Controlling a Rolling Ball On a Tilting Plane - STEM2

5 The Plane Normal as a Function of Actuator
Lengths
The tilt of the table is controlled by two actuator lengths, as specified in the
previous section. Let us first consider the x actuator with length α.
We want to compute a vector u 1 that lies in the intersection of the y =0
plane and the table plane. This vector is given by
u 1 =(cos(θ x ), 0,sin(θ x )),
where θ x is the angle between a line that is the intersection of the table plane
with the y = 0 plane, and the horizontal plane. This is
θ x = φ x − π/2,
where angle φ x is given by the law of Cosines
cos(φ x )= c2 x + t 2 x − α 2
.
2c x
Similarly, we compute a vector u 2 that lies in the intersection of the x =0
plane and the table plane. This vector is given by
u 2 =(0, cos(θ y ), sin(θ y )),
where θ y is the angle between a line that is the intersection of the table plane
with the x = 0 plane, and the horizontal plane. This is
θ y = φ y − π/2,
where angle φ y is given by the law of Cosines
cos(φ y )= c2 y + t2 y − β2
.
2c y
Now the plane normal is
n = u 1 × u 2 .
The equation of our table plane then is
n · p = h,
where h is the vertical distance from the floor to the table pivot point. We
will use this equation to compute the z coordinate of the center of the ball
rolling on the tilted plane, where the x and y coordinates are specified by the
camera image.
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