Page 90 - Introduction to Autonomous Mobile Robots

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Mobile Robot Kinematics
In contrast to a bicycle, consider an omnibot, a robot with three Swedish wheels. We
know that in this case there are zero standard wheels and therefore
δ = δ + δ = 3 + 0 = 3 . So, the omnibot has three differential degrees of freedom.
M m s
This is appropriate, given that because such a robot has no kinematic motion constraints, it
· ·
is able to independently set all three pose variables: x y θ,, · .
Given the difference in DDOF between a bicycle and an omnibot, consider the overall
degrees of freedom in the workspace of each configuration. The omnibot can achieve any
pose x( ,, in its environment and can do so by directly achieving the goal positions of
y θ)
all three axes simultaneously because DDOF = 3 . Clearly, it has a workspace with
DOF = . 3
Can a bicycle achieve any pose x( ,, in its environment? It can do so, but achieving
y θ)
some goal points may require more time and energy than an equivalent omnibot. For exam-
ple, if a bicycle configuration must move laterally 1 m, the simplest successful maneuver
would involve either a spiral or a back-and-forth motion similar to parallel parking of auto-
y θ)
mobiles. Nevertheless, a bicycle can achieve any x( ,, and therefore the workspace of
a bicycle has DOF =3 as well.
Clearly, there is an inequality relation at work: DDOF ≤ δ ≤ DOF . Although the
M
dimensionality of a robot’s workspace is an important attribute, it is clear from the example
above that the particular paths available to a robot matter as well. Just as workspace DOF
governs the robot’s ability to achieve various poses, so the robot’s DDOF governs its abil-
ity to achieve various paths.

3.4.2 Holonomic robots
In the robotics community, when describing the path space of a mobile robot, often the con-
cept of holonomy is used. The term holonomic has broad applicability to several mathemat-
ical areas, including differential equations, functions and constraint expressions. In mobile
robotics, the term refers specifically to the kinematic constraints of the robot chassis. A
holonomic robot is a robot that has zero nonholonomic kinematic constraints. Conversely,
a nonholonomic robot is a robot with one or more nonholonomic kinematic constraints.
A holonomic kinematic constraint can be expressed as an explicit function of position
variables only. For example, in the case of a mobile robot with a single fixed standard
wheel, a holonomic kinematic constraint would be expressible using α β l r ϕ ,, 1 , , 1 , 1
1
1
·
ϕ
·
,,
ξ
xy θ only. Such a constraint may not use derivatives of these values, such as or . A
nonholonomic kinematic constraint requires a differential relationship, such as the deriva-
tive of a position variable. Furthermore, it cannot be integrated to provide a constraint in
terms of the position variables only. Because of this latter point of view, nonholonomic sys-
tems are often called nonintegrable systems.