294 lines
12 KiB
Markdown
294 lines
12 KiB
Markdown
# Kinematics
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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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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 leaving
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the extruder can be quite fragile - rapid jerks and/or extruder flow
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changes lead to poor quality and poor bed adhesion. Even when not
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extruding, if the print head is at the same level as the print then
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rapid jerking of the head can cause disruption of recently deposited
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filament. Limiting speed changes of the print head (relative to the
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print) reduces risks of disrupting the print.
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It is also important to limit acceleration so that the stepper motors
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do not skip or put excessive stress on the machine. Klipper limits the
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torque on each stepper by virtue of limiting the acceleration of the
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print head. Enforcing acceleration at the print head naturally also
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limits the torque of the steppers that move the print head (the
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inverse is not always true).
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Klipper implements constant acceleration. The key formula for constant
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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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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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## Look-ahead
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The "look-ahead" 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 "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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speed that can be obtained during the "junction" between two moves. If
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the next move is nearly in the same direction then the head need only
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slow down a little (if at all).
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However, if the next move forms an acute angle (the head is going to
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travel in nearly a reverse direction on the next move) then only a
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small junction speed is permitted.
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The junction speeds are determined using "approximated centripetal
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acceleration". Best
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[described by the author](https://onehossshay.wordpress.com/2011/09/24/improving_grbl_cornering_algorithm/).
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However, in Klipper, junction speeds are configured by specifying the
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desired speed that a 90° corner should have (the "square corner
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velocity"), and the junction speeds for other angles are derived from
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that.
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Key formula for look-ahead:
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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 look-ahead
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Klipper also implements a mechanism for smoothing out the motions of
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short "zigzag" 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 "zigzag" moves
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are limited to smooth out the printer motion:
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Specifically, the code calculates what the velocity of each move would
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be if it were limited to this virtual "acceleration to deceleration"
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rate (half the normal acceleration rate by default). In the above
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picture the dashed gray lines represent this virtual acceleration rate
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for the first move. If a move can not reach its full cruising speed
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using this virtual acceleration rate then its top speed is reduced to
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the maximum speed it could obtain at this virtual acceleration
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rate. For most moves the limit will be at or above the move's existing
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limits and no change in behavior is induced. For short zigzag moves,
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however, this limit reduces the top speed. Note that it does not
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change the actual acceleration within the move - the move continues to
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use the normal acceleration scheme up to its adjusted top-speed.
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## Generating steps
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Once the look-ahead 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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Klipper uses an
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[iterative solver](https://en.wikipedia.org/wiki/Root-finding_algorithm)
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to generate the step times for each stepper. The code contains the
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formulas to calculate the ideal cartesian coordinates of the head at
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each moment in time, and it has the kinematic formulas to calculate
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the ideal stepper positions based on those cartesian coordinates. With
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these formulas, Klipper can determine the ideal time that the stepper
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should be at each step position. The given steps are then scheduled at
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these calculated times.
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The key formula to determine how far a move should travel under
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constant acceleration is:
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```
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move_distance = (start_velocity + .5 * accel * move_time) * move_time
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```
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and the key formula for movement with constant velocity is:
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```
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move_distance = cruise_velocity * move_time
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```
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The key formulas for determining the cartesian coordinate of a move
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given a move distance is:
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```
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cartesian_x_position = start_x + move_distance * total_x_movement / total_movement
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cartesian_y_position = start_y + move_distance * total_y_movement / total_movement
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cartesian_z_position = start_z + move_distance * total_z_movement / total_movement
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```
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### Cartesian Robots
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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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Key formulas:
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```
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stepper_x_position = cartesian_x_position
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stepper_y_position = cartesian_y_position
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stepper_z_position = cartesian_z_position
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```
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### CoreXY Robots
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Generating steps on a CoreXY machine is only a little more complex
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than basic cartesian robots. The key formulas are:
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```
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stepper_a_position = cartesian_x_position + cartesian_y_position
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stepper_b_position = cartesian_x_position - cartesian_y_position
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stepper_z_position = cartesian_z_position
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```
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### Delta Robots
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Step generation on a delta robot is based on Pythagoras's theorem:
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```
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stepper_position = (sqrt(arm_length^2
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- (cartesian_x_position - tower_x_position)^2
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- (cartesian_y_position - tower_y_position)^2)
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+ cartesian_z_position)
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```
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### Stepper motor acceleration limits
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With delta kinematics it is possible for a move that is accelerating
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in cartesian space to require an acceleration on a particular stepper
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motor greater than the move's acceleration. This can occur when a
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stepper arm is more horizontal than vertical and the line of movement
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passes near that stepper's tower. Although these moves could require a
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stepper motor acceleration greater than the printer's maximum
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configured move acceleration, the effective mass moved by that stepper
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would be smaller. Thus the higher stepper acceleration does not result
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in significantly higher stepper torque and it is therefore considered
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harmless.
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However, to avoid extreme cases, Klipper enforces a maximum ceiling on
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stepper acceleration of three times the printer's configured maximum
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move acceleration. (Similarly, the maximum velocity of the stepper is
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limited to three times the maximum move velocity.) In order to enforce
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this limit, moves at the extreme edge of the build envelope (where a
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stepper arm may be nearly horizontal) will have a lower maximum
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acceleration and velocity.
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### Extruder kinematics
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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 formulas that cartesian robots use:
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```
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stepper_position = requested_e_position
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```
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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'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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rate can be modeled using a linear coefficient:
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```
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pa_position = nominal_position + pressure_advance_coefficient * nominal_velocity
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```
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See the [pressure advance](Pressure_Advance.md) document for
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information on how to find this pressure advance coefficient.
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The basic pressure advance formula can cause the extruder motor to
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make sudden velocity changes. Klipper implements "smoothing" of the
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extruder movement to avoid this.
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The above graph shows an example of two extrusion moves with a
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non-zero cornering velocity between them. Note that the pressure
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advance system causes additional filament to be pushed into the
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extruder during acceleration. The higher the desired filament flow
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rate, the more filament must be pushed in during acceleration to
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account for pressure. During head deceleration the extra filament is
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retracted (the extruder will have a negative velocity).
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The "smoothing" is implemented using a weighted average of the
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extruder position over a small time period (as specified by the
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`pressure_advance_smooth_time` config parameter). This averaging can
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span multiple g-code moves. Note how the extruder motor will start
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moving prior to the nominal start of the first extrusion move and will
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continue to move after the nominal end of the last extrusion move.
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Key formula for "smoothed pressure advance":
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```
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smooth_pa_position(t) =
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( definitive_integral(pa_position(x) * (smooth_time/2 - abs(t - x)) * dx,
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from=t-smooth_time/2, to=t+smooth_time/2)
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/ (smooth_time/2)^2 )
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```
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