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Collision Avoidance for Urban Air Mobility using Markov Decision ProcessesSydney M. Katz, Stanford University, Department of Aeronautics and Astronautics, Stanford, CA 94305
AIRCRAFT COLLISION AVOIDANCE
• As Urban Air Mobility (UAM) vehicles become reality, we need to develop robust collision avoidance systems to ensure their safe integration into the airspace
• The Traffic Alert and Collision Avoidance System (TCAS) has been the primary collision avoidance logic mandated by the Federal Aviation Administration (FAA) since the 1980s [1]• Relies on complicated heuristic logic rules
• In the past decade, a new approach to collision avoidance called the Airborne Collision Avoidance System X (ACAS X) has been under development to account for NextGen airspace procedures [2]• Formulated as a partially observable Markov decision process (POMDP)
• Algorithm easily adapts to new environments and aircraft types• Versions for both manned and unmanned aircraft [2-4]• Primarily differ in operational considerations and the type of collision avoidance maneuvers they command
Level Off, Level Off!
PROBLEM SETUP AND METHODS
• The UAM collision avoidance problem is formulated as a Markov decision process (MDP) with a discretized state and action space• Easily extended to the partially observable case using the QMDP method [5]
• As in ACAS X, the MDP is solved using offline methods to output a lookup table of action costs at each state that can be loaded onto to aircraft to provide alerts
• With a discrete state and action space, the MDP can be solved using a form of dynamic programming called value iteration• Bellman equation used to perform iterative updates to the value function
Q⇤(s, a) = R(s, a) +P
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• The MDP is implemented using POMDPs.jl, a package written for the Julia language that provides a general framework for defining and solving POMDPs
• With a parallelized implementation the MDP can be solved in under 95 seconds on five Intel Core i7 processors operating at 4.20 GHz.
MDP FORMULATION
State Space
ℎ
ℎ#
ℎ$
𝑡 = 𝜏Horizontal Closest Point of Approach
• Need to choose both state variables and discretizations
• States are chosen to encode the current relative positions of the aircraft as well as how the encounter will evolve
• Discretizations are chosen to be fine enough to accurately encode the vehicle state while still allowing the problem to remain tractable• Values are chosen based on states expected for a UAM vehicle flying at low altitudes
Action Space
• The actions are the maneuvers, called Resolution Advisories or RAs, that the collision avoidance system tells the vehicle to execute
• Because this system is meant to operation at low altitudes where obstacles may be present, it is desirable to execute only vertical collision avoidance maneuvers with descents inhibited in order to minimize lateral deviation from the flight path
• Advisories are defined by a range of vertical rates in which the aircraft is considered compliant with the advisory
Action Description Min (ft/min) Max (ft/min)
COC clear of conflict �1 1DNC do not climb �1 0DND do not descend 0 1CL250 climb 250 1SCL450 strong climb 450 1
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Transition Model
• Transitions can be estimated using simple kinematics• It is assumed that the the ownship aircraft responds to its advisory with a noisy
acceleration and that the intruder noisily continues on its current path• These transitions are approximated using sigma-point sampling
Reward Model
• Rewards balance between safety and alert rate• There is a large penalty for a near midair collision (NMAC), defined as any state in which 𝜏 =0 and ℎ < 100 ft• In order to prevent excessive alerting, there are small penalties associated with particular alert events
Event Reward
NMAC �1
alert �0.01COC 0.0001strengthening �0.009reversal �0.1invalid transition �10
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An MDP is defined by the tuple 𝑆, 𝐴, 𝑇, 𝑅, 𝛾 where 𝑆 defines the state space, 𝐴 specifies the space of possible actions, 𝑇 encodes the probabilistic transition
model, 𝑅 represents the reward function, and 𝛾 is the discount factor.
State Variable Description Units Range (low:high) Size
h relative altitude ft -600:600 39h0 ownship vertical rate ft/min -500:500 41h1 intruder vertical rate ft/min -500:500 41aprev previous action N/A N/A 5⌧ time to horizontal closest point of approach s 0:100 101
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POLICY ANALYSIS
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Figure 1. Plots of optimal policy for various ownship vertical rates (in ft/min). The plots assume the intruder is flying level (ℎ = 0). The top row represents the policy for a previous RA of COC and the bottom row represents the policy for a previous RA of CL250.
Figure 2. Optimal policy for different reward function structures. Both aircraft are in level flight and the previous RA is COC.
Figure 2 shows policy plots for various reward function structures. Each structure progressively results in a larger penalty for the aircraft being in close proximity with one another• As the penalty increases, the size of the alert
region also increases• This represents the tradeoff between safety and
alert rate
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Figure 1 shows plots of the optimal policy for various slices of the state space. These plots allow for multiple key observations:• When the previous RA is COC, there is a large
gap in alerting from ~50-100 ft• Descent inhibits make it so that no action will be able to resolve the conflict so the system chooses not to alert• This is a potentially undesirable side effect of the reward structure
• As expected, the policy adjusts if the ownship is descending
SIMULATION RESULTS
Simulation results were obtained by simulating 1,000 encounters between straight, level flying aircraft. Encounters were generated by sampling various trajectory features such as airspeed, horizontal and vertical miss distances, and heading from pre-specified distributions.
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Figure 3. Examples of simulated encounters. The markers represent alerts and follow the same color scheme as Figures 1 and 2.
Figure 4. Vertical profile of the same encounter simulated using the tables corresponding to the policies in Figure 2. The leftmost policy alerts too late and results in an NMAC, while the other policies are able to resolve the conflict.
Ultimately, the simulations show that the implementation successfully resolves potential collisions without causing new ones. Additionally, Figure 4 illustrates the tradeoff between safety and alert rate.
FUTURE WORK
This work leaves multiple avenues for future work. Many of the unresolved NMACs in the encounter set are due to the gap in alerting identified in Figure 1. Future work should determine a method to address this issue, and further analysis of the policy plots should be conducted. Additionally, the output should be tested on more encounter sets that realistically encode UAM vehicle behavior. Based on the results of this further testing, the reward function will likely need to be retuned.
Equipage NMACs Alerts Induced Resolved
unequipped 472 - - -⌧ = 0 217 611 0 255⌧ < 5 211 635 0 261closeness 197 754 0 275
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REFERENCES
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