions to different task modes 4:45-5:00pm Fina open-foor discussion FB, EF, MP, KS, SLS UCSB, MIT Dynamic Vehice Routing Lecture 7/8 Batimore, ACC 36 / 36