# Principle and Requirements

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## 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]
VertCarriage[n] = Sqrt ( RodLength² - (DiffX[n]² + DiffY[n]² ) ) + PredNozzleZ + ZeroOffsetZ[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).

In result the precision of the printer depends very much on joint precision!

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# Joint Variations

## Cardan Joint

### DIY / RepRap Cardan Joint

Most RepRap delta design uses also printed Cardan 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
• Rotational stiffness add stability on effector

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

USED BY:

### 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.

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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

CHECK:

• Check other manufacturers for wider tilt angle

## Rod End Bearing

Rod End Bearing
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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

CHECK:

• Check other manufacturers for wider tilt angle

USED BY:

## Ball-and-Socket Joint

Other variation of the ‘Rod End Bearing’ but with a more limited pivot angle of 25° (Source: igus.com). The Igus ball and socket articulation have a mechanism which fold sort of socket 'petals' to retain the ball and they does not need spring or other retainers. However, this system creates important friction which may make this kind of articulations unusable for a common printer.

There is another type of ball and socket, with springs retainer is very frequently used in 'pick and place' deltas and have been experienced on a few printers.

## Ball in socket with wire tensioners

The ball is set in sockets, generally printed with retaining wire(s) parallels to arms

USED BY:

• Cherry-Pi IIIS

## Fork and ball Joint

A fork pinching a ball. The spring affect of the fork is maintaining it on the ball in all axis. The advantage of this system is that the fork is attached on the arm, which can rotate if the maximum angle of the articulation is reached, so this system does not have any angle limit for a printer.

USED BY:

## Magnetic Joint

Magnetic Joint Example 1
main article: magnet joint
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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 Variations

#### Magnet in Tube

Magnet in Tube Variant

A cylindrical magnet is glued and/or pressed in a tube. The magnet should not touch the steel ball but should also be as near as possible to the ball to get maximum force. The edge of the tube is sliding over the steel ball.

The ‘tube’ can also be drill in the construction.

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• Precision depends on tube cutting, drilling and deflashing

#### Moving Ring Magnet

Moving Ring Magnet Variant

The steel balls are mounted (glued) to the lift and effector. The ring magnets are part of the diagonal rods.

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• -

CHECK:

• Long time abrasion of magnets is unknown

#### Fixed Ring Magnet

Fixed Ring Magnet Variant

The ring magnets are glued and/or pressed in to the lift and effector. The steel balls are part of the diagonal rods.

• -

CHECK:

• Long time abrasion of magnets is unknown

USED BY:

### Ball Variations

Note: Stainless steel (V2A) balls are not magnetic and can not be used!

#### Ball from Ball-and-Socket Joint (DIN 71802)

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• Screw end

• Flat top - limited tilt angle

#### Bearing Ball

Bearing balls are produced with a very high precision. Even if you get class B merchandise the precision much higher than you need for the delta joints.

Note: Look for “slingshot balls” on eBay.

• Low cost (100 pieces for about 10 Euro)
• Implicit high precision
• Very smooth surface – less friction, no abrasion
• Available in all sizes

• Have to be glued

USED BY:

#### Ball from Photo Tripod Ball Head

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• Screw end
• Big balls with big magnets for high force

• Expensive

CHECK:

• Is ball material magnetic? (Could be chromed brass)

## Other Joints

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# Friction in articulations

## What are the causes and effects

Most articulations not based on ball bearing do have some friction.

Such friction will drive to have moments on the arms, which will be transferred to the effector and the carriages. That may drive to slight imprecision if there are plays in carriage bearings or in the articulation themselves. Other imprecision due to friction moment is also related to all components stiffness, including motor torque and belt drive.

However, this friction will have a dampening effect on structural and motor vibrations. This dampening could reduce or completely neutralize ringing effects and a controlled friction may be desirable. It shall be associated with a stiff machine design.

Generally, system which will create friction will also reduce articulation play, which may counterbalance positively the defaults caused by friction moments.

Lubrication of the articulation is required to avoid noises and squeaks and will reduce somewhat the friction, but may help maintaining it relatively constant in time.

## How to control the friction ?

• drilled ball articulation ("Traxxas" type)
On classic ball articulations, it is common to use spring/rubber tensioner between the two arms to control play. It will have a side effect of maintaining relatively constant friction while the articulations wears.
• Ball and cup
Ball and cups articulation (as found on pick and place machines and a few printers) are also using using springs to maintain cups in place.
Some industrial ball and cup articulations (Igus) do have axial retain effect and does not need springs. However, the system which grip the ball does have a lot of friction, which may be excessive for most printers.
• "Fork" and ball
A system with forks pinching the balls is used on the Fisher delta. It does have a friction which depends from manufacturing tolerances and is not then easily adjustable but may be relatively constant during the printer lifetime.
• Ball and printed cups maintained by tensioning wire
Good wire tension will reduce the system play and create sufficient friction to cancel ringing effects.