New Inverted (Grounded) Delta Design

Howdy folks!

I've been a regular in the #RepRap IRC channel for a few months now and caught the design/building bug which matches my skillset well. I got interested in the GUS Simpson Nicholas Seward spearheaded a few years ago and Solidus Labs' GDR Dogleg variation. Reading through the design evolution and analysis threads, I saw the potential for solving some of the early problems in a different way than with the GUS. The cantilever math and simplicity appeals to me more than the DCLJ approach, and I also wanted to try solving it without a string-n-spring actuation so here's how I approached it.

As I see it, the main challenge of the simple cantilever is driving the "elbow" without introducing a lot of inertia from motors placed high or a mechanism that severely restricts the build volume when the arm extends over it. To that end, I started thinking about this



iteration of Seward's design, and the relationship between cantilevers and similar linkages. As is, this setup is similar to a scissor-linkage, and we could drive it by operating the links on the "back" of the arms (past the pivot), but with poor mechanical advantage and high inertia. Unfortunately, any kind of belting has to deal with the fact that as a triangle with a fixed side traverses a circle, its perimeter isn't constant, so driving it is hard. I worked towards figuring out a linkage that would work (while being elegant and simple!!) for quite a while, and just a few days ago realized there's an obvious solution!

Here's an early diagram of the idea (ignore the NEMA 14, didn't work out lol):


The important points:

• Use a 4-bar parallel linkage to move the actuation to the bottom -- less inertia and no impingement on build volume
• Use gearing (or belting) to increase torque on the parallel link for greater motor resolution and mechanical advantage
• Same operation as the scissor, just driven remotely and with the superfluous vertical extension removed

I felt like I was really onto something, so I started testing the linkage idea. Here's a short video with a Herringbone gearset I made:
YouTube Link

Seemed like the attachment and gearing was sound, so I modeled it up to test the whole arm assembly, which you can see here:
YouTube Link

Nothing left to do but print out a prototype and test it with a motor. I borrowed one from another printer and it performed amazingly well!
YouTube Link

(pay no attention to the vaper wafting around, I promise nothing was on fire ;P)

This was with a 45Ncm or so NEMA 17 driven at F500, 1k 2k 4k 6k 8k 10k 12k. I didn't capture audio but there's no skipping or lurching (and it's pretty quiet too!). I did use a belt and pulley for this particular test, but gears work just as well. At this point I'm printing out the other two arms and have a full CAD model built and jointed. You can view the model here and I uploaded the current part STL files here for now.

I figured it's about time to get some feedback from folks on here, and maybe we can even find where Seward's been hiding and get him to chime in The work he did on GUS is fantastic and has greatly informed many of my design choices. There's obviously a lot left to do, but I wanted to share the work I've done because there are several workable variations of linkage actuation that are dead-easy and may have other benefits over this one, but it's a great start and I feel a vastly simpler system to design around and calculate math for.

Pretty interesting design!
What steps/mm could you achieve with the 20/60 gears and a common 1.8°stepper?

Quote
o_lampe
Pretty interesting design!
What steps/mm could you achieve with the 20/60 gears and a common 1.8°stepper?

Thanks, I and a lot of other folks think so too! Quick update and I'll try to answer your question.

I have three arm assemblies built and ready to go as of this morning:



And my motors arrive today, so I'm pretty excited. Still have to figure out the "wrist" design a little better, but we're almost to a moving prototype

For the math, a common 1.8º/step stepper is 200 steps/rev, and the 3:1 ratio makes that 600 steps/rev, or 0.6º/step. In operation, where the shoulder joint will be free to move, I'm using Seward's simple inverse-kinematics seen here:





So for my case, if we assume y and z are held constant at 0, this simplifies to x = sqrt(2*radius^2*(1-cos(theta))). My arm radius from the elbow is 150mm currently. Considering a change in 0.6º for x, from an angle of 45º to 45.6º, that should work out to 1.45mm/step.

It's not great, and I have some ideas on how to improve the resolution without sacrificing the design criteria. A 0.9º/step motor would result in 0.73mm/step, for a start.

It lives!




Short Video

At least an early alpha incarnation of it

Lots of work left to do, but my controller works, motors work, old ATX PSU I chopped into a bench supply 10 years ago works... off to a pretty good start for first prototype build!

Also, I was off on the IK simplification above. We can only consider one axis for an arm, yes, but that ignores the shoulder-arm and shoulder-base rotations that occur in conjunction with the elbow angles as the arms move together. I took a stab at the correct IK in some code last night but didn't flesh it out fully. I'll be working on that today and will update with video of some linear motion soon. I'm just writing a Gcode generator in Python until I get the math right, then I'll try to implement it in Smoothieware, so the real fun can begin

What if you'd skip the parallelogram arms and use a leadscrew to move the upper vertical arm directly?
Don't know about the added complexity involved, but the resolution would be better.

Well the primary reason for the parallelogram/parallel linkage configuration is so the actuator doesn't have to deal with triangularly-varying angles and distances as the elbow angle changes. I'm not sure I see how to attach a leadscrew in a way that wouldn't physically bind as the angle varies without mounting the motor on a proportionally-varying link. Care to diagram what you're suggesting if you think you see something novel?

Quote
o_lampe
What if you'd skip the parallelogram arms and use a leadscrew to move the upper vertical arm directly?
Don't know about the added complexity involved, but the resolution would be better.

Yeah I was thinking of this for the scara, which is similar to the arms here if flipped 90 deg

Quote
MechaBits

Quote
o_lampe
What if you'd skip the parallelogram arms and use a leadscrew to move the upper vertical arm directly?
Don't know about the added complexity involved, but the resolution would be better.

Yeah I was thinking of this for the scara, which is similar to the arms here if flipped 90 deg

Sure, for something with the orientation flipped and SCARA-like where the arm extension is in the plane perpindicular to the hotend, that'd make sense. Unfortunately I don't see a great way to actuate this elbow with the motor low, without using a winch/stiff-cables, and without encountering triangle-tracing-a-semicircle type geometries, unless you designed and printed elliptical wheels/cams for belts to differentially roll on... something goofy like that might get around it through geometry.

Otherwise, we want to have the actuation match the elbow joint, but without putting it *on* the elbow joint. A parallel linkage seems like the elegant way to do it, and it's certainly possible to create gear trains with compound gears/pulleys to up the resolution, or a rack/pinion like Solidus GDR for much higher tooth counts in the same relative space.

Another quick update, I completed the Inverse Kinematics using some very clean Python code to generate alpha/beta/gamma "distance" gcode which corresponds to arm angles. Once I get the physical prototype a little more refined in design, I'll figure out how to port it to Smoothieware.

The code is here at the moment. There's some vestigial scaling trappings in there too, trying to figure out how to translate to a cartesian-configured firmware, and for the most part it's working fine. Just having some minor binding problems due to the bearing interfaces in the prototype parts, so I'm focusing on fixing that now that I completed IK code.

Feel free to ask questions if you don't understand what I'm doing exactly, but I comment my code pretty thoroughly so hopefully it's clear enough to glean the overall strategy.

Is there anything on this page that can help [www.thingiverse.com]
doubt it being delta based, you seem to have things figured out, cool anyway.
I do like parallelograms from my time on a real drawing board
like the arms, i have a half dismantled tripod which looks pretty similar
and yours provides a great backbeat for my robotic band

Edited 1 time(s). Last edit at 10/29/2016 05:58AM by MechaBits.

Added a page on thingiverse to track the project as well, and so others can see the media and model files more easily.

Maybe the cycloidal gearbox would be helpful here? You might end up with three NEMA14 pancakes like you wanted to in the beginning...

Quote
o_lampe
Maybe the cycloidal gearbox would be helpful here? You might end up with three NEMA14 pancakes like you wanted to in the beginning...

Oh man I love cycloidal gearboxes and eccentric drive systems. But as far as I know, there's not a great/cheap way to fix these disadvantages: [en.wikipedia.org].

Great idea though! Maybe a compact, compound planetary gearbox might be doable. I'm thinking about possibilities for sure, but I want to keep vitamin count low and maximize printability without sacrificing ease of math, assembly, part-sourcing, etc.

Thinking about it, I *do* see a way to add a planetary gearbox in here, and I think I can do it in the same space the current spur gear is using!

Theory is to halve the height of the spur and pinion, make a ring fixed to the lower link underneath it, attach the sun to the current spur, and attach the planet carrier to the parallel arm. If I keep the same 20T count on sun and planets, and 60T on ring, driving the sun and fixing the ring should get me 4:1 reduction ratio on the planets. The compound ratio therefore should be 12:1 reduction from motor to arm. That's pretty damn good! Think I'll work on trying out that design and see if I can come up with something sturdy and workable.

Ok, new video! After tuning steps/mm and accel settings in conjunction with my IK code that pretends to be Cartesian, I was able to vastly increase the smoothness and behavior of the prototype.

YouTube Link

Please note that without endstops, I'm telling the IK where the arms are to start, but I'm manually positioning them to roughly that spot, which appears to account for the minor Z variations during X/Y moves visible in the video. That and the fact that these long, 20% infill Carbon Fiber PETG arms have a bit of flex to them, which might account even more for the variation as the arms are constantly pushing and pulling against each other during moves.

I plan to reprint the entire assembly in a stiff PLA now that I have the dimensions/clearance/tolerance dialed in, and see where that gets me on the IK side. Gotta figure out an endstop design too and wire that up for extra credit!

Quote
Apsu
Please note that without endstops, I'm telling the IK where the arms are to start, but I'm manually positioning them to roughly that spot, which appears to account for the minor Z variations during X/Y moves visible in the video. That and the fact that these long, 20% infill Carbon Fiber PETG arms have a bit of flex to them, which might account even more for the variation as the arms are constantly pushing and pulling against each other during moves.

Nope, I was wrong. My IK isn't taking into account the radial changes in the arm shoulders as they translate out of the home position. That's why the Z deviations are arcs. The circularity got me thinking about it more carefully and it's easy to see. I think I may redesign the shoulders to not be offset matching the wrists, because the coaxial effector and the DOF restriction due to the intersection of all three arms means the effector is always a simple fixed X/Y offset from the forearm joints (as a vector in the same Z rotation as the arm).

As a bonus, with the 305mm 2020 extrusions I have, that should expand the build envelope since those extrusions are too big for the rest of the arm dimensions I chose. Just a side effect of my printer's 210x250 bed and what 2020 I had on hand.

Alright, I came up with a two-piece shoulder design and a new frame bracket for it to sit in.







The frame bracket has open ends now so the extrusions can be slid farther through the bracket to resize the arm locations as necessary given what extrusion you have on hand, or just want to resize it to adjust build envelope.

Alright, after setting up the new brackets and shoulders I've realized that my dimensions aren't quite right for the general size of frame I want to achieve, and for the arm angle limitations due to my current motor placement. Here's a picture of the current setup



The arms are just too long given the -40º to +40º arm angle range the motor placement limits me to, so I think I'm going to model up a simplified version (so the CAD joint solver doesn't lose its mind like it does in the full model with gears and motors and whatnot ) and see what kind of proportions and angle ranges make sense. I'm also not *positive* that the IK with coaxial shoulder joints will be easier, but I can always switch back to offset pretty easily.

Any opinions or analysis or alternate ideas are fully welcome! I'm considering moving the motors to free up the arm angles a bit, too, but not sure where to put them yet.

Yep, it's all about the elbow/shoulder angles. I whipped up a quick motion simulation with the relevant joints in place and it moves as expected over a relatively large volume (compared to the base).


YouTube Link

As you can see, plenty of clearance for a 100mm tall 100mm radius cylinder. The trick is that the elbow and shoulder joints need to hit up in the 60-70º deflection range. Which means I definitely need to move the motors somehow.

(EDIT: Uploaded a better quality video)

Edited 1 time(s). Last edit at 10/31/2016 06:39PM by Apsu.

Alright, did some rendered animated joint studies. These clearly show that the basic idea of the shoulders and straight cantilever can reach a significant portion of its volume. It's nearly an armlength-radius x armlength-height cylinder.

Study 1
Study 2

Then I thought about the problem a lot more, and I think I've come up with a really good solution to motor placement and actuation system, allowing for much sharper arm angles.



The idea being to change the parallel linkage to have equal arms on all sides, then run a single belt in a pattern that actuates it in both directions, using it as a scissor link. It will require some biplanar belt paths and/or twists in the crossing... but it's simple, clean, and gains mechanical advantage through double lever arms without gears and big wheels, or block and tackle! I mocked it up on my current arms and the theory/geometry is sound. I'll have to alter and print some parts to really test it, but this seems completely reasonable to me.

The last sketch is pretty simple and straight forward, but it also has a drawback:
Lengths of the hypotenuse of P_P triangle specifies the "gear ratio" and thereby accuracy, but it is very low when you print on the bed and the arms are almost folded completely. ( Jeez, I hope you got my point? )
Also the longer it is, the more inertia the motor will produce.

With a ( planetary ) gearbox you can choose the ratio but keep the stepper as close as possible to the twist point.
I've printed some planetary gears in the past and they don't run very well. Milling the gears would be a nightmare, but the cycloid gears are pretty easy to mill.

The drawback about cycloid gears you mentioned earlier are IMHO non relevant, like "the gears can't be driven backwards" actually is an advantage here. You might be able to idle the steppers between moves, which keeps drivers and motors happy.
The cycloid gears I linked comes with dual eccentric gears and I didn't feel any significant vibrations. ( They might reduce max. RPM though )

.

Edited 3 time(s). Last edit at 11/01/2016 06:38AM by o_lampe.

Quote
o_lampe
The last sketch is pretty simple and straight forward, but it also has a drawback:
Lengths of the hypotenuse of P_P triangle specifies the "gear ratio" and thereby accuracy, but it is very low when you print on the bed and the arms are almost folded completely. ( Jeez, I hope you got my point? )
Also the longer it is, the more inertia the motor will produce.

I thought through this pretty carefully and tried to confirm with some physical mockups as well, and I think you are incorrect about the resolution being low, but explaining why I don't think it's an issue is going to take some words and discussion. Let me try to walk through it clearly

First, the reason why my diagram points out that the relationship between motor and pulley never changes is not just because it's part of understanding that a fixed-length belt will work, but also because it's key to understanding the relationship of the scissor to the forearm/effector motion. If you think about it in terms of relative forces and driven vs fixed members, we can see that the vertical main arm is the most "fixed" by the wrist/frame/surface attachments. The driven members are the parallel vertical arm and the back portion of the forearm -- the top-left triangle.

This is important because we can see a few things immediately. We know we're driving the top-left joint of the scissor at a constant rate, because the belting is parallel with the linkage and at a fixed length, but more importantly, the belt is *not* then attached at the hypotenuse of that triangle, but rather the apex of the opposite triangle formed in the bottom-right joint. So the heights of both triangles are changing at the same time, at the same rate.

EDIT: This might not be clear enough why this rate changes constantly. It's because of the parallel linkages. Pulling opposite corners together distributes the forces to the adjacent joints, and because the links are all equal length, the distribution is equal, causing the corners to move towards each other in a straight line at a constant rate when pulled at a constant rate, because that force drives the adjacent links at whatever rate you're pulling. It's the result of the rigid jointing and link length ratios.

Now, you're correct that this causes the hypotenuses (hypotenii? lol) to change at a varying rate, but we don't care about that. The reason why we don't care about it is because we can show that due to the scissor jointing (P = P), the driven links are changing angles at a constant rate, which means the arm/forearm "elbow" angle will change at a constant rate as well. You have to keep in mind that as a hypotenuse changes, the distance between motor and corner pulley stays the same, and also that there's two hypotenii which change equal and opposite amounts.

This is the essence of the scissor, and you can see it in scissor lifts. They usually have "legs" extending past the last joint, and drive them by pushing/pulling those legs towards each other, but there are many scissor lifts driven by pushing/pulling on the joint itself, which is exactly what we're doing here. The rate of actuation is constant, and results in constant motion of the scissor. It's really cool

So, tl;dr, we should get constant angular rates on the elbow, constant mechanical advantage, and fixed belt lengths at all scissor extensions.

EDIT: I should note that constant angular rate doesn't mean that we can ignore the ratio of driven-link length to forearm length. A 3:1 forearm length ratio definitely means 3x less resolution at the effector joint, so that length matters and deg/step matter and so on, *BUT*, the constant angular rate does mean that the resolution doesn't change regardless of the angle. That gives us the opportunity to adjust proportions and ratios and add gearsets or [insert cleverness here] to affect the angular resolution, without having to calculate and compensate for varying angular rates.

Quote

With a ( planetary ) gearbox you can choose the ratio but keep the stepper as close as possible to the twist point.
I've printed some planetary gears in the past and they don't run very well. Milling the gears would be a nightmare, but the cycloid gears are pretty easy to mill.

I've had great success with printing gears I've designed, like the ones in the prototype I've built so far, and I suspect they'd make great planetary gears, if a little tricky to insert without printing-in-place (the source of printed planetary sets not running great, imo ), but that's solvable with creativity (like splitting the annular/ring gear into two halves and clamping.

Quote

The drawback about cycloid gears you mentioned earlier are IMHO non relevant, like "the gears can't be driven backwards" actually is an advantage here. You might be able to idle the steppers between moves, which keeps drivers and motors happy.
The cycloid gears I linked comes with dual eccentric gears and I didn't feel any significant vibrations. ( They might reduce max. RPM though )

That's a fair point. I was thinking about the drawbacks in a different light but I think you're right on here.

Overall, I'm going to pursue this belting since it's pretty painless to setup and actually test. I had to get some 5mm idler/motor pulleys in correct tooth counts, so those will arrive tomorrow. I'll print some replacement vertical arms and mock up a prototype mechanism in the next day or two and post a video to demonstrate. That should hopefully answer any questions about the whole thing

Edited 5 time(s). Last edit at 11/01/2016 10:47AM by Apsu.

Thanks for taking the time to explain it to a limited german mind

But you mentioned the scissor lift which was actually what I had in mind too, when I wrote my last reply.
A car jack is similar to a scissor lift, right?
When you try to lift a heavy car ( and they are all overweight nowadays ) you feel it very difficult to turn the crank at the beginning. Later, when the jack is almost fully extended, you have to make several turns to achieve the same lift like before with only one turn. This is what made me think the "gear ratio " and thereby the accuracy is not constant in your design.

Quote
o_lampe
Thanks for taking the time to explain it to a limited german mind

But you mentioned the scissor lift which was actually what I had in mind too, when I wrote my last reply.
A car jack is similar to a scissor lift, right?
When you try to lift a heavy car ( and they are all overweight nowadays ) you feel it very difficult to turn the crank at the beginning. Later, when the jack is almost fully extended, you have to make several turns to achieve the same lift like before with only one turn. This is what made me think the "gear ratio " and thereby the accuracy is not constant in your design.

Sure, that's an easy system to analyze and consider the behavior of. I got a little worried about my assumptions so I went ahead and did some math and got some numbers to look at. You're right that there is some angular rate variation, but it's actually very minimal except in the last few degrees of full extensions in either direction, which I won't be seeing anyway without pantograph style overlapping links. But, in the interest of accuracy, here's what I calculated and how:

I wrote some code to "drive" the triangle height at a fixed rate, used the height to obtain the isosceles hypotenuse, then used the hypotenuse to obtain the angle. I kept track of previous angles to get the difference (angular step) for each height step. Then I computed the difference between angular steps, to get deviation over the range of angles produced. I captured the data over 70 degrees of extension from vertical (90 deg), from 20 to 160, to match the motion studies, then put it into Wolfram Alpha for analysis.

Here's a parametric plot



And the numerical analysis



First it's important to note in the plot the scale of angular step deviation is incredibly small. The largest deviation is 0.0018 degrees, or < 0.2%.

Second, the standard deviation is overall about the same, 0.0018.

These numbers get larger near the extremes as mentioned, but in the "middle" it's actually quite small. I'm not sure this will be a practical issue or can't be compensated for with minor math based on the desired angle. I'll have to do some more thinking, math and building to see what shakes out. Thanks for the discussion

Another way to think about this without compensating is to compute the height for the angle, and factor in the deg/step of the motor and teeth/radius of the drive pulley, to see what the minimum step/effector angle is.

Thinking it through out loud, I think it looks something like this:

1.8deg/step motor with 20T pulley with 2mm pitch GT2 belt, using standard steps/mm formulas with worst-case full steps is 0.2mm/step, or 5 steps/mm. Gets even better with 12T pulleys, but we'll start on the "worse" end to be conservative.

We can get the scissor angle from the height by using the following two formulas:

hypotenuse = 2*sqrt(sides^2 - height^2)
angle = arccos(1 - (hypotenuse^2 / (2*sides^2)))

We can also use this formula for height from the angle:

height = sides * cos(1/2*angle)

Lets say the side (fixed links) are 100mm, and we start in the worst-case area for the angle, 20deg:

height = 100 * cos(1/2*20) = 98.481mm

Then we'll decrease the height by a full step of 0.2mm, to 98.461mm, and compute the angle change:

hypotenuse = 2*sqrt(100^2 - 98.461^2) = 36.926mm
angle = arccos(1 - (36.926^2 / (2*100^2))) = 21.279 deg

So the worst-case change in angle is 1.3 deg for a 1.8 deg motor full step, meaning the angular resolution on the elbow is at worst better than the motor directly.

The best case is moving near the 160 deg angle, so I ran the same calculation for the height at 160 + a full step, and the result is:

159.233 deg, for a difference of 0.767 deg, better than a 0.9 deg/step motor.

Seems like this is worth doing, and since the angular error is less than the positioning error of common stepper motors (5% angular deviation), I think this is perfectly reasonable. The fun part is we can use 12T pulleys and microstepping, and try variations like moving the motor to the arm center with 60T pulleys on the joints to make use of a 5:1 gearing without changing any other behavior. Something like this



There's lots of improvements that can be made, but since the worst case is still "decent", I'm going to start building it and testing

Edited 3 time(s). Last edit at 11/01/2016 02:45PM by Apsu.

Well, that actuation attempt was a bust I'm not that great at geometry, at least in an intuitive sense, and I totally forgot/missed the differing rates of hypotenuse expansion/collapse for the opposite corner pairs. Belts do not, in fact, work

I think the next step would be to go back to gears and switch the motor position to the back of the parallel arm, much like the belted layout, but there are still potential clearance issues between a gear/pulley and the other arms, as well as the attachment point on bottom/top link restricting angular max/min. Damn you triangles!

I might take a break from this for the moment and see about working through a Tripteron printer model. Still has cantilevered arms but all linear actuation and beautiful kinematics/build volume. I'll start a new thread if I get somewhere. In the meantime, feel free to let me know if you see a better way to actuate these delta arms without giant wheels or DCLJs shaped like wheels. I'll even consider clever or elegant winch systems if they make sense.

To bad it didn't work out

TBH, I understood only 10% of the math you used ( but blamed it entirely on my bad english_tech-talk skills ) I can only judge geometry by gut feeling and it failed me more than once!
I wish you'd have shown a video of the belt failing, to understand what went wrong and to keep other viewers from following a ( possible ) dead end street.

I could make a video if necessary, but if you look at the last diagram I posted, with the crossed belts, what happens is very simple. As one belt moves closer (and the other end of it moves away), the two speeds they each move is not the same, because they're crossing the hypotenuse of each triangle half inside the scissor linkage. This means the belt either keeps tightening until it binds, or it loosens and slips, thus unusable.

I somehow didn't realize they don't change at the same rates as each other, not sure why I didn't think about that fact, but there you have it

Edited 1 time(s). Last edit at 11/03/2016 07:28PM by Apsu.