kin_extruder: Convert pressure advance to use "weighted average"

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

1 files changed, 5 insertions, 4 deletions

diff --git a/docs/Kinematics.md b/docs/Kinematics.md
index f4b84550..ee501dff 100644
--- a/docs/Kinematics.md
+++ b/docs/Kinematics.md
@@ -284,8 +284,8 @@ rate, the more filament must be pushed in during acceleration to
 account for pressure. During head deceleration the extra filament is
 retracted (the extruder will have a negative velocity).
 
-The "smoothing" is implemented by averaging the extruder position over
-a small time period (as specified by the
+The "smoothing" is implemented using a weighted average of the
+extruder position over a small time period (as specified by the
 `pressure_advance_smooth_time` config parameter). This averaging can
 span multiple g-code moves. Note how the extruder motor will start
 moving prior to the nominal start of the first extrusion move and will
@@ -294,6 +294,7 @@ continue to move after the nominal end of the last extrusion move.
 Key formula for "smoothed pressure advance":
 ```
 smooth_pa_position(t) =
-    ( definitive_integral(pa_position, from=t-smooth_time/2, to=t+smooth_time/2)
-     / smooth_time )
+    ( definitive_integral(pa_position(x) * (smooth_time/2 - abs(t - x)) * dx,
+                          from=t-smooth_time/2, to=t+smooth_time/2)
+     / (smooth_time/2)^2 )
 ```