Improved Inertial/Odometry/GPS Positioning of Wheeled Robots Even in GPS-Denied Environments - Global Navigation Satellite Systems: Signal,Theory and Applications - page 260

l b

R

=

⎡
⎣

⎤
⎦

ψ θ − ψ ρ θ − ψ ρ ψ θ + ψ ρ θ

cos cos sin sin sin sin cos cos sin sin sin sin

sin ψ cos θ + cos ψ sin ρ sin θ cos ψ cos ρ sin ψ sin θ − cos ψ sin ρ cos θ

(1)

−

cos ρ sin θ sin ρ cos ρ cos θ

2.3 Mobile robot odometry equation
The conventions and notation presented in (Chong & Kleeman, 1997) are used to create
a kinematic model for the mobile robot. In this work, a simple model for mobile robot
kinematics is considered. The wheels must be as thin as possible (one rolling point-of-contact
between the terrain and each wheel). Also, there must not be any slipping along the
longitudinal direction. There must not be any sliding along the transverse direction.

Define the instantaneous center of curvature (ICC) as a means of describing the curvilinear
motion that the mobile robot makes on a plane. In a two-dimensional environment the plane
that the robot travels on remains fixed for all possible positions and orientations of the mobile
robot (some authors refer to the reference frame enclosed in this plane as the "global reference
frame"(Chong & Kleeman, 1997). In this work, motion of the mobile robot on possibly distinct
planes for each time interval is considered. The mobile robot travels on a plane that is fixed
from time k − ≤ ≤ + 1 t k 1.

r R ω R k + r L ω L k

V T k =

(2)

2

where:

V T k is the velocity of the robot measured from its center and tangent to the circular path
contained on a plane from time k

−

1

≤

t

≤

k

+

1;

r R is the radius of the right drive wheel;
r L is the radius of the left drive wheel;
ω R k is the angular velocity of the right drive wheel from time k

−

1

≤

t

≤

k

+

1;

ω L k is the angular velocity of the left drive wheel from time k

−

1

≤

t

≤

k

+

1;

k represents discrete time epochs; and
Δ

t is the sampling time.

Rotational speeds of the left and right drive wheels are measured using encoders which are
used to calculate the forward velocity of the robot. The forward velocity is transformed into
velocities in the local frame using the equations below. Equation 2 is expressed in the mobile
robot frame. In order for us to use the velocities of the robot's wheels as a measurement update
we must transform these quantities to the local frame. Using the following transformation we
have:

V odo
e k

=

V T k cos

( ρ k )

sin

( α k )

(3)

V odo
n k

= ( ρ k ) ( α k ) V T k cos cos (4)

V odo

u k =

V T k sin

( ρ k )

(5)