Page 86 - Introduction to Autonomous Mobile Robots

71
Mobile Robot Kinematics
dent constraints. In fact, when rank C β() = 3 , then the robot is completely constrained
1 s
in all directions and is, therefore, degenerate since motion in the plane is totally impossible.
Now we are ready to formally define a robot’s degree of mobility δ :
m
δ m = dimN C β() = 3 – rank C β() (3.40)
s
1
1
s
The dimensionality of the null space (dimN ) of matrix C β() is a measure of the
1
s
number of degrees of freedom of the robot chassis that can be immediately manipulated
through changes in wheel velocity. It is logical therefore that δ must range between 0
m
and 3.
Consider an ordinary differential-drive chassis. On such a robot there are two fixed stan-
dard wheels sharing a common horizontal axis. As discussed above, the second wheel adds
no independent kinematic constraints to the system. Therefore, rank C β() = 1 and
1
s
δ = 2 . This fits with intuition: a differential drive robot can control both the rate of its
m
change in orientation and its forward/reverse speed, simply by manipulating wheel veloci-
ties. In other words, its ICR is constrained to lie on the infinite line extending from its
wheels’ horizontal axles.
In contrast, consider a bicycle chassis. This configuration consists of one fixed standard
wheel and one steerable standard wheel. In this case, each wheel contributes an indepen-
dent sliding constraint to C β() . Therefore, δ m = 1 . Note that the bicycle has the same
s
1
total number of nonomidirectional wheels as the differential-drive chassis, and indeed one
of its wheels is steerable. Yet it has one less degree of mobility. Upon reflection this is
appropriate. A bicycle only has control over its forward/reverse speed by direct manipula-
tion of wheel velocities. Only by steering can the bicycle change its ICR .
As expected, based on equation (3.40) any robot consisting only of omnidirectional
wheels such as Swedish or spherical wheels will have the maximum mobility, δ = . 3
m
Such a robot can directly manipulate all three degrees of freedom.

3.3.2 Degree of steerability
The degree of mobility defined above quantifies the degrees of controllable freedom based
on changes to wheel velocity. Steering can also have an eventual impact on a robot chassis
ξ
pose , although the impact is indirect because after changing the angle of a steerable stan-
dard wheel, the robot must move for the change in steering angle to have impact on pose.
As with mobility, we care about the number of independently controllable steering
parameters when defining the degree of steerability δ s :

β
δ = rank C () (3.41)
s 1s s