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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 additional challenge, I wanted one motor to drive forward or
backward, one motor to drive sideways (at 90 degrees to the first) and a third
motor to rotate on the spot. This is converted to the correct motion of the
wheels using an arrangement of four differentials (similar to an add-subtract
mechanism but a little more complicated - is there a simpler way?), so any of
these motions can also be performed simultaneously.
http://www.brickshelf.com/cgi-bin/gallery.cgi?f=284996
It works reasonably well. Inevitably it is a little bumpy because of the
wheels. Large power functions motors might work better, but of course they are
harder to mount.
Comments welcome. Video and more details of the mechanism to follow.
cheers, Alexander
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Very, very impressive! Am I the only one who always feels like a simpleton when
some of the technique/motor lads post? ‘Just for the challenge...’ ; ) Oh well,
we can’t all build castle : ) Great work and I love the rotating of three sets
of wheels solution (I think I see how this working).
Thanks for sharing and God Bless!
Nathan
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Comments welcome. Video and more details of the mechanism to follow.
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Congratulations, Alexander. A wonderful piece of mechanics! Eagerly waiting for
a video ;o)
Philo
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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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In lugnet.technic, Alexander Holroyd wrote:
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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!
Enjoy!
Alexander
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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?
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In lugnet.technic, Timothy P. Smith wrote:
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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?
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No - the mechanism with the differentials does precisely this conversion for
you! As you say, there is one IR channel for each motor. I am controlling it
using this: http://www.brickshelf.com/cgi-bin/gallery.cgi?i=2830880 The
joystick on the left (inspired by this one
http://news.lugnet.com/technic/?n=15622 from Mark Bellis) moves it around in
any direction without turning in the natural way. The lever on the right makes
it rotate.
Thanks for the interest! cheers, Alexander
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In lugnet.technic, Timothy P. Smith wrote:
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In lugnet.technic, Alexander Holroyd wrote:
| |
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!
Enjoy!
Alexander
|
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?
|
Looks like the tricky gear train handles a lot of it
http://www.brickshelf.com/cgi-bin/gallery.cgi?i=2823577
ROSCO
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In lugnet.technic, Timothy P. Smith wrote:
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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?
|
Here are some building instructions (or at least an MLCad file). The PF parts
are not shown, but it should be clear where they go:
http://www.brickshelf.com/gallery/aeh/Killough/ldraw/killough.mpd For anyone
planning to build one, there is a warning: the way the wheels are put together
may cause some damage to the 8L axles (18 of them), as there is a moderate
bending force on them. (I haven’t taken any of them out so I can’t tell for
sure). However, at 1-2 cents each from bricklink this seems an acceptable
sacrifice!
I’ve also added another video which should make it clearer how the controls
work: http://www.brickshelf.com/gallery/aeh/Killough/video/controls.avi
Many thanks for the encouragement! Alexander
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In lugnet.technic, Alexander Holroyd wrote:
Thanks a lot, Alexander! Now I have to find enough ballon wheels... or
substitute Rama wheels ;o)
Two little remarks about the mpd:
- The “whole” model should be the first in the MPD (Multipart > Model Sequence)
- You can add all PF parts that I modelled... get them here:
http://philohome.com/pf/pf.htm
Philo
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In lugnet.technic, Philippe Hurbain wrote:
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Two little remarks about the mpd:
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Done - thanks!
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Thanks for the videos and even more for the detailed and chear instructions!
Coming up with this complex geartrain must have been quite tough ;o)
Philo
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