# Principle and Requirements

TBD

## Math / Inverse Kinematic

The predicted nozzle position is given by the software (slicer, G-code) in Cartesian coordinates (X, Y, Z). The firmware has to calculate the position of the 3 vertical carriages so that the nozzle moves to the predicted position. The calculation can be done with 3 “theorem of Pythagoras” (in space). Math example (simplified)

DiffX[n] = PredNozzleX - TowerX[n]
DiffY[n] = PredNozzleY - TowerY[n]
DiffZ[n] = PredNozzleZ + ZeroTowerZ[n]
VertCarriage[n] = Sqrt ( RodLength² - (DiffX[n]² + DiffY[n]²) ) + DiffZ[n]

## Error Analysis

The precision of nozzle positioning depends on rod length and a correct nodal point.

In contrast to a cartesian printer the error on a delta printer dependent on the nozzle X/Y position and movement direction. This can result in a curved space and bent printed surfaces.

E.g. a backlash of 0.1 mm in joints can misalign the nozzle up to 1 mm (PositioningError = 2 x Backlash * RodLength / JointDistance)

Other (angle) errors are multiplied by the length of the nozzle (below joints nodal plane).

TBD

# Joint Variants

## Cardan Joint

### DIY / RepRap Cardan Joint

Most RepRap delta design uses also printed cardian joints. Based on this technique the horizontal drills leaks often on precision. Little variation in joints results in much higher nozzle positioning error.

PROs:

• Very low cost
• Can be printed

CONs:

• Precision depends on manufacturing and assembling. Little manufacturing and assembling error results in a much higher nozzle positioning error
• Design is prone to backlash and misaligned nodal points for vertical and horizontal axis
• Needs more space in construction

### Industrial Cardan Joint

Industrial Cardan

There are industrial cardan joints for R/C cars available. Some of them has also plugin adapters with M4 or M5 threads.

TBD

Main Problem: In only 4 directions the joint can be tilt up to 90°. In directions between the usable angle is limited to about 30° (depends on design).

PROs:

• Less friction
• M4 or M5 mounting threads
• Threaded rods can be used in between

CONs:

• Cost per joint about 12 Euro
• Have to be constructed in e.g. 45° angle to minimize limits
• Have to be mounted in optimal working direction

## Rod End Bearing

Rod End Bearing

TBD

Main Problem: At rotation axis the joint is designed for endless rotation. But at the tilt axis the joint is limited by construction. As manufacturer documentation tilt is limited to ±30° (Source: igus.com). To use the maximum (theoretical) reaching area as printing area a tilt angle of 35…40° is necessary. In result you can use only a limited printing area.

PROs:

• Threaded rods can be used in between

CONs:

• Limited printing area
• Conical spacer required
• TBD

## Magnetic Joint

Magnetic Joint Example 1
main article: magnet joint

TBD

Note: V2A balls are not magnetic and can not be used!

PROs:

• Precision by design
• No backlash
• Implicit correct and well known nodal point
• Low cost. About 1 Euro per joint
• Simple construction
• Easy assembly
• Easy disconnecting for service and transport

CONs:

• Limited holding force
• In most designs parts have to be glued
• Magnets are sensitive to shock and high temperatures (>80°C)

### Magnet in Tube Variant

Magnet in Tube Variant

TBD

• TBD

• Precision depends on tube cutting and deflashing
• TBD

### Moving Ring Magnet Variant

Moving Ring Magnet Variant

The bearing balls are glued to the lift and carriage. The ring magnets are part of the rods.

TBD

• TBD

• TBD

### Fixed Ring Magnet Variant

Fixed Ring Magnet Variant

The ring magnets are pressed in and glued to the lift and carriage. The bearing balls are part of the rods.

TBD