Globally Consistent Range Scan Alignment for Environment Mapping

F. LU AND E. MILIOS∗Department of Computer Science, York University, North York, Ontario, Canada

eem@cs.yorku.ca

Problem Definition

“In this paper, we address the issue of consistent alignment of data frames so that they can be integrated to form a world model.”

Problem Definition Odometry information is not sufficient to determine relative pose

scans. Incrementally integrating new frames of data to the global model ,

may result in an inconsistent model.

Related Work

HILARE project (Chatila and Laumond, 1985). Moutarlier and Chatila, 1989. Durrant-Whyte (1987, 1988a,1988b) Tang and Lee (1992)

Approach to the Problem Statement A framework to continuously register range scans. Maintain all local frames of data as well as a network of spatial

relationships amongst them. Use pose relations as constraints, all pose as variables and solve the

final optimization problem.

Deriving Pose Relations Pose relations can be directly obtained from odometry or aligning

pairwise scans of points.

The corresponding points from the two scans will form a constraint between the two poses.

The pose relation is derived from the overlapping potion of the two scans.

Constructing and Combining Relations in a Network

A node of a network is a pose of the robot which can be defined as a 3 dimensional vector (x, y, θ)t

Links are determined by considering the amount of overlap between the range scans at two poses.

Considering the pose as a variable we build an objective function and solve by minimizing the function.

Optimal Estimation Formulation

Mahalanobis Distance

X0,X1,....,Xn : n+1 nodes

Dij = Xi -Xj .

D = Dij + Δ Dij

Δ Dij= Random Gaussian Error

Solution

We can express the eqn. As follows:D = HXW = (D-HX)t .C.-1 (D-HX)Then the solution for X which minimizes W is given by:X = (Ht C-1 H)-1.Ht .C-1D.

Cx =(Ht.C-1.H)-1 [Covariance of X ]

Solution Continues..

Denoting HtC-1H by G, and expanding the matrix multiplicationswe can obtain d*d sub matrices of G as:

Special Networks:A serial and a Parallel network

For the serial one , the derived estimate and covariance matrix are given by:

X2 = D01+D12

C2 = C01 + C12

For the parallel one:

X1 = (C'-1 +C''-1)-1(C'-1D' + C''-1D'')

C = (C'-1 + C''-1)-1

Wheatstone Bridge Network

Pose Compounding Operation

Assume that the robot starts at a pose Vb = (xb , yb , θb )t and it then

changes its pose by D = (x, y, θ)t relative to V b , ending up at a new pose Va = (xa , ya, θa )

t .

Then we say that pose Va is the compounding of Vb and D.

We denote it as:

Va = Vb D.⊕The coordinates of the poses are related by:

xa = xb + x cos θb − y sin θb

ya = yb + x sin θb + y cos θb

θa = θb + θ.

Compounding Continues

It is also useful to define the inverse of compounding which takes two poses and gives the relative pose:

D = Va V⊕ b .

The coordinates are related by the following equations:

x = (xa − xb ) cos θb + (ya − yb ) sin θb

y = −(xa − xb ) sin θb + (ya − yb ) cos θb

θ = θa − θb .

We also want to define a compounding operation between a full 3D pose V b = (xb , yb , θb ) and a 2D position vector u = (x, y)t . The result is another 2D vector u' = (x' , y' )t . We still denote the operation as:

u = Vb u.⊕

Pose Relations from Matched Scans Let Va and Vb be two nodes in the network .From the pairwise scan matching algorithm, we get a set

of pairs of corresponding points: uak , u

bk , k = 1, . . . , m,where the 2D non-oriented points ua

k , ub

k are from scan Sa and Sb , respectively.

ΔZ k = Va u⊕ ak − Vb u⊕ b

k = 0.

m

Fab(Va,Vb)= Σ || (Va u⊕ ak) - (Vb u⊕ b

k)||2

K= 1

Pose Relations Continues In order to reduce Fab into

the Mahalanobis distance form, we linearize each term ΔZk . Let Va =( xa , ya , θa )

t , Vb = ( xb , yb, θb )t be

some close estimates of Va and Vb . Denote

ΔVa = Va − Va and ΔVb = Vb −Vb

Let uk = (xk , yk )t

= Va u⊕ ak ≈Vb u⊕ b

k

Pose Relation Continues

Pose Relation Continues

Pose Relation Continues

Pose Relations From Odometry

Pose Relation Continues

Pose Relation Continues

Optimal Pose Estimation

Optimal Pose Estimation

Sequential Estimation

Sequential Estimation

Conclusion

In this paper, we formulated the problem of consistent range data registration as one of optimal pose estimation from a network of relations. Although we develop our method for mapping a 2D environment using 2D range scans, our formulation is general and it can be applied to the 3D case as well, by generalizing pose composition an linearization .