1. Everything You Need to Know about
Mixed Integer Linear Programming
Ben Gardiner, Travis Cooper,
Matthew Haveard, Waseem Ahmad

2. Outline
I. Overview of MILP
II. Existing UAV path planning Implementations
III. Our Implementation
a. Common Constraints
b. Significant Variables
c. Improvements
IV. Results
V. Future Improvements

3. Mixed Integer Linear Programming
● Used to solve problems formulated as
systems of linear constraints
● A solver, like Gurobi, can find the most
optimal solution if given enough time
● Originally designed for offline problems, like
crop planning

4. Prior Usages with UAVs
● Plan most effective con­colliding paths for
UAVs
● Computationally intensive
● More than a minute to find the best path

5. MILP Collision Avoidance
t=72.713 dt=2

6. Receding Horizon Control (RHC)
● Reduce computation time by having multiple
smaller calculations
● Get a portion of a path solved
● Re­solve new path by the time UAVs have
reached end of previous path.

11. Connected Components
● Creates multiple problems that can be solved
simultaneously
● Select a subset of UAVs that are close to each
other
● Solve that subset as a component that has no
relevance to other components

13. Commonly Used Constraints
● Dynamics Constraint – so UAVs conform to
real world physics
● Maximum Speed and Force – physical
constraints specific to UAVs
● Separation Distance – Distance that must be
maintained between UAVs

14. Physical Dynamics Constraint
∀ p∈ [ 1... N ] ∀ i∈ [ 0 ... T −1 ]
si1 p= A p sip B p f ip
The next state must always be equal to force
and velocity vectors added to current state

15. Maximum Speed and Force
The next state must always be inside of a polygon
which represents maximum speed and force

16. Separation Distance
UAVs must always maintain a minimum
distance from each other

17. New Constraint: Minimum Speed
statet1
statet

18. Significant Variables
● Standard Implementation
● RHC
● Connected Components

19. Variables from Prior
Implementations
● Size of timesteps (dt)
● Number of timesteps (Nt)
● Separation Distance (d)
● Time to solve (T)

20. RHC/Connected Components
Variables
● Horizon Length
● Component Threshold Distance
● Recalculation Point
● Number of waypoints given

21. Paths Planned Fully

22. Paths Planned Fully
Computation Time: 865.9s
Vehicle 1 Completion Time: 159
Vehicle 2 Completion Time: 138
Vehicle 3 Completion Time: 118

23. Paths Planned using RHC

24. Paths Planned using RHC
Vehicle 1 Completion Time: 160
Vehicle 2 Completion Time: 143
Vehicle 3 Completion Time: 125
Number of Infeasible Solutions Returned: 20
Number of Feasible Solutions Returned: 139
Mean Iteration Computation Time: 0.0676489431153
Net Computation Time: 10.7561819553

25. Comparison
Paths Planned Fully Paths Planned using RHC
Total Computation Total Computation
time: 865.9s time: 10.75618s

26. Results
Real­time Flight Simulated
using ROS
Number of UAVs: 3
Number of Waypoints per UAV:
Unlimited Random (only
achieved shown)
Field Size: 1000m x 1000m
(Flight Map scaled to 10m/unit)
Flight Time: 300s
UAV Speed: 11.76 m/s
Mean Iteration Computation Time:
0.0573s
Distance actual/distance
minimum: 1.046

27. Results

28. Results
Real­time Flight Simulated
using ROS
Number of UAVs: 6
Number of Waypoints per UAV::
Unlimited Random (only
achieved shown)
Field Size: 1000m x 1000m
(Flight Map scaled to 10m/unit)
Flight Time: 300s
UAV Speed: 11.76 m/s
Mean Iteration Computation Time:
0.107s
Distance actual/distance
minimum: 1.063

29. Results

30. Results

31. Results

32. Conclusions
● MILP implemented through RHC with
Connected Components works very well
● The interactions between the ROS
framework and the MILP solver become
unstable with large amounts of UAVs
● There is a lack of concurrency between
the information from ROS and the found
solutions

33. Possible Future Research
● Improve relations between MILP and ROS
● Dynamic RHC
● Smart Connected Components

34. The End