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docs: Add initial Kinematics document
Signed-off-by: Kevin O'Connor <kevin@koconnor.net>
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docs/Kinematics.md
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docs/Kinematics.md
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This document provides an overview of how Klipper implements robot
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motion (its [kinematics](https://en.wikipedia.org/wiki/Kinematics)).
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The contents may be of interest to both developers interested in
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working on the Klipper software as well as users interested in better
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understanding the mechanics of their machines.
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Acceleration
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============
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Klipper implements a constant acceleration scheme whenever the print
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head changes velocity - the velocity is gradually changed to the new
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speed instead of suddenly jerking to it. Klipper always enforces
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acceleration between the tool head and the print. The filament
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leaving the extruder can be quite fragile - rapid jerks and/or
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extruder flow changes lead to poor quality and poor bed adhesion. Even
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when not extruding, if the print head is at the same level as the
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print then rapid jerking of the head can cause disruption of recently
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deposited filament. Limiting speed changes of the print head (relative
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to the print) reduces risks of disrupting the print.
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It is also important to enforce a maximum acceleration of the stepper
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motors to ensure they do not skip or put excessive stress on the
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machine. Klipper limits the acceleration of each stepper by virtue of
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limiting the acceleration of the print head. Enforcing acceleration at
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the print head naturally also enforces acceleration at the steppers
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that control that print head (the inverse is not always true).
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Klipper implements constant acceleration. The key formula for
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constant acceleration is:
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```
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velocity(time) = start_velocity + accel*time
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```
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Trapezoid generator
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===================
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Klipper uses a traditional "trapezoid generator" to model the motion
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of each move - each move has a start speed, it accelerates to a
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cruising speed at constant acceleration, it cruises at a constant
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speed, and then decelerates to the end speed using constant
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acceleration.
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It's called a "trapezoid generator" because a velocity diagram of the
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move looks like a trapezoid.
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The cruising speed is always greater than or equal to both the start
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speed and the end speed. The acceleration phase may be of zero
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duration (if the start speed is equal to the cruising speed), the
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cruising phase may be of zero duration (if the move immediately starts
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decelerating after acceleration), and/or the deceleration phase may be
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of zero duration (if the end speed is equal to the cruising speed).
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Lookahead
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=========
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The "lookahead" system is used to determine cornering speeds between
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moves.
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Consider the following two moves contained on an XY plane:
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In the above situation it is possible to fully decelerate after the
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first move and then fully accelerate at the start of the next move,
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but that is not ideal as all that acceleration and deceleration would
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greatly increase the print time and the frequent changes in extruder
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flow would result in poor print quality.
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To solve this, the "lookahead" mechanism queues multiple incoming
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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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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. 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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The junction speeds are determined using "approximated centripetal
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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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by the author.
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Klipper implements lookahead 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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implementing lookahead between them is unnecessary.
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Key formula for lookahead:
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```
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end_velocity^2 = start_velocity^2 + 2*accel*move_distance
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```
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Smoothed lookahead
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------------------
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Klipper also implements a mechanism for smoothing out the motions of
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short "zig-zag" moves. Consider the following moves:
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In the above, the frequent changes from acceleration to deceleration
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can cause the machine to vibrate which causes stress on the machine
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and increases the noise. To reduce this, Klipper tracks both regular
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move acceleration as well as a virtual "acceleration to deceleration"
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rate. Using this system, the top speed of these short "zig zag" moves
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are limited to smooth out the printer motion:
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In the above, note the dashed gray lines - this is a graphical
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representation of the "pseudo acceleration". Where the two dashed
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lines meet enforces a limit on the move's top speed. For most moves
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the limit will be at or above the move's existing limits and no change
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in behavior is induced. However, for short "zig-zag" moves the limit
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comes into play and it reduces the top speed. Note that the grey lines
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represent a pseudo-acceleration to limit top speed only - the move
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continues to use it's normal acceleration scheme up to its adjusted
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top-speed.
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Generating steps
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================
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Once the lookahead process completes, the print head movement for the
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given move is fully known (time, start position, end position,
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velocity at each point) and it is possible to generate the step times
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for the move. This process is done within "kinematic classes" in the
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Klipper code. Outside of these kinematic classes, everything is
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tracked in millimeters, seconds, and in cartesian coordinate space.
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It's the task of the kinematic classes to convert from this generic
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coordinate system to the hardware specifics of the particular printer.
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In general, the code determines each step time by first calculating
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where along the line of movement the head would be if a step is
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taken. It then calculates what time the head should be at that
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position. Determining the time along the line of movement can be done
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using the formulas for constant acceleration and constant velocity:
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```
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time = sqrt(2*distance/accel + (start_velocity/accel)^2) - start_velocity/accel
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time = distance/cruise_velocity
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```
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Cartesian Robots
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----------------
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Generating steps for cartesian printers is the simplest case. The
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movement on each axis is directly related to the movement in cartesian
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space.
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Delta Robots
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------------
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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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tower.
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To simplify the math, for each move contained in an XY plane, the code
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calculates the location of a "virtual tower" that is along the line of
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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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It is then possible to calculate where the head will be along the line
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of movement after each step is taken on the virtual tower. The key
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formula is Pythagorean's formula:
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```
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distance_to_tower^2 = arm_length^2 - tower_height^2
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```
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One complexity is that if the print head passes the virtual tower
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location then the stepper direction must be reversed. In this case
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forward steps will be taken at the start of the move and reverse steps
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will be taken at the end of the move.
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### Delta movements beyond simple XY plane ###
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Movement calculation is a little more complicated if the move is not
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fully contained within a simple XY plane. A virtual tower along the
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line of movement is still calculated, but in this case the tower is
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not at a 90 degree angle relative to the line of movement:
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The code continues to calculate step times using the same general
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scheme as delta moves within an XY plane, but the slope of the tower
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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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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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Extruder kinematics
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-------------------
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Klipper implements extruder motion in its own kinematic class. Since
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the timing and speed of each print head movement is fully known for
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each move, it's possible to calculate the step times for the extruder
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independently from the step time calculations of the print head
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movement.
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Basic extruder movement is simple to calculate. The step time
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generation uses the same constant acceleration and constant velocity
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formulas that cartesian robots use.
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### Pressure advance ###
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Experimentation has shown that it's possible to improve the modeling
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of the extruder beyond the basic extruder formula. In the ideal case,
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as an extrusion move progresses, the same volume of filament should be
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deposited at each point along the move and there should be no volume
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extruded after the move. Unfortunately, it's common to find that the
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basic extrusion formulas cause too little filament to exit the
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extruder at the start of extrusion moves and for excess filament to
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extrude after extrusion ends. This is often referred to as "ooze".
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The "pressure advance" system attempts to account for this by using a
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different model for the extruder. Instead of naively believing that
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each mm^3 of filament fed into the extruder will result in that amount
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of mm^3 immediately exiting the extruder, it uses a model based on
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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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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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[Poiseuille law](https://en.wikipedia.org/wiki/Poiseuille_law)). The
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details of the above physics are not important - only that the
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relationship between pressure and flow rate is linear. It is expected
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that an appropriate "pressure advance" value for a particular filament
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and extruder will be determined experimentally.
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Once configured, Klipper will push in an additional amount of filament
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during acceleration and retract that additional filament during
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deceleration. The higher the desired filament flow rate, the more
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filament must be pushed in during acceleration to account for
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pressure. Any additional filament pushed in during head acceleration
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is retracted during head deceleration (the extruder will have a
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negative velocity).
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One may notice that the pressure advance algorithm can cause the
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extruder motor to make sudden velocity changes. This is tolerated
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based on the idea that the majority of the inertia in the system is in
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changing the extruder pressure. As long as the extruder pressure does
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not change rapidly it is okay to make some sudden changes in extruder
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motor velocity.
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One area where sudden velocity changes become problematic is during
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small changes in head speed due to cornering.
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To prevent this, the Klipper pressure advance code utilizes the move
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lookahead queue to detect intermittent speed changes. In these cases
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the amount of pressure increased and decreased will be reduced or
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eliminated.
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