This is project presentation for AAE-568 (Optimal Control) at Purdue University. The whole Project Report is available at https://arxiv.org/abs/1708.03055

1. OPTIMAL CONTROL FOR
CONSTRAINED COVERAGE PATH PLANNING

2. PROBLEM DESCRIPTION
 Generating a coverage path for a mobile robot
on a 2-D surface with obstacles
 Goals –
 Maximum coverage area
 Minimum overlap area
 Minimum time
 Minimum energy
Use Heuristics
Use Optimal Control

3. LINE SWEEP BASED METHODS
 Decompose coverage region into sub-regions (cells) using single line-sweep
 Back-and-forth motions perpendicular to sweep direction in each cell
 All cells use same sweep direction
 Boustrophedon Decomposition
IN event OUT event Boustrophedon Path

4. LINE SWEEP BASED METHODS (contd.)
 Boustrophedon generates complete coverage with zero overlap*
 Minimum time and minimum energy performance not considered
 Number of turns is the main factor in coverage cost**
 Apply optimal control to each slice to reduce overall coverage cost
**Huang, Wesley H. "Optimal line-sweep-based decompositions for coverage algorithms." Robotics and Automation, 2001. Proceedings 2001 ICRA. IEEE International Conference on. Vol. 1. IEEE, 2001.
*Choset, Howie, and Philippe Pignon. "Coverage path planning: The boustrophedon cellular decomposition." Field and Service Robotics. Springer London, 1998.,
Sweep
Sweep
Need to decide
optimal sweep
direction

5. PROBLEM STATEMENT
Minimize
Subject to

6. PONTRYAGIN PRINCIPLE
Hamiltonian
Necessary Conditions
for all time

7. CAVEATS
• Analytically intractable with increasing number of obstacles. (Hamiltonian
becomes complicated)
• Very sensitive to initial guess for TPBVP
(Non-intuitive and non-physical co-states)
SOLUTION*
• Fast Convergence irrespective of no. of obstacles
• Alleviate Curse of Sensitivity (Not Sensitive to initial
guess)
• Implementation should be easy.
*Garg, Divya. Advances in global pseudospectral methods for optimal control. Diss. University of Florida, 2011.
Pseudospectral
Theory

8. PSEUDO-SPECTRAL OPTIMAL CONTROL
**
• Discretizes time into non-uniform
collocation points. *
• Constraints satisfied only at interpolated
Points
• Non-linear static optimization after
discretization
• Stone – Weierstrass Theorem ensures
Shaping Functions are not restrictive.
* Using Legendre-Gauss-Lobatto Nodes ** Ross. A primer on pontryagin's principle in optimal control

9. Assumptions:
•Circular, Random, Disconnected
Obstacles.
•Turning cost not considered –
Main objective is to minimize the
number of turns *.
•MSA (Minimum Sum of Altitudes) –
to find optimal direction of sweep †
.
ALGORITHM – IMPLEMENTATION
* Choset, Howie. "Coverage of known spaces: The boustrophedon cellular decomposition." Autonomous Robots 9.3 (2000): 247-253.
†
Huang, Wesley H. "The minimal sum of altitudes decomposition for coverage algorithms." Rensselaer Polytechnic Institute Computer Science Technical Report 6 (2000).

10. COVERAGE PATH SIMULATION – ROBOT PATH
Fixed Area, No Obstacles Fixed Area, 10 Obstacles
Total Coverage Area = 100.000
Total Coverage Time = 234.6085
Total Control Energy = 16.0006
Total Coverage Area = 92.2575
Total Coverage Time = 236.1654
Total Control Energy = 16.0442
Direction of
Sweep,
θ* = 90 o
Coverage
Radius,
rrob = 0.25

11. COVERAGE PATH SIMULATION – ANALYSIS
Coverage Path – Fixed Area, No
Obstacles
Coverage Path – Fixed Area, 10
Obstacles
• Almost a linear response for
Acov, tcov and Ecov
• Note: Turning Cost is not
considered
• Almost a similar Response
• Difference in total area
covered Acov – due to
obstacle area + overlap
Acov
tcov
Ecov
Acov
tcov
Ecov
∆A = 7.7425,
Aobs = 1.133

12. COVERAGE PATH SIMULATION – ANALYSIS
• rrob = 0.25
• Response for each iteration
exhibits minimum time / energy
optimality
Coverage Path – Fixed Area, No
Obstacles
Coverage Path – Fixed Area, 10
Obstacles
Obstacles
• States change only in the
vicinity of the obstacles
• Severity of deviation - more for
clustered obstacles

13. Comments:
•Total response also displays minimum time/energy optimality – implies additive optimality for response
•Change of weights does not affect total area coverage
RESPONSE TO PARAMETER VARIATION:
Weight, w

14. RESPONSE TO PARAMETER VARIATION:
Obstacle Number, Nobs
Comments:
•Time and Energy response are acceptable for upto 14 obstacles
•Efficiency of Coverage gradually degrades with increasing number of obstacles

15. RESPONSE TO PARAMETER VARIATION:
Obstacle Size, robs
Comments:
•Performance degrades drastically for obstacle radii above 2rbot
•Modification of the algorithm for larger obstacles – subject of future work

16. FUTURE WORK
• Closed form Expression for Coverage Cost
• Unstructured Environment (Dynamic Obstacles, Multiple Robots, Uneven
Surfaces)
• Extension to Larger Radius Obstacles
CONCLUSIONS
• Trajectory assumes linear sweep form in absence of obstacles - desirable feature.
• Satisfactory response subject to constraints/ assumptions made.
• Area coverage independent of variation in weight, but has other parameter
constraints.

17. QUESTIONS
?