Controlling a Rolling Ball On a Tilting Plane - STEM2

3 The Equation of a Plane
Suppose a plane has a unit normal n. Let a point on the plane be given by
a vector p. Then
n · p = ‖p‖ cos(θ),
where θ is the angle between n and p. This is the perpendicular distance d
from the origin to the plane. Therefore the equation of the plane is
n · p = d,
where
that is
p =(x, y, z).
n 1 x + n 2 y + n 3 z = d.
4 A Description of the Tilted Plane
We consider a table elevated from the floor by a distance h. That is the pivot
point on the table plane is located at a distance h. This pivot point in the
center of the table is supported by a column of length h. We assume here
for simplicity that the pivot point is on the surface of the table, neglecting
certain offsets that will be needed later. The table is tilted by two actuators.
We take our coordinate system so that the xy plane is the floor, and the
origin is directly below the table pivot point. An x actuator is a line segment
of variable length that always lies in the y = 0 plane. It is connected to the
center supporting column at a point a distance c x from the pivot point in
the table plane. It is connected to the table plane at a point that is at a
distance t x from the pivot point. Similarly, a y actuator always lies in the
x = 0 plane. It is connected to the center supporting column at a point a
distance c y from the pivot point in the table plane. It is connected to the
table plane at a point that is at a distance t y from the pivot point. Let the x
actuator have variable length α, andthey actuator have variable length β.
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