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Short-Term Conflict Resolution for
Unmanned Aircraft Traffic Management
Hao Yi Ong and Mykel J. Kochenderfer
Stanford Univers...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Unmanned aircraft traffic management
automating conflict avoidance is critical to integrating small
unmanned aerial systems (...
Conflict avoidance
given potential conflicts between drones, find the best set of advisories
focus on horizontal or co-altitu...
Complications
multiagent problem
– system must coordinate between many aircraft
– large search space for solution
uncertai...
Goals
robust and efficient method
tractable approach to complex stochastic problem
– account for uncertainty in large proble...
Previous approaches
mathematical programming (MIP, SCP) [SVFH05, ASD12]
– can work well for simple vehicle networks
distri...
Overview
idea: decomposition + coordination
multi-agent pairwise joint policy
decompose
pairwise solutions
+
coordination
...
Conflict Resolution as a UTM service
UTM server client side
resolution server
filtered sector
status updates
advisories
adv...
Conflict Resolution as a Standalone service
derivative UTM client service
– subscribes to UTM server to track aircraft and ...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Markov decision process (MDP)
defined by the tuple (S, A, T, R)
S and A are the sets of all possible states and actions, re...
Value iteration
agent
π⋆(s)"
action
a"
state
s"
want to find the optimal policy π (s)
– gives action that maximizes the uti...
Multiagent MDP
extension of MDP to cooperative multiagent setting
similar to case where centralized planner has access to ...
Short-term conflict resolution
horizontal conflict resolution between n aircraft
– conflict defined as loss of minimum separat...
Resolution advisories
at each time step T = 5 s, system issues a joint advisory
– joint advisory φ chosen out of a finite s...
Dynamics
ith aircraft state si = (xi, yi, ψi, vi, pi)
– latitude and longitude xi and yi
– heading ψi
– groundspeed vi
– r...
Pilot model
advisory response determined stochastically by Bernoulli process
5 s average response delay to first corrective...
Reward function
balance competing objectives
maximize safety
loss of
separation
cost
separation
closeness
penalty
minimize...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Curse of dimensionality
in an MMDP, S and A are all possible joint states and actions
problem: S and A scale exponentially...
Decompose and coordinate
multi-agent pairwise joint policy
decompose
pairwise solutions
+
coordination
=
ac1: left
ac2: ri...
MMDP decomposition
multi-agent pairwise joint policy
decompose
ac1: left
ac2: right
ac3: straight
pairwise solutions
+
coo...
Pairwise conflict resolution
decompose full MMDP into O n2
pairwise encounters
action set, pilot response model, and reward...
Discretization
in general, no analytical solution to problem with continuous state
space
approximate value function ˆQ ove...
Approximating environment noise
sigma-point sampling
speed bank
weights
sigma-point
sample
Approach 26
Implementation
discretize continuous state space into 9.6 million states
– expensive but one-off computation
ˆQ converges i...
Pairwise policy visualization
passing intrusion (240◦
bearing)
COC
−2,000−1,000 0 1,000 2,000
−2,000
−1,000
0
1,000
2,000
...
Example advisory execution
passing intrusion at (1000 m, 0 m)
ownship: 10° left bank intruder: 10° left bank
Approach 29
Accounting for pilot response
pilots responsive (top) and non-responsive (bottom)
COC
−2,000−1,000 0 1,000 2,000
−2,000
−1...
Accounting for pilot response
ownship responsive (top) and intruder responsive (bottom)
COC
−2,000−1,000 0 1,000 2,000
−2,...
Multithreat coordination
multi-agent pairwise joint policy
decompose
pairwise solutions
+
coordination
=
ac1: left
ac2: ri...
Utility fusion
idea: combine pairwise utilities Qij to form proxy for global utility Q
Qij operates on the pairwise MDP st...
Search heuristic: Alternating maximization
maximize global utility by varying one action and fixing the rest
e.g., fix drone...
Distributed variant I: Parallel search
greedy local maximization
0 L 0
0 0 L
R 0 0
broadcast action
R L L
R L L
R L L
repe...
Distributed variant II: Consensus search
parallel alt. maximization
R R L
R L L
R L R
broadcast joint action
R R L
R L L
R...
Consensus search message
parallel search could incur large communication cost
messages reduce communication between comput...
Joint policy visualization
triple aircraft encounter max-min (top) and max-sum policies (bottom)
COC
−2,000−1,000 0 1,000 ...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Encounter model
generate uniformly random starting aircraft states in annulus
– speeds uniformly random
– headings initial...
Aircraft model
ODE solver for dynamics
– continuous Dubin’s kinematic model
– Gaussian noise in acceleration and banking
c...
Baseline methods
closest threat
– multithreat scenario decomposed into pairwise encounters
– aircraft execute solution to ...
Experiment
>1 million simulations from 2 to 10 aircraft
– code: Julia implementation
– machine: 3.4 GHz Intel i7 processor...
Safety performance
max-min utility fusion
4 5 6 7 8 9 10
6
7
8
9
×10−3
number of aircraft
conflictprobability
uncoord
centr...
Safety performance takeaways
no near mid-air collisions (NMACs) in simulations
– NMAC threshold distance defined at 30 m (1...
Trade-off: Safety vs. alert rate
vary relative penalty between loss of separation and disruption
best performance at “knee”...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Practical conflict avoidance
recall: pub-sub system that UTM clients subscribe to for advisories
– standalone system that s...
System design
UTM-α: A distributed framework 49
System design
why Kafka?
– designed as unified, high-throughput, low-latency platform for
handling real-time data feeds
– o...
Driver-worker model
driver-worker model for conflict resolution
– driver loads up policy object that contains lookup table ...
Pub-sub model
ingestor publishes conflict objects to conflict Kafka topic
– subscribes to and ingests UTM server stream and...
Working model
Spark driver launches workers/executors and broadcasts policy object
workers receive conflict data in paralle...
Outline
Introduction
Mathematical formulation
Approach
Numerical experiments
UTM-α: A distributed framework
Conclusion and...
Summary
three variants of coordination-based conflict resolution algorithm
– 1 centralized/serial and 2 distributed variant...
Implementation
policy generation on Julia scientific programming language
https://github.com/sisl/ConflictAvoidanceDASC
dis...
Further research
tracking aircraft via belief states
– use the UTM client server to track reported aircraft states
– gener...
Further software development
standardize resolution advisory and advisory receipt formats with
focus on minimizing message...
References
Federico Augugliaro, Angela P Schoellig, and Raffaello D’Andrea.
Generation of collision-free trajectories for a...
References
M.J. Kochenderfer and J.P. Chryssanthacopoulos.
Robust airborne collision avoidance through dynamic programming...
Impact of search timeout
conflict decreases with iteration limit but with diminishing returns
2 3 4 5 6
4.8
4.9
5
5.1
5.2
5...
Restriction messages for consensus search
2 4 6 8 10
0.5
1
1.5
2
2.5
3
3.5
×10−3
number of uavs
conflictprobability
consens...
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Short-Term Conflict Resolution for Unmanned Aircraft Traffic Management

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We propose a set of controllers to solve the stochastic short-term conflict avoidance problem, in which a traffic management system supervises a collection of aircraft. The controllers generate advisories for each aircraft, and are based on decomposing a Markov decision process and fusing solutions to its subproblems online.

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Short-Term Conflict Resolution for Unmanned Aircraft Traffic Management

  1. 1. Short-Term Conflict Resolution for Unmanned Aircraft Traffic Management Hao Yi Ong and Mykel J. Kochenderfer Stanford University September 22, 2015
  2. 2. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work Introduction 2
  3. 3. Unmanned aircraft traffic management automating conflict avoidance is critical to integrating small unmanned aerial systems (UAS) to civil airspace automation to augment controllers Introduction 3
  4. 4. Conflict avoidance given potential conflicts between drones, find the best set of advisories focus on horizontal or co-altitude conflict resolution – NASA’s proposed airspace for UAS is under 150 m (500 ft) – flight altitudes need to avoid disruption and buildings conflict defined as loss of minimum separation between aircraft Introduction 4
  5. 5. Complications multiagent problem – system must coordinate between many aircraft – large search space for solution uncertainty in system – imperfect sensor measurements of current state – variable pilot response, vehicle performance, etc. affect future path appropriate trade-off between safety and efficiency not obvious Introduction 5
  6. 6. Goals robust and efficient method tractable approach to complex stochastic problem – account for uncertainty in large problem with reasonable compute resources balances between airspace safety and efficiency – ensure safety and provide timely conflict alerts to aircraft arbitrary-scale optimization – real-time conflict resolution for large airspace Introduction 6
  7. 7. Previous approaches mathematical programming (MIP, SCP) [SVFH05, ASD12] – can work well for simple vehicle networks distributed convex optimization [OG15] – hard to incorporate stochastic objectives/constraints Markov decision process formulation (ACAS X) [KC11, CK12] – assumes “white noise” accelerations for intruder aircraft and many more. . . [KY00] Introduction 7
  8. 8. Overview idea: decomposition + coordination multi-agent pairwise joint policy decompose pairwise solutions + coordination = ac1: left ac2: right ac3: straight Introduction 8
  9. 9. Conflict Resolution as a UTM service UTM server client side resolution server filtered sector status updates advisories advisories status updates Introduction 9
  10. 10. Conflict Resolution as a Standalone service derivative UTM client service – subscribes to UTM server to track aircraft and deconflict routes – pub-sub system that operators subscribe to to receive advisories current implementation uses standalone model – still unclear about end architecture for resolution service – clean approach to get started with prototype system – implemented with scalability and modularity in mind Introduction 10
  11. 11. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work Mathematical formulation 11
  12. 12. Markov decision process (MDP) defined by the tuple (S, A, T, R) S and A are the sets of all possible states and actions, respectively T (s, a, s ) gives the probability of transitioning into state s by taking action a at the current state s R (s, a) gives the reward for taking action a at the current state s environment T(s,a,s’)! agent action a! state s! reward R(s,a)! Mathematical formulation 12
  13. 13. Value iteration agent π⋆(s)" action a" state s" want to find the optimal policy π (s) – gives action that maximizes the utility Q (s, a) from any given state π (s) = argmax a∈A Q (s, a) value iteration updates value function guess ˆQ until convergence – expensive one-off compute but cheap policy extraction ( ˆQ lookup) ˆQ (s, a) := R (s, a) + s ∈S T (s, a, s ) max a ∈A ˆQ (s , a ) Mathematical formulation 13
  14. 14. Multiagent MDP extension of MDP to cooperative multiagent setting similar to case where centralized planner has access to system state also defined by the tuple (S, A, T, R), except – S and A are all possible joint states and actions – T and R operate on elements of S and A Mathematical formulation 14
  15. 15. Short-term conflict resolution horizontal conflict resolution between n aircraft – conflict defined as loss of minimum separation distance, 500 m – aircraft at risk if it could experience a conflict within two minutes alert aircraft that need corrective maneuvers – but not to the extent that alerts become a nuisance Mathematical formulation 15
  16. 16. Resolution advisories at each time step T = 5 s, system issues a joint advisory – joint advisory φ chosen out of a finite set of corrective bank angles for each aircraft action set A = {−20◦ , −10◦ , 0◦ , 10◦ , 20◦ , COC} n – positive and negative angles correspond to left and right banks – COC is a clear-of-conflict status advisory—nothing needs to be done higher resolution comes at the cost of computational complexity Mathematical formulation 16
  17. 17. Dynamics ith aircraft state si = (xi, yi, ψi, vi, pi) – latitude and longitude xi and yi – heading ψi – groundspeed vi – responsiveness indicator pi aircraft follow Dubin’s kinematic model – simplicity avoids risk of overfitting complicated models ˙xi = vi cos ψi ˙yi = vi sin ψi ˙ψi = g tan φi vi ψi! bank: φi! (xi ,yi)! vi! Mathematical formulation 17
  18. 18. Pilot model advisory response determined stochastically by Bernoulli process 5 s average response delay to first corrective advisory as recommended by ICAO [ICA07] – when responding, the pilot executes the advisory for 5 s time step non-responsive: white noise path responsive: corrective action COC 5 s average response lag Mathematical formulation 18
  19. 19. Reward function balance competing objectives maximize safety loss of separation cost separation closeness penalty minimize disruption cost advisoryCOC ±0° ±10° ±20° conflict penalty Mathematical formulation 19
  20. 20. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work Approach 20
  21. 21. Curse of dimensionality in an MMDP, S and A are all possible joint states and actions problem: S and A scale exponentially with number of agents 3 drones: 1000 states 2 drones: 100 states 1 drone: 10 states Approach 21
  22. 22. Decompose and coordinate multi-agent pairwise joint policy decompose pairwise solutions + coordination = ac1: left ac2: right ac3: straight Approach 22
  23. 23. MMDP decomposition multi-agent pairwise joint policy decompose ac1: left ac2: right ac3: straight pairwise solutions + coordination = Approach 23
  24. 24. Pairwise conflict resolution decompose full MMDP into O n2 pairwise encounters action set, pilot response model, and reward function unchanged use relative positions and bearing to reduce state space – arbitrarily set one aircraft as ownship, and the other as intruder – speeds remain in global reference frame s = xrel , yrel , ψrel , vownship , vintruder , pownship , pintruder (0,0)! vownship ! ψrel (xrel ,yrel)! vintruder Approach 24
  25. 25. Discretization in general, no analytical solution to problem with continuous state space approximate value function ˆQ over discretized state space – discretize state space via multilinear interpolation – iteratively update ˆQ until convergence ˆQ (s, a) := R (s, a) + s ∈S T (s, a, s ) max a ∈A ˆQ (s , a ) Approach 25
  26. 26. Approximating environment noise sigma-point sampling speed bank weights sigma-point sample Approach 26
  27. 27. Implementation discretize continuous state space into 9.6 million states – expensive but one-off computation ˆQ converges in 7 hours (40 iterations over S × A) – solver: parallel implementation in Julia – machine: 20 2.3 GHz Intel Xeon cores with 125 GB RAM Variable Minimum Maximum Number of values xrel , yrel −3000 m 3000 m 51 ψrel 0◦ 360◦ 37 vownship , vintruder 10 m s−1 20 m s−1 5 pownship , pintruder - - 2 Approach 27
  28. 28. Pairwise policy visualization passing intrusion (240◦ bearing) COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) y(m) Ownship action COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) Intruder action −20 −10 0 10 20 Approach 28
  29. 29. Example advisory execution passing intrusion at (1000 m, 0 m) ownship: 10° left bank intruder: 10° left bank Approach 29
  30. 30. Accounting for pilot response pilots responsive (top) and non-responsive (bottom) COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) y(m) Ownship action COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) Intruder action −20 −10 0 10 20 COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) y(m) Ownship action COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) Intruder action −20 −10 0 10 20 Approach 30
  31. 31. Accounting for pilot response ownship responsive (top) and intruder responsive (bottom) COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) y(m) Ownship action COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) Intruder action −20 −10 0 10 20 COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) y(m) Ownship action COC −2,000−1,000 0 1,000 2,000 −2,000 −1,000 0 1,000 2,000 x (m) Intruder action −20 −10 0 10 20 Approach 31
  32. 32. Multithreat coordination multi-agent pairwise joint policy decompose pairwise solutions + coordination = ac1: left ac2: right ac3: straight Approach 32
  33. 33. Utility fusion idea: combine pairwise utilities Qij to form proxy for global utility Q Qij operates on the pairwise MDP state-action pair (sij, aij) for aircraft i and j fusion strategies π(s) = argmax a∈A Q(s, a) – max-min: “worst-case” Qmin (s, a) = min i<j {Uij(sij, aij)} – max-sum: “average” Qsum (s, a) = i<j Uij(sij, aij) problem: action space A scales exponentially with number of aircraft Approach 33
  34. 34. Search heuristic: Alternating maximization maximize global utility by varying one action and fixing the rest e.g., fix drone 1 and 2’s actions (R and L) and vary drone 3’s action 1.1 1.3 0.9 2.1 1.1 0.9 1.3 2.3 0.8 2.3 1.4 0.3 0.9 1.7 0.8 0.8 2.2 1.4 2.1 1.1 0.3 argmax pair 1 pair 2 L R S L R S L R S L R S aircraft 3 action: L Q13(s,a)! Q23(s,a)! min Approach 34
  35. 35. Distributed variant I: Parallel search greedy local maximization 0 L 0 0 0 L R 0 0 broadcast action R L L R L L R L L repeat until convergence Approach 35
  36. 36. Distributed variant II: Consensus search parallel alt. maximization R R L R L L R L R broadcast joint action R R L R L L R L R greedy local maximization R L L R R L R R L incorporate msgs R ? L R R ? ? R L Approach 36
  37. 37. Consensus search message parallel search could incur large communication cost messages reduce communication between compute nodes, but risk yielding unsynchronized joint action across aircraft incorporate messages by “averaging” what intruders think ownship should do to approximate consensus broadcast joint action R R L R L L R L R incorporate msgs R L? L R R 0? R? R L Approach 37
  38. 38. Joint policy visualization triple aircraft encounter max-min (top) and max-sum policies (bottom) COC −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) y(m) First aircraft action −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) Second aircraft action −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) Third aircraft action −20 −10 0 10 20 COC −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) y(m) First aircraft action COC −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) Second aircraft action COC −2,000−1,000 0 1,000 2,000 −2,000 0 2,000 x (m) Third aircraft action −20 −10 0 10 20 Approach 38
  39. 39. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work Numerical experiments 39
  40. 40. Encounter model generate uniformly random starting aircraft states in annulus – speeds uniformly random – headings initialized to point to annulus center – resample if loss of separation Numerical experiments 40
  41. 41. Aircraft model ODE solver for dynamics – continuous Dubin’s kinematic model – Gaussian noise in acceleration and banking corrective bank angles mapped to PID control policies Numerical experiments 41
  42. 42. Baseline methods closest threat – multithreat scenario decomposed into pairwise encounters – aircraft execute solution to encounter with closest intruder uncoordinated – aircraft assumes all other aircraft are white-noise intruders – executes greedy local maximization to find own action Numerical experiments 42
  43. 43. Experiment >1 million simulations from 2 to 10 aircraft – code: Julia implementation – machine: 3.4 GHz Intel i7 processor with 32 GB RAM ∼10 ms decision times (vs. 5 s decision period) – serial solve times of <10 ms for up to 10 aircraft sufficient for server-based resolution system – unoptimized code that doesn’t account for communication costs for distributed variants Numerical experiments 43
  44. 44. Safety performance max-min utility fusion 4 5 6 7 8 9 10 6 7 8 9 ×10−3 number of aircraft conflictprobability uncoord centralized parallel consensus max-sum utility fusion 4 5 6 7 8 9 10 3 4 5 6 ×10−2 number of aircraft conflictprobability uncoord centralized parallel consensus distributed variants perform as well as serial version coordinated methods >10 times better than closest threat heuristic coordinated methods ∼10% better than uncoordinated method Numerical experiments 44
  45. 45. Safety performance takeaways no near mid-air collisions (NMACs) in simulations – NMAC threshold distance defined at 30 m (100 ft) – due to penalty on smaller separation distances even in conflict simulations validate all algorithmic variants – distributed variants perform as well as serial version – coordinated methods significantly better than baselines Numerical experiments 45
  46. 46. Trade-off: Safety vs. alert rate vary relative penalty between loss of separation and disruption best performance at “knee” of Pareto frontier 0 2 4 6 ×10−2 1.8 1.9 2 2.1 2.2 ×10−2 conflict probability alertrate centralized parallel consensus Numerical experiments 46
  47. 47. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work UTM-α: A distributed framework 47
  48. 48. Practical conflict avoidance recall: pub-sub system that UTM clients subscribe to for advisories – standalone system that subscribes to UTM server for flight tracking goals: robustness, scalability, and modularity – cluster computing framework for large-scale data processing – streaming analytics for real-time conflict resolution – built-in system fault-tolerance UTM-α: A distributed framework 48
  49. 49. System design UTM-α: A distributed framework 49
  50. 50. System design why Kafka? – designed as unified, high-throughput, low-latency platform for handling real-time data feeds – open source with active industry use (originally developed by LinkedIn) why Spark? – easy, high-level API – ease of operations with minimal fuss over fault-tolerance – enormous, active community and industry deployments (CISCO, Netflix, Intel,. . . ) UTM-α: A distributed framework 50
  51. 51. Driver-worker model driver-worker model for conflict resolution – driver loads up policy object that contains lookup table and algorithm for conflict resolution – workers receives policy object via broadcast and subsequently delegated tasks by driver formatted conflict data stored as an “infinite” stream of resilient distributed datasets (RDDs) – distributed, partitioned collection of conflict objects (JSON) – automatically rebuilt on failure UTM-α: A distributed framework 51
  52. 52. Pub-sub model ingestor publishes conflict objects to conflict Kafka topic – subscribes to and ingests UTM server stream and crudely identifies potential conflicts by proximity – formats conflict data as JSON strings advisor publishes advisories to advisory Kafka topic – producers are worker nodes in cluster generating advisories – consumers are UTM operator clients flying drones UTM-α: A distributed framework 52
  53. 53. Working model Spark driver launches workers/executors and broadcasts policy object workers receive conflict data in parallel from conflict Kafka topic as RDD stream workers publish advisories to advisory Kafka topic for UTM operator clients UTM-α: A distributed framework 53
  54. 54. Outline Introduction Mathematical formulation Approach Numerical experiments UTM-α: A distributed framework Conclusion and future work Conclusion and future work 54
  55. 55. Summary three variants of coordination-based conflict resolution algorithm – 1 centralized/serial and 2 distributed variants millisecond solve times for real-time application for multiple aircraft robust distributed system for practical conflict avoidance Conclusion and future work 55
  56. 56. Implementation policy generation on Julia scientific programming language https://github.com/sisl/ConflictAvoidanceDASC distributed system with Apache Kafka and Spark Streaming https://bitbucket.org/sisl/utm-alpha Conclusion and future work 56
  57. 57. Further research tracking aircraft via belief states – use the UTM client server to track reported aircraft states – generate local set of belief states for all aircraft to better estimate state uncertainty in real-time best action for drone in case of communication loss – figure out the response state of the aircraft via belief state tracking – where to go in case of communication loss, what flight profile to take, and what UTM should assume about the drones flight dealing with adversarial flights Conclusion and future work 57
  58. 58. Further software development standardize resolution advisory and advisory receipt formats with focus on minimizing message size quantify delay times for sending packets between advisory server and clients in order to model it into the problem Conclusion and future work 58
  59. 59. References Federico Augugliaro, Angela P Schoellig, and Raffaello D’Andrea. Generation of collision-free trajectories for a quadrocopter fleet: A sequential convex programming approach. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1917–1922, October 2012. J. P. Chryssanthacopoulos and M. J. Kochenderfer. Decomposition methods for optimized collision avoidance with multiple threats. Journal of Guidance, Control, and Dynamics, 35(2):398–405, 2012. International Civil Aviation Organization ICAO. Surveillance, radar and collision avoidance. International Standards and Recommended Practices, 4, 2007. 59
  60. 60. References M.J. Kochenderfer and J.P. Chryssanthacopoulos. Robust airborne collision avoidance through dynamic programming. Project Report ATC-371, MIT Lincoln Laboratory, 2011. J. K. Kuchar and L. C. Yang. A review of conflict detection and resolution modeling methods. IEEE Trans. Intell. Transp. Syst., 1(4):179–189, 2000. H. Y. Ong and J. C. Gerdes. Cooperative collision avoidance via proximal message passing. In American Control Conference (ACC), July 2015. Tom Schouwenaars, Mario Valenti, Eric Feron, and Jonathan How. Implementation and flight test results of MILP-based UAV guidance. In IEEE Aerospace Conference, pages 1–13. IEEE, 2005. 60
  61. 61. Impact of search timeout conflict decreases with iteration limit but with diminishing returns 2 3 4 5 6 4.8 4.9 5 5.1 5.2 5.3 ×10−3 iteration limit conflictprobability 61
  62. 62. Restriction messages for consensus search 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 ×10−3 number of uavs conflictprobability consensus (maxmin) restriction (maxmin) instead of broadcasting joint action, aircraft broadcast restriction messages that indicate what other aircraft shouldn’t do – e.g., reward straight path when it receives no left and no right messages at a decision period 62

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