Status: inverse kinematics resolved and implemented in the firmware as well

Updates:

• 2021/07/14: PAX inverse kinematics finally working on firmware level too
• 2021/05/25: using worm gear with 28BYJ-48 stepper motor to drive tilt rotation
• 2021/05/14: adding Duet 3 Mini 5+ Setup details (inverse kinematic not yet done in firmware), first motion tests made with 2x NEMA 17 (40mm)
• 2021/02/28: added animation PAX printing a 4-axis/RTN sliced overhang model
• 2021/02/17: added two brief animations of inverse kinematics
• 2021/02/08: inverse kinematics working, printhead mounted on Ashtar K as draft
• 2021/02/06: machine vs nozzle coordinates, forward/reverse transformation requirements, heatsink fan and part cooler added
• 2021/02/04: starting with collecting ideas and first drafts

Introduction

After I saw 5- and 6-axis printers at Formnext 2019 and particularly seeing the belt printers able to print 90° overhangs in one direction without support, and then RotBot by ZHAW, a Rotating Tilted Nozzle (RTN) printer, where the 90° overhangs in different directions (given some conditions) can be printed without support – so it was more natural to consider to make the tilting nozzle angle (t_rot) flexible as well, 0 – 180°.

• 0°: nozzle looking down, ordinary orientation with Z sliced layers 3D printing
• 45°: belt printer or rotating tilted nozzle (RTN) printer, printing 90° overhangs in one or more directions ^*)
• 90°: printing horizontally outward
• 135°: printing upward 45°
• 180°: printing upward entirely

*) Rotating Tilted Nozzle capability printing 90° overhangs depends on location and symmetric alignments, no slicing software yet available.

Flexible Tilting

2 Additional Axes

• 2x NEMA 17 37mm/263g (Z rotation (A) + Tilt (B)): strong enough, but too heavy, my X gantry rattles and introduces ~ 1 to 2mm errors before swinging in; verdict: unusable unless entire X gantry/carriage is strengthened
• 2x NEMA 17 23mm/131g (Z rotation (A) + Tilt (B)): too weak, can’t hold position Z rotation (A) or tilt (B) on PAX 180/90 due PTFE with filament stiffness; verdict: unusable
• 1x NEMA 17 37mm/263g (Z rotation (A)): strong enough to position with PTFE with filament and cables + 1x 28BYJ-48 (Tilt (B)):
  • direct drive
    • PAX 90 arm (dual hinge): PTFE with filament stiffness prevents 0° tilt, but 15° at minimum: unreliable
    • PAX 180 arm (dual hinge): a bit better, but still problem to reliable position tilt from 0° to 90° and back to 0°: still unreliable
  • worm gear
    • PAX 90 arm (dual hinge): barely works (tilt: -5° to 85°)
    • PAX 180 arm (dual hinge): works so far (tilt: -20° to 170°)

NEMA 17 (23mm & 37mm)

Let’s look at bit closer to the new rotating axes implemented with direct drive NEMA 17 stepper motors:

• Z rotation (aka Z_rot) with
  • NEMA 17 37/40mm/263g long (45Ncm) or
  • NEMA 17 23mm/131g (13Ncm)
• Tilt rotation (aka T_rot) with
  • NEMA 17 20mm/140g (16Ncm) or
  • NEMA 17 23mm/131g (13Ncm)

both provide 1.8° (or optionally 0.9°) resolution per full step, or

• 8 microsteps (20% force) 0.225° @1.8°step (0.1125° @0.9°step) per microstep
  • Z rotation 9 Ncm (or 2.6Ncm)
  • T rotation 3.2 Ncm
• 16 microsteps (10% force) 0.1125° @1.8°step (0.0565° @0.9°step) per microstep
  • Z rotation 4.5 Ncm (or 1.3Ncm)
  • T rotation 1.6 Ncm

For now, for sake of simplicity, both axes are in direct drive – if precision requirements dictate a simple reduction gear 1:4 or 1:5 (see below for some more details).

28BYJ-48

With just 33g weight and easy to source this stepper motor is quite remarkable. With a small modification, the 28BYJ-48 can be driven in bi-polar mode and then connected with the existing stepper drivers.

More in-depth information can be found at Control 28BYJ-48 Stepper Motor with ULN2003 Driver & Arduino.

• introduces significant margins via the shaft:
  • apprx. 2-3° rotation backlash (last gear of the shaft has 31 tooths, 360° / 31 = 11.6°)
  • apprx. 0.5mm tolerance of the shaft itself, causing at the tip of the shaft > 1mm tolerance or margin
  • possible solutions:
    • add another gearbox in front and introduce more stable (longer) shaft (pros: still light solution, cons: more mechanical complexity)
      • worm gear: works

Even though it’s a low-cost stepper motor and introduces backlash and tolerances to take care of, the lightness of just 33g which is about 12.5% of the NEMA 17 40mm or 25% of NEMA 17 23mm while actually provide sufficient holding torque due the gears.

Duet 3 Mini 5+ config.g changes, using it just the tilt rotation (B):

M906 B200   ; set motor current (mA)
M360 B1 I0  ; turn off microstepping without interpolation
M92 5.66    ; 32 steps / revolution x 63.68395 = 2038 => 2038 / 360 deg => 5.66 steps / deg

The 200mA setting caused the motor to warm up to apprx. 50°C, so likely to reduce the current to 100mA – if a gearbox with a worm is used, likely the reduced current still provides sufficient torque (or alternatively add a small fan to cool the motor). So I went ahead to use a worm gear setup, as direct drive was too weak – for now I use PAX 180 (long arm) to have sufficient space to flex PTFE/filament on top of the printhead.

; 24BYJ-48 
M350 B1 I0
M92 B5.6                                      ; direct drive: 32 steps / revolution x 63.68395 = 2038 => 2038 / 360 deg => 5.66 steps / deg
M92 B90.56                                    ; worm gear: 5.66 steps / deg x 16 = 90.56 steps / degrees

N20 BLDC with gearbox & encoder

Not yet tested.

Taking Advantage of 5 Axis

In 5 axis CNC context there are multiple configurations possible such as table/table, head/head and table/head – as I came from the Rotating Tilted Nozzle (RTN) the extra 2 axis are added to the head, hence head/head configuration.

The T_rot of 0° is the equivalent of ordinary 3D printer with top/down oriented nozzle, the 45° the belt printer or RotBot/RTN, and 90° printing vertical walls with the nozzle perpendicular, and 90-135° printing up-side down – I’m not sure if 135-180° is that useful, perhaps making underlying structures really smooth. For now I keep the tilt rotation from 0° to 180° even though I think I’m going to use 0-135° in real life application when the actual print procedure is developed, planar or non-planar slicing.

Dual Hinge

Mathematics

5 Axis Kinematics

LinuxCNC 5 Axis kinematics describes the notations and provides a starting point and I realized quickly that in CNC context the 5 axis operations is quite thoroughly explored, but much fewer focus on 5 axis additive manufacturing yet (see below at References).

In order to reflect the rotation per axis, the notion A, B and C are adopted, which can be described in OpenSCAD as

translate([X,Y,Z]) rotate([A,B,C]) ...

Further the CNC notion I, J and K are used as tool vector, the way the nozzle points away. G-code supports natively G1 X Y Z I J K which is machine independent. The machine specific firmware then computes the machine coordinates and rotating angles so the machine tool tip reaches those coordinates with that particular tool vector.

Tool/Nozzle vs Machine Coordinates

Absolute coordinates X, Y, Z of the nozzle tip and angles Z_rot, T_rot and given the Z rotation arm there is a mapping required to X_m, Y_m, Z_m with same Z_rot, T_rot.

The forward transformation in OpenSCAD:

translate([Xm,Ym,Zm]) rotate([0,0,Zrot]) rotate([Trot,0,0]) translate([0,0,-45]) sphere(0.1); // nozzle tip at X, Y, Z with I, J, K

or expressing it as a list:

1. at 0, 0, 0 (origin)
2. translate([Xm,Ym,Zm])
3. rotate([0,0,Zrot])
4. rotate([Trot,0,0])
5. translate([0,0,-45])
6. at X, Y, Z

The -45 is the 45mm vertical offset of the tilt rotation to the tip of the nozzle.

In OpenSCAD it’s the reverse order, and enumerate matrices (used further below):

1. at X, Y, Z
2. translate([0,0,45]) aka M1
3. rotate([-Trot,0,0]) aka M2
4. rotate([0,0,-Zrot]) aka M3
5. translate([ Xm, Ym, Zm ]) aka MR, by inverting matrix the X_m, Y_m, Z_m can be extracted
6. at 0, 0, 0 (origin)

Each transformation can be represented by a 4×4 matrix, the sequence of transformations are the multiplication of such, and when multiplying symbolical a single 4×4 matrix will result. When keeping the symbolical notion the inverse transformation can be obtained with one operation and getting X_m, Y_m, Z_m from X, Y, Z, Z_rot, T_rot, and that very computation needs to be done in the firmware and controller in order to process X, Y, Z, Z_rot, T_rot as G1 X.. Y.. Z.. A.. B.. and the internally X_m, Y_m, Z_m is processed to achieve that tool coordinate; A = Z_rot, B = T_rot .

Numerical Inverse Transformation

Just for sake of confirming the reverse/inverse transformation utilizing m4.scad:

include <m4.scad>

Zrot = 45;
Trot = 45;

M1 =       [ [ 1, 0, 0, 0 ],
             [ 0, 1, 0, 0 ],
             [ 0, 0, 1, 45 ],
             [ 0, 0, 0, 1 ] ];

M2 =       [ [ 1, 0, 0, 0 ],
             [ 0, cos(-Trot), -sin(-Trot), 0 ],
             [ 0, sin(-Trot), cos(-Trot), 0 ],
             [ 0, 0, 0, 1 ] ];

M3 =       [ [ cos(-Zrot), -sin(-Zrot), 0, 0 ],
             [ sin(-Zrot), cos(-Zrot), 0, 0 ],
             [ 0, 0, 1, 0 ],
             [ 0, 0, 0, 1 ] ];

MR = m4inv(M1 * M2 * M3);
// or MR = m4inv(m4tr([0,0,45])*m4rx(-Trot)*m4rz(-Zrot));
echo(MR);

which outputs

ECHO: [[1, 0, 0, 0], [0, 0.707107, -0.707107, 31.8198], [0, 0.707107, 0.707107, -31.8198], [0, 0, 0, 1]]

so MR contains the inverse matrix of the previous transformations, so I can extract the translation vector, the 4th column [ MR[0][3], MR[1][3], MR[2][3] ] or m4trv(MR) to compensate X, Y, Z, Z_rot, T_rot -> X_m, Y_m, Z_m:

Testing Inverse Kinematics (IK)

Perhaps it’s worth to actually calculate symbolical forward and inverse transformation matrix using e.g. Matlab or alike to have transformations in one operation instead multiplying three matrices and inverting it – depending what hardware is used as controller multiplying individual matrices is faster than trying to have a complex single step matrix construct.

Although Marlin firmware supports Delta printer with complex delta inverse kinematics, not sure I can add mine as well, or I have to go with Duet RepRap firmware which seems more suitable (see notes below).

Nozzle Precision from T_rot

The 45mm offset of the last rotation massivly contributes to the loss of nozzle position resolution:

s = 2 π * r / 360 = section length per degree

s = π * 45mm / 180 = 0.785mm/° which means:

• 8 microsteps with 1.8° full step: 0.225°/microstep => 176.7μm/microstep
• 16 microsteps with 1.8° full step: 0.1125°/microstep => 88.3μm/microstep
• 8 microsteps with 0.9° full step: 0.1125°/microstep => 88.3μm/microstep
• 16 microsteps with 0.9° full step: 0.0565°/microstep => 44.4μm/microstep

so the direct drive using the shafts of the NEMA 17 will provide OK resolution, but for anything a bit more serious, a reduction gear might be worth to use.

G-code & Firmware

As I likely will generate machine independent G-code as G1 X Y Z I J K, the slicer stage will likely operate in X, Y, Z and Z_rot and T_rot as well – so we end up with a data pipeline like this:

1. Slicer: X, Y, Z, Z_rot, T_rot
2. G-code: G1 X Y Z I J K
3. Firmware: X Y Z I J K => X_m, Y_m, Z_m and I J K => Z_rot, T_rot

Alternatively, instead using I J K notion use the G-code A and B as two rotational axes as Duet RepRap Firmware offers G1 X Y Z A B then it’s a bit simpler:

1. Slicer: X, Y, Z, Z_rot=>A, T_rot=>B
2. G-code G1 X Y Z A B
3. Firmware: X Y Z A B => X_m, Y_m, Z_m and A=>Z_rot, B=>T_rot

Reviewing existing slicer/printing software to see which notion is more suitable, and perhaps cover both to stay flexible.

Issues to Resolve

• Bowden tube- & cable management: properly resolve it, guides etc.
• Develop the transformation/inverse kinematics X, Y, Z, Z_rot, T_rot <=> X_m, Y_m, Z_m, Z_rot, T_rot as well X, Y, Z, I, J, K <=> X_m, Y_m, Z_m, Z_rot, T_rot as required for slicing, and firmware stage, done
• Calculate the precision of X, Y, Z in relation of X_m, Y_m, Z_m and Z_rot, T_rot, whether or how the motor resolution affect axes => getting a grasp how the overall precision of the final setup
• Direct drive mechanical precision with 8/16 microsteps, repeatability, and heat dissipation from motor (see tweet)
• Slicer/print software supporting 5 axis nozzle
  • Duet RepRap Firmware: supports ABCD rotational axes
    • Supporting firmware calculation of X_m, Y_m, Z_m (keeping G-code machine independent) inverse kinematic: e.g. adding new kinematics to Duet, and 5 Axis Robot configuration might help as a start
  • Marlin firmware capabilities, as it support Delta printers, inverse kinematics (IK) calculations seem supported well, question is how simple to add my own custom IK as well
  • printing: recognize model features or base design on seams or boundaries, like OPENCASCADE (STEP/IGES support)
  • printing: use vertical slicing as fallback, given no other printing method is suitable
  • explore different print modes like RotBot/RTN
    • recognize rotational objects: print from inside out like with a lath but adding material
    • detect steep overhangs, find proper way to print them
  • collision detection, e.g. tilt rotation becomes available at a certain Z level as below the upside looking nozzle will touch the build plate with the opposite end – the slicing/print software must be aware of the printhead geometry to calculate what’s possible to print and how

Considerations

Pros

• many new ways to print in (almost) all directions
• hopefully print 90° and more overhangs without support at all

Cons

• significant mechanical complexity
  • mechanical limitations arise with new freedom of rotation of printhead
  • collision detection becomes essential which in Z sliced layers is not an issue at all
• significant software complexity, no current 3D printing software available to take advantage of 5 axis printing

Ashtar K with 5 Axis (PAX) Printhead

Experimental mounting on Ashtar K to see how it looks, including display showing rotations of Z_rot (A), T_rot (B) too.

Note: the display shows original tool coordinates, whereas the firmware does tool translation aka inverse kinematics as shown in the screenshots with the display readable.

With all the freedom to angle the nozzle, all of the sudden the part cooler air nozzle shape becomes an issue, and has to become narrow as well; and overall geometry of the printhead becomes quite relevant when planning print sequences (collision detection).

PAX can operate in 4-axis mode, e.g. printing a conic sliced (4-axis RTN) overhang model (without support structure) in 4-axis Rotating Tilted Nozzle (RTN) mode (45° titled nozzle):

PAX Tilt 180, 135 and 90

Printable Parts

Print with 0.4-0.5mm nozzle @ 0.20 – 0.25mm layer height:

• pax_arm: connects motor A / Z_rotation with motor B/tilt
  • pax_arm_90 & pax_arm_90_dualh
  • pax_arm_135 & pax_arm_135_dualh (recommended)
  • pax_arm_180 & pax_arm_180_dualh
• pax_carriage_mount: holds motor A / Z_rotation connects to X carriage
• pax_printhead: connects motor B with printhead, @ 0.1mm layer height
  • pax_printhead
  • pax_printhead-dualh: for NEMA 17 for tilt
  • pax-printhead_dualh_xl: for 28BYJ-48 direct drive or tilt gearbox (worm gear)
• pax_tilt_gearbox_28byj48:
  • pax_tilt_gearbox_28byj48: PAX 180
  • pax_tilt_gearbox_28byj48-tilt=true: use with PAX 90/135/190 (recommended)
• pax_partcooler: connects part cooler fan with printhead
• pax_fan_nose: part cooler fan nose, @ 0.1mm layer height
• 28byj_cooler
  • 28byj_cooler-type=single: PAX 90/135
  • 28byj_cooler-type=dual: PAX 90/135/180

Hardware

NEMA 17 23mm Datasheet (17HS4023)

For for my own reference, a few details of stepper motor for Z-rotation (“A” axis) and tilt (“B” axis):

Note: early tests have shown the NEMA 17 23mm “thin” stepper motor has too weak holding torque (without gear reduction) with the stiffness of PTFE/filament and cables for Z rotation (A) and tilt rotation (B) – so I switched back to heavy NEMA 17 40mm “long” stepper motor – but that meant the mounting pieces need to be stronger.

If I find a low-overhead and space efficient gear reduction perhaps the NEMA 17 23mm may still have a chance to be used in this setup.

28BYJ-48 Datasheet

Note: early tests have shown for tilt rotation (B) in direct drive it’s too weak, but with worm gear setup it seems to work, though quite slow, and with just 33g weight it’s quite an advantage in this regard compared to heavy NEMA17.

Duet 3 Mini 5+ & Mini 2 Expansion

Duet 3 Mini 5+ & Expansion Mini 2 provides 7 stepper motors drivers (TMC 2209) with microstep interpolation:

• Stepper 0: X-axis
• Stepper 1: Y-axis
• Stepper 2/3: 2x Z-axis (I could runs 2 Steppers on on the same driver, but since I run quick Z changes, I keep it dedicated driver for each)
• Stepper 4: E0
• Stepper 5 (Expansion 0): A (Z rotation)
• Stepper 6 (Expansion 1): B (tilt rotation)

config.g

; Default config.g template for Duet 3 Mini 5+
; Replace this with a proper configuration file (e.g from https://configtool.reprapfirmware.org)

M575 P1 B57600 S1                             ; Enable UART 

; test network enable
;M552 P192.168.2.14
M552 P10.0.0.100
M552 S1

M550 P"Ashtar K2 PAX"                         ; Name the board/machine

; config drivers -> motors: X, Y, Z(dual), E0, A & B
M584 X0 Y1 Z2:3 E4 
M584 A5 B6 S1

M208 X-10 Y-10 Z0 A-185 B-40 S1               ; Set axis minima
M208 X320 Y320 Z300 A185 B100 S0              ; Set axis maxima

M92 X100 Y100 Z3200 E95                       ; Set axis steps/mm
M92 A9 B9                                     ; Set axis steps/degrees

M350 X16 Y16 Z16 E16 I1                       ; Set 16x microstepping with interpolation
M350 A4 B4 I1                                 ; Set 4/8/16x microstepping with(out) interpolation

M906 X800 Y1000 Z800 E800 A1000 B1000         ; Set motor currents (mA)

M201 X400 Y400 Z15 E1000                      ; Accelerations (mm/s^2)
M201 A800 B800

M203 X10000 Y10000 Z360 E3600                 ; Maximum speeds (mm/min)
M203 A20000 B20000                            ; Maximum speed (mm/min)

M566 X600 Y600 Z30 E20                        ; Maximum jerk speeds (mm/min)
M566 A300 B300

; Define end-stops:
M574 X1 S1 P"io0.in"
M574 Y1 S1 P"io1.in"
M574 Z1 S1 P"io2.in"
;M574 A1 S1 P"io3.in"
;M574 B1 S1 P"io4.in"
M564 H0                       ; Allow (all) axis without homing (required for A/B)

M918 P1                       ; Set 12864 display (doesn't work yet)

M569 P0 S1                    ; Set motor drive P0 (X)
M569 P1 S1                    ; Set motor drive P1 (Y)
M569 P2 S1                    ; Set motor drive P2 (Z1)
M569 P3 S1                    ; Set motor drive P3 (Z2)
M569 P4 S0                    ; Set motor drive P4 (E0)
M569 P5 S0                    ; Set motor drive P5 (A/Zrot)
M569 P6 S0                    ; Set motor drive P6 (B/tilt)

M563 P0 D0 H1 F0       		  ; Define tool 0
G10 P0 X0 Y0 Z0       		  ; Set tool 0 axis offsets
G10 P0 R0 S0                      ; Set initial tool 0 active and standby temperatures to 0C
; M572 D0 S0.06			      ; Set pressure Advance

; Bed Heater
; M308 S0 P"temp0" Y"thermistor" B4725 C7.060000e-8 					; configure sensor 0 as thermistor on pin temp0
; M950 H0 C"out0" Q25 T0                           					; create bed heater output on out0 and map it to sensor 0, PWM frequency: 25Hz
; M307 H0 R0.262 C338.0 D10.52 S1.00 V11.8 B0							; Bed tuning values, enable PID
; M140 H0                                                    			; Bed uses Heater 0
; M143 H0 S120                                               			; Set temperature limit for heater 0 to 120C Bed

; Hotend heater
M308 S1 P"temp1" Y"thermistor" T100000 B4725 C7.060000e-8 					; configure sensor 1 as thermistor on pin temp1
M950 H1 C"out1" T1                           						; create nozzle heater output on out1 and map it to sensor 1
M307 H1 B0 S1.00                             						; disable bang-bang mode for heater  and set PWM limit
M143 H1 S295                                 						; set temperature limit for heater 1 to 295C
M302 S170 R170                                         				; allow extrusion starting from 170°C and retractions already from 170°C

; Part cooling fan
M950 F0 C"out3" Q100          ; Create fan 0 part cooler on pin out3 and set its frequency
M106 P0 S0 H-1                ; Set fan 0 value. Thermostatic control is turned off

; Heatsink cooling fan
M950 F1 C"out4" Q1000		  ; Create fan 1 heatsink fan an pin out4 and set its frequency
M106 P1 T45 S255 H1  	      ; Set fan 1 value. Thermostatic control is turned on > 45C turn on   
;M106 P1 S255 H-1

M302 P1                       ; Allow cold extrusion (for testing)
T0                            ; Set tool 0 (default extruder)

G21                           ; Work in millimetres
G90                           ; Send absolute coordinates...

; pre 3.3 (3.2.x)
G92 A-1 B-1                   ; Make Zrot & Tilt active
G1 A0 B0
; 3.3 or later
;M17 A B

G92 A0 B0
G1 A0 B0 F20000

Endstops:

• X: io_0 (Pin 2/3)
• Y: io_1 (Pin 2/3)
• Z: io_2 (Pin 2/3)
• [optional] A: io_3 (Pin 2/3)
• [optional] B: io_4 (Pin 2/3)

Thermistors:

• temp0: bed
• temp1: hotend

Heating:

• out0: bed (max 15A)
• out1: hotend (max 5A, 60W@12V)

Fans:

• out3: part cooling fan (Pin 2/4)
• out4: heatsink fan (Pin 2/4) or 12V: heatsink fan (always on)

So far Duet 3 Mini 5+ with RepRap Firmware V3.2 works well with the first motion tests. It took me an afternoon to configure the board without the web configurator but direct editing config.g on the web-console and reading documentation for each individual G-code configuration as part of getting to know the Duet 3D approach using G-code to configure the entire firmware (except the inverse kinematic).

The inverse kinematic with adjustable offsets comes next.

In order to be able to set microstepping and motor current via firmware, the driver should operate in UART mode.

Test Setup

Duet 3 Mini 5+ with Expansion Mini 2 with 2x NEMA 17 (40mm) without gear reduction:

and first movement with NEMA 17 (40mm) with enhanced carriage mount, manually entered G1 A B motions, a shorter yet stiffer PAX 90 arm (means tilt only 0..90): A=-180..180 and B=0..90:

Closer look:

And alternatively 24-BYJ-48 stepper motor direct drive or with worm gear, with a dual hinge arm:

RepRapFirmware (RRF) vs Marlin 2.x

Duet 3D RepRapFirnware (RRF) natively supports multiple axis, and since RRF V3.2 also treats axis A-D as rotational axis by default. So far Marlin 2.x does not support multiple axis yet, but there is a fork / PR available (reviewed 2021/05/14) which introduces the functionality and be available soon.

RRF Inverse Kinematics (PAXKinematics)

The key element of course is the dedicated inverse kinematics, which I implemented and finally works rudimentary within RRF (2021/07/14), still some small things to take care:

The nozzle tip stays in place, but Z_rot and tilt_rot is changed; this allows machine independent G-code to be processed.

References

5 Axis

• Pentarod, 5 axis RepRap, 2 axis added to bed (not head), master thesis (see also paper), most useful
• Multi-Axis 3D Printing Technique Improves FDM Strength Over 2X (see also paper)
• Three-Dimensional Printing of Continuous Flax Fiber-Reinforced Thermoplastic Composites by Five-Axis Machine with paper (PDF)
• 5 Axis Maker, former Kickstarter, multi-tool CNC & 3D Printer
• 3D Digital Manufacturing via Synchronous 5-Axes Printing for Strengthening Printing Parts (paper, PDF)
• 5-Axis 3D Printing paper (PDF), it gives some information on
  • Volume decomposition
  • Build orientation
  • Sub-volume sequencing
  • Collision detection
  • 5-axis G-code generation
  • with some sort of details via pseudo-code, no software released
• How To Fully Leverage 5-Axis 3D Printing (article)
• Epit3D Delta with rotating & tilting bed (5 Axis)
  • goosli (Slicer, @github)
  • spycer (G-code visualizer, @github)
• DotXControl: 5 Axis Slicer (closed source), uses I J K tool vector notion (machine independent positioning)

4 Axis

• RotBot by ZHAW, rotating 45° tilted nozzle, little details released but informative paper (paywalled)
• Rotating Tilted Nozzle Option (RTN), my own approach of RotBot
  • Conic Slicing for 4-axis/RTN

As I progress I will update this blog-post.

That’s it.