Last week, I built a mobile robot in Gazebo that uses ROS2 for message passing. I wanted the robot to go to predefined goal coordinates inside a maze. In order to do that, I needed to create a Publisher node that published the points to a topic.
Here is the ROS2 node code (in Python) that publishes a list of target coordinates (i.e. waypoints) for the mobile robot to go to. Feel free to use this code for your ROS2 projects:
''' ####################
Publish (x,y) coordinate goals for a differential drive robot in a
Gazebo maze.
==================================
Author: Addison Sears-Collins
Date: November 21, 2020
#################### '''
import rclpy # Import the ROS client library for Python
from rclpy.node import Node # Enables the use of rclpy's Node class
from std_msgs.msg import Float64MultiArray # Enable use of std_msgs/Float64MultiArray message
class GoalPublisher(Node):
"""
Create a GoalPublisher class, which is a subclass of the Node class.
The class publishes the goal positions (x,y) for a mobile robot in a Gazebo maze world.
"""
def __init__(self):
"""
Class constructor to set up the node
"""
# Initiate the Node class's constructor and give it a name
super().__init__('goal_publisher')
# Create publisher(s)
self.publisher_goal_x_values = self.create_publisher(Float64MultiArray, '/goal_x_values', 10)
self.publisher_goal_y_values = self.create_publisher(Float64MultiArray, '/goal_y_values', 10)
# Call the callback functions every 0.5 seconds
timer_period = 0.5
self.timer1 = self.create_timer(timer_period, self.get_x_values)
self.timer2 = self.create_timer(timer_period, self.get_y_values)
# Initialize a counter variable
self.i = 0
self.j = 0
# List the goal destinations
# We create a list of the (x,y) coordinate goals
self.goal_x_coordinates = [ 0.0, 3.0, 0.0, -1.5, -1.5, 4.5, 0.0]
self.goal_y_coordinates = [-4.0, 1.0, 1.5, 1.0, -3.0, -4.0, 0.0]
def get_x_values(self):
"""
Callback function.
"""
msg = Float64MultiArray()
msg.data = self.goal_x_coordinates
# Publish the x coordinates to the topic
self.publisher_goal_x_values.publish(msg)
# Increment counter variable
self.i += 1
def get_y_values(self):
"""
Callback function.
"""
msg = Float64MultiArray()
msg.data = self.goal_y_coordinates
# Publish the y coordinates to the topic
self.publisher_goal_y_values.publish(msg)
# Increment counter variable
self.j += 1
def main(args=None):
# Initialize the rclpy library
rclpy.init(args=args)
# Create the node
goal_publisher = GoalPublisher()
# Spin the node so the callback function is called.
# Publish any pending messages to the topics.
rclpy.spin(goal_publisher)
# Destroy the node explicitly
# (optional - otherwise it will be done automatically
# when the garbage collector destroys the node object)
goal_publisher.destroy_node()
# Shutdown the ROS client library for Python
rclpy.shutdown()
if __name__ == '__main__':
main()
In this tutorial, we will learn about the Bug2 algorithm for robot motion planning. The Bug2 algorithm is used when you have a mobile robot:
With a known starting location
With a known goal location
Inside an unexplored environment
Contains a distance sensor that can detect the distances to objects and walls in the environment (e.g. like an ultrasonic sensor or a laser distance sensor.)
Contains an encoder that the robot can use to estimate how far the robot has traveled from the starting location.
Here is a video of a simulated robot I developed in Gazebo and ROS2 that uses the Bug2 algorithm to move from a starting point to a goal point, avoiding walls along the way.
Real-World Applications
Imagine an automatic vacuum cleaner like the one below. It vacuums a room and then needs to move autonomously to another room so that it can clean that room. Along the way, the robot must avoid bumping into walls.
Algorithm Description
On a high level, the Bug2 algorithm has two main modes:
Go to Goal Mode: Move from the current location towards the goal (x,y) coordinate.
Wall Following Mode: Move along a wall.
Here is pseudocode for the algorithm:
1. Calculate a start-goal line. The start-goal line is an imaginary line that connects the starting position to the goal position.
2. While Not at the Goal
Move towards the goal along the start-goal line.
If a wall is encountered:
Remember the location where the wall was first encountered. This is the “hit point.”
Follow the wall until you encounter the start-goal line. This point is known as the “leave point.”
If the leave point is closer to the goal than the hit point, leave the wall, and move towards the goal again.
Otherwise, continue following the wall.
That’s the algorithm. In the image below, I have labeled the hit points and leave points.
Python Implementation (ROS2)
The robot you see in the video at the beginning of this tutorial has a laser distance scanner mounted on top of it that enables it to detect distances from 0 degrees (right side of the robot) to 180 degrees (left side of the robot). It is using three Python functions, bug2, follow_wall, and go_to_goal.
Below is my complete code. It runs in ROS2 Foxy Fitzroy and Gazebo and makes use of the Differential Drive plugin.
Since we’re just focusing on the algorithms in this post, I won’t go into the details of how I created the robot, built the simulated maze, and created publisher and subscriber nodes for the goal locations, robot pose, and laser distance sensor information (I might cover this in a future post).
Don’t be intimidated by the code. It is long but well-commented. Go through each piece and function one step at a time. Focus only on the following three methods:
go_to_goal(self)method
follow_wall(self) method
bug2(self) method
Those three methods encapsulate the implementation of Bug2.
I created descriptive variable names so that you can compare the code to the pseudocode I wrote earlier in this tutorial.
# Author: Addison Sears-Collins
# Date: December 17, 2020
# ROS Version: ROS 2 Foxy Fitzroy
############## IMPORT LIBRARIES #################
# Python math library
import math
# ROS client library for Python
import rclpy
# Enables pauses in the execution of code
from time import sleep
# Used to create nodes
from rclpy.node import Node
# Enables the use of the string message type
from std_msgs.msg import String
# Twist is linear and angular velocity
from geometry_msgs.msg import Twist
# Handles LaserScan messages to sense distance to obstacles (i.e. walls)
from sensor_msgs.msg import LaserScan
# Handle Pose messages
from geometry_msgs.msg import Pose
# Handle float64 arrays
from std_msgs.msg import Float64MultiArray
# Handles quality of service for LaserScan data
from rclpy.qos import qos_profile_sensor_data
# Scientific computing library
import numpy as np
class PlaceholderController(Node):
"""
Create a Placeholder Controller class, which is a subclass of the Node
class for ROS2.
"""
def __init__(self):
"""
Class constructor to set up the node
"""
####### INITIALIZE ROS PUBLISHERS AND SUBSCRIBERS##############
# Initiate the Node class's constructor and give it a name
super().__init__('PlaceholderController')
# Create a subscriber
# This node subscribes to messages of type Float64MultiArray
# over a topic named: /en613/state_est
# The message represents the current estimated state:
# [x, y, yaw]
# The callback function is called as soon as a message
# is received.
# The maximum number of queued messages is 10.
self.subscription = self.create_subscription(
Float64MultiArray,
'/en613/state_est',
self.state_estimate_callback,
10)
self.subscription # prevent unused variable warning
# Create a subscriber
# This node subscribes to messages of type
# sensor_msgs/LaserScan
self.scan_subscriber = self.create_subscription(
LaserScan,
'/en613/scan',
self.scan_callback,
qos_profile=qos_profile_sensor_data)
# Create a subscriber
# This node subscribes to messages of type geometry_msgs/Pose
# over a topic named: /en613/goal
# The message represents the the goal position.
# The callback function is called as soon as a message
# is received.
# The maximum number of queued messages is 10.
self.subscription_goal_pose = self.create_subscription(
Pose,
'/en613/goal',
self.pose_received,
10)
# Create a publisher
# This node publishes the desired linear and angular velocity
# of the robot (in the robot chassis coordinate frame) to the
# /en613/cmd_vel topic. Using the diff_drive
# plugin enables the basic_robot model to read this
# /end613/cmd_vel topic and execute the motion accordingly.
self.publisher_ = self.create_publisher(
Twist,
'/en613/cmd_vel',
10)
# Initialize the LaserScan sensor readings to some large value
# Values are in meters.
self.left_dist = 999999.9 # Left
self.leftfront_dist = 999999.9 # Left-front
self.front_dist = 999999.9 # Front
self.rightfront_dist = 999999.9 # Right-front
self.right_dist = 999999.9 # Right
################### ROBOT CONTROL PARAMETERS ##################
# Maximum forward speed of the robot in meters per second
# Any faster than this and the robot risks falling over.
self.forward_speed = 0.035
# Current position and orientation of the robot in the global
# reference frame
self.current_x = 0.0
self.current_y = 0.0
self.current_yaw = 0.0
# By changing the value of self.robot_mode, you can alter what
# the robot will do when the program is launched.
# "obstacle avoidance mode": Robot will avoid obstacles
# "go to goal mode": Robot will head to an x,y coordinate
# "wall following mode": Robot will follow a wall
self.robot_mode = "go to goal mode"
############# OBSTACLE AVOIDANCE MODE PARAMETERS ##############
# Obstacle detection distance threshold
self.dist_thresh_obs = 0.25 # in meters
# Maximum left-turning speed
self.turning_speed = 0.25 # rad/s
############# GO TO GOAL MODE PARAMETERS ######################
# Finite states for the go to goal mode
# "adjust heading": Orient towards a goal x, y coordinate
# "go straight": Go straight towards goal x, y coordinate
# "goal achieved": Reached goal x, y coordinate
self.go_to_goal_state = "adjust heading"
# List the goal destinations
# We create a list of the (x,y) coordinate goals
self.goal_x_coordinates = False # [ 0.0, 3.0, 0.0, -1.5, -1.5, 4.5, 0.0]
self.goal_y_coordinates = False # [-4.0, 1.0, 1.5, 1.0, -3.0, -4.0, 0.0]
# Keep track of which goal we're headed towards
self.goal_idx = 0
# Keep track of when we've reached the end of the goal list
self.goal_max_idx = None # len(self.goal_x_coordinates) - 1
# +/- 2.0 degrees of precision
self.yaw_precision = 2.0 * (math.pi / 180)
# How quickly we need to turn when we need to make a heading
# adjustment (rad/s)
self.turning_speed_yaw_adjustment = 0.0625
# Need to get within +/- 0.2 meter (20 cm) of (x,y) goal
self.dist_precision = 0.2
############# WALL FOLLOWING MODE PARAMETERS ##################
# Finite states for the wall following mode
# "turn left": Robot turns towards the left
# "search for wall": Robot tries to locate the wall
# "follow wall": Robot moves parallel to the wall
self.wall_following_state = "turn left"
# Set turning speeds (to the left) in rad/s
# These values were determined by trial and error.
self.turning_speed_wf_fast = 1.0 # Fast turn
self.turning_speed_wf_slow = 0.125 # Slow turn
# Wall following distance threshold.
# We want to try to keep within this distance from the wall.
self.dist_thresh_wf = 0.45 # in meters
# We don't want to get too close to the wall though.
self.dist_too_close_to_wall = 0.15 # in meters
################### BUG2 PARAMETERS ###########################
# Bug2 Algorithm Switch
# Can turn "ON" or "OFF" depending on if you want to run Bug2
# Motion Planning Algorithm
self.bug2_switch = "ON"
# Start-Goal Line Calculated?
self.start_goal_line_calculated = False
# Start-Goal Line Parameters
self.start_goal_line_slope_m = 0
self.start_goal_line_y_intercept = 0
self.start_goal_line_xstart = 0
self.start_goal_line_xgoal = 0
self.start_goal_line_ystart = 0
self.start_goal_line_ygoal = 0
# Anything less than this distance means we have encountered
# a wall. Value determined through trial and error.
self.dist_thresh_bug2 = 0.15
# Leave point must be within +/- 0.1m of the start-goal line
# in order to go from wall following mode to go to goal mode
self.distance_to_start_goal_line_precision = 0.1
# Used to record the (x,y) coordinate where the robot hit
# a wall.
self.hit_point_x = 0
self.hit_point_y = 0
# Distance between the hit point and the goal in meters
self.distance_to_goal_from_hit_point = 0.0
# Used to record the (x,y) coordinate where the robot left
# a wall.
self.leave_point_x = 0
self.leave_point_y = 0
# Distance between the leave point and the goal in meters
self.distance_to_goal_from_leave_point = 0.0
# The hit point and leave point must be far enough
# apart to change state from wall following to go to goal
# This value helps prevent the robot from getting stuck and
# rotating in endless circles.
# This distance was determined through trial and error.
self.leave_point_to_hit_point_diff = 0.25 # in meters
def pose_received(self,msg):
"""
Populate the pose.
"""
self.goal_x_coordinates = [msg.position.x]
self.goal_y_coordinates = [msg.position.y]
self.goal_max_idx = len(self.goal_x_coordinates) - 1
def scan_callback(self, msg):
"""
This method gets called every time a LaserScan message is
received on the /en613/scan ROS topic
"""
# Read the laser scan data that indicates distances
# to obstacles (e.g. wall) in meters and extract
# 5 distinct laser readings to work with.
# Each reading is separated by 45 degrees.
# Assumes 181 laser readings, separated by 1 degree.
# (e.g. -90 degrees to 90 degrees....0 to 180 degrees)
self.left_dist = msg.ranges[180]
self.leftfront_dist = msg.ranges[135]
self.front_dist = msg.ranges[90]
self.rightfront_dist = msg.ranges[45]
self.right_dist = msg.ranges[0]
# The total number of laser rays. Used for testing.
#number_of_laser_rays = str(len(msg.ranges))
# Print the distance values (in meters) for testing
#self.get_logger().info('L:%f LF:%f F:%f RF:%f R:%f' % (
# self.left_dist,
# self.leftfront_dist,
# self.front_dist,
# self.rightfront_dist,
# self.right_dist))
if self.robot_mode == "obstacle avoidance mode":
self.avoid_obstacles()
def state_estimate_callback(self, msg):
"""
Extract the position and orientation data.
This callback is called each time
a new message is received on the '/en613/state_est' topic
"""
# Update the current estimated state in the global reference frame
curr_state = msg.data
self.current_x = curr_state[0]
self.current_y = curr_state[1]
self.current_yaw = curr_state[2]
# Wait until we have received some goal destinations.
if self.goal_x_coordinates == False and self.goal_y_coordinates == False:
return
# Print the pose of the robot
# Used for testing
#self.get_logger().info('X:%f Y:%f YAW:%f' % (
# self.current_x,
# self.current_y,
# np.rad2deg(self.current_yaw))) # Goes from -pi to pi
# See if the Bug2 algorithm is activated. If yes, call bug2()
if self.bug2_switch == "ON":
self.bug2()
else:
if self.robot_mode == "go to goal mode":
self.go_to_goal()
elif self.robot_mode == "wall following mode":
self.follow_wall()
else:
pass # Do nothing
def avoid_obstacles(self):
"""
Wander around the maze and avoid obstacles.
"""
# Create a Twist message and initialize all the values
# for the linear and angular velocities
msg = Twist()
msg.linear.x = 0.0
msg.linear.y = 0.0
msg.linear.z = 0.0
msg.angular.x = 0.0
msg.angular.y = 0.0
msg.angular.z = 0.0
# Logic for avoiding obstacles (e.g. walls)
# >d means no obstacle detected by that laser beam
# <d means an obstacle was detected by that laser beam
d = self.dist_thresh_obs
if self.leftfront_dist > d and self.front_dist > d and self.rightfront_dist > d:
msg.linear.x = self.forward_speed # Go straight forward
elif self.leftfront_dist > d and self.front_dist < d and self.rightfront_dist > d:
msg.angular.z = self.turning_speed # Turn left
elif self.leftfront_dist > d and self.front_dist > d and self.rightfront_dist < d:
msg.angular.z = self.turning_speed
elif self.leftfront_dist < d and self.front_dist > d and self.rightfront_dist > d:
msg.angular.z = -self.turning_speed # Turn right
elif self.leftfront_dist > d and self.front_dist < d and self.rightfront_dist < d:
msg.angular.z = self.turning_speed
elif self.leftfront_dist < d and self.front_dist < d and self.rightfront_dist > d:
msg.angular.z = -self.turning_speed
elif self.leftfront_dist < d and self.front_dist < d and self.rightfront_dist < d:
msg.angular.z = self.turning_speed
elif self.leftfront_dist < d and self.front_dist > d and self.rightfront_dist < d:
msg.linear.x = self.forward_speed
else:
pass
# Send the velocity commands to the robot by publishing
# to the topic
self.publisher_.publish(msg)
def go_to_goal(self):
"""
This code drives the robot towards to the goal destination
"""
# Create a geometry_msgs/Twist message
msg = Twist()
msg.linear.x = 0.0
msg.linear.y = 0.0
msg.linear.z = 0.0
msg.angular.x = 0.0
msg.angular.y = 0.0
msg.angular.z = 0.0
# If Bug2 algorithm is activated
if self.bug2_switch == "ON":
# If the wall is in the way
d = self.dist_thresh_bug2
if ( self.leftfront_dist < d or
self.front_dist < d or
self.rightfront_dist < d):
# Change the mode to wall following mode.
self.robot_mode = "wall following mode"
# Record the hit point
self.hit_point_x = self.current_x
self.hit_point_y = self.current_y
# Record the distance to the goal from the
# hit point
self.distance_to_goal_from_hit_point = (
math.sqrt((
pow(self.goal_x_coordinates[self.goal_idx] - self.hit_point_x, 2)) + (
pow(self.goal_y_coordinates[self.goal_idx] - self.hit_point_y, 2))))
# Make a hard left to begin following wall
msg.angular.z = self.turning_speed_wf_fast
# Send command to the robot
self.publisher_.publish(msg)
# Exit this function
return
# Fix the heading
if (self.go_to_goal_state == "adjust heading"):
# Calculate the desired heading based on the current position
# and the desired position
desired_yaw = math.atan2(
self.goal_y_coordinates[self.goal_idx] - self.current_y,
self.goal_x_coordinates[self.goal_idx] - self.current_x)
# How far off is the current heading in radians?
yaw_error = desired_yaw - self.current_yaw
# Adjust heading if heading is not good enough
if math.fabs(yaw_error) > self.yaw_precision:
if yaw_error > 0:
# Turn left (counterclockwise)
msg.angular.z = self.turning_speed_yaw_adjustment
else:
# Turn right (clockwise)
msg.angular.z = -self.turning_speed_yaw_adjustment
# Command the robot to adjust the heading
self.publisher_.publish(msg)
# Change the state if the heading is good enough
else:
# Change the state
self.go_to_goal_state = "go straight"
# Command the robot to stop turning
self.publisher_.publish(msg)
# Go straight
elif (self.go_to_goal_state == "go straight"):
position_error = math.sqrt(
pow(
self.goal_x_coordinates[self.goal_idx] - self.current_x, 2)
+ pow(
self.goal_y_coordinates[self.goal_idx] - self.current_y, 2))
# If we are still too far away from the goal
if position_error > self.dist_precision:
# Move straight ahead
msg.linear.x = self.forward_speed
# Command the robot to move
self.publisher_.publish(msg)
# Check our heading
desired_yaw = math.atan2(
self.goal_y_coordinates[self.goal_idx] - self.current_y,
self.goal_x_coordinates[self.goal_idx] - self.current_x)
# How far off is the heading?
yaw_error = desired_yaw - self.current_yaw
# Check the heading and change the state if there is too much heading error
if math.fabs(yaw_error) > self.yaw_precision:
# Change the state
self.go_to_goal_state = "adjust heading"
# We reached our goal. Change the state.
else:
# Change the state
self.go_to_goal_state = "goal achieved"
# Command the robot to stop
self.publisher_.publish(msg)
# Goal achieved
elif (self.go_to_goal_state == "goal achieved"):
self.get_logger().info('Goal achieved! X:%f Y:%f' % (
self.goal_x_coordinates[self.goal_idx],
self.goal_y_coordinates[self.goal_idx]))
# Get the next goal
self.goal_idx = self.goal_idx + 1
# Do we have any more goals left?
# If we have no more goals left, just stop
if (self.goal_idx > self.goal_max_idx):
self.get_logger().info('Congratulations! All goals have been achieved.')
while True:
pass
# Let's achieve our next goal
else:
# Change the state
self.go_to_goal_state = "adjust heading"
# We need to recalculate the start-goal line if Bug2 is running
self.start_goal_line_calculated = False
else:
pass
def follow_wall(self):
"""
This method causes the robot to follow the boundary of a wall.
"""
# Create a geometry_msgs/Twist message
msg = Twist()
msg.linear.x = 0.0
msg.linear.y = 0.0
msg.linear.z = 0.0
msg.angular.x = 0.0
msg.angular.y = 0.0
msg.angular.z = 0.0
# Special code if Bug2 algorithm is activated
if self.bug2_switch == "ON":
# Calculate the point on the start-goal
# line that is closest to the current position
x_start_goal_line = self.current_x
y_start_goal_line = (
self.start_goal_line_slope_m * (
x_start_goal_line)) + (
self.start_goal_line_y_intercept)
# Calculate the distance between current position
# and the start-goal line
distance_to_start_goal_line = math.sqrt(pow(
x_start_goal_line - self.current_x, 2) + pow(
y_start_goal_line - self.current_y, 2))
# If we hit the start-goal line again
if distance_to_start_goal_line < self.distance_to_start_goal_line_precision:
# Determine if we need to leave the wall and change the mode
# to 'go to goal'
# Let this point be the leave point
self.leave_point_x = self.current_x
self.leave_point_y = self.current_y
# Record the distance to the goal from the leave point
self.distance_to_goal_from_leave_point = math.sqrt(
pow(self.goal_x_coordinates[self.goal_idx]
- self.leave_point_x, 2)
+ pow(self.goal_y_coordinates[self.goal_idx]
- self.leave_point_y, 2))
# Is the leave point closer to the goal than the hit point?
# If yes, go to goal.
diff = self.distance_to_goal_from_hit_point - self.distance_to_goal_from_leave_point
if diff > self.leave_point_to_hit_point_diff:
# Change the mode. Go to goal.
self.robot_mode = "go to goal mode"
# Exit this function
return
# Logic for following the wall
# >d means no wall detected by that laser beam
# <d means an wall was detected by that laser beam
d = self.dist_thresh_wf
if self.leftfront_dist > d and self.front_dist > d and self.rightfront_dist > d:
self.wall_following_state = "search for wall"
msg.linear.x = self.forward_speed
msg.angular.z = -self.turning_speed_wf_slow # turn right to find wall
elif self.leftfront_dist > d and self.front_dist < d and self.rightfront_dist > d:
self.wall_following_state = "turn left"
msg.angular.z = self.turning_speed_wf_fast
elif (self.leftfront_dist > d and self.front_dist > d and self.rightfront_dist < d):
if (self.rightfront_dist < self.dist_too_close_to_wall):
# Getting too close to the wall
self.wall_following_state = "turn left"
msg.linear.x = self.forward_speed
msg.angular.z = self.turning_speed_wf_fast
else:
# Go straight ahead
self.wall_following_state = "follow wall"
msg.linear.x = self.forward_speed
elif self.leftfront_dist < d and self.front_dist > d and self.rightfront_dist > d:
self.wall_following_state = "search for wall"
msg.linear.x = self.forward_speed
msg.angular.z = -self.turning_speed_wf_slow # turn right to find wall
elif self.leftfront_dist > d and self.front_dist < d and self.rightfront_dist < d:
self.wall_following_state = "turn left"
msg.angular.z = self.turning_speed_wf_fast
elif self.leftfront_dist < d and self.front_dist < d and self.rightfront_dist > d:
self.wall_following_state = "turn left"
msg.angular.z = self.turning_speed_wf_fast
elif self.leftfront_dist < d and self.front_dist < d and self.rightfront_dist < d:
self.wall_following_state = "turn left"
msg.angular.z = self.turning_speed_wf_fast
elif self.leftfront_dist < d and self.front_dist > d and self.rightfront_dist < d:
self.wall_following_state = "search for wall"
msg.linear.x = self.forward_speed
msg.angular.z = -self.turning_speed_wf_slow # turn right to find wall
else:
pass
# Send velocity command to the robot
self.publisher_.publish(msg)
def bug2(self):
# Each time we start towards a new goal, we need to calculate the start-goal line
if self.start_goal_line_calculated == False:
# Make sure go to goal mode is set.
self.robot_mode = "go to goal mode"
self.start_goal_line_xstart = self.current_x
self.start_goal_line_xgoal = self.goal_x_coordinates[self.goal_idx]
self.start_goal_line_ystart = self.current_y
self.start_goal_line_ygoal = self.goal_y_coordinates[self.goal_idx]
# Calculate the slope of the start-goal line m
self.start_goal_line_slope_m = (
(self.start_goal_line_ygoal - self.start_goal_line_ystart) / (
self.start_goal_line_xgoal - self.start_goal_line_xstart))
# Solve for the intercept b
self.start_goal_line_y_intercept = self.start_goal_line_ygoal - (
self.start_goal_line_slope_m * self.start_goal_line_xgoal)
# We have successfully calculated the start-goal line
self.start_goal_line_calculated = True
if self.robot_mode == "go to goal mode":
self.go_to_goal()
elif self.robot_mode == "wall following mode":
self.follow_wall()
def main(args=None):
# Initialize rclpy library
rclpy.init(args=args)
# Create the node
controller = PlaceholderController()
# Spin the node so the callback function is called
# Pull messages from any topics this node is subscribed to
# Publish any pending messages to the topics
rclpy.spin(controller)
# Destroy the node explicitly
# (optional - otherwise it will be done automatically
# when the garbage collector destroys the node object)
controller.destroy_node()
# Shutdown the ROS client library for Python
rclpy.shutdown()
if __name__ == '__main__':
main()
Here is the state estimator that goes with the code above. I used an Extended Kalman Filter to filter the odometry measurements from the mobile robot.
# Author: Addison Sears-Collins
# Date: December 8, 2020
# ROS Version: ROS 2 Foxy Fitzroy
# Python math library
import math
# ROS client library for Python
import rclpy
# Used to create nodes
from rclpy.node import Node
# Twist is linear and angular velocity
from geometry_msgs.msg import Twist
# Position, orientation, linear velocity, angular velocity
from nav_msgs.msg import Odometry
# Handles laser distance scan to detect obstacles
from sensor_msgs.msg import LaserScan
# Used for laser scan
from rclpy.qos import qos_profile_sensor_data
# Enable use of std_msgs/Float64MultiArray message
from std_msgs.msg import Float64MultiArray
# Scientific computing library for Python
import numpy as np
class PlaceholderEstimator(Node):
"""
Class constructor to set up the node
"""
def __init__(self):
############## INITIALIZE ROS PUBLISHERS AND SUBSCRIBERS ######
# Initiate the Node class's constructor and give it a name
super().__init__('PlaceholderEstimator')
# Create a subscriber
# This node subscribes to messages of type
# geometry_msgs/Twist.msg
# The maximum number of queued messages is 10.
self.velocity_subscriber = self.create_subscription(
Twist,
'/en613/cmd_vel',
self.velocity_callback,
10)
# Create a subscriber
# This node subscribes to messages of type
# nav_msgs/Odometry
self.odom_subscriber = self.create_subscription(
Odometry,
'/en613/odom',
self.odom_callback,
10)
# Create a publisher
# This node publishes the estimated position (x, y, yaw)
# The type of message is std_msgs/Float64MultiArray
self.publisher_state_est = self.create_publisher(
Float64MultiArray,
'/en613/state_est',
10)
############# STATE TRANSITION MODEL PARAMETERS ###############
# Time step from one time step t-1 to the next time step t
self.delta_t = 0.002 # seconds
# Keep track of the estimate of the yaw angle
# for control input vector calculation.
self.est_yaw = 0.0
# A matrix
# 3x3 matrix -> number of states x number of states matrix
# Expresses how the state of the system [x,y,yaw] changes
# from t-1 to t when no control command is executed. Typically
# a robot on wheels only drives when the wheels are commanded
# to turn.
# For this case, A is the identity matrix.
# A is sometimes F in the literature.
self.A_t_minus_1 = np.array([ [1.0, 0, 0],
[ 0,1.0, 0],
[ 0, 0, 1.0]])
# The estimated state vector at time t-1 in the global
# reference frame
# [x_t_minus_1, y_t_minus_1, yaw_t_minus_1]
# [meters, meters, radians]
self.state_vector_t_minus_1 = np.array([0.0,0.0,0.0])
# The control input vector at time t-1 in the global
# reference frame
# [v,v,yaw_rate]
# [meters/second, meters/second, radians/second]
# In the literature, this is commonly u.
self.control_vector_t_minus_1 = np.array([0.001,0.001,0.001])
# Noise applied to the forward kinematics (calculation
# of the estimated state at time t from the state transition
# model of the mobile robot. This is a vector with the
# number of elements equal to the number of states.
self.process_noise_v_t_minus_1 = np.array([0.096,0.096,0.032])
############# MEASUREMENT MODEL PARAMETERS ####################
# Measurement matrix H_t
# Used to convert the predicted state estimate at time t=1
# into predicted sensor measurements at time t=1.
# In this case, H will be the identity matrix since the
# estimated state maps directly to state measurements from the
# odometry data [x, y, yaw]
# H has the same number of rows as sensor measurements
# and same number of columns as states.
self.H_t = np.array([ [1.0, 0, 0],
[ 0,1.0, 0],
[ 0, 0, 1.0]])
# Sensor noise. This is a vector with the
# number of elements as the number of sensor measurements.
self.sensor_noise_w_t = np.array([0.07,0.07,0.04])
############# EXTENDED KALMAN FILTER PARAMETERS ###############
# State covariance matrix P_t_minus_1
# This matrix has the same number of rows (and columns) as the
# number of states (i.e. 3x3 matrix). P is sometimes referred
# to as Sigma in the literature. It represents an estimate of
# the accuracy of the state estimate at t=1 made using the
# state transition matrix. We start off with guessed values.
self.P_t_minus_1 = np.array([ [0.1, 0, 0],
[ 0,0.1, 0],
[ 0, 0, 0.1]])
# State model noise covariance matrix Q_t
# When Q is large, the Kalman Filter tracks large changes in
# the sensor measurements more closely than for smaller Q.
# Q is a square matrix that has the same number of rows as
# states.
self.Q_t = np.array([ [1.0, 0, 0],
[ 0, 1.0, 0],
[ 0, 0, 1.0]])
# Sensor measurement noise covariance matrix R_t
# Has the same number of rows and columns as sensor measurements.
# If we are sure about the measurements, R will be near zero.
self.R_t = np.array([ [1.0, 0, 0],
[ 0, 1.0, 0],
[ 0, 0, 1.0]])
def velocity_callback(self, msg):
"""
Listen to the velocity commands (linear forward velocity
in the x direction in the robot's reference frame and
angular velocity (yaw rate) around the robot's z-axis.
Convert those velocity commands into a 3-element control
input vector ...
[v,v,yaw_rate]
[meters/second, meters/second, radians/second]
"""
# Forward velocity in the robot's reference frame
v = msg.linear.x
# Angular velocity around the robot's z axis
yaw_rate = msg.angular.z
# [v,v,yaw_rate]
self.control_vector_t_minus_1[0] = v
self.control_vector_t_minus_1[1] = v
self.control_vector_t_minus_1[2] = yaw_rate
def odom_callback(self, msg):
"""
Receive the odometry information containing the position and orientation
of the robot in the global reference frame.
The position is x, y, z.
The orientation is a x,y,z,w quaternion.
"""
roll, pitch, yaw = self.euler_from_quaternion(
msg.pose.pose.orientation.x,
msg.pose.pose.orientation.y,
msg.pose.pose.orientation.z,
msg.pose.pose.orientation.w)
obs_state_vector_x_y_yaw = [msg.pose.pose.position.x,msg.pose.pose.position.y,yaw]
# These are the measurements taken by the odometry in Gazebo
z_t_observation_vector = np.array([obs_state_vector_x_y_yaw[0],
obs_state_vector_x_y_yaw[1],
obs_state_vector_x_y_yaw[2]])
# Apply the Extended Kalman Filter
# This is the updated state estimate after taking the latest
# sensor (odometry) measurements into account.
updated_state_estimate_t = self.ekf(z_t_observation_vector)
# Publish the estimate state
self.publish_estimated_state(updated_state_estimate_t)
def publish_estimated_state(self, state_vector_x_y_yaw):
"""
Publish the estimated pose (position and orientation) of the
robot to the '/en613/state_est' topic.
:param: state_vector_x_y_yaw [x, y, yaw]
x is in meters, y is in meters, yaw is in radians
"""
msg = Float64MultiArray()
msg.data = state_vector_x_y_yaw
self.publisher_state_est.publish(msg)
def euler_from_quaternion(self, x, y, z, w):
"""
Convert a quaternion into euler angles (roll, pitch, yaw)
roll is rotation around x in radians (counterclockwise)
pitch is rotation around y in radians (counterclockwise)
yaw is rotation around z in radians (counterclockwise)
"""
t0 = +2.0 * (w * x + y * z)
t1 = +1.0 - 2.0 * (x * x + y * y)
roll_x = math.atan2(t0, t1)
t2 = +2.0 * (w * y - z * x)
t2 = +1.0 if t2 > +1.0 else t2
t2 = -1.0 if t2 < -1.0 else t2
pitch_y = math.asin(t2)
t3 = +2.0 * (w * z + x * y)
t4 = +1.0 - 2.0 * (y * y + z * z)
yaw_z = math.atan2(t3, t4)
return roll_x, pitch_y, yaw_z # in radians
def getB(self,yaw,dt):
"""
Calculates and returns the B matrix
3x3 matix -> number of states x number of control inputs
The control inputs are the forward speed and the rotation
rate around the z axis from the x-axis in the
counterclockwise direction.
[v,v,yaw_rate]
Expresses how the state of the system [x,y,yaw] changes
from t-1 to t
due to the control commands (i.e. inputs).
:param yaw: The yaw (rotation angle around the z axis) in rad
:param dt: The change in time from time step t-1 to t in sec
"""
B = np.array([ [np.cos(yaw) * dt, 0, 0],
[ 0, np.sin(yaw) * dt, 0],
[ 0, 0, dt]])
return B
def ekf(self,z_t_observation_vector):
"""
Extended Kalman Filter. Fuses noisy sensor measurement to
create an optimal estimate of the state of the robotic system.
INPUT
:param z_t_observation_vector The observation from the Odometry
3x1 NumPy Array [x,y,yaw] in the global reference frame
in [meters,meters,radians].
OUTPUT
:return state_estimate_t optimal state estimate at time t
[x,y,yaw]....3x1 list --->
[meters,meters,radians]
"""
######################### Predict #############################
# Predict the state estimate at time t based on the state
# estimate at time t-1 and the control input applied at time
# t-1.
state_estimate_t = self.A_t_minus_1 @ (
self.state_vector_t_minus_1) + (
self.getB(self.est_yaw,self.delta_t)) @ (
self.control_vector_t_minus_1) + (
self.process_noise_v_t_minus_1)
# Predict the state covariance estimate based on the previous
# covariance and some noise
P_t = self.A_t_minus_1 @ self.P_t_minus_1 @ self.A_t_minus_1.T + (
self.Q_t)
################### Update (Correct) ##########################
# Calculate the difference between the actual sensor measurements
# at time t minus what the measurement model predicted
# the sensor measurements would be for the current timestep t.
measurement_residual_y_t = z_t_observation_vector - (
(self.H_t @ state_estimate_t) + (
self.sensor_noise_w_t))
# Calculate the measurement residual covariance
S_t = self.H_t @ P_t @ self.H_t.T + self.R_t
# Calculate the near-optimal Kalman gain
# We use pseudoinverse since some of the matrices might be
# non-square or singular.
K_t = P_t @ self.H_t.T @ np.linalg.pinv(S_t)
# Calculate an updated state estimate for time t
state_estimate_t = state_estimate_t + (K_t @ measurement_residual_y_t)
# Update the state covariance estimate for time t
P_t = P_t - (K_t @ self.H_t @ P_t)
#### Update global variables for the next iteration of EKF ####
# Update the estimated yaw
self.est_yaw = state_estimate_t[2]
# Update the state vector for t-1
self.state_vector_t_minus_1 = state_estimate_t
# Update the state covariance matrix
self.P_t_minus_1 = P_t
######### Return the updated state estimate ###################
state_estimate_t = state_estimate_t.tolist()
return state_estimate_t
def main(args=None):
"""
Entry point for the progam.
"""
# Initialize rclpy library
rclpy.init(args=args)
# Create the node
estimator = PlaceholderEstimator()
# Spin the node so the callback function is called.
# Pull messages from any topics this node is subscribed to.
# Publish any pending messages to the topics.
rclpy.spin(estimator)
# Destroy the node explicitly
# (optional - otherwise it will be done automatically
# when the garbage collector destroys the node object)
estimator.destroy_node()
# Shutdown the ROS client library for Python
rclpy.shutdown()
if __name__ == '__main__':
main()
Wall Following Robot Mode Demo
Just for fun, I created a switch in the first section of the code (see previous section) that enables you to have the robot do nothing but follow walls. Here is a video demonstration of that.
Obstacle Avoidance Robot Mode Demo
Here is a demo of what the robot looks like when it does nothing but wander around the room, avoiding obstacles and walls.
I recently built a project to combine the Extended Kalman Filter (for state estimation) and the Linear Quadratic Regulator (for control) to drive a simulated path-following race car (which is modeled as a point) around a track that is the same shape as the Formula 1 Singapore Grand Prix Marina Bay Street Circuit race track.
Output of the program I developed (see the Python code later in this post).
In a real-world application, it is common for a robot to use the Extended Kalman Filter to calculate near-optimal estimates of the state of a robotic system and to use LQR to generate the control values that move the robot from one state to the next.
In my application, I assume there are sensors mounted on the race car that help it to localize itself in the x-y coordinate plane. These sensors measure the range (r) and bearing (b) from the race car to each landmark. Each landmark has a known x and y position.
The programs I developed were written in Python and use the same sequence of actions as indicated in the block diagram (in the first part of this post) in order to follow the race track (which is a list of waypoints).
Code
You will need to put these five Python (3.7 or higher) programs into a single directory.
cubic_spine_planner.py
"""
Cubic spline planner
Program: cubic_spine_planner.py
Author: Atsushi Sakai(@Atsushi_twi)
Source: https://github.com/AtsushiSakai/PythonRobotics
This program implements a cubic spline. For more information on
cubic splines, check out this link:
https://mathworld.wolfram.com/CubicSpline.html
"""
import math
import numpy as np
import bisect
class Spline:
"""
Cubic Spline class
"""
def __init__(self, x, y):
self.b, self.c, self.d, self.w = [], [], [], []
self.x = x
self.y = y
self.nx = len(x) # dimension of x
h = np.diff(x)
# calc coefficient c
self.a = [iy for iy in y]
# calc coefficient c
A = self.__calc_A(h)
B = self.__calc_B(h)
self.c = np.linalg.solve(A, B)
# print(self.c1)
# calc spline coefficient b and d
for i in range(self.nx - 1):
self.d.append((self.c[i + 1] - self.c[i]) / (3.0 * h[i]))
tb = (self.a[i + 1] - self.a[i]) / h[i] - h[i] * \
(self.c[i + 1] + 2.0 * self.c[i]) / 3.0
self.b.append(tb)
def calc(self, t):
"""
Calc position
if t is outside of the input x, return None
"""
if t < self.x[0]:
return None
elif t > self.x[-1]:
return None
i = self.__search_index(t)
dx = t - self.x[i]
result = self.a[i] + self.b[i] * dx + \
self.c[i] * dx ** 2.0 + self.d[i] * dx ** 3.0
return result
def calcd(self, t):
"""
Calc first derivative
if t is outside of the input x, return None
"""
if t < self.x[0]:
return None
elif t > self.x[-1]:
return None
i = self.__search_index(t)
dx = t - self.x[i]
result = self.b[i] + 2.0 * self.c[i] * dx + 3.0 * self.d[i] * dx ** 2.0
return result
def calcdd(self, t):
"""
Calc second derivative
"""
if t < self.x[0]:
return None
elif t > self.x[-1]:
return None
i = self.__search_index(t)
dx = t - self.x[i]
result = 2.0 * self.c[i] + 6.0 * self.d[i] * dx
return result
def __search_index(self, x):
"""
search data segment index
"""
return bisect.bisect(self.x, x) - 1
def __calc_A(self, h):
"""
calc matrix A for spline coefficient c
"""
A = np.zeros((self.nx, self.nx))
A[0, 0] = 1.0
for i in range(self.nx - 1):
if i != (self.nx - 2):
A[i + 1, i + 1] = 2.0 * (h[i] + h[i + 1])
A[i + 1, i] = h[i]
A[i, i + 1] = h[i]
A[0, 1] = 0.0
A[self.nx - 1, self.nx - 2] = 0.0
A[self.nx - 1, self.nx - 1] = 1.0
# print(A)
return A
def __calc_B(self, h):
"""
calc matrix B for spline coefficient c
"""
B = np.zeros(self.nx)
for i in range(self.nx - 2):
B[i + 1] = 3.0 * (self.a[i + 2] - self.a[i + 1]) / \
h[i + 1] - 3.0 * (self.a[i + 1] - self.a[i]) / h[i]
return B
class Spline2D:
"""
2D Cubic Spline class
"""
def __init__(self, x, y):
self.s = self.__calc_s(x, y)
self.sx = Spline(self.s, x)
self.sy = Spline(self.s, y)
def __calc_s(self, x, y):
dx = np.diff(x)
dy = np.diff(y)
self.ds = [math.sqrt(idx ** 2 + idy ** 2)
for (idx, idy) in zip(dx, dy)]
s = [0]
s.extend(np.cumsum(self.ds))
return s
def calc_position(self, s):
"""
calc position
"""
x = self.sx.calc(s)
y = self.sy.calc(s)
return x, y
def calc_curvature(self, s):
"""
calc curvature
"""
dx = self.sx.calcd(s)
ddx = self.sx.calcdd(s)
dy = self.sy.calcd(s)
ddy = self.sy.calcdd(s)
k = (ddy * dx - ddx * dy) / ((dx ** 2 + dy ** 2)**(3 / 2))
return k
def calc_yaw(self, s):
"""
calc yaw
"""
dx = self.sx.calcd(s)
dy = self.sy.calcd(s)
yaw = math.atan2(dy, dx)
return yaw
def calc_spline_course(x, y, ds=0.1):
sp = Spline2D(x, y)
s = list(np.arange(0, sp.s[-1], ds))
rx, ry, ryaw, rk = [], [], [], []
for i_s in s:
ix, iy = sp.calc_position(i_s)
rx.append(ix)
ry.append(iy)
ryaw.append(sp.calc_yaw(i_s))
rk.append(sp.calc_curvature(i_s))
return rx, ry, ryaw, rk, s
def main():
print("Spline 2D test")
import matplotlib.pyplot as plt
x = [-2.5, 0.0, 2.5, 5.0, 7.5, 3.0, -1.0]
y = [0.7, -6, 5, 6.5, 0.0, 5.0, -2.0]
ds = 0.1 # [m] distance of each intepolated points
sp = Spline2D(x, y)
s = np.arange(0, sp.s[-1], ds)
rx, ry, ryaw, rk = [], [], [], []
for i_s in s:
ix, iy = sp.calc_position(i_s)
rx.append(ix)
ry.append(iy)
ryaw.append(sp.calc_yaw(i_s))
rk.append(sp.calc_curvature(i_s))
flg, ax = plt.subplots(1)
plt.plot(x, y, "xb", label="input")
plt.plot(rx, ry, "-r", label="spline")
plt.grid(True)
plt.axis("equal")
plt.xlabel("x[m]")
plt.ylabel("y[m]")
plt.legend()
flg, ax = plt.subplots(1)
plt.plot(s, [math.degrees(iyaw) for iyaw in ryaw], "-r", label="yaw")
plt.grid(True)
plt.legend()
plt.xlabel("line length[m]")
plt.ylabel("yaw angle[deg]")
flg, ax = plt.subplots(1)
plt.plot(s, rk, "-r", label="curvature")
plt.grid(True)
plt.legend()
plt.xlabel("line length[m]")
plt.ylabel("curvature [1/m]")
plt.show()
if __name__ == '__main__':
main()
kinematics.py
"""
Implementation of the two-wheeled differential drive robot car
and its controller.
Our goal in using LQR is to find the optimal control inputs:
[linear velocity of the car, angular velocity of the car]
We want to both minimize the error between the current state
and a desired state, while minimizing actuator effort
(e.g. wheel rotation rate). These are competing objectives because a
large u (i.e. wheel rotation rates) expends a lot of
actuator energy but can drive the state error to 0 really fast.
LQR helps us balance these competing objectives.
If a system is linear, LQR gives the optimal control inputs that
takes a system's state to 0, where the state is
"current state - desired state".
Implemented by Addison Sears-Collins
Date: December 10, 2020
"""
# Import required libraries
import numpy as np
import scipy.linalg as la
class DifferentialDrive(object):
"""
Implementation of Differential Drive kinematics.
This represents a two-wheeled vehicle defined by the following states
state = [x,y,theta] where theta is the yaw angle
and accepts the following control inputs
input = [linear velocity of the car, angular velocity of the car]
"""
def __init__(self):
"""
Initializes the class
"""
# Covariance matrix representing action noise
# (i.e. noise on the control inputs)
self.V = None
def get_state_size(self):
"""
The state is (X, Y, THETA) in global coordinates, so the state size is 3.
"""
return 3
def get_input_size(self):
"""
The control input is ([linear velocity of the car, angular velocity of the car]),
so the input size is 2.
"""
return 2
def get_V(self):
"""
This function provides the covariance matrix V which
describes the noise that can be applied to the forward kinematics.
Feel free to experiment with different levels of noise.
Output
:return: V: input cost matrix (3 x 3 matrix)
"""
# The np.eye function returns a 2D array with ones on the diagonal
# and zeros elsewhere.
if self.V is None:
self.V = np.eye(3)
self.V[0,0] = 0.01
self.V[1,1] = 0.01
self.V[2,2] = 0.1
return 1e-5*self.V
def forward(self,x0,u,v,dt=0.1):
"""
Computes the forward kinematics for the system.
Input
:param x0: The starting state (position) of the system (units:[m,m,rad])
np.array with shape (3,) ->
(X, Y, THETA)
:param u: The control input to the system
2x1 NumPy Array given the control input vector is
[linear velocity of the car, angular velocity of the car]
[meters per second, radians per second]
:param v: The noise applied to the system (units:[m, m, rad]) ->np.array
with shape (3,)
:param dt: Change in time (units: [s])
Output
:return: x1: The new state of the system (X, Y, THETA)
"""
u0 = u
# If there is no noise applied to the system
if v is None:
v = 0
# Starting state of the vehicle
X = x0[0]
Y = x0[1]
THETA = x0[2]
# Control input
u_linvel = u0[0]
u_angvel = u0[1]
# Velocity in the x and y direction in m/s
x_dot = u_linvel * np.cos(THETA)
y_dot = u_linvel * np.sin(THETA)
# The new state of the system
x1 = np.empty(3)
# Calculate the new state of the system
# Noise is added like in slide 34 in Lecture 7
x1[0] = x0[0] + x_dot * dt + v[0] # X
x1[1] = x0[1] + y_dot * dt + v[1] # Y
x1[2] = x0[2] + u_angvel * dt + v[2] # THETA
return x1
def linearize(self, x, dt=0.1):
"""
Creates a linearized version of the dynamics of the differential
drive robotic system (i.e. a
robotic car where each wheel is controlled separately.
The system's forward kinematics are nonlinear due to the sines and
cosines, so we need to linearize
it by taking the Jacobian of the forward kinematics equations with respect
to the control inputs.
Our goal is to have a discrete time system of the following form:
x_t+1 = Ax_t + Bu_t where:
Input
:param x: The state of the system (units:[m,m,rad]) ->
np.array with shape (3,) ->
(X, Y, THETA) ->
X_system = [x1, x2, x3]
:param dt: The change in time from time step t to time step t+1
Output
:return: A: Matrix A is a 3x3 matrix (because there are 3 states) that
describes how the state of the system changes from t to t+1
when no control command is executed. Typically,
a robotic car only drives when the wheels are turning.
Therefore, in this case, A is the identity matrix.
:return: B: Matrix B is a 3 x 2 matrix (because there are 3 states and
2 control inputs) that describes how
the state (X, Y, and THETA) changes from t to t + 1 due to
the control command u.
Matrix B is found by taking the The Jacobian of the three
forward kinematics equations (for X, Y, THETA)
with respect to u (3 x 2 matrix)
"""
THETA = x[2]
####### A Matrix #######
# A matrix is the identity matrix
A = np.array([[1.0, 0, 0],
[ 0, 1.0, 0],
[ 0, 0, 1.0]])
####### B Matrix #######
B = np.array([[np.cos(THETA)*dt, 0],
[np.sin(THETA)*dt, 0],
[0, dt]])
return A, B
def dLQR(F,Q,R,x,xf,dt=0.1):
"""
Discrete-time linear quadratic regulator for a non-linear system.
Compute the optimal control given a nonlinear system, cost matrices,
a current state, and a final state.
Compute the control variables that minimize the cumulative cost.
Solve for P using the dynamic programming method.
Assume that Qf = Q
Input:
:param F: The dynamics class object (has forward and linearize functions
implemented)
:param Q: The state cost matrix Q -> np.array with shape (3,3)
:param R: The input cost matrix R -> np.array with shape (2,2)
:param x: The current state of the system x -> np.array with shape (3,)
:param xf: The desired state of the system xf -> np.array with shape (3,)
:param dt: The size of the timestep -> float
Output
:return: u_t_star: Optimal action u for the current state
[linear velocity of the car, angular velocity of the car]
[meters per second, radians per second]
"""
# We want the system to stabilize at xf,
# so we let x - xf be the state.
# Actual state - desired state
x_error = x - xf
# Calculate the A and B matrices
A, B = F.linearize(x, dt)
# Solutions to discrete LQR problems are obtained using dynamic
# programming.
# The optimal solution is obtained recursively, starting at the last
# time step and working backwards.
N = 50
# Create a list of N + 1 elements
P = [None] * (N + 1)
# Assume Qf = Q
Qf = Q
# 1. LQR via Dynamic Programming
P[N] = Qf
# 2. For t = N, ..., 1
for t in range(N, 0, -1):
# Discrete-time Algebraic Riccati equation to calculate the optimal
# state cost matrix
P[t-1] = Q + A.T @ P[t] @ A - (A.T @ P[t] @ B) @ la.pinv(
R + B.T @ P[t] @ B) @ (B.T @ P[t] @ A)
# Create a list of N elements
K = [None] * N
u = [None] * N
# 3 and 4. For t = 0, ..., N - 1
for t in range(N):
# Calculate the optimal feedback gain K_t
K[t] = -la.pinv(R + B.T @ P[t+1] @ B) @ B.T @ P[t+1] @ A
for t in range(N):
# Calculate the optimal control input
u[t] = K[t] @ x_error
# Optimal control input is u_t_star
u_t_star = u[N-1]
# Return the optimal control inputs
return u_t_star
def get_R():
"""
This function provides the R matrix to the lqr_ekf_control simulator.
Returns the input cost matrix R.
Experiment with different gains.
This matrix penalizes actuator effort
(i.e. rotation of the motors on the wheels).
The R matrix has the same number of rows as are actuator states
[linear velocity of the car, angular velocity of the car]
[meters per second, radians per second]
This matrix often has positive values along the diagonal.
We can target actuator states where we want low actuator
effort by making the corresponding value of R large.
Output
:return: R: Input cost matrix
"""
R = np.array([[0.01, 0], # Penalization for linear velocity effort
[0, 0.01]]) # Penalization for angular velocity effort
return R
def get_Q():
"""
This function provides the Q matrix to the lqr_ekf_control simulator.
Returns the state cost matrix Q.
Experiment with different gains to see their effect on the vehicle's
behavior.
Q helps us weight the relative importance of each state in the state
vector (X, Y, THETA).
Q is a square matrix that has the same number of rows as there are states.
Q penalizes bad performance.
Q has positive values along the diagonal and zeros elsewhere.
Q enables us to target states where we want low error by making the
corresponding value of Q large.
We can start with the identity matrix and tweak the values through trial
and error.
Output
:return: Q: State cost matrix (3x3 matrix because the state vector is
(X, Y, THETA))
"""
Q = np.array([[0.4, 0, 0], # Penalize X position error (global coordinates)
[0, 0.4, 0], # Penalize Y position error (global coordinates)
[0, 0, 0.4]])# Penalize heading error (global coordinates)
return Q
lqr_ekf_control.py
"""
A robot will follow a racetrack using an LQR controller and EKF (with landmarks)
to estimate the state (i.e. x position, y position, yaw angle) at each timestep
####################
Adapted from code authored by Atsushi Sakai (@Atsushi_twi)
Source: https://github.com/AtsushiSakai/PythonRobotics
"""
# Import important libraries
import numpy as np
import math
import matplotlib.pyplot as plt
from kinematics import *
from sensors import *
from utils import *
from time import sleep
show_animation = True
def closed_loop_prediction(desired_traj, landmarks):
## Simulation Parameters
T = desired_traj.shape[0] # Maximum simulation time
goal_dis = 0.1 # How close we need to get to the goal
goal = desired_traj[-1,:] # Coordinates of the goal
dt = 0.1 # Timestep interval
time = 0.0 # Starting time
## Initial States
state = np.array([8.3,0.69,0]) # Initial state of the car
state_est = state.copy()
sigma = 0.1*np.ones(3) # State covariance matrix
## Get the Cost-to-go and input cost matrices for LQR
Q = get_Q() # Defined in kinematics.py
R = get_R() # Defined in kinematics.py
## Initialize the Car and the Car's landmark sensor
DiffDrive = DifferentialDrive()
LandSens = LandmarkDetector(landmarks)
# Process noise and sensor measurement noise
V = DiffDrive.get_V()
W = LandSens.get_W()
## Create objects for storing states and estimated state
t = [time]
traj = np.array([state])
traj_est = np.array([state_est])
ind = 0
while T >= time:
## Point to track
ind = int(np.floor(time))
goal_i = desired_traj[ind,:]
## Generate noise
# v = process noise, w = measurement noise
v = np.random.multivariate_normal(np.zeros(V.shape[0]),V)
w = np.random.multivariate_normal(np.zeros(W.shape[0]),W)
## Generate optimal control commands
u_lqr = dLQR(DiffDrive,Q,R,state_est,goal_i[0:3],dt)
## Move forwad in time
state = DiffDrive.forward(state,u_lqr,v,dt)
# Take a sensor measurement for the new state
y = LandSens.measure(state,w)
# Update the estimate of the state using the EKF
state_est, sigma = EKF(DiffDrive,LandSens,y,state_est,sigma,u_lqr,dt)
# Increment time
time = time + dt
# Store the trajectory and estimated trajectory
t.append(time)
traj = np.concatenate((traj,[state]),axis=0)
traj_est = np.concatenate((traj_est,[state_est]),axis=0)
# Check to see if the robot reached goal
if np.linalg.norm(state[0:2]-goal[0:2]) <= goal_dis:
print("Goal reached")
break
## Plot the vehicles trajectory
if time % 1 < 0.1 and show_animation:
plt.cla()
plt.plot(desired_traj[:,0], desired_traj[:,1], "-r", label="course")
plt.plot(traj[:,0], traj[:,1], "ob", label="trajectory")
plt.plot(traj_est[:,0], traj_est[:,1], "sk", label="estimated trajectory")
plt.plot(goal_i[0], goal_i[1], "xg", label="target")
plt.axis("equal")
plt.grid(True)
plt.title("SINGAPORE GRAND PRIX\n" + "speed[m/s]:" + str(
round(np.mean(u_lqr), 2)) +
",target index:" + str(ind))
plt.pause(0.0001)
#input()
return t, traj
def main():
# Countdown to start
print("\n*** SINGAPORE GRAND PRIX ***\n")
print("Start your engines!")
print("3.0")
sleep(1.0)
print("2.0")
sleep(1.0)
print("1.0\n")
sleep(1.0)
print("LQR + EKF steering control tracking start!!")
# Create the track waypoints
ax = [8.3,8.0, 7.2, 6.5, 6.2, 6.5, 1.5,-2.0,-3.5,-5.0,-7.9,
-6.7,-6.7,-5.2,-3.2,-1.2, 0.0, 0.2, 2.5, 2.8, 4.4, 4.5, 7.8, 8.5, 8.3]
ay = [0.7,4.3, 4.5, 5.2, 4.0, 0.7, 1.3, 3.3, 1.5, 3.0,-1.0,
-2.0,-3.0,-4.5, 1.1,-0.7,-1.0,-2.0,-2.2,-1.2,-1.5,-2.4,-2.7,-1.7,-0.1]
# These landmarks help the mobile robot localize itself
landmarks = [[4,3],
[8,2],
[-1,-4]]
# Compute the desired trajectory
desired_traj = compute_traj(ax,ay)
t, traj = closed_loop_prediction(desired_traj,landmarks)
# Display the trajectory that the mobile robot executed
if show_animation:
plt.close()
flg, _ = plt.subplots(1)
plt.plot(ax, ay, "xb", label="input")
plt.plot(desired_traj[:,0], desired_traj[:,1], "-r", label="spline")
plt.plot(traj[:,0], traj[:,1], "-g", label="tracking")
plt.grid(True)
plt.axis("equal")
plt.xlabel("x[m]")
plt.ylabel("y[m]")
plt.legend()
plt.show()
if __name__ == '__main__':
main()
sensors.py
"""
Extended Kalman Filter implementation using landmarks to localize
Implemented by Addison Sears-Collins
Date: December 10, 2020
"""
import numpy as np
import scipy.linalg as la
import math
class LandmarkDetector(object):
"""
This class represents the sensor mounted on the robot that is used to
to detect the location of landmarks in the environment.
"""
def __init__(self,landmark_list):
"""
Calculates the sensor measurements for the landmark sensor
Input
:param landmark_list: 2D list of landmarks [L1,L2,L3,...,LN],
where each row is a 2D landmark defined by
Li =[l_i_x,l_i_y]
which corresponds to its x position and y
position in the global coordinate frame.
"""
# Store the x and y position of each landmark in the world
self.landmarks = np.array(landmark_list)
# Record the total number of landmarks
self.N_landmarks = self.landmarks.shape[0]
# Variable that represents landmark sensor noise (i.e. error)
self.W = None
def get_W(self):
"""
This function provides the covariance matrix W which describes the noise
that can be applied to the sensor measurements.
Feel free to experiment with different levels of noise.
Output
:return: W: measurement noise matrix (4x4 matrix given two landmarks)
In this implementation there are two landmarks.
Each landmark has an x location and y location, so there are
4 individual sensor measurements.
"""
# In the EKF code, we will condense this 4x4 matrix so that it is a 4x1
# landmark sensor measurement noise vector (i.e. [x1 noise, y1 noise, x2
# noise, y2 noise]
if self.W is None:
self.W = 0.0095*np.eye(2*self.N_landmarks)
return self.W
def measure(self,x,w=None):
"""
Computes the landmark sensor measurements for the system.
This will be a list of range (i.e. distance)
and relative angles between the robot and a
set of landmarks defined with [x,y] positions.
Input
:param x: The state [x_t,y_t,theta_t] of the system -> np.array with
shape (3,). x is a 3x1 vector
:param w: The noise (assumed Gaussian) applied to the measurement ->
np.array with shape (2 * N_Landmarks,)
w is a 4x1 vector
Output
:return: y: The resulting observation vector (4x1 in this
implementation because there are 2 landmarks). The
resulting measurement is
y = [h1_range_1,h1_angle_1, h2_range_2, h2_angle_2]
"""
# Create the observation vector
y = np.zeros(shape=(2*self.N_landmarks,))
# Initialize state variables
x_t = x[0]
y_t = x[1]
THETA_t = x[2]
# Fill in the observation model y
for i in range(0, self.N_landmarks):
# Set the x and y position for landmark i
x_l_i = self.landmarks[i][0]
y_l_i = self.landmarks[i][1]
# Set the noise value
w_r = w[i*2]
w_b = w[i*2+1]
# Calculate the range (distance from robot to the landmark) and
# bearing angle to the landmark
range_i = math.sqrt(((x_t - x_l_i)**2) + ((y_t - y_l_i)**2)) + w_r
angle_i = math.atan2(y_t - y_l_i,x_t - x_l_i) - THETA_t + w_b
# Populate the predicted sensor observation vector
y[i*2] = range_i
y[i*2+1] = angle_i
return y
def jacobian(self,x):
"""
Computes the first order jacobian around the state x
Input
:param x: The starting state (position) of the system -> np.array
with shape (3,)
Output
:return: H: The resulting Jacobian matrix H for the sensor.
The number of rows is equal to two times the number of
landmarks H. 2 rows (for the range (distance) and bearing
to landmark measurement) x 3 columns for the number of
states (X, Y, THETA).
In other words, the number of rows is equal to the total
number of individual sensor measurements.
Since we have 2 landmarks, H will be a 4x3 matrix
(2 * N_Landmarks x Number of states)
"""
# Create the Jacobian matrix H
# For example, the first two rows of the Jacobian matrix H pertain to
# the first landmark:
# dh_range/dx_t dh_range/dy_t dh_range/dtheta_t
# dh_bearing/dx_t dh_bearing/dy_t dh_bearing/dtheta_t
# Note that 'bearing' and 'angle' are synonymous in this program.
H = np.zeros(shape=(2 * self.N_landmarks, 3))
# Extract the state
X_t = x[0]
Y_t = x[1]
THETA_t = x[2]
# Fill in H, the Jacobian matrix with the partial derivatives of the
# observation (measurement) model h
# Partial derivatives were calculated using an online partial derivatives
# calculator.
# atan2 derivatives are listed here https://en.wikipedia.org/wiki/Atan2
for i in range(0, self.N_landmarks):
# Set the x and y position for landmark i
x_l_i = self.landmarks[i][0]
y_l_i = self.landmarks[i][1]
dh_range_dx_t = (X_t-x_l_i) / math.sqrt((X_t-x_l_i)**2 + (Y_t-y_l_i)**2)
H[i*2][0] = dh_range_dx_t
dh_range_dy_t = (Y_t-y_l_i) / math.sqrt((X_t-x_l_i)**2 + (Y_t-y_l_i)**2)
H[i*2][1] = dh_range_dy_t
dh_range_dtheta_t = 0
H[i*2][2] = dh_range_dtheta_t
dh_bearing_dx_t = -(Y_t-y_l_i) / ((X_t-x_l_i)**2 + (Y_t-y_l_i)**2)
H[i*2+1][0] = dh_bearing_dx_t
dh_bearing_dy_t = (X_t-x_l_i) / ((X_t-x_l_i)**2 + (Y_t-y_l_i)**2)
H[i*2+1][1] = dh_bearing_dy_t
dh_bearing_dtheta_t = -1.0
H[i*2+1][2] = dh_bearing_dtheta_t
return H
def EKF(DiffDrive,Sensor,y,x_hat,sigma,u,dt,V=None,W=None):
"""
Some of these matrices will be non-square or singular. Utilize the
pseudo-inverse function la.pinv instead of inv to avoid these errors.
Input
:param DiffDrive: The DifferentialDrive object defined in kinematics.py
:param Sensor: The Landmark Sensor object defined in this class
:param y: The observation vector (4x1 in this implementation because
there are 2 landmarks). The measurement is
y = [h1_range_1,h1_angle_1, h2_range_2, h2_angle_2]
:param x_hat: The starting estimate of the state at time t ->
np.array with shape (3,)
(X_t, Y_t, THETA_t)
:param sigma: The state covariance matrix at time t -> np.array with
shape (3,1) initially, then 3x3
:param u: The input to the system at time t -> np.array with shape (2,1)
These are the control inputs to the system.
[left wheel rotational velocity, right wheel rotational
velocity]
:param dt: timestep size delta t
:param V: The state noise covariance matrix -> np.array with shape (3,3)
:param W: The measurment noise covariance matrix -> np.array with shape
(2*N_Landmarks,2*N_Landmarks)
4x4 matrix
Output
:return: x_hat_2: The estimate of the state at time t+1
[X_t+1, Y_t+1, THETA_t+1]
:return: sigma_est: The new covariance matrix at time t+1 (3x3 matrix)
"""
V = DiffDrive.get_V() # 3x3 matrix for the state noise
W = Sensor.get_W() # 4x4 matrix for the measurement noise
## Generate noise
# v = process noise, w = measurement noise
v = np.random.multivariate_normal(np.zeros(V.shape[0]),V) # 3x1 vector
w = np.random.multivariate_normal(np.zeros(W.shape[0]),W) # 4x1 vector
##### Prediction Step #####
# Predict the state estimate based on the previous state and the
# control input
# x_predicted is a 3x1 vector (X_t+1, Y_t+1, THETA_t+1)
x_predicted = DiffDrive.forward(x_hat,u,v,dt)
# Calculate the A and B matrices
A, B = DiffDrive.linearize(x=x_hat)
# Predict the covariance estimate based on the
# previous covariance and some noise
# A and V are 3x3 matrices
sigma_3by3 = None
if (sigma.size == 3):
sigma_3by3 = sigma * np.eye(3)
else:
sigma_3by3 = sigma
sigma_new = A @ sigma_3by3 @ A.T + V
##### Correction Step #####
# Get H, the 4x3 Jacobian matrix for the sensor
H = Sensor.jacobian(x_hat)
# Calculate the observation model
# y_predicted is a 4x1 vector
y_predicted = Sensor.measure(x_predicted, w)
# Measurement residual (delta_y is a 4x1 vector)
delta_y = y - y_predicted
# Residual covariance
# 4x3 @ 3x3 @ 3x4 -> 4x4 matrix
S = H @ sigma_new @ H.T + W
# Compute the Kalman gain
# The Kalman gain indicates how certain you are about
# the observations with respect to the motion
# 3x3 @ 3x4 @ 4x4 -> 3x4
K = sigma_new @ H.T @ la.pinv(S)
# Update the state estimate
# 3x1 + (3x4 @ 4x1 -> 3x1)
x_hat_2 = x_predicted + (K @ delta_y)
# Update the covariance estimate
# 3x3 - (3x4 @ 4x3) @ 3x3
sigma_est = sigma_new - (K @ H @ sigma_new)
return x_hat_2 , sigma_est
utils.py
"""
Program: utils.py
This program helps calculate the waypoints along the racetrack
that the robot needs to follow.
Modified from code developed by Atsushi Sakai
Source: https://github.com/AtsushiSakai/PythonRobotics
"""
import numpy as np
import math
import cubic_spline_planner
def pi_2_pi(angle):
return (angle + math.pi) % (2*math.pi) - math.pi
def compute_traj(ax,ay):
cx, cy, cyaw, ck, s = cubic_spline_planner.calc_spline_course(
ax, ay, ds=0.1)
target_speed = 1 # simulation parameter km/h -> m/s
sp = calc_speed_profile(cx, cy, cyaw, target_speed)
desired_traj = np.array([cx,cy,cyaw,sp]).T
return desired_traj
def calc_nearest_index(state, traj):
cx = traj[:,0]
cy = traj[:,1]
cyaw = traj[:,2]
dx = [state[0] - icx for icx in cx]
dy = [state[1] - icy for icy in cy]
d = [idx ** 2 + idy ** 2 for (idx, idy) in zip(dx, dy)]
mind = min(d)
ind = d.index(mind)
mind = math.sqrt(mind)
dxl = cx[ind] - state[0]
dyl = cy[ind] - state[1]
angle = pi_2_pi(cyaw[ind] - math.atan2(dyl, dxl))
if angle < 0:
mind *= -1
return ind, mind
def calc_speed_profile(cx, cy, cyaw, target_speed):
speed_profile = [target_speed] * len(cx)
direction = 1.0
# Set stop point
for i in range(len(cx) - 1):
dyaw = abs(cyaw[i + 1] - cyaw[i])
switch = math.pi / 4.0 <= dyaw < math.pi / 2.0
if switch:
direction *= -1
if direction != 1.0:
speed_profile[i] = - target_speed
else:
speed_profile[i] = target_speed
if switch:
speed_profile[i] = 0.0
speed_profile[-1] = 0.0
return speed_profile
To run the code, navigate to the directory and run the lqr_ekf_control.py program. I’m using Anaconda to run my Python programs, so I’ll type:
