Spherical robot arm

On top of the base (the omniwheel platform and the cabinet on top of it), there will be two robot arms.

These will be standard spherical robot arms, finally at the end of them there will be the gimbal and connect and onto that any tool head can be attached and all objects can be moved with the connect.

The Blender file on IPFS: https://ipfs.io/ipfs/QmZuAkDBxG8QeotCv5by1R8GSXcZZE1X8NAzURLvg7iwqt?filename=complete7.2.0221%20-%20render.blend

Design
The spherical coordinate system is described here: https://en.wikipedia.org/wiki/Spherical_coordinate_system

With the transformation matrix being:

$$\begin{array}{lcl} x = Radius * cos(\theta_{angleaboutzaxis}) * sin(\theta_{angleaboutxaxis}) \\ y = Radius * sin(\theta_{angleaboutzaxis}) * sin(\theta_{angleaboutxaxis}) \\ z = Radius * cos(\theta_{angleaboutxaxis}) \end{array}$$

And the reverse transformation being:

$$\begin{array}{lcl} Radius = \sqrt{x^2 + y^2 + z^2}\\ \theta_{anglewithxaxis} = arccos(z/r)\\ \theta_{anglewithzaxis} = arctan(y/x) \end{array}$$

The last one is the angle between the segments. There will be a two segment version and three segment version. In either case the Radius = hyp of the triangle formed:

law of cosines https://en.wikipedia.org/wiki/Law_of_cosines and the law of sines https://en.wikipedia.org/wiki/Law_of_sines

We use the law of cosines here:

s1 is known, s2 is known, only hyp is not known

We control the angle between s1 and s2 and we get:

$$hyp^2 = s1^2 + s2^2 - 2 * s1 * s2 * cos(anglebetweens1ands2)$$

If we add another segment it is the same thing except that we now have two angles and more control over the arm, not needed but can give more flexibility, especially in hard to reach places.

Details
There will be three stepper motors for operating each of the axis, the first one will be attached to the base gear it will have a 5:1 gear ratio.

The second will be on the base and will operate the first segment of the arm, this one will have a 5:1 planetary gearbox and will have a 2:1 pulley ratio.

Finally the last one will be operating the last segment and will have a 5:1 planetary gearbox and a 1:1 pulley mechanism.

The pulley will be operated by ropes and will be a cheap and hopefully effective way of operating the segments of the robotic arm.

At the end of the arm will be the last piece that will connect to the gimbal which will rotate the connect which will hold the piece or the toolhead.

Electrical motor

Gearbox

Torque transfer and pivot

idler pivot

Base rotation