A few notes:
I like simplicity so I have done a few things in my design to simplify the math. What Annirak has written above will cover any generic Simpson with a concentric extruder. (Off axis extruders aren't that bad but let's not confuse the issue.)
For the current iteration...
The distance from the shoulder axis to the humorous connection is equal to the distance from the extruder axis to the forearm connection.
sx=-nx
All the forearms connect at the same plane
nva=nvb=nvc=nv
You can pretend that the shoulders are lower by the z distance from the tip of the nozzle to the forearm connection plane.
Ba_z'=Ba_z-nv
Bb_z'=Bb_z-nv
Bc_z'=Bc_z-nv
Ha' = P-Ba'
Hb' = P-Bb'
Hc' = P-Bc'
That leaves you with...
la = len(Ha')= sqrt( (Ha_x')^2 + (Ha_y')^2 + (Ha_z')^2)
lb = len(Hb')= sqrt( (Hb_x')^2 + (Hb_y')^2 + (Hb_z')^2)
lc = len(Hc')= sqrt( (Hc_x')^2 + (Hc_y')^2 + (Hc_z')^2)
Given that Ba', Bb', and Bc' are all constants I get a total of 15 add, 9 multiply, and 3 sqrt. That is a bit of an improvement.
The Annirak Drive or the Proportional Gear Joint Drive both make the relative rotation of one of the steppers proportional to la, lb, or lc.
My plan is for all of this to be programmatically solved for after calibration. Calibration will consist of pushing a few buttons on the screen to get the nozzle to a position indicated by a picture on the screen. (But if you can't wait, the math is above and I have a rough gcode preprocessor on thingiverse.)
Not that you can't fit this in firmware but I personally do all the inverse kinematics with a gcode preprocessor. The reason for this is that is I am already planning a monolithic software program for Simpson and Wally. (Calibrate, run, slice, etc all from one program.) This program will make the gcode preprocessing invisible to a user and it will make it easy for me to adapt to crazy new designs that I have. The firmware that I use is vanilla Repetier. This also has the benefit letting Simpson and Wally share the same firmware. (One downside is that the calibration data is stored on the host. However, one could modify the firmware to store calibration data or you could put it on a flash drive. I imagine that most people don't typically move a 3D printer from computer to computer.)
I like simplicity so I have done a few things in my design to simplify the math. What Annirak has written above will cover any generic Simpson with a concentric extruder. (Off axis extruders aren't that bad but let's not confuse the issue.)
For the current iteration...
The distance from the shoulder axis to the humorous connection is equal to the distance from the extruder axis to the forearm connection.
sx=-nx
All the forearms connect at the same plane
nva=nvb=nvc=nv
You can pretend that the shoulders are lower by the z distance from the tip of the nozzle to the forearm connection plane.
Ba_z'=Ba_z-nv
Bb_z'=Bb_z-nv
Bc_z'=Bc_z-nv
Ha' = P-Ba'
Hb' = P-Bb'
Hc' = P-Bc'
That leaves you with...
la = len(Ha')= sqrt( (Ha_x')^2 + (Ha_y')^2 + (Ha_z')^2)
lb = len(Hb')= sqrt( (Hb_x')^2 + (Hb_y')^2 + (Hb_z')^2)
lc = len(Hc')= sqrt( (Hc_x')^2 + (Hc_y')^2 + (Hc_z')^2)
Given that Ba', Bb', and Bc' are all constants I get a total of 15 add, 9 multiply, and 3 sqrt. That is a bit of an improvement.
The Annirak Drive or the Proportional Gear Joint Drive both make the relative rotation of one of the steppers proportional to la, lb, or lc.
My plan is for all of this to be programmatically solved for after calibration. Calibration will consist of pushing a few buttons on the screen to get the nozzle to a position indicated by a picture on the screen. (But if you can't wait, the math is above and I have a rough gcode preprocessor on thingiverse.)
Not that you can't fit this in firmware but I personally do all the inverse kinematics with a gcode preprocessor. The reason for this is that is I am already planning a monolithic software program for Simpson and Wally. (Calibrate, run, slice, etc all from one program.) This program will make the gcode preprocessing invisible to a user and it will make it easy for me to adapt to crazy new designs that I have. The firmware that I use is vanilla Repetier. This also has the benefit letting Simpson and Wally share the same firmware. (One downside is that the calibration data is stored on the host. However, one could modify the firmware to store calibration data or you could put it on a flash drive. I imagine that most people don't typically move a 3D printer from computer to computer.)
