Automated Guided Vehicle Optimal Control Problem Preliminary Research Reijer Idema

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

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

FROG Navigation Systems

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.

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

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

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

Constraint Projection

Problem Formulation (Mathematical)

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

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

B-Spline Definition

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

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

Cubic Basis Functions

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

NURBS Definition

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

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

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

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

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.)

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

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

Q & A ?