Page - 196 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

Byadding thechangeofx,y andθ inone timestep to thepreviouspose, the robotsposeatanygiven timecanbecalculated recursively. xt(xt−1,u) =   xt=xt−1 +v ·cos(θt−1) ·∆tyt=yt−1 +v ·sin(θt−1) ·∆t θt= θt−1 + v·tan(ϕ) wwheelbase ·∆t   (1) The change of the pose is represented by the jacobian matrixG(xt−1,u), which is the derivative of theposext−1 with respect to theposext−1. G= ∂xt(xt−1,u) ∂xt−1 =   1 0 −v ·sin(θt−1) ·∆t0 1 v ·cos(θt−1) ·∆t 0 0 1   (2) The jacobian matrixV (xt,u) is the derivative of the posext−1 with respect to the motion command u. This equals thechangeof themotion. V = ∂xt(xt−1,u) ∂u =   cos(θt−1) ·∆t 0sin(θt−1) ·∆t 0 tan(θt−1)·∆t wwheelbase v·∆t wwheelbase·cos2(θt−1)   (3) TheerrormatrixM (u,α)considers theeffectofmotionerrors,while theparameterαconsiders their severity. M= ( α1v 2 +α2ϕ 2 0 0 α3v 2 +α4ϕ 2 ) (4) ThecovariancematrixPt contains the uncertaintyof thepose. Pt=G ·Pt−1 ·GT+V ·M ·VT (5) The first term of calculation 5 represents the pose prediction and the second term the uncertainty in the accuracy of the motion. The covariance can be visualized by plotting the ellipse defined by the eigen-vectors and the eigen-values of this matrix. Without any correction, the covariance ellipse will growwhenever the robotmoves. Thegrowthrateof thisellipse isdefinedbyαwhichdependson the robot and its environment. (a) (b) Figure3: The (a) real robot and its (b) simulation. 196