Page 77 - Introduction to Autonomous Mobile Robots

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ϕ t() Chapter 3
f
ϕ t() = (3.23)
ϕ t()
s
The rolling constraints of all wheels can now be collected in a single expression:

·
·
J β()R θ()ξ – J ϕ = 0 (3.24)
1 s I 2
This expression bears a strong resemblance to the rolling constraint of a single wheel,
but substitutes matrices in lieu of single values, thus taking into account all wheels. J 2 is a
constant diagonal N × N matrix whose entries are radii of all standard wheels. J β()
r
1 s
denotes a matrix with projections for all wheels to their motions along their individual
wheel planes:

J 1f
J β() = (3.25)
s
1
β
J ()
1s
s
Note that J β() is only a function of β and not β . This is because the orientations of
1 s s f
steerable standard wheels vary as a function of time, whereas the orientations of fixed stan-
dard wheels are constant. J 1f is therefore a constant matrix of projections for all fixed stan-
dard wheels. It has size (N × 3 ), with each row consisting of the three terms in the three-
f
β
matrix from equation (3.12) for each fixed standard wheel. J () is a matrix of size
s
1s
(N × 3 ), with each row consisting of the three terms in the three-matrix from equation
s
(3.15) for each steerable standard wheel.
In summary, equation (3.24) represents the constraint that all standard wheels must spin
around their horizontal axis an appropriate amount based on their motions along the wheel
plane so that rolling occurs at the ground contact point.
We use the same technique to collect the sliding constraints of all standard wheels into
a single expression with the same structure as equations (3.13) and (3.16):
·
C β()R θ()ξ = 0 (3.26)
1
s
I
C
C β() = 1f (3.27)
1
s
β
C ()
1s s
C 1f and C 1s are (N × 3 ) and (N × 3 ) matrices whose rows are the three terms in the
s
f
three-matrix of equations (3.13) and (3.16) for all fixed and steerable standard wheels. Thus