Page 85 - Introduction to Autonomous Mobile Robots

Chapter 3
70
Now let us add an additional fixed standard wheel to create a differential-drive robot by
constraining the second wheel to be aligned with the same horizontal axis as the original
P
wheel. Without loss of generality, we can place point at the midpoint between the centers
of the two wheels. Given α β l, 1 , 1 for wheel w 1 and α β l, 2 , 2 for wheel w 2 , it
1
2
(
,
(
,
holds geometrically that ({ l = l ) β = β = 0) α + π = α )} . Therefore, in this
2
1
1
2
1
2
case, the matrix C β() has two constraints but a rank of one:
s
1
cos ( α ) sin ( α ) 0
1
1
C β() = C = (3.38)
1 s 1f
cos ( α + π) sin ( α + π) 0
1 1
Alternatively, consider the case when w 2 is placed in the wheel plane of w 1 but with
the same orientation, as in a bicycle with the steering locked in the forward position. We
P
again place point between the two wheel centers, and orient the wheels such that they lie
,
(
(
(
⁄
,
,
on axis x . This geometry implies that ({ l = l ) β = β = π 2) α = 0) α = π)}
1 1 2 1 2 1 2
and, therefore, the matrix C β() retains two independent constraints and has a rank of two:
s
1
⁄
⁄
⁄
cos ( π 2) sin ( π 2) l sin ( π 2) 01 l 1
1
C β() = C = = (3.39)
1 s 1f
⁄
⁄
⁄
–
cos ( 3π 2) sin ( 3π 2) l sin ( π 2) 01 l 1
1
In general, if rank C > 1 then the vehicle can, at best, only travel along a circle or
1f
along a straight line. This configuration means that the robot has two or more independent
constraints due to fixed standard wheels that do not share the same horizontal axis of rota-
tion. Because such configurations have only a degenerate form of mobility in the plane, we
do not consider them in the remainder of this chapter. Note, however, that some degenerate
configurations such as the four-wheeled slip/skid steering system are useful in certain envi-
ronments, such as on loose soil and sand, even though they fail to satisfy sliding constraints.
Not surprisingly, the price that must be paid for such violations of the sliding constraints is
that dead reckoning based on odometry becomes less accurate and power efficiency is
reduced dramatically.
In general, a robot will have zero or more fixed standard wheels and zero or more steer-
able standard wheels. We can therefore identify the possible range of rank values for any
robot: 0 ≤ rank C β() ≤ 3 . Consider the case rank C β() = 0 . This is only possible
1 s 1 s
if there are zero independent kinematic constraints in C β() . In this case there are neither
1 s
fixed nor steerable standard wheels attached to the robot frame: N = N = . 0
s
f
Consider the other extreme, rank C β() = 3 . This is the maximum possible rank
1 s
since the kinematic constraints are specified along three degrees of freedom (i.e., the con-
straint matrix is three columns wide). Therefore, there cannot be more than three indepen-