### Wednesday, August 16, 2006

## Just for fun...

I've been kicking around the threaded rod conundrum for a little bit. The plastic gear motors that we can get cheaply give us good torque at about 50 - 60 rpm. We need about 300-360 rpm for a little reprap that can extrude at the rates that the Mk II extruder is prepared to work at. Running them at those speeds eats torque and causes vibration, though, which gives us all sorts of other problems to deal with.

I've made a rack and pinion script but a reprap will have a hard time making a gear tooth much smaller than about 2 mm according to Adrian. That means that you need something like a 25 mm radius gear to engage the rack and then you have to get a 12-15:1 ratio for your gear motor to slow it down enough to give you a translation rate in the 4-10 mm/sec range. That's either a lot of little gears or a few really big ones.

Looking at Kiplinger's linkages page got me to thinking about Joseph Whitworth and his genius for precision machinery during the 19th century.

Sure enough, Kiplinger had one of Whitworth's linkages. That one wasn't as interesting as Peaucelliers straight line generating linkage, though. I thought for a while of using a threaded rod to push that linkage to generate longer straight line movement. The only problem with the idea is that whereas a threaded rod gives a very linear input vis a vis rotation, Peaucellier's linkage wouldn't use it like that. Then I thought... why not use a bloody pantograph to multiply the translation speed of a threaded rod?

We'll lose force and resolution by the same proportions as we increase the travel distance, but it should be easy enough to build and test. We've also got resolution on the threaded rod to burn.

With a pitch of 1 thread/mm, a very common one, we get a resolution with a 256 pulse encoder of 0.0039 mm/pulse. Pushing that up to six times that will still give us just over 0.02 mm/pulse which is a factor of 4-5 better than our target.

It's not like others haven't done similar things before, but it might solve our problem with wanting to use threaded rods with slow gear motors. :-)

CLICK!

You know what? If we hooked Peaucellier linkages on the short and long stroke nodes of the pantograph we'd have a hell of a stable motion amplifier. Wowzie!

Hmmm... If we looked at the pantograph as not being basically a stick linkages but rather extended those sticks on the vertical axis and made them planes and then slid the Mk II up and down on the vertical edge of the long stroke node plane you'd have a very different kind of reprap machine.

That's enough creativity for one evening... or morning, I see. :-s I need to catch a little sleep.

Nite all! :-D

I've made a rack and pinion script but a reprap will have a hard time making a gear tooth much smaller than about 2 mm according to Adrian. That means that you need something like a 25 mm radius gear to engage the rack and then you have to get a 12-15:1 ratio for your gear motor to slow it down enough to give you a translation rate in the 4-10 mm/sec range. That's either a lot of little gears or a few really big ones.

Looking at Kiplinger's linkages page got me to thinking about Joseph Whitworth and his genius for precision machinery during the 19th century.

Sure enough, Kiplinger had one of Whitworth's linkages. That one wasn't as interesting as Peaucelliers straight line generating linkage, though. I thought for a while of using a threaded rod to push that linkage to generate longer straight line movement. The only problem with the idea is that whereas a threaded rod gives a very linear input vis a vis rotation, Peaucellier's linkage wouldn't use it like that. Then I thought... why not use a bloody pantograph to multiply the translation speed of a threaded rod?

We'll lose force and resolution by the same proportions as we increase the travel distance, but it should be easy enough to build and test. We've also got resolution on the threaded rod to burn.

With a pitch of 1 thread/mm, a very common one, we get a resolution with a 256 pulse encoder of 0.0039 mm/pulse. Pushing that up to six times that will still give us just over 0.02 mm/pulse which is a factor of 4-5 better than our target.

It's not like others haven't done similar things before, but it might solve our problem with wanting to use threaded rods with slow gear motors. :-)

CLICK!

You know what? If we hooked Peaucellier linkages on the short and long stroke nodes of the pantograph we'd have a hell of a stable motion amplifier. Wowzie!

Hmmm... If we looked at the pantograph as not being basically a stick linkages but rather extended those sticks on the vertical axis and made them planes and then slid the Mk II up and down on the vertical edge of the long stroke node plane you'd have a very different kind of reprap machine.

That's enough creativity for one evening... or morning, I see. :-s I need to catch a little sleep.

Nite all! :-D

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This would be identical using two rotating disks, one mounted on the circumferance of the other (the pivot arm could be considered two radius arms of these disks). I thought about doing something like this after the idea for the '2 motor X/Y/Z axis' using a circular work surface mounted at the top of a vertical thread.

The equations to turn line segments you wish to extrude into a form that can be consumed by a 2-disk platform is more complicated than using linear drives, but should be manageble; worst case, it can be solved numerically from the equations defining an (x,y) coordinate from the two disk angles (a,b):

x = r0 cos(a) + r1 cos(a+b)

y = r0 sin(a) + r1 sin(a+b)

dx/dt = -r0 cos(a) sin(a)(da/dt) - r1 cos(a+b) sin(a+b) (da/dt + db/dt)

dy/dt = r0 sin(a)cos(a) + r1 sin(a+b) cos(a+b) (da/dt + db/dt)

==> solve for da/dt, db/dt

==> use iterative numerical approximation to get (a0,b0),(a1,b1),... for t0, t1, t2.

There might be a way to solve these equations exactly too; I've not spent a lot of time examining them.

The equations to turn line segments you wish to extrude into a form that can be consumed by a 2-disk platform is more complicated than using linear drives, but should be manageble; worst case, it can be solved numerically from the equations defining an (x,y) coordinate from the two disk angles (a,b):

x = r0 cos(a) + r1 cos(a+b)

y = r0 sin(a) + r1 sin(a+b)

dx/dt = -r0 cos(a) sin(a)(da/dt) - r1 cos(a+b) sin(a+b) (da/dt + db/dt)

dy/dt = r0 sin(a)cos(a) + r1 sin(a+b) cos(a+b) (da/dt + db/dt)

==> solve for da/dt, db/dt

==> use iterative numerical approximation to get (a0,b0),(a1,b1),... for t0, t1, t2.

There might be a way to solve these equations exactly too; I've not spent a lot of time examining them.

I've always had the idea of making a 'robotic arm' to hold the extruder. Something similar to this (but a little more complex) to reach the "entire work surface".

I was figuring either mounting it to the 'wall' with a vertical axis to it, or mounting it to the 'floor' with the first 'segment' being long enough to ensure that the arm doesn't violate the 'extruded space' as it moves up the layers. You might be able to 'gear it up' so that it runs on a single motor (a second to pivot the arm), but I was planning on putting a motor at each 'elbo' and controlling it all via the computer... I'm essentially envisioning that welding arm you see welding car frames in that stock car frame welding robot arm footage that everyone uses...

You could take it a step further and either have multiple arms for different extruder heads (at first) or (with some design and testing) give it a 'rack' of extruder heads that it can 'pick from' as it works (much like a plotter picks the color it's going to use)

That's my 3rd reprap that I'm planning to make... the first two will run slower (though the second will be rack and pinion so it'll hopefully be fast enough), but they won't be as complex... I'm looking at them as building blocks, not only for proving the technology along the way, but to build the increasingly complex versions.

I was figuring either mounting it to the 'wall' with a vertical axis to it, or mounting it to the 'floor' with the first 'segment' being long enough to ensure that the arm doesn't violate the 'extruded space' as it moves up the layers. You might be able to 'gear it up' so that it runs on a single motor (a second to pivot the arm), but I was planning on putting a motor at each 'elbo' and controlling it all via the computer... I'm essentially envisioning that welding arm you see welding car frames in that stock car frame welding robot arm footage that everyone uses...

You could take it a step further and either have multiple arms for different extruder heads (at first) or (with some design and testing) give it a 'rack' of extruder heads that it can 'pick from' as it works (much like a plotter picks the color it's going to use)

That's my 3rd reprap that I'm planning to make... the first two will run slower (though the second will be rack and pinion so it'll hopefully be fast enough), but they won't be as complex... I'm looking at them as building blocks, not only for proving the technology along the way, but to build the increasingly complex versions.

What kind of wiggle would a robot arm holding the extruder head have, when the arm is fully extended? It would seem that the part would need to be made with very strong material (metal?)

Multiple heads can be added without 'swapping' them out, as long as X/Y axis has extra room. For the double disk solution, you could mount 6 heads in a hexagonal shape around the primary disk, and simply rotate the part to the proper head. Of course, this layout would eat up a bit of floor space. Actually, if you had removable nozzles, this arrangement would allow you to make 6 copies at once, if you simply added 5 outer disk platforms, and 5 extra extruder heads! Just think, instead of making one reprap in 20 days, you could make 6 at the same time (a spare, another spare, one for the guy down the street, one to mail to your kids/parents, ...!!) Okay, maybe that is a bit silly, but it might make sense to have two or three, to allow different color threads, different nozzle sizes, or to allow printing multiple copies of a part more quickly. Two fine nozzles, and one course nozzle would be great for this (make 2 parts at the same time, and do it quickly by using the course nozzle for infilling the big voids in the middle.)

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Multiple heads can be added without 'swapping' them out, as long as X/Y axis has extra room. For the double disk solution, you could mount 6 heads in a hexagonal shape around the primary disk, and simply rotate the part to the proper head. Of course, this layout would eat up a bit of floor space. Actually, if you had removable nozzles, this arrangement would allow you to make 6 copies at once, if you simply added 5 outer disk platforms, and 5 extra extruder heads! Just think, instead of making one reprap in 20 days, you could make 6 at the same time (a spare, another spare, one for the guy down the street, one to mail to your kids/parents, ...!!) Okay, maybe that is a bit silly, but it might make sense to have two or three, to allow different color threads, different nozzle sizes, or to allow printing multiple copies of a part more quickly. Two fine nozzles, and one course nozzle would be great for this (make 2 parts at the same time, and do it quickly by using the course nozzle for infilling the big voids in the middle.)

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