Delta geometry

Geometry

A 'delta' robot or printer is a robot where a platform is maintained by three pair of arms set in a triangle. The pairs of parallel arms maintain the horizontality of the platform and the movement of these arms displace the platform in the three dimensions. Many solutions exists, but a few are used practically.

There are two principles to displace the arms pairs: 1/ Each arms pairs are installed on a main articulated arm, the movement being generated be the rotation of this main arm. The geometry is the rotational delta or 'Clavel delta' from the name of its inventor. The principle is widely used on 'pick and place' machine in a wide range of industries, from electronic to food. This system can be found in the Delta by 'Energetic' or on the Firepick printer.

2/ Arms pairs are attached to carriage sliding along parallel rails. This geometry is called 'linear delta' and is the most frequent type used in 3D printer world, the machines originating the movement being the 'Rostock' and the 'Kossel'. The term 'parallel delta' shall not be used as all robots with parallel arms are called [parallel robots].

There is another solution without rigid mechanics which is to suspend the platform to wires. There are a few examples and notably the 'Skydelta' or this one

The linear delta kinematic calculation is simple because the carriage follow a straight line, so the horizontal movement of the platform is linked to the vertical movement of the carriage by Pythagorean theorem (which say that the square of the diagonal of a rectangle triangle is equal to the sum of the square of its sides). Here the diagonal is the arm length, constant, the vertical branch is the relative vertical position of the platform and carriage, the horizontal branch is the relative horizontal position of the platform and carriage. The math is not difficult, but for a printer a lot of square roots shall be done and control board based on 8 bit processors are struggling doing this calculation, so a lot of fine software optimization were done for the delta geometry for these processors.

It shall be noted that the sliders columns can be on a not equilateral triangle and asymmetric dispositions were experienced, notably the 'Square' delta with angles of 90°,90° and 180°.

Interactive web simulation

Linear deltas

• GeoGebra dSim graphic simulator for linear delta, use sliders to adjust parameters. Can be viewed in 3D with red/blue glasses.
• Thinkyhead Deltabot calculator for linear delta, enter numbers for simulation.

Rotational deltas

• http://arvc.umh.es/parola/delta.html Parola: Java based simulator, rotational delta with many options

http://arvc.umh.es/label/delta.html

Simulation on software

OpenScad delta simulator for linear delta with active simulation, with common printers datasets predefined. Change variables values for simulation. You can see a film here : https://github.com/PRouzeau/OpenSCAD-Delta-Simulator/blob/master/Delta_simulator.avi?raw=true or here https://github.com/PRouzeau/OpenSCAD-Delta-Simulator/blob/master/FisherDelta/film_FisherDelta.avi?raw=true (Fisher Delta)

https://www.youtube.com/watch?v=OwI6hYHWbjw A python simulator, undisclosed program.

Delta calculators

Linear deltas

http://www.heliumfrog.com/deltarobot/details/details.html Excel sheet calculator for linear delta https://github.com/Jaydmdigital/mk_visual_calc OpenScad calculator with visualization dedicated to Kossel

Rotational deltas

https://www.marginallyclever.com/other/samples/fk-ik-test.html Online calculator for rotational delta using info from the Trossen robotics forum : http://forums.trossenrobotics.com/tutorials/introduction-129/delta-robot-kinematics-3276/ (rotational delta).

Math and research papers

http://robinsonia.com/wp/?p=161 math explanations for linear delta http://scholar.rose-hulman.edu/cgi/viewcontent.cgi? article=1000&context=mechanical_engineering_grad_theses Linear delta research paper, but with the sliders parallel to the effector plan.

Rotational delta are a quite common research topic, notably for university students. http://www.ohio.edu/people/williar4/html/pdf/DeltaKin.pdf Rotational delta kinematics.

Calculation links

https://hackaday.io/project/963-firepick-delta-the-open-source-microfactory/log/3588-6152014-delta-mechanism-simulation-and-accuracy-determination (Rotational delta)

Simulation without source/access

Some people have done Delta simulation on CAD/Math software, but not publicly release them. Simulation on Matlab film here: https://www.youtube.com/watch?v=8_6QfZ6DJfU https://www.youtube.com/watch?v=K2zxvHq3iC8