1. Automated Guided Vehicle Optimal Control Problem
Preliminary Research
Reijer Idema

2. Automated Guided Vehicle Optimal Control Problem
Supervisors:
prof.dr.ir. P. Wesseling
dr.ir. Kees Vuik
ir. Patrick H.F. Segeren
dr.ir. René Jager

3. FROG Navigation Systems
AGV : Automated Guided Vehicle
industrial transport
public transport
entertainment
FROG : Free Ranging On Grid
grid of magnets

4. FROG Navigation Systems

5. Problem Formulation (Textual)
Suppose an AGV has to perform an action in the world.
Find the best control input for the AGV to achieve the action.

6. Example Problem 1/3
Vehicle model:
point mass
throttle on/off
instant steering
orientation = path direction
Vehicle task:
hallway (2D)
collect at A
deliver at B

7. throttle: steering wheel: collect/deliver: N/A position: orientation:
Example Problem 2/3
Internal state space:
throttle:
steering wheel:
collect/deliver: N/A
External state space:
position:
orientation:
velocity:
collect/deliver: N/A

8. begin position/orientation end position/orientation
Example Problem 3/3
Task:
begin position/orientation
end position/orientation
Internal Constraints:
[no jumping]
[no sharp corners]
External Constraints:
keep clear of obstacles

9. Constraint Projection

10. Problem Formulation (Mathematical)

11. NURBS curves Local search methods Tools Path description:
Solver Algorithm:
Local search methods

12. parametric curve: C(u) = (x(u),y(u)) piecewise rational function
NURBS Curves
NURBS: Non-Uniform Rational B-Spline:
parametric curve: C(u) = (x(u),y(u))
piecewise rational function

13. B-Spline Definition

14. Knot Vector Properties
Knot vector: U = {u0,…,um}
knot multiplicity k
non-periodic:
u0 = … = up = a
um-p = … = um = b
consequence:
C(u0)=P0, C(um)=Pn

15. Basis Function Properties
Basis function: Ni,p(u;U)
depends only on degree p and knot vector U
local support property:
partition of unity
non-negativity
polynomial on each knot span
C on the interior of a knot span
Cp-k at a knot with multiplicity k

16. Cubic Basis Functions

17. C on the interior of a knot span Cp-k at a knot with multiplicity k
B-Spline Properties
B-Spline: C(u;P,U)
piecewise polynomial
C on the interior of a knot span
Cp-k at a knot with multiplicity k
local modification scheme
moving Pi modifies C(u) only on [ui,ui+p+1)
strong convex hull property

18. NURBS Definition

19. C on the interior of a knot span Cp-k at a knot with multiplicity k
NURBS Properties
NURBS: C(u;P,U,W)
piecewise rational
C on the interior of a knot span
Cp-k at a knot with multiplicity k
local modification scheme
moving Pi or changing wi
modifies C(u) only on [ui,ui+p+1)
strong convex hull property

20. Homogeneous Representation
A rational curves of dimension d can be represented
by a non-rational curve of dimension d+1
using homogeneous coordinates.

21. knot insertion and knot refinement knot removal degree elevation
B-Spline Operations 1/2
knot insertion and knot refinement
knot removal
degree elevation
degree reduction
point inversion and point projection
reparameterization
conversion to and from piecewise power basis form

22. control point and weight modification relatively easy calculations
B-Spline Operations 2/2
control point and weight modification
relatively easy calculations
high level shaping tools
warping, flattening, bending and
constraint based curve shaping
curve fitting
interpolation and approximation

23. Research Planning 1/3
Task:
path from A to B
Vehicle model:
point mass
2D work area
orientation = path direction
disconnect velocity
Extensions:
vehicle body
higher dimensions (crabbing, etc.)

24. Research Planning 2/3
Internal constraints:
general description
constraint preserving operations
External constraints:
hull curve
collision free path

25. Research Planning 3/3
Cost and heuristics:
cost function
time, energy consumption
heuristics
straight, circular corner
Solver algorithm:
highly constructive
local search

26. Q & A ?