Subject:
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Re: Killough platform with power functions
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Newsgroups:
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lugnet.technic
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Date:
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Sat, 24 Nov 2007 21:50:53 GMT
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Viewed:
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9650 times
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Thank you very much for the nice comments! Here at last are some videos of the
Killough platform in action (sorry for poor quality). The first three show
various motions, starting with the three basic ones: backwards/forwards,
sideways, and turning on the spot, and then in various combinations. The last
shows what the wheels are doing for the basic motions.
http://www.brickshelf.com/gallery/aeh/Killough/video/move1.avi
http://www.brickshelf.com/gallery/aeh/Killough/video/move2.avi
http://www.brickshelf.com/gallery/aeh/Killough/video/move3.avi
http://www.brickshelf.com/gallery/aeh/Killough/video/test.avi
I hope to post instructions at some point.
At the risk of boring all but the most committed technic/math geeks, here is a
bit more explanation!
First, how does a Killough platform work? The three wheels are of a special
type that allows free motion in the direction of the axle. Various lego designs
for such wheels, and explanations, can be seen e.g. at Philo’s site:
http://www.philohome.com/rama/rama.htm and the links there. To make it turn
on the spot, all three wheels should turn in the same direction (when viewed
from above) at the same speed. To make it go “forward” (in the direction of one
of the axles), two wheels should turn in opposite directions while the third
doesn’t turn. To make it go “sideways”, two wheels should turn at the same
speed in the same direction, while the other should turn in the opposite
direction, at exactly twice the speed (twice because cos(60 deg) = 1/2). You
can see all these things happening (in the order: forward; sideways; turn) at
the beginning of “test.avi” and “move.avi”.
The idea with this design is to have three motors control these three basic
motions via differentials. The mechanism is shown here (the internal gears in
the differentials are not shown):
http://www.brickshelf.com/cgi-bin/gallery.cgi?i=2823577 The three wheels are
connected to the purple, orange and yellow axles. The three motors drive the
red (turning on the spot), blue (forward/backward) and green (sideways) gears.
To understand how it works, we can let X, Y and Z be the angular velocities of
the three axles connected to the wheels, and A, B and C for the three motor
connections, as shown. Angular velocity means speed of rotation with a sign -
positive for clockwise, negative anticlockwise (as seen from the point of view
in the diagram). We can work out the velocities of the other gears because
meshing gears turn in opposite directions, so the gear next to X gets “-X” etc.
We also need another label G for the 3 grey gears.
In a differential, the velocity of the housing is always the average of the two
axles, so we get, for the four differentials:
B=(X+Y)/2
G=(X-Y)/2
-G=(A+Z)/2
C=(G-Z)/2
Eliminating G and solving these equations for X,Y,Z gives:
X = (-A+3B+2C)/3
Y = ( A+3B-2C)/3
Z = (-A-4C)/3
So if (A,B,C)=(1,0,0) (running only motor A), then (X,Y,Z)=(-1/3,1/3,-1/3),
which gives turning of the spot. (To understand that the signs are correct one
needs to think about what “clockwise” means from this viewpoint..)
Similaly, (A,B,C)=(0,1,0) gives (X,Y,Z)=(1,1,0), so motor B drives forward.
And (A,B,C)=(0,0,1) gives (X,Y,Z)=(2/3,-2/3,-4/3) (the important thing is that
4/3 is twice 2/3), so motor C moves it sideways.
(Actually, the three motors are geared differently, so the speeds are a bit
different). Now the nice thing is that all these basic motions can be combined.
E.g. running motors A and B together moves it diagonally. 3x3x3=27 different
motions (including standing still) are possible!
Enjoy!
Alexander
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Message has 2 Replies:
 | | Re: Killough platform with power functions
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| (...) I'm enjoying it- I'd enjoy instructions even more! How are your controls working? You control each motor with one IR channel, right? So you have to convert linear-to-holonomic in your head? In real time? (16 years ago, 24-Nov-07, to lugnet.technic, FTX)
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Message is in Reply To:
| | Killough platform with power functions
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| A Killough platform is a triangular vehicle with three "omni-directional" wheels that can move or rotate in any direction. Here is my version using Power Functions motors. The obvious arrangement is to have one motor for each wheel. Instead, for an (...) (16 years ago, 13-Nov-07, to lugnet.technic, lugnet.announce.moc, FTX) !
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