docs: Kinematics document image updates
Signed-off-by: Kevin O'Connor <kevin@koconnor.net>
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@ -73,21 +73,24 @@ flow would result in poor print quality.
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To solve this, the "look-ahead" mechanism queues multiple incoming
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To solve this, the "look-ahead" mechanism queues multiple incoming
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moves and analyzes the angles between moves to determine a reasonable
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moves and analyzes the angles between moves to determine a reasonable
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speed that can be obtained during the "junction" between two moves. If
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speed that can be obtained during the "junction" between two moves. If
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the next move forms an acute angle (the head is going to travel in
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the next move is nearly in the same direction then the head need only
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nearly a reverse direction on the next move) then only a small
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slow down a little (if at all).
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junction speed is permitted. If the next move is nearly in the same
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direction then the head need only slow down a little (if at all).
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![lookahead](img/lookahead.svg.png)
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![lookahead](img/lookahead.svg.png)
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If the next move forms an acute angle (the head is going to travel in
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nearly a reverse direction on the next move) then only a small
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junction speed is permitted.
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![lookahead](img/lookahead-slow.svg.png)
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The junction speeds are determined using "approximated centripetal
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The junction speeds are determined using "approximated centripetal
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acceleration". Best
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acceleration". Best
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[described](https://onehossshay.wordpress.com/2011/09/24/improving_grbl_cornering_algorithm/)
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[described by the author](https://onehossshay.wordpress.com/2011/09/24/improving_grbl_cornering_algorithm/).
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by the author.
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Klipper implements look-ahead between moves contained in the XY plane
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Klipper implements look-ahead between moves contained in the XY plane
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that have similar extruder flow rates. Other moves are rare and
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that have similar extruder flow rates. Other moves are relatively rare
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implementing look-ahead between them is unnecessary.
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and implementing look-ahead between them is unnecessary.
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Key formula for look-ahead:
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Key formula for look-ahead:
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```
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```
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@ -158,20 +161,20 @@ To generate step times on Delta printers it is necessary to correlate
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the movement in cartesian space with the movement on each stepper
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the movement in cartesian space with the movement on each stepper
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tower.
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tower.
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To simplify the math, for each stepper tower, the code calculates the
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location of a "virtual tower" that is along the line of movement.
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This virtual tower is chosen at the point where the line of movement
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(extended infinitely in both directions) would be closest to the
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||||||
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actual tower.
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![delta-tower](img/delta-tower.svg.png)
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![delta-tower](img/delta-tower.svg.png)
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To simplify the math, for each move contained in an XY plane, the code
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It is then possible to calculate where the head will be along the line
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calculates the location of a "virtual tower" that is along the line of
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of movement after each step is taken on the virtual tower.
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movement. This virtual tower is chosen at the point where the line of
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movement (extended infinitely in both directions) would be closest to
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the actual tower.
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![virtual-tower](img/virtual-tower.svg.png)
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![virtual-tower](img/virtual-tower.svg.png)
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It is then possible to calculate where the head will be along the line
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The key formula is Pythagoras's theorem:
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||||||
of movement after each step is taken on the virtual tower. The key
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formula is Pythagoras's theorem:
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```
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```
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||||||
distance_to_tower^2 = arm_length^2 - tower_height^2
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distance_to_tower^2 = arm_length^2 - tower_height^2
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||||||
```
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```
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||||||
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@ -196,7 +199,7 @@ must also be used in the calculations.
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Should the move contain only Z movement (ie, no XY movement at all)
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Should the move contain only Z movement (ie, no XY movement at all)
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||||||
then the same math is used - just in this case the tower is parallel
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then the same math is used - just in this case the tower is parallel
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||||||
to the line of movement (its slope is 1.0).
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to the line of movement.
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||||||
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||||||
### Stepper motor acceleration limits ###
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### Stepper motor acceleration limits ###
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@ -248,7 +251,7 @@ pressure. Pressure increases when filament is pushed into the extruder
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||||||
(as in [Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law)) and
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(as in [Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law)) and
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||||||
the pressure necessary to extrude is dominated by the flow rate
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the pressure necessary to extrude is dominated by the flow rate
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||||||
through the nozzle orifice (as in
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through the nozzle orifice (as in
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||||||
[Poiseuille law](https://en.wikipedia.org/wiki/Poiseuille_law)). The
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[Poiseuille's law](https://en.wikipedia.org/wiki/Poiseuille_law)). The
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||||||
key idea is that the relationship between filament, pressure, and flow
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key idea is that the relationship between filament, pressure, and flow
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||||||
rate can be modeled using a linear coefficient:
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rate can be modeled using a linear coefficient:
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||||||
```
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```
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@ -133,7 +133,7 @@
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