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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
Copyright (c) 2011 Sungeun K. Jeon
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. */
#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"
// The number of linear motions that can be in the plan at any give time
#ifdef __AVR_ATmega328P__
static block_t block_buffer[BLOCK_BUFFER_SIZE]; // A ring buffer for motion instructions
static volatile uint8_t block_buffer_head; // Index of the next block to be pushed
static volatile uint8_t block_buffer_tail; // Index of the block to process now
static int32_t position[3]; // The current position of the tool in absolute steps
static double previous_unit_vec[3]; // Unit vector of previous path line segment
static double previous_nominal_speed; // Nominal speed of previous path line segment
// Returns the index of the next block in the ring buffer
// NOTE: Removed modulo (%) operator, which uses an expensive divide and multiplication.
static uint8_t next_block_index(uint8_t block_index)
if (block_index == BLOCK_BUFFER_SIZE) { block_index = 0; }
// Returns the index of the previous block in the ring buffer
static uint8_t prev_block_index(uint8_t block_index)
if (block_index == 0) { block_index = BLOCK_BUFFER_SIZE; }
// Calculates the distance (not time) it takes to accelerate from initial_rate to target_rate using the
// given acceleration:
static double estimate_acceleration_distance(double initial_rate, double target_rate, double acceleration)
return( (target_rate*target_rate-initial_rate*initial_rate)/(2*acceleration) );
/* + <- some maximum rate we don't care about
/ | \
/ | + <- final_rate
/ | |
initial_rate -> +----+--+
^ ^
| |
intersection_distance distance */
// 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)
static double intersection_distance(double initial_rate, double final_rate, double acceleration, double distance)
return( (2*acceleration*distance-initial_rate*initial_rate+final_rate*final_rate)/(4*acceleration) );
// Calculates the maximum allowable speed at this point when you must be able to reach target_velocity
// using the acceleration within the allotted distance.
// NOTE: sqrt() reimplimented here from prior version due to improved planner logic. Increases speed
// in time critical computations, i.e. arcs or rapid short lines from curves. Guaranteed to not exceed
// BLOCK_BUFFER_SIZE calls per planner cycle.
static double max_allowable_speed(double acceleration, double target_velocity, double distance)
return( sqrt(target_velocity*target_velocity-2*acceleration*distance) );
// The kernel called by planner_recalculate() when scanning the plan from last to first entry.
static void planner_reverse_pass_kernel(block_t *previous, block_t *current, block_t *next)
if (!current) { return; } // Cannot operate on nothing.
if (next) {
// If entry speed is already at the maximum entry speed, no need to recheck. Block is cruising.
// If not, block in state of acceleration or deceleration. Reset entry speed to maximum and
// check for maximum allowable speed reductions to ensure maximum possible planned speed.
if (current->entry_speed != current->max_entry_speed) {
// If nominal length true, max junction speed is guaranteed to be reached. Only compute
// for max allowable speed if block is decelerating and nominal length is false.
if ((!current->nominal_length_flag) && (current->max_entry_speed > next->entry_speed)) {
current->entry_speed = min( current->max_entry_speed,
} else {
current->entry_speed = current->max_entry_speed;
current->recalculate_flag = true;
} // Skip last block. Already initialized and set for recalculation.
// planner_recalculate() needs to go over the current plan twice. Once in reverse and once forward. This
// implements the reverse pass.
static void planner_reverse_pass()
uint8_t block_index = block_buffer_head;
block_t *block[3] = {NULL, NULL, NULL};
while(block_index != block_buffer_tail) {
block_index = prev_block_index( block_index );
block[2]= block[1];
block[1]= block[0];
block[0] = &block_buffer[block_index];
planner_reverse_pass_kernel(block[0], block[1], block[2]);
// Skip buffer tail/first block to prevent over-writing the initial entry speed.
// The kernel called by planner_recalculate() when scanning the plan from first to last entry.
static void planner_forward_pass_kernel(block_t *previous, block_t *current, block_t *next)
if(!previous) { return; } // Begin planning after buffer_tail
// 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 the entry speed accordingly. Entry
// speeds have already been reset, maximized, and reverse planned by reverse planner.
// If nominal length is true, max junction speed is guaranteed to be reached. No need to recheck.
if (!previous->nominal_length_flag) {
if (previous->entry_speed < current->entry_speed) {
double entry_speed = min( current->entry_speed,
max_allowable_speed(-settings.acceleration,previous->entry_speed,previous->millimeters) );
// Check for junction speed change
if (current->entry_speed != entry_speed) {
current->entry_speed = entry_speed;
current->recalculate_flag = true;
// planner_recalculate() needs to go over the current plan twice. Once in reverse and once forward. This
// implements the forward pass.
static void planner_forward_pass()
uint8_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 = next_block_index( block_index );
planner_forward_pass_kernel(block[1], block[2], NULL);
+--------+ <- nominal_rate
/ \
nominal_rate*entry_factor -> + \
| + <- nominal_rate*exit_factor
time -->
// 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.
// This converts the planner parameters to the data required by the stepper controller.
// NOTE: Final rates must be computed in terms of their respective blocks.
static void calculate_trapezoid_for_block(block_t *block, double entry_factor, double exit_factor)
block->initial_rate = ceil(block->nominal_rate*entry_factor); // (step/min)
block->final_rate = ceil(block->nominal_rate*exit_factor); // (step/min)
int32_t acceleration_per_minute = block->rate_delta*ACCELERATION_TICKS_PER_SECOND*60.0; // (step/min^2)
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));
accelerate_steps = max(accelerate_steps,0); // Check limits due to numerical round-off
accelerate_steps = min(accelerate_steps,block->step_event_count);
plateau_steps = 0;
block->accelerate_until = accelerate_steps;
block->decelerate_after = accelerate_steps+plateau_steps;
+--------+ <- current->nominal_speed
/ \
current->entry_speed -> + \
| + <- next->entry_speed
time -->
// Recalculates the trapezoid speed profiles for flagged blocks in the plan according to the
// entry_speed for each junction and the entry_speed of the next junction. Must be called by
// planner_recalculate() after updating the blocks. Any recalulate flagged junction will
// compute the two adjacent trapezoids to the junction, since the junction speed corresponds
// to exit speed and entry speed of one another.
static void planner_recalculate_trapezoids()
uint8_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) {
// Recalculate if current block entry or exit junction speed has changed.
if (current->recalculate_flag || next->recalculate_flag) {
// NOTE: Entry and exit factors always > 0 by all previous logic operations.
calculate_trapezoid_for_block(current, current->entry_speed/current->nominal_speed,
current->recalculate_flag = false; // Reset current only to ensure next trapezoid is computed
block_index = next_block_index( block_index );
// Last/newest block in buffer. Exit speed is set with MINIMUM_PLANNER_SPEED. Always recalculated.
calculate_trapezoid_for_block(next, next->entry_speed/next->nominal_speed,
next->recalculate_flag = false;
// 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_speed)
// so that:
// a. The junction speed is equal to or less than the maximum junction speed 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 values if
// a. The speed increase within one block would require faster acceleration than the one, true
// constant acceleration.
// When these stages are complete all blocks have an entry speed that will allow all speed changes to
// be performed using only the one, true constant acceleration, and where no junction speed is greater
// than the max limit. Finally it will:
// 3. Recalculate trapezoids for all blocks using the recently updated junction speeds. Block trapezoids
// with no updated junction speeds will not be recalculated and assumed ok as is.
// All planner computations are performed with doubles (float on Arduinos) to minimize numerical round-
// off errors. Only when planned values are converted to stepper rate parameters, these are integers.
static void planner_recalculate()
void plan_init()
block_buffer_head = 0;
block_buffer_tail = 0;
previous_nominal_speed = 0.0;
void plan_discard_current_block()
if (block_buffer_head != block_buffer_tail) {
block_buffer_tail = next_block_index( block_buffer_tail );
block_t *plan_get_current_block()
if (block_buffer_head == block_buffer_tail) { return(NULL); }
// Add a new linear movement to the buffer. x, y and z is the signed, absolute target position in
// millimeters. Feed rate specifies the speed of the motion. If feed rate is inverted, the feed
// rate is taken to mean "frequency" and would complete the operation in 1/feed_rate minutes.
void plan_buffer_line(double x, double y, double z, double feed_rate, uint8_t invert_feed_rate)
// 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
uint8_t next_buffer_head = next_block_index( block_buffer_head );
// 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];
// 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); }
// 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; };
// Compute path vector in terms of absolute step target and current positions
double delta_mm[3];
delta_mm[X_AXIS] = (target[X_AXIS]-position[X_AXIS])/settings.steps_per_mm[X_AXIS];
delta_mm[Y_AXIS] = (target[Y_AXIS]-position[Y_AXIS])/settings.steps_per_mm[Y_AXIS];
delta_mm[Z_AXIS] = (target[Z_AXIS]-position[Z_AXIS])/settings.steps_per_mm[Z_AXIS];
block->millimeters = sqrt(square(delta_mm[X_AXIS]) + square(delta_mm[Y_AXIS]) +
double inverse_millimeters = 1.0/block->millimeters; // Inverse millimeters to remove multiple divides
// Calculate speed in mm/minute for each axis. No divide by zero due to previous checks.
// NOTE: Minimum stepper speed is limited by MINIMUM_STEPS_PER_MINUTE in stepper.c
double inverse_minute;
if (!invert_feed_rate) {
inverse_minute = feed_rate * inverse_millimeters;
} else {
inverse_minute = 1.0 / feed_rate;
block->nominal_speed = block->millimeters * inverse_minute; // (mm/min) Always > 0
block->nominal_rate = ceil(block->step_event_count * inverse_minute); // (step/min) Always > 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:
// Convert universal acceleration for direction-dependent stepper rate change parameter
block->rate_delta = ceil( block->step_event_count*inverse_millimeters *
settings.acceleration / (60 * ACCELERATION_TICKS_PER_SECOND )); // (step/min/acceleration_tick)
// Compute path unit vector
double unit_vec[3];
unit_vec[X_AXIS] = delta_mm[X_AXIS]*inverse_millimeters;
unit_vec[Y_AXIS] = delta_mm[Y_AXIS]*inverse_millimeters;
unit_vec[Z_AXIS] = delta_mm[Z_AXIS]*inverse_millimeters;
// Compute maximum allowable entry speed at junction by centripetal acceleration approximation.
// Let a circle be tangent to both previous and current path line segments, where the junction
// deviation is defined as the distance from the junction to the closest edge of the circle,
// colinear with the circle center. The circular segment joining the two paths represents the
// path of centripetal acceleration. Solve for max velocity based on max acceleration about the
// radius of the circle, defined indirectly by junction deviation. This may be also viewed as
// path width or max_jerk in the previous grbl version. This approach does not actually deviate
// from path, but used as a robust way to compute cornering speeds, as it takes into account the
// nonlinearities of both the junction angle and junction velocity.
double vmax_junction = MINIMUM_PLANNER_SPEED; // Set default max junction speed
// Skip first block or when previous_nominal_speed is used as a flag for homing and offset cycles.
if ((block_buffer_head != block_buffer_tail) && (previous_nominal_speed > 0.0)) {
// Compute cosine of angle between previous and current path. (prev_unit_vec is negative)
// NOTE: Max junction velocity is computed without sin() or acos() by trig half angle identity.
double cos_theta = - previous_unit_vec[X_AXIS] * unit_vec[X_AXIS]
- previous_unit_vec[Y_AXIS] * unit_vec[Y_AXIS]
- previous_unit_vec[Z_AXIS] * unit_vec[Z_AXIS] ;
// Skip and use default max junction speed for 0 degree acute junction.
if (cos_theta < 0.95) {
vmax_junction = min(previous_nominal_speed,block->nominal_speed);
// Skip and avoid divide by zero for straight junctions at 180 degrees. Limit to min() of nominal speeds.
if (cos_theta > -0.95) {
// Compute maximum junction velocity based on maximum acceleration and junction deviation
double sin_theta_d2 = sqrt(0.5*(1.0-cos_theta)); // Trig half angle identity. Always positive.
vmax_junction = min(vmax_junction,
sqrt(settings.acceleration * settings.junction_deviation * sin_theta_d2/(1.0-sin_theta_d2)) );
block->max_entry_speed = vmax_junction;
// Initialize block entry speed. Compute based on deceleration to user-defined MINIMUM_PLANNER_SPEED.
double v_allowable = max_allowable_speed(-settings.acceleration,MINIMUM_PLANNER_SPEED,block->millimeters);
block->entry_speed = min(vmax_junction, v_allowable);
// Initialize planner efficiency flags
// Set flag if block will always reach maximum junction speed regardless of entry/exit speeds.
// If a block can de/ac-celerate from nominal speed to zero within the length of the block, then
// the current block and next block junction speeds are guaranteed to always be at their maximum
// junction speeds in deceleration and acceleration, respectively. This is due to how the current
// block nominal speed limits both the current and next maximum junction speeds. Hence, in both
// the reverse and forward planners, the corresponding block junction speed will always be at the
// the maximum junction speed and may always be ignored for any speed reduction checks.
if (block->nominal_speed <= v_allowable) { block->nominal_length_flag = true; }
else { block->nominal_length_flag = false; }
block->recalculate_flag = true; // Always calculate trapezoid for new block
// Update previous path unit_vector and nominal speed
memcpy(previous_unit_vec, unit_vec, sizeof(unit_vec)); // previous_unit_vec[] = unit_vec[]
previous_nominal_speed = block->nominal_speed;
// Move buffer head
block_buffer_head = next_buffer_head;
// Update position
memcpy(position, target, sizeof(target)); // position[] = target[]
// Reset the planner position vector and planner speed
void plan_set_current_position(double x, double y, double z) {
position[X_AXIS] = lround(x*settings.steps_per_mm[X_AXIS]);
position[Y_AXIS] = lround(y*settings.steps_per_mm[Y_AXIS]);
position[Z_AXIS] = lround(z*settings.steps_per_mm[Z_AXIS]);
previous_nominal_speed = 0.0; // Resets planner junction speeds. Assumes start from rest.