Page 70 - Introduction to Autonomous Mobile Robots

55
Mobile Robot Kinematics
The first term of the sum denotes the total motion along the wheel plane. The three ele-
· ·
ments of the vector on the left represent mappings from each of x y θ,, · to their contri-
butions for motion along the wheel plane. Note that the R θ()ξ · I term is used to transform
the motion parameters ξ · I that are in the global reference frame { X Y, I I } into motion
parameters in the local reference frame X Y,{ R R } as shown in example equation (3.5). This
is necessary because all other parameters in the equation, α β l,, , are in terms of the robot’s
local reference frame. This motion along the wheel plane must be equal, according to this
constraint, to the motion accomplished by spinning the wheel, rϕ · .
The sliding constraint for this wheel enforces that the component of the wheel’s motion
orthogonal to the wheel plane must be zero:

·
cos ( α + β) sin ( α + β) lsin β R θ()ξ = 0 (3.13)
I
For example, suppose that wheel is in a position such that ({ α = 0) β = 0)} . This
(
,
A
would place the contact point of the wheel on X with the plane of the wheel oriented par-
I
allel to Y I . If θ = 0 , then the sliding constraint [equation (3.13)] reduces to
100 x · x ·
· ·
100 010 y = 100 y = 0 (3.14)
001 θ · θ ·

This constrains the component of motion along X to be zero and since X and X are
I I R
parallel in this example, the wheel is constrained from sliding sideways, as expected.
3.2.3.2 Steered standard wheel
The steered standard wheel differs from the fixed standard wheel only in that there is an
additional degree of freedom: the wheel may rotate around a vertical axis passing through
the center of the wheel and the ground contact point. The equations of position for the
steered standard wheel (figure 3.5) are identical to that of the fixed standard wheel shown
in figure 3.4 with one exception. The orientation of the wheel to the robot chassis is no
β
longer a single fixed value, , but instead varies as a function of time: β t() . The rolling
and sliding constraints are

·
·
l
sin ( α + β) – cos ( α + β) –()cos β R θ()ξ – rϕ = 0 (3.15)
I
·
cos ( α + β) sin ( α + β) lsin β R θ()ξ I = 0 (3.16)