Basics 6 DoF Stewart platform

Since i didn't find a topic on this in this forum, and i don't wanna hijack wannabeaflyer2s thread i'll start a new one here.

I’ll try to explain how you can influence the rotational and translational movement of the upper platform.
Let's start with the beginning. Since we live in a 3 dimensional world, we can describe every point with 3 axes. And i think it's also important to have the same basics, in order to minimize misunderstandings.
the easiest to use is the cartesian coordinate system

There are 3 axes. x, y and z.
If we want to describe the Point P we need the values for x, y and z.

Let's change to the cylindrical system.

Here again we have 3DoF. The axis z stays the same, but than we have the radius r and the angle phi.
The same point P can be described in a cylindrical coordinate system:

z gives the height of the point P. r the distance from the z-axis and phi is the angle between point p and a reference layer.

This system is nice to use for describing the steward platform.

If we want to discuss the actuators, the spherical system match better:

We have the radius r, which is the distance to the base. than the same angle phi, which rotates around the z-axis and an additional angle theta, which gives us the angle between the axis z and the line to the point P.

Now let's come the the platform itself.

For all considerations we have identical actuators with a defined lengths and the same stroke.
With the stewart platform we can move in all 6 degrees of freedom:
- translational movement: heave sway and surge
- and rotational movement: pitch roll and yaw.
Due to the construction (symmetrical arrangement of the actuators which are fixed on 2-dimensional platforms) we can manipulate sway and surge only together as well as pitch and roll.
So we have to consider only 4 states.
With given lengths for the actuators we can change the following values:
- Radius r1 of the bottom platform
- radius r2 of the top platform
- angle theta between the fixing points of one actuator on the top and the bottom platform (and also the angle between the fixing points on the base between 2 actuators)

Lets start with 2 platforms with identical radii. The fixing points of the actuators can now only be moved with the angle phi on the bottom and the top platform. We choose the an angle of 0°:

r=r1=r2
red dots are the fixing points of the actuators.
Sideview:

With this arrangement we'll have:
- actuators parallel to the z-axis
- the most heave possible
- a good pitch&roll
- no sway, surge and yaw

So this would be only a 3DoF rig although you use 6 actuators. And if you don’t fix it in y- and x- direction you’ll soon have a 0DoF rig, due to the fact that i will collapse.
If you raise the angle phi or use different radii (r1≠r2) you will decrease the possible heave!!

Next step is maximize the pitch/roll. The actuators are still fixed with the same angle phi 0°. If you reduce the radii r1 and r2 together you will increase the pitch/roll. You'll end up with all actuators being next to each other in the middle. This is the best way of having the most roll and pitch. with an infinitesimal thin actuator you could tilt the platform with 90° ^^ But after a certain radius it's a stupid way to increase roll and pitch further. Your rig will get less stable and the motors will consume more power, remember: W=F*a, the distance a will get small, so the force F will be bigger in order to achieve the same tilt angles of the platform (work W).
One other way to increase the possible roll and pitch is to decrease the radius r2, while using the same radius r1.

But this will work only unto a certain angle theta. than the roll and pitch will decrease again, we'll come to that later.
Increasing the radius r2(>r1) and the angle phi will result in a lower roll/pitch movement.

How to maximize sway and surge? If you look from the top to the rig it's the easiest way to imagine. You'll have the maximal surge and sway if your upper platform lies IN the bottom platform in the x-y-plane

view from the side:

The radii have to be chosen depending to the length of the actuators. You won't be able to heave and roll and pitch your platform, since you work in two dimensions. This is, because the angle theta is 90°, and cos 90°=0, so now pitch and no heave. I know, if all actuators will move out, it will heave, but maybe downwards. If you now raise the radius r of the upper platform, your upper platform will be higher and you'll reduce the possible sway and surge. Sway and surge is independent from the angle phi, if symmetrically arranged. You can rotate them around there fixing points on the base and so change r2.
Simply said:
The higher the angle theta is, the more sway and surge you'll have, but it'll reduce the heave. At first increase roll pitch and than decrease it again.

Let's introduce the 6th DoF, yaw. Since we didn't use the angle phi yet, this is the one we need to rotate our platform.

Example with same radii.
View from the top:

The higher the angle phi, the higher the rotation yaw of the upper platform, easily said.

Here an example with different radii r1 and r2 and the angle phi.

just to illustrate it. but be aware, the angle theta is 'wrong' in this pic i mean i explained it different. it should be between the actuator and the axis z, not the base. but of course you can use it this way...

i still may be wrong somewhere so no guarantee

Good writeup

Thanks @Tim McGuire

maybe one more point, since this can also become important. Raising the angle theta results in a higher power consumption.
The resulting force F_res=F/cos theta. (If you take 'the other' angle theta--> F_res=F/sin theta)
With a steep actuator the resulting force will be around the the same as the force from the top (weight, acceleration, deceleration)

If you have a bigger angle theta

you'll have more work for the motors to do.
An angle of 60° results in the double load.

yaw should pivot from the center and traction loss should pivot from the front. although yaw can be set up to mimic tl by reducing the movement at the front of the rig thus forcing the pivot (center of rotation) forward. this also creates a larger radius of movement.
edit:

@bruce stephen
Thanks for pointing this out. As long as i haven't my own rig, this will all stay a little theoretical^^ Deleted the point, don't want to confuse somebody

great info , a lot of guys heading towards 6 dof platforms and this info fills some areas that we normally guess at , so aJust Saying thanks for a well written explanation of some of the consideration required for platform design .. I don't profess to understand it all Trigonometry scared crap outta me at school but a few re-reads and things will fall into place Cheers

I have found a few nice papers on the subject. (Warning: Lots of Calculus, mind melting stuff. I'm digging through trying to find the meat of it that we can use for our purposes...)

http://robotics.caltech.edu/~jwb/courses/ME115/handouts/ReviewStewartPlatform.pdf Caltech paper on Stewart platforms

http://image.slidesharecdn.com/g363...-parallel-manipulator-3-638.jpg?cb=1440225870
I don't know who made this paper but these formulas look useful.

http://dnc.tamu.edu/drjunkins/yearwise/2006/conference/GNC_2006_stewart_platform_bai.pdf
A paper dealing with solving for error... this would be more useful for hyper advanced mathypants users.


https://www.researchgate.net/public...trol_System_of_Parallel_Kinematic_Manipulator
"Design and Control System of Parallel Kinematic Manipulators" Looking good! I bet there's something in here that can be boiled down to some sort of code that can be used for all of us. I'll keep us posted on what I find out...

Great stuff
is already printed when i finished my first rig, i will definetly dive into this matter again and i'm sure your links will be usefull

Thanks for starting this Thread!

I have to say, I am pretty surprised how many great platforms were being built without ever fully considering the effect of the different parameters

Maybe we can come up with a few simple "rules" for the builder to better analyse misbehavior in their platforms.

Like...

1. More Theta --> more power required for heave(especially when platform is at lower edge of the envelope).

2. Pitch-roll-yaw capabilities get better with the ratio upper_platform_diameter divided by stroke.

3. To reduce footprint you can make the lower platform smaller but be aware of stability problems. --> bolt actuators to the ground

4. If unintended axis-coupling occurs, the reason might be that either the upper and/or lower platform might be un-equilateral.

...something like this.

we are also still missing a 6dof maths plugin for stewart platforms. we tune to correct at this point. its close but still incorrect.

There is no 6DOF plugin for SimTools? Really? I am surprised. I am still in a rather early stage of my build, but I simply assumed that one was available. Didn't Thanos publish one for use with his AMC1280?

Back on Topic:
Does anyone know, if a Stewart platform is an over-determined system? I am almost(!) sure an "ideal" Stewart platform is neutrally determined.

- "Neutrally determined" meaning that for every possible actuator position there is a valid solution to the (presumably) 6 equations of position.
- "Ideal" in this case meaning that the connecting points of two neighbouring actuators onto the platforms (top and bottom) are in the same position. This is not possible in real life. Like here:



I can virtually see @My.stAr scratching his head when he reads this... But I really DO think these are the Basics of a 6DOF Stewart-Platform.

Dirty

Thanos controller isn't simtools. Im not sure about his controller. I use arduino and smc3.

Simtools can do a good job of moving a 6DOF rig as is, but is not as exact as using software with the proper math. I have access to BFF software and purchased a 1280 as it is compatible with it. BFF uses full 6DOF math and some excellent mixing and washout algorithms. Since I added a G-System to my rig all the shortcoming of Simtools are not an issue to me. Unfortunately 6DOF math is just one part of the simulation puzzle and IMO really is not as important as many may think. From my experience, good mixing (Surge/pitch and Sway/roll) and washout algorithms are more important if you don't have a G-system fitted. Note that BFF has limited games support and can only support six axis's (I need more than six to run my G-systems), which is why I moved away from it.

First, this is a very helpful thread for newbies such as myself. Thanks for all the details.

I had a quick question - what would a "good" angle (theta) be? I understand that "good" is relative, and would depend on a multitude of factors. What I am more interested in is what ranges of theta have worked best for 6 DOF builds based on the Stewart platform.

Thanks in advance!

Hello guys,

I can understand some aspects of this post, but I would like to have a better understanding through visualization of the 6DOF mathematical schema. Is there something available on the internet or with in this community in the form of videos, animations, or GIFs that could help with visualization?