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Simen Svale Skogsrud
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doc/planner-maths.txt
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30
doc/planner-maths.txt
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Reasoning behind the mathematics in 'planner' module (in the key of 'Mathematica')
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==================================================================================
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s == speed, a == acceleration, t == time, d == distance
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Basic definitions:
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Speed[s_, a_, t_] := s + (a*t)
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Travel[s_, a_, t_] := Integrate[Speed[s, a, t], t]
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Distance to reach a specific speed with a constant acceleration:
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Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, d, t]
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d -> (m^2 - s^2)/(2 a) --> estimate_acceleration_distance()
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Speed after a given distance of travel with constant acceleration:
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Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, m, t]
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m -> Sqrt[2 a d + s^2]
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DestinationSpeed[s_, a_, d_] := Sqrt[2 a d + s^2]
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When to start braking (di) to reach a specified destionation speed (s2) after accelerating
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from initial speed s1 without ever stopping at a plateau:
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Solve[{DestinationSpeed[s1, a, di] == DestinationSpeed[s2, a, d - di]}, di]
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di -> (2 a d - s1^2 + s2^2)/(4 a) --> intersection_distance()
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IntersectionDistance[s1_, s2_, a_, d_] := (2 a d - s1^2 + s2^2)/(4 a)
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@ -3,7 +3,7 @@ The general structure of Grbl
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The main processing stack:
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'protocol' : Accepts command lines from the serial port and passes them to 'gcode' for execution.
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'protocol' : Accepts command lines from the serial port and passes them to 'gcode' for execution.
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Provides status responses for each command.
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'gcode' : Recieves gcode from 'protocol', parses it according to the current state
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@ -27,14 +27,16 @@
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// Execute linear motion in absolute millimeter coordinates. Feed rate given in millimeters/second
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// unless invert_feed_rate is true. Then the feed_rate means that the motion should be completed in
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// (1 minute)/feed_rate time.
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#define mc_line(x, y, z, feed_rate, invert_feed_rate) plan_buffer_line(x, y, z, feed_rate, invert_feed_rate)
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// NOTE: Although this function structurally belongs in this module, there is nothing to do but
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// to forward the request to the planner. For efficiency the function is implemented with a macro.
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#define mc_line(x, y, z, feed_rate, invert_feed_rate) plan_buffer_line(x, y, z, feed_rate, invert_feed_rate)
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// Execute an arc. theta == start angle, angular_travel == number of radians to go along the arc,
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// positive angular_travel means clockwise, negative means counterclockwise. Radius == the radius of the
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// circle in millimeters. axis_1 and axis_2 selects the circle plane in tool space. Stick the remaining
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// axis in axis_l which will be the axis for linear travel if you are tracing a helical motion.
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void mc_arc(double theta, double angular_travel, double radius, double linear_travel, int axis_1, int axis_2,
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int axis_linear, double feed_rate, int invert_feed_rate, double *position);
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55
planner.c
55
planner.c
@ -20,38 +20,6 @@
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/* The ring buffer implementation gleaned from the wiring_serial library by David A. Mellis. */
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/*
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Reasoning behind the mathematics in this module (in the key of 'Mathematica'):
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s == speed, a == acceleration, t == time, d == distance
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Basic definitions:
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Speed[s_, a_, t_] := s + (a*t)
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Travel[s_, a_, t_] := Integrate[Speed[s, a, t], t]
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Distance to reach a specific speed with a constant acceleration:
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Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, d, t]
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d -> (m^2 - s^2)/(2 a) --> estimate_acceleration_distance()
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Speed after a given distance of travel with constant acceleration:
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Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, m, t]
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m -> Sqrt[2 a d + s^2]
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DestinationSpeed[s_, a_, d_] := Sqrt[2 a d + s^2]
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When to start braking (di) to reach a specified destionation speed (s2) after accelerating
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from initial speed s1 without ever stopping at a plateau:
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Solve[{DestinationSpeed[s1, a, di] == DestinationSpeed[s2, a, d - di]}, di]
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di -> (2 a d - s1^2 + s2^2)/(4 a) --> intersection_distance()
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IntersectionDistance[s1_, s2_, a_, d_] := (2 a d - s1^2 + s2^2)/(4 a)
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*/
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#include <inttypes.h>
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#include <math.h>
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#include <stdlib.h>
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@ -67,8 +35,8 @@
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#define BLOCK_BUFFER_SIZE 16
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static block_t block_buffer[BLOCK_BUFFER_SIZE]; // A ring buffer for motion instructions
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static volatile uint8_t block_buffer_head; // Index of the next block to be pushed
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static volatile uint8_t block_buffer_tail; // Index of the block to process now
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static volatile uint8_t block_buffer_head; // Index of the next block to be pushed
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static volatile uint8_t block_buffer_tail; // Index of the block to process now
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// The current position of the tool in absolute steps
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static int32_t position[3];
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@ -86,11 +54,6 @@ double estimate_acceleration_distance(double initial_rate, double target_rate, d
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);
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}
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// This function gives you the point at which you must start braking (at the rate of -acceleration) if
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// you started at speed initial_rate and accelerated until this point and want to end at the final_rate after
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// a total travel of distance. This can be used to compute the intersection point between acceleration and
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// deceleration in the cases where the trapezoid has no plateau (i.e. never reaches maximum speed)
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/* + <- some maximum rate we don't care about
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/|\
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/ | \
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@ -101,6 +64,12 @@ double estimate_acceleration_distance(double initial_rate, double target_rate, d
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intersection_distance distance */
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// This function gives you the point at which you must start braking (at the rate of -acceleration) if
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// you started at speed initial_rate and accelerated until this point and want to end at the final_rate after
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// a total travel of distance. This can be used to compute the intersection point between acceleration and
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// deceleration in the cases where the trapezoid has no plateau (i.e. never reaches maximum speed)
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double intersection_distance(double initial_rate, double final_rate, double acceleration, double distance) {
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return(
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(2*acceleration*distance-initial_rate*initial_rate+final_rate*final_rate)/
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@ -108,10 +77,6 @@ double intersection_distance(double initial_rate, double final_rate, double acce
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);
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}
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// Calculates trapezoid parameters so that the entry- and exit-speed is compensated by the provided factors.
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// The factors represent a factor of braking and must be in the range 0.0-1.0.
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/*
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+--------+ <- nominal_rate
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/ \
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@ -121,6 +86,9 @@ double intersection_distance(double initial_rate, double final_rate, double acce
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time -->
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*/
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// Calculates trapezoid parameters so that the entry- and exit-speed is compensated by the provided factors.
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// The factors represent a factor of braking and must be in the range 0.0-1.0.
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void calculate_trapezoid_for_block(block_t *block, double entry_factor, double exit_factor) {
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block->initial_rate = ceil(block->nominal_rate*entry_factor);
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block->final_rate = ceil(block->nominal_rate*exit_factor);
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@ -418,4 +386,3 @@ void plan_buffer_line(double x, double y, double z, double feed_rate, int invert
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if (acceleration_manager_enabled) { planner_recalculate(); }
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st_wake_up();
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}
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