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planner.c - buffers movement commands and manages the acceleration profile plan
Part of Grbl
Copyright (c) 2009-2011 Simen Svale Skogsrud
Grbl is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
Grbl is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with Grbl. If not, see <>.
/* The ring buffer implementation gleaned from the wiring_serial library by David A. Mellis. */
Reasoning behind the mathematics in this module (in the key of 'Mathematica'):
s == speed, a == acceleration, t == time, d == distance
Basic definitions:
Speed[s_, a_, t_] := s + (a*t)
Travel[s_, a_, t_] := Integrate[Speed[s, a, t], t]
Distance to reach a specific speed with a constant acceleration:
Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, d, t]
d -> (m^2 - s^2)/(2 a) --> estimate_acceleration_distance()
Speed after a given distance of travel with constant acceleration:
Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, m, t]
m -> Sqrt[2 a d + s^2]
DestinationSpeed[s_, a_, d_] := Sqrt[2 a d + s^2]
When to start braking (di) to reach a specified destionation speed (s2) after accelerating
from initial speed s1 without ever stopping at a plateau:
Solve[{DestinationSpeed[s1, a, di] == DestinationSpeed[s2, a, d - di]}, di]
di -> (2 a d - s1^2 + s2^2)/(4 a) --> intersection_distance()
IntersectionDistance[s1_, s2_, a_, d_] := (2 a d - s1^2 + s2^2)/(4 a)
#include <inttypes.h>
#include <math.h>
#include <stdlib.h>
#include "planner.h"
#include "nuts_bolts.h"
#include "stepper.h"
#include "settings.h"
#include "config.h"
#include "wiring_serial.h"
// The number of linear motions that can be in the plan at any give time
#ifdef __AVR_ATmega328P__
// The current position of the tool in absolute steps
static int32_t position[3];
static block_t block_buffer[BLOCK_BUFFER_SIZE]; // A ring buffer for motion instructions
static volatile int block_buffer_head; // Index of the next block to be pushed
static volatile int block_buffer_tail; // Index of the block to process now
static uint8_t acceleration_manager_enabled; // Acceleration management active?
// Calculates the distance (not time) it takes to accelerate from initial_rate to target_rate using the
// given acceleration:
inline double estimate_acceleration_distance(double initial_rate, double target_rate, double acceleration) {
// This function gives you the point at which you must start braking (at the rate of -acceleration) if
// you started at speed initial_rate and accelerated until this point and want to end at the final_rate after
// a total travel of distance. This can be used to compute the intersection point between acceleration and
// deceleration in the cases where the trapezoid has no plateau (i.e. never reaches maximum speed)
/* + <- some maximum rate we don't care about
/ | \
/ | + <- final_rate
/ | |
initial_rate -> +----+--+
^ ^
| |
intersection_distance distance */
inline double intersection_distance(double initial_rate, double final_rate, double acceleration, double distance) {
// Calculates trapezoid parameters so that the entry- and exit-speed is compensated by the provided factors.
// The factors represent a factor of braking and must be in the range 0.0-1.0.
+--------+ <- nominal_rate
/ \
nominal_rate*entry_factor -> + \
| + <- nominal_rate*exit_factor
time -->
void calculate_trapezoid_for_block(block_t *block, double entry_factor, double exit_factor) {
block->initial_rate = ceil(block->nominal_rate*entry_factor);
block->final_rate = ceil(block->nominal_rate*exit_factor);
int32_t acceleration_per_minute = block->rate_delta*ACCELERATION_TICKS_PER_SECOND*60.0;
int32_t accelerate_steps =
ceil(estimate_acceleration_distance(block->initial_rate, block->nominal_rate, acceleration_per_minute));
int32_t decelerate_steps =
floor(estimate_acceleration_distance(block->nominal_rate, block->final_rate, -acceleration_per_minute));
// Calculate the size of Plateau of Nominal Rate.
int32_t plateau_steps = block->step_event_count-accelerate_steps-decelerate_steps;
// Is the Plateau of Nominal Rate smaller than nothing? That means no cruising, and we will
// have to use intersection_distance() to calculate when to abort acceleration and start braking
// in order to reach the final_rate exactly at the end of this block.
if (plateau_steps < 0) {
accelerate_steps = ceil(
intersection_distance(block->initial_rate, block->final_rate, acceleration_per_minute, block->step_event_count));
plateau_steps = 0;
block->accelerate_until = accelerate_steps;
block->decelerate_after = accelerate_steps+plateau_steps;
// Calculates the maximum allowable speed at this point when you must be able to reach target_velocity using the
// acceleration within the allotted distance.
inline double max_allowable_speed(double acceleration, double target_velocity, double distance) {
// "Junction jerk" in this context is the immediate change in speed at the junction of two blocks.
// This method will calculate the junction jerk as the euclidean distance between the nominal
// velocities of the respective blocks.
inline double junction_jerk(block_t *before, block_t *after) {
pow(before->speed_x-after->speed_x, 2)+
pow(before->speed_y-after->speed_y, 2)+
pow(before->speed_z-after->speed_z, 2))
// Calculate a braking factor to reach baseline speed which is max_jerk/2, e.g. the
// speed under which you cannot exceed max_jerk no matter what you do.
double factor_for_safe_speed(block_t *block) {
// The kernel called by planner_recalculate() when scanning the plan from last to first entry.
void planner_reverse_pass_kernel(block_t *previous, block_t *current, block_t *next) {
if(!current) { return; }
double entry_factor = 1.0;
double exit_factor;
if (next) {
exit_factor = next->entry_factor;
} else {
exit_factor = factor_for_safe_speed(current);
// Calculate the entry_factor for the current block.
if (previous) {
// Reduce speed so that junction_jerk is within the maximum allowed
double jerk = junction_jerk(previous, current);
if (jerk > settings.max_jerk) {
entry_factor = (settings.max_jerk/jerk);
// If the required deceleration across the block is too rapid, reduce the entry_factor accordingly.
if (entry_factor > exit_factor) {
double max_entry_speed = max_allowable_speed(-settings.acceleration,current->nominal_speed*exit_factor,
double max_entry_factor = max_entry_speed/current->nominal_speed;
if (max_entry_factor < entry_factor) {
entry_factor = max_entry_factor;
} else {
entry_factor = factor_for_safe_speed(current);
// Store result
current->entry_factor = entry_factor;
// planner_recalculate() needs to go over the current plan twice. Once in reverse and once forward. This
// implements the reverse pass.
void planner_reverse_pass() {
auto int8_t block_index = block_buffer_head;
block_t *block[3] = {NULL, NULL, NULL};
while(block_index != block_buffer_tail) {
if(block_index < 0) {
block_index = BLOCK_BUFFER_SIZE-1;
block[2]= block[1];
block[1]= block[0];
block[0] = &block_buffer[block_index];
planner_reverse_pass_kernel(block[0], block[1], block[2]);
planner_reverse_pass_kernel(NULL, block[0], block[1]);
// The kernel called by planner_recalculate() when scanning the plan from first to last entry.
void planner_forward_pass_kernel(block_t *previous, block_t *current, block_t *next) {
if(!current) { return; }
// If the previous block is an acceleration block, but it is not long enough to
// complete the full speed change within the block, we need to adjust out entry
// speed accordingly. Remember current->entry_factor equals the exit factor of
// the previous block.
if(previous->entry_factor < current->entry_factor) {
double max_entry_speed = max_allowable_speed(-settings.acceleration,
current->nominal_speed*previous->entry_factor, previous->millimeters);
double max_entry_factor = max_entry_speed/current->nominal_speed;
if (max_entry_factor < current->entry_factor) {
current->entry_factor = max_entry_factor;
// planner_recalculate() needs to go over the current plan twice. Once in reverse and once forward. This
// implements the forward pass.
void planner_forward_pass() {
int8_t block_index = block_buffer_tail;
block_t *block[3] = {NULL, NULL, NULL};
while(block_index != block_buffer_head) {
block[0] = block[1];
block[1] = block[2];
block[2] = &block_buffer[block_index];
block_index = (block_index+1) % BLOCK_BUFFER_SIZE;
planner_forward_pass_kernel(block[1], block[2], NULL);
// Recalculates the trapezoid speed profiles for all blocks in the plan according to the
// entry_factor for each junction. Must be called by planner_recalculate() after
// updating the blocks.
void planner_recalculate_trapezoids() {
int8_t block_index = block_buffer_tail;
block_t *current;
block_t *next = NULL;
while(block_index != block_buffer_head) {
current = next;
next = &block_buffer[block_index];
if (current) {
calculate_trapezoid_for_block(current, current->entry_factor, next->entry_factor);
block_index = (block_index+1) % BLOCK_BUFFER_SIZE;
calculate_trapezoid_for_block(next, next->entry_factor, factor_for_safe_speed(next));
// Recalculates the motion plan according to the following algorithm:
// 1. Go over every block in reverse order and calculate a junction speed reduction (i.e. block_t.entry_factor)
// so that:
// a. The junction jerk is within the set limit
// b. No speed reduction within one block requires faster deceleration than the one, true constant
// acceleration.
// 2. Go over every block in chronological order and dial down junction speed reduction values if
// a. The speed increase within one block would require faster accelleration than the one, true
// constant acceleration.
// When these stages are complete all blocks have an entry_factor that will allow all speed changes to
// be performed using only the one, true constant acceleration, and where no junction jerk is jerkier than
// the set limit. Finally it will:
// 3. Recalculate trapezoids for all blocks.
void planner_recalculate() {
void plan_init() {
block_buffer_head = 0;
block_buffer_tail = 0;
void plan_set_acceleration_manager_enabled(int enabled) {
if ((!!acceleration_manager_enabled) != (!!enabled)) {
acceleration_manager_enabled = !!enabled;
int plan_is_acceleration_manager_enabled() {
inline void plan_discard_current_block() {
if (block_buffer_head != block_buffer_tail) {
block_buffer_tail = (block_buffer_tail + 1) % BLOCK_BUFFER_SIZE;
inline block_t *plan_get_current_block() {
if (block_buffer_head == block_buffer_tail) { return(NULL); }
// Add a new linear movement to the buffer. steps_x, _y and _z is the absolute position in
// mm. Microseconds specify how many microseconds the move should take to perform. To aid acceleration
// calculation the caller must also provide the physical length of the line in millimeters.
void plan_buffer_line(double x, double y, double z, double feed_rate, int invert_feed_rate) {
// The target position of the tool in absolute steps
// Calculate target position in absolute steps
int32_t target[3];
target[X_AXIS] = lround(x*settings.steps_per_mm[X_AXIS]);
target[Y_AXIS] = lround(y*settings.steps_per_mm[Y_AXIS]);
target[Z_AXIS] = lround(z*settings.steps_per_mm[Z_AXIS]);
// Calculate the buffer head after we push this byte
int next_buffer_head = (block_buffer_head + 1) % BLOCK_BUFFER_SIZE;
// If the buffer is full: good! That means we are well ahead of the robot.
// Rest here until there is room in the buffer.
while(block_buffer_tail == next_buffer_head) { sleep_mode(); }
// Prepare to set up new block
block_t *block = &block_buffer[block_buffer_head];
// Number of steps for each axis
block->steps_x = labs(target[X_AXIS]-position[X_AXIS]);
block->steps_y = labs(target[Y_AXIS]-position[Y_AXIS]);
block->steps_z = labs(target[Z_AXIS]-position[Z_AXIS]);
block->step_event_count = max(block->steps_x, max(block->steps_y, block->steps_z));
// Bail if this is a zero-length block
if (block->step_event_count == 0) { return; };
double delta_x_mm = (target[X_AXIS]-position[X_AXIS])/settings.steps_per_mm[X_AXIS];
double delta_y_mm = (target[Y_AXIS]-position[Y_AXIS])/settings.steps_per_mm[Y_AXIS];
double delta_z_mm = (target[Z_AXIS]-position[Z_AXIS])/settings.steps_per_mm[Z_AXIS];
block->millimeters = sqrt(square(delta_x_mm) + square(delta_y_mm) + square(delta_z_mm));
uint32_t microseconds;
if (!invert_feed_rate) {
microseconds = lround((block->millimeters/feed_rate)*1000000);
} else {
microseconds = lround(ONE_MINUTE_OF_MICROSECONDS/feed_rate);
// Calculate speed in mm/minute for each axis
double multiplier = 60.0*1000000.0/microseconds;
block->speed_x = delta_x_mm * multiplier;
block->speed_y = delta_y_mm * multiplier;
block->speed_z = delta_z_mm * multiplier;
block->nominal_speed = block->millimeters * multiplier;
block->nominal_rate = ceil(block->step_event_count * multiplier);
block->entry_factor = 0.0;
// Compute the acceleration rate for the trapezoid generator. Depending on the slope of the line
// average travel per step event changes. For a line along one axis the travel per step event
// is equal to the travel/step in the particular axis. For a 45 degree line the steppers of both
// axes might step for every step event. Travel per step event is then sqrt(travel_x^2+travel_y^2).
// To generate trapezoids with contant acceleration between blocks the rate_delta must be computed
// specifically for each line to compensate for this phenomenon:
double travel_per_step = block->millimeters/block->step_event_count;
block->rate_delta = ceil(
((settings.acceleration*60.0)/(ACCELERATION_TICKS_PER_SECOND))/ // acceleration mm/sec/sec per acceleration_tick
travel_per_step); // convert to: acceleration steps/min/acceleration_tick
if (acceleration_manager_enabled) {
// compute a preliminary conservative acceleration trapezoid
double safe_speed_factor = factor_for_safe_speed(block);
calculate_trapezoid_for_block(block, safe_speed_factor, safe_speed_factor);
} else {
block->initial_rate = block->nominal_rate;
block->final_rate = block->nominal_rate;
block->accelerate_until = 0;
block->decelerate_after = block->step_event_count;
block->rate_delta = 0;
// Compute direction bits for this block
block->direction_bits = 0;
if (target[X_AXIS] < position[X_AXIS]) { block->direction_bits |= (1<<X_DIRECTION_BIT); }
if (target[Y_AXIS] < position[Y_AXIS]) { block->direction_bits |= (1<<Y_DIRECTION_BIT); }
if (target[Z_AXIS] < position[Z_AXIS]) { block->direction_bits |= (1<<Z_DIRECTION_BIT); }
// Move buffer head
block_buffer_head = next_buffer_head;
// Update position
memcpy(position, target, sizeof(target)); // position[] = target[]
if (acceleration_manager_enabled) { planner_recalculate(); }
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