1. P. Ögren (KTH) N. Leonard (Princeton University)
A Convergent Dynamic Window Approach to Obstacle Avoidance & Obstacle Avoidance in Formation
P. Ögren (KTH)
N. Leonard (Princeton University)

2. Problem Formulation Drive a robot from A to B through a partially
unknown environment without collisions. B
Differential drive robots can be feedback linearized to this. A

3. Background: The Dynamic Unicycle (or a Tank?)q

4. Desirable Properties
No collisions
Convergence to goal position
Efficient, large inputs
‘Real time’
‘Reactive’, to changes

5. Background: Two main Obstacle Avoidance approaches
Reactive/Behavior Based
Biologically motivated
Fast, local rules.
‘The world is the map’
No proofs.
Changing environment not a problem
Deliberative/Sense-Plan-Act
Trajectory planning/tracking
Navigation function (Koditschek ’92).
Provable features.
Changes are a problem
Combine the two?

6. Background: The Navigation Function (NF) tool
One local/global min at goal.
Gradient gives direction to goal.
Solves ‘maze’ problems.
Obstacles and NF level curves
Goal

7. Basic Idea DWA, Fox et. al. Exact Navigation, Convergent DWA
Control Lyapunov
Function (CLF)
DWA, Fox et. al.
and Brock et al
Model Predictive
Control (MPC)
MPC/CLF Framework,
Primbs ’99
Convergent DWA
Exact Navigation,
using Art. Pot. Fcn.
Koditscheck ’92
‘Real time’
Efficient, large inputs
‘Reactive’, to changes
Convergence proof.
No collisions

8. Background: Model Predictive Control (MPC)
Idea: Given a good model, we can simulate the result of different control choices (over time T) and apply the best.
Feedback: repeat simulation every t<T seconds.
How is this used in the Dynamic Window Approach?

9. Global Dynamic Window Approach (Brock and Khatib ‘99)
Robot
Cirular arc pseudo-trajectories Vx Vy
Dynamic Window
Control Options
Obstacles
Vmax
Current Velocity
Velocity Space

10. Global Dynamic Window Approach (continued)
Check arcs for collision free length.
Chose control by optimization of the heuristic utility function:
Speeds up to 1m/s indoors with XR 4000 robot (Good!).
No proofs. (Counter example!)
Idea:
See as Model Predictive Control (MPC)
Use navigation function as CLF

11. Background: Control Lyapunov Function (CLF)
Idea: If the energy of a system decreases all the time, it will eventually “stop”.
A CLF, V, is an “energy-like” function such that V x

12. Exact Robot Navigation using Artificial Potential Functions, (Rimon and Koditscheck ‘92)
C1 Navigation Function NF(p) constructed.
NF(p)=NFmax at obstacles of Sphere and Star worlds.
Control:
Features:
Lyapunov function:
=> No collisions.
Bounded Control.
Convergence Proof
Drawbacks
Hard to (re)calculate.
Inefficient
Idea: Use C0 Control Lyapunov Function.

13. Our Navigation Function (NF)
One local/global min at goal.
Calculate shortest path in discretization.
Make continuous surface by careful interpolation using triangles.
Provable properties.
The discretization

14. MPC/CLF framework
Primbs general form:
Here we write:

15. The resulting scheme: Lyapunov Function and Control
Lyapunov function candidate:
gives the following set of controls, incl.
Compare: Acceleration of down hill skier.

16. Safety and Discretization
The CLF gives stability, what about safety?
In MPC, consider controls stop without collision.
Plan to first accelerate:
then brake:
Apply first part and replan.
Compare: Being able to stop in visible part of road ) safety

17. Evaluated MPC Trajectories

18. Simulation Trajectory

19. Single Vehicle Conclusions
Properties:
No collisions (stop safely option)
Convergence to goal position (CLF)
Efficient (MPC).
Reactive (MPC).
Real time (?), small discretized control set, formalizing earlier approach.
Can this scheme be extended to the multi vehicle case?

20. Why Multi Agent Robotics?
Applications:
Search and Rescue missions
Carry large/awkward objects
Adaptive sensing
Satellite imaging in formation
Motivations:
Flexibility
Robustness
Performance
Price

21. Obstacle Avoidance in Formation
How do we use singel vehicle Obstacle Avoidance?

22. Desirable properties No collisions Convergence to goal position
Efficient, large inputs
‘Real time’
‘Reactive’, to changes
&
Distributed/Local information

23. A Leader-Follower Structure
Two Cases:
No explicit information exchange ) leader acceleration, u1, is a disturbance
Feedforward of u1) time delays and calibration errors are disturbances
Information flow
How big deviations will the disturbances cause?

24. Background: Input to State Stability (ISS)
We will use the ISS to calculate ”Uncertainty Regions”

25. ISS ) Uncertainty Region

26. How do we calculate a map of ”free” leader positions?
Formation Leader Obstacles, an extension of Configuration Space Obstacles
”Occupied” leader pos.
Obstacle
”Free” leader pos.
How do we calculate a map of ”free” leader positions?

27. Formation Leader Map Unc. Region and Obstacles Formation Obstacles
Computable by conv2 (matlab).
Leader does obstacle avoidance in new map.
Followers do formation keeping under disturbance.

28. Simulation Trajectories

29. Final Conclusions
Obstacle Avoidance extended to formations by assuming leader-follower structure and ISS.
Future directions
Rotations
Expansions
Braking formation
) ¸ 3 dim NF

30. Comparison