Carnegie Mellon University TRAJECTORY MODIFICATION TECHNIQUES IN COVERAGE PLANNING By - Sanjiban Choudhury (Indian Institute of Technology, Kharagpur, India) Sanjiv Singh (Carnegie Mellon University, USA)

About the Institute…

… and the robots

Motivation

Problem At Hand NO – GO AREAS New outgrowths

Sample Field

Main aspects of work  Handling new obstacles without re-planning.  Ensuring all routes respect curvature constraints.  Optimal turnings without entering “no-go” areas.

Current Approaches  Exact Cellular Decomposition  - Trapezoidal Decomposition (F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, 1991)  - Boustrophedon Decomposition (H. Choset and P. Pignon, Coverage path planning: The boustrophedon decomposition, 1997)  - Split and Merge Method (T.Oksanen and A.Visala, Path Planning Algorithms for Agricultural Field Machines, 2007)

Split and Merge Method

Problems with the approach  Obstacles are not handled efficiently.  Requires extra pre-determined spaces for turning, known as headlands.  Headlands have to be provided externally by operator.

Problem with obstacles  Re-planning required  Distribution of cells completely changed  Trivial cases may have over-complicated solutions.

Headlands  They are pre-determined and taken to be a considerably large enough value to allow turning.  No optimization of headland width with respect to – Curvature Constraints – Obstacle profile

Modified Solution

‘Bending’ Lines  Each track is represented as a B-spline  Tracks are subjected to forces by 1. Obstacles 2. Curvature of track 3. Interaction forces between tracks

Vector fields

Modification of curves  Splines are used because of their local modification property.  Knot intervals containing the obstacle are modified.

Algorithm  1. A track passing through the obstacle is taken.  2. Knots are inserted appropriately.  3. Energy associated with each knot interval is calculated.  4. Points corresponding to the knots are acted upon by forces given by the energy gradient.

Knot insertion This part frozen These points span over a distance of (π/2)*R to alter INITIAL heading These points span over a distance of (π/2)*R to alter FINAL heading These points carry the curve over obstacles

Knot insertion This part frozen These points span over a distance of (π/2)*R to alter INITIAL heading These points span over a distance of (π/2)*R to alter FINAL heading These points carry the curve over obstacles

Energy of tracks  Each track has an energy associated with it.

Movements of spline points

Head to head comparisons

MethodRe-PlanningBending Paths Distance Travelled m m Moving Time s s Turning Time s s Total Time s s Moving speed = 1m/s, Turning speed = 0.5 m/s, improvement = 9.6%

Headlands

Turning at headlands The vehicle turns at 1. Field edges which are un-restricted. 2. Field edges which are restricted, i.e, “No Go Areas” 3. Edges of obstacles

Solution for the problem  The problem is framed as an “Optimal Control Problem” (OCP) (Oksanen, T., & Visala, A. (2004, October). Optimal control of tractor-trailer system in headlands)

Vehicle Model Used x and y are the coordinates of the rear axle center θ is the heading angle α is the steering angle u is the steering rate v is the velocity L is the wheel base

The OCP x is a state vector f dynamics function t0 and tf the time limits J the cost functional F the cost function C the path constraint function E the boundary condition function

Case 1 : No restriction on headlands d (0,0) (0,d)

Case 2: “No Go” Area  For such cases, solutions have been derived only for given headland width.  The OCP problem is reframed to solve for optimal headland width and time. d (0,0) (0,d) No go

Case 2: “No Go” Area Initial Values : Path Constraints : d (0,0) (0,d) No go φ Cost Function :

Case 3: Obstacles  Obstacles are also “no-go” areas inside the field.  In a similar fashion, the optimal turn is calculated.

Fitting the turns  Two sets of turns, one for no restriction, and another for restriction is stored for 10< φ <170 degrees, where φ is the angle between adjacent tracks.  The angle between tracks is rounded of to nearest multiple of 10, and the corresponding turn route is applied.

Local adjustment of turns φ Turn in no go area

Local adjustment of turns φ Turn in no go area Trajectory is converted to a spline and “bent” to respect both curvature as well as experience repulsion by the restricted areas.

Optimisation of Cellular Traversal

 Currently all cellular traversal is boustrophedon. (back and forth)  However a circle takes more time back and forth than just spiraling down from its edges.  A rectangle would take more time spiraling than just covering it back and forth.

Algorithm  Choose the cell to optimise.  Offset the polygon inside by track width. Compute traversal time of the spiral and add it with traversal time of the remaining polygon.  Repeat till offsetting not possible.  Take the best solution.

Some optimal solution

Results  Guaranteed better solutions than Split and Merge method.  No significant increase in computation time.

Results Total Time = 4965Total Time = 4270 Improvement in time = 14%

Results Total time = 7060Total Time = 7951 Improvement in time = 11.1%

Some more results

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