docs: Add initial Kinematics document

Signed-off-by: Kevin O'Connor <kevin@koconnor.net>

27 files changed, 2324 insertions, 3 deletions

diff --git a/docs/Code_Overview.md b/docs/Code_Overview.md
index 011de21a..07df5599 100644
--- a/docs/Code_Overview.md
+++ b/docs/Code_Overview.md
@@ -113,7 +113,8 @@ Code flow of a move command
 A typical printer movement starts when a "G1" command is sent to the
 Klippy host and it completes when the corresponding step pulses are
 produced on the micro-controller. This section outlines the code flow
-of a typical move command.
+of a typical move command. The [kinematics](Kinematics.md) document
+provides further information on the mechanics of moves.
 
 * Processing for a move command starts in gcode.py. The goal of
   gcode.py is to translate G-code into internal calls. Changes in
diff --git a/docs/Kinematics.md b/docs/Kinematics.md
new file mode 100644
index 00000000..924d1291
--- /dev/null
+++ b/docs/Kinematics.md
@@ -0,0 +1,266 @@
+This document provides an overview of how Klipper implements robot
+motion (its [kinematics](https://en.wikipedia.org/wiki/Kinematics)).
+The contents may be of interest to both developers interested in
+working on the Klipper software as well as users interested in better
+understanding the mechanics of their machines.
+
+Acceleration
+============
+
+Klipper implements a constant acceleration scheme whenever the print
+head changes velocity - the velocity is gradually changed to the new
+speed instead of suddenly jerking to it.  Klipper always enforces
+acceleration between the tool head and the print.  The filament
+leaving the extruder can be quite fragile - rapid jerks and/or
+extruder flow changes lead to poor quality and poor bed adhesion. Even
+when not extruding, if the print head is at the same level as the
+print then rapid jerking of the head can cause disruption of recently
+deposited filament. Limiting speed changes of the print head (relative
+to the print) reduces risks of disrupting the print.
+
+It is also important to enforce a maximum acceleration of the stepper
+motors to ensure they do not skip or put excessive stress on the
+machine. Klipper limits the acceleration of each stepper by virtue of
+limiting the acceleration of the print head. Enforcing acceleration at
+the print head naturally also enforces acceleration at the steppers
+that control that print head (the inverse is not always true).
+
+Klipper implements constant acceleration.  The key formula for
+constant acceleration is:
+```
+velocity(time) = start_velocity + accel*time
+```
+
+Trapezoid generator
+===================
+
+Klipper uses a traditional "trapezoid generator" to model the motion
+of each move - each move has a start speed, it accelerates to a
+cruising speed at constant acceleration, it cruises at a constant
+speed, and then decelerates to the end speed using constant
+acceleration.
+
+![trapezoid](img/trapezoid.svg.png)
+
+It's called a "trapezoid generator" because a velocity diagram of the
+move looks like a trapezoid.
+
+The cruising speed is always greater than or equal to both the start
+speed and the end speed. The acceleration phase may be of zero
+duration (if the start speed is equal to the cruising speed), the
+cruising phase may be of zero duration (if the move immediately starts
+decelerating after acceleration), and/or the deceleration phase may be
+of zero duration (if the end speed is equal to the cruising speed).
+
+![trapezoids](img/trapezoids.svg.png)
+
+Lookahead
+=========
+
+The "lookahead" system is used to determine cornering speeds between
+moves.
+
+Consider the following two moves contained on an XY plane:
+
+![corner](img/corner.svg.png)
+
+In the above situation it is possible to fully decelerate after the
+first move and then fully accelerate at the start of the next move,
+but that is not ideal as all that acceleration and deceleration would
+greatly increase the print time and the frequent changes in extruder
+flow would result in poor print quality.
+
+To solve this, the "lookahead" mechanism queues multiple incoming
+moves and analyzes the angles between moves to determine a reasonable
+speed that can be obtained during the "junction" between two moves. If
+the next move forms an acute angle (the head is going to travel in
+nearly a reverse direction on the next move) then only a small
+junction speed is permitted. If the next move is nearly in the same
+direction then the head need only slow down a little (if at all).
+
+![lookahead](img/lookahead.svg.png)
+
+The junction speeds are determined using "approximated centripetal
+acceleration". Best
+[described](https://onehossshay.wordpress.com/2011/09/24/improving_grbl_cornering_algorithm/)
+by the author.
+
+Klipper implements lookahead between moves contained in the XY plane
+that have similar extruder flow rates. Other moves are rare and
+implementing lookahead between them is unnecessary.
+
+Key formula for lookahead:
+```
+end_velocity^2 = start_velocity^2 + 2*accel*move_distance
+```
+
+Smoothed lookahead
+------------------
+
+Klipper also implements a mechanism for smoothing out the motions of
+short "zig-zag" moves.  Consider the following moves:
+
+![zigzag](img/zigzag.svg.png)
+
+In the above, the frequent changes from acceleration to deceleration
+can cause the machine to vibrate which causes stress on the machine
+and increases the noise.  To reduce this, Klipper tracks both regular
+move acceleration as well as a virtual "acceleration to deceleration"
+rate.  Using this system, the top speed of these short "zig zag" moves
+are limited to smooth out the printer motion:
+
+![smoothed](img/smoothed.svg.png)
+
+In the above, note the dashed gray lines - this is a graphical
+representation of the "pseudo acceleration".  Where the two dashed
+lines meet enforces a limit on the move's top speed.  For most moves
+the limit will be at or above the move's existing limits and no change
+in behavior is induced. However, for short "zig-zag" moves the limit
+comes into play and it reduces the top speed. Note that the grey lines
+represent a pseudo-acceleration to limit top speed only - the move
+continues to use it's normal acceleration scheme up to its adjusted
+top-speed.
+
+Generating steps
+================
+
+Once the lookahead process completes, the print head movement for the
+given move is fully known (time, start position, end position,
+velocity at each point) and it is possible to generate the step times
+for the move.  This process is done within "kinematic classes" in the
+Klipper code.  Outside of these kinematic classes, everything is
+tracked in millimeters, seconds, and in cartesian coordinate space.
+It's the task of the kinematic classes to convert from this generic
+coordinate system to the hardware specifics of the particular printer.
+
+In general, the code determines each step time by first calculating
+where along the line of movement the head would be if a step is
+taken. It then calculates what time the head should be at that
+position. Determining the time along the line of movement can be done
+using the formulas for constant acceleration and constant velocity:
+
+```
+time = sqrt(2*distance/accel + (start_velocity/accel)^2) - start_velocity/accel
+time = distance/cruise_velocity
+```
+
+Cartesian Robots
+----------------
+
+Generating steps for cartesian printers is the simplest case. The
+movement on each axis is directly related to the movement in cartesian
+space.
+
+Delta Robots
+------------
+
+To generate step times on Delta printers it is necessary to correlate
+the movement in cartesian space with the movement on each stepper
+tower.
+
+![delta-tower](img/delta-tower.svg.png)
+
+To simplify the math, for each move contained in an XY plane, the code
+calculates the location of a "virtual tower" that is along the line of
+movement.  This virtual tower is chosen at the point where the line of
+movement (extended infinitely in both directions) would be closest to
+the actual tower.
+
+![virtual-tower](img/virtual-tower.svg.png)
+
+It is then possible to calculate where the head will be along the line
+of movement after each step is taken on the virtual tower. The key
+formula is Pythagorean's formula:
+
+```
+distance_to_tower^2 = arm_length^2 - tower_height^2
+```
+
+One complexity is that if the print head passes the virtual tower
+location then the stepper direction must be reversed. In this case
+forward steps will be taken at the start of the move and reverse steps
+will be taken at the end of the move.
+
+### Delta movements beyond simple XY plane ###
+
+Movement calculation is a little more complicated if the move is not
+fully contained within a simple XY plane. A virtual tower along the
+line of movement is still calculated, but in this case the tower is
+not at a 90 degree angle relative to the line of movement:
+
+![xy+z-tower](img/xy+z-tower.svg.png)
+
+The code continues to calculate step times using the same general
+scheme as delta moves within an XY plane, but the slope of the tower
+must also be used in the calculations.
+
+Should the move contain only Z movement (ie, no XY movement at all)
+then the same math is used - just in this case the tower is parallel
+to the line of movement (its slope is 1.0).
+
+Extruder kinematics
+-------------------
+
+Klipper implements extruder motion in its own kinematic class.  Since
+the timing and speed of each print head movement is fully known for
+each move, it's possible to calculate the step times for the extruder
+independently from the step time calculations of the print head
+movement.
+
+Basic extruder movement is simple to calculate. The step time
+generation uses the same constant acceleration and constant velocity
+formulas that cartesian robots use.
+
+### Pressure advance ###
+
+Experimentation has shown that it's possible to improve the modeling
+of the extruder beyond the basic extruder formula. In the ideal case,
+as an extrusion move progresses, the same volume of filament should be
+deposited at each point along the move and there should be no volume
+extruded after the move. Unfortunately, it's common to find that the
+basic extrusion formulas cause too little filament to exit the
+extruder at the start of extrusion moves and for excess filament to
+extrude after extrusion ends. This is often referred to as "ooze".
+
+![ooze](img/ooze.svg.png)
+
+The "pressure advance" system attempts to account for this by using a
+different model for the extruder. Instead of naively believing that
+each mm^3 of filament fed into the extruder will result in that amount
+of mm^3 immediately exiting the extruder, it uses a model based on
+pressure. Pressure increases when filament is pushed into the extruder
+(as in [Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law)) and
+the pressure necessary to extrude is dominated by the flow rate
+through the nozzle orifice (as in
+[Poiseuille law](https://en.wikipedia.org/wiki/Poiseuille_law)). The
+details of the above physics are not important - only that the
+relationship between pressure and flow rate is linear. It is expected
+that an appropriate "pressure advance" value for a particular filament
+and extruder will be determined experimentally.
+
+Once configured, Klipper will push in an additional amount of filament
+during acceleration and retract that additional filament during
+deceleration. The higher the desired filament flow rate, the more
+filament must be pushed in during acceleration to account for
+pressure. Any additional filament pushed in during head acceleration
+is retracted during head deceleration (the extruder will have a
+negative velocity).
+
+![pressure-advance](img/pressure-advance.svg.png)
+
+One may notice that the pressure advance algorithm can cause the
+extruder motor to make sudden velocity changes. This is tolerated
+based on the idea that the majority of the inertia in the system is in
+changing the extruder pressure. As long as the extruder pressure does
+not change rapidly it is okay to make some sudden changes in extruder
+motor velocity.
+
+One area where sudden velocity changes become problematic is during
+small changes in head speed due to cornering.
+
+![pressure-cornering](img/pressure-cornering.svg.png)
+
+To prevent this, the Klipper pressure advance code utilizes the move
+lookahead queue to detect intermittent speed changes. In these cases
+the amount of pressure increased and decreased will be reduced or
+eliminated.
diff --git a/docs/Overview.md b/docs/Overview.md
index 8b221763..14fb5bd3 100644
--- a/docs/Overview.md
+++ b/docs/Overview.md
@@ -1,7 +1,9 @@
 See [installation](Installation.md) for information on compiling,
 installing, and running Klipper. Read [features](Features.md) for a
-high-level description of useful capabilities. The history of releases
-is available at [releases](Releases.md).
+high-level description of useful capabilities. The
+[kinematics](Kinematics.md) document provides information on how
+movement is implemented. The history of releases is available at
+[releases](Releases.md).
 
 There are also several documents available for developers interested
 in understanding how Klipper works:
diff --git a/docs/img/corner.svg b/docs/img/corner.svg
new file mode 100644
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diff --git a/docs/img/corner.svg.png b/docs/img/corner.svg.png
new file mode 100644
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diff --git a/docs/img/delta-tower.svg b/docs/img/delta-tower.svg
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