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A simple working solution is the following:

#!/usr/bin/env python
import rospy
from geometry_msgs.msg import Twist
import time

def move(velocity_publisher):
    vel_msg = Twist()

    #Receiveing the user's input
    print("Let's move your robot")
    speed = input("Input your speed:")
    distance = input("Type your distance:")
    isForward = input("Foward?: ")#True or False

    #Checking if the movement is forward or backwards
    if(isForward):
        vel_msg.linear.x = abs(speed)
    else:
        vel_msg.linear.x = -abs(speed)
    #Since we are moving just in x-axis
    vel_msg.linear.y = 0
    vel_msg.linear.z = 0
    vel_msg.angular.x = 0
    vel_msg.angular.y = 0
    vel_msg.angular.z = 0

    #Setting the current time for distance calculus
    t0 = rospy.Time.now().to_sec()
    current_distance = 0

    #Loop to move the turtle in an specified distance
    while(current_distance < distance):
        #Publish the velocity
        velocity_publisher.publish(vel_msg)
        #Takes actual time to velocity calculus
        t1=rospy.Time.now().to_sec()
        #Calculates distancePoseStamped
        current_distance= speed*(t1-t0)
    #After the loop, stops the robot
    vel_msg.linear.x = 0
    #Force the robot to stop
    velocity_publisher.publish(vel_msg)


if __name__ == '__main__':
    # Starts a new node
    rospy.init_node('robot_cleaner', anonymous=True)
    velocity_publisher = rospy.Publisher('/turtle1/cmd_vel', Twist, queue_size=10)
    try:
     timeout =  time.time() + 60
     while time.time() < timeout:
         move(velocity_publisher)

    except rospy.ROSInterruptException:
     pass

As of why your implementation is not working, you use while not rospy.is_shutdown(): inside the move function and you are preventing it from returning, since this while loop will stop only when the node is killed.

A simple working solution is the following:

#!/usr/bin/env python
import rospy
from geometry_msgs.msg import Twist
import time

def move(velocity_publisher):
    vel_msg = Twist()

    #Receiveing the user's input
    print("Let's move your robot")
    speed = input("Input your speed:")
    distance = input("Type your distance:")
    isForward = input("Foward?: ")#True or False

    #Checking if the movement is forward or backwards
    if(isForward):
        vel_msg.linear.x = abs(speed)
    else:
        vel_msg.linear.x = -abs(speed)
    #Since we are moving just in x-axis
    vel_msg.linear.y = 0
    vel_msg.linear.z = 0
    vel_msg.angular.x = 0
    vel_msg.angular.y = 0
    vel_msg.angular.z = 0

    #Setting the current time for distance calculus
    t0 = rospy.Time.now().to_sec()
    current_distance = 0

    #Loop to move the turtle in an specified distance
    while(current_distance < distance):
        #Publish the velocity
        velocity_publisher.publish(vel_msg)
        #Takes actual time to velocity calculus
        t1=rospy.Time.now().to_sec()
        #Calculates distancePoseStamped
        current_distance= speed*(t1-t0)
    #After the loop, stops the robot
    vel_msg.linear.x = 0
    #Force the robot to stop
    velocity_publisher.publish(vel_msg)


if __name__ == '__main__':
    # Starts a new node
    rospy.init_node('robot_cleaner', anonymous=True)
    velocity_publisher = rospy.Publisher('/turtle1/cmd_vel', Twist, queue_size=10)
    try:
     timeout =  time.time() + 60
     while time.time() < timeout:
         move(velocity_publisher)

    except rospy.ROSInterruptException:
     pass

As of why your implementation is not working, you use while not rospy.is_shutdown(): inside the move function and you are preventing it from returning, since this while loop will stop only when the node is killed.killed. Also notice that the node and publisher definitions are in main and not in the move function. We do not want to define those everytime we call the move function.