Page 75 - Introduction to Autonomous Mobile Robots

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Y R Chapter 3

β

Robot chassis A
ϕ, r
l v A
α

P X R

Figure 3.9
A spherical wheel and its parameters.

The behavior of this constraint and thereby the Swedish wheel changes dramatically as
γ
the value varies. Consider γ = 0 . This represents the swedish 90-degree wheel. In this
case, the zero component of velocity is in line with the wheel plane and so equation (3.19)
reduces exactly to equation (3.12), the fixed standard wheel rolling constraint. But because
of the rollers, there is no sliding constraint orthogonal to the wheel plane [see equation
·
ϕ
(3.20)]. By varying the value of , any desired motion vector can be made to satisfy equa-
tion (3.19) and therefore the wheel is omnidirectional. In fact, this special case of the Swed-
ish design results in fully decoupled motion, in that the rollers and the main wheel provide
orthogonal directions of motion.
⁄
At the other extreme, consider γ = π 2 . In this case, the rollers have axes of rotation
that are parallel to the main wheel axis of rotation. Interestingly, if this value is substituted
γ
for in equation (3.19) the result is the fixed standard wheel sliding constraint, equation
(3.13). In other words, the rollers provide no benefit in terms of lateral freedom of motion
since they are simply aligned with the main wheel. However, in this case the main wheel
never needs to spin and therefore the rolling constraint disappears. This is a degenerate
form of the Swedish wheel and therefore we assume in the remainder of this chapter that
γ ≠ π . 2 ⁄

3.2.3.5 Spherical wheel
The final wheel type, a ball or spherical wheel, places no direct constraints on motion (fig-
ure 3.9). Such a mechanism has no principal axis of rotation, and therefore no appropriate
rolling or sliding constraints exist. As with castor wheels and Swedish wheels, the spherical