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Quaternion fundamentals
Goal: Learn the basics of quaternion usage in ROS 2.
Tutorial level: Intermediate
Time: 10 minutes
Contents
Background
A quaternion is a 4-tuple representation of orientation, which is more concise than a rotation matrix. Quaternions are very efficient for analyzing situations where rotations in three dimensions are involved. Quaternions are used widely in robotics, quantum mechanics, computer vision, and 3D animation.
You can learn more about the underlying mathematical concept on Wikipedia. You can also take a look at an explorable video series Visualizing quaternions made by 3blue1brown.
In this tutorial, you will learn how quaternions and conversion methods work in ROS 2.
Prerequisites
For this tutorial you may need to install the tf_transformations ROS 2 package to follow the Python code parts.
You can find the source code of the tf_transformations package here.
However, this is not a hard requirement and you can stick to any other geometric transfromation library that suit you best. You can take a look at libraries like transforms3d, scipy.spatial.transform, pytransform3d, numpy-quaternion or blender.mathutils.
Components of a quaternion
ROS 2 uses quaternions to track and apply rotations.
A quaternion has 4 components (x, y, z, w).
In ROS 2, w is last, but in some libraries like Eigen, w can be placed at the first position.
The commonly-used unit quaternion that yields no rotation about the x/y/z axes is (0, 0, 0, 1), and can be created in a following way:
#include <tf2/LinearMath/Quaternion.h>
...
tf2::Quaternion q;
// Create a quaternion from roll/pitch/yaw in radians (0, 0, 0)
q.setRPY(0, 0, 0);
// Print the quaternion components (0, 0, 0, 1)
RCLCPP_INFO(this->get_logger(), "%f %f %f %f",
q.x(), q.y(), q.z(), q.w());
The magnitude of a quaternion should always be one. If numerical errors cause a quaternion magnitude other than one, ROS 2 will print warnings. To avoid these warnings, normalize the quaternion:
q.normalize();
Quaternion types in ROS 2
ROS 2 uses two quaternion datatypes: tf2::Quaternion and its equivalent geometry_msgs::msg::Quaternion.
To convert between them in C++, use the methods of tf2_geometry_msgs.
C++
#include <tf2_geometry_msgs/tf2_geometry_msgs.hpp>
...
tf2::Quaternion tf2_quat, tf2_quat_from_msg;
tf2_quat.setRPY(roll, pitch, yaw);
// Convert tf2::Quaternion to geometry_msgs::msg::Quaternion
geometry_msgs::msg::Quaternion msg_quat = tf2::toMsg(tf2_quat);
// Convert geometry_msgs::msg::Quaternion to tf2::Quaternion
tf2::convert(msg_quat, tf2_quat_from_msg);
// or
tf2::fromMsg(msg_quat, tf2_quat_from_msg);
Python
from geometry_msgs.msg import Quaternion
...
# Create a list of floats, which is compatible with tf2
# Quaternion methods
quat_tf = [0.0, 1.0, 0.0, 0.0]
# Convert a list to geometry_msgs.msg.Quaternion
msg_quat = Quaternion(x=quat_tf[0], y=quat_tf[1], z=quat_tf[2], w=quat_tf[3])
Quaternion operations
1 Think in RPY then convert to quaternion
It’s easy for us to think of rotations about axes, but hard to think in terms of quaternions. A suggestion is to calculate target rotations in terms of roll (about an X-axis), pitch (about the Y-axis), and yaw (about the Z-axis), and then convert to a quaternion.
import tf_transformations
...
q = tf_transformations.quaternion_from_euler(1.5707, 0, -1.5707)
print(f'The quaternion representation is x: {q[0]} y: {q[1]} z: {q[2]} w: {q[3]}.')
2 Applying a quaternion rotation
To apply the rotation of one quaternion to a pose, simply multiply the previous quaternion of the pose by the quaternion representing the desired rotation. The order of this multiplication matters.
C++
#include <tf2_geometry_msgs/tf2_geometry_msgs.hpp>
...
tf2::Quaternion q_orig, q_rot, q_new;
q_orig.setRPY(0.0, 0.0, 0.0);
// Rotate the previous pose by 180* about X
q_rot.setRPY(3.14159, 0.0, 0.0);
q_new = q_rot * q_orig;
q_new.normalize();
Python
import tf_transformations
...
q_orig = tf_transformations.quaternion_from_euler(0, 0, 0)
# Rotate the previous pose by 180* about X
q_rot = tf_transformations.quaternion_from_euler(3.14159, 0, 0)
q_new = tf_transformations.quaternion_multiply(q_rot, q_orig)
3 Inverting a quaternion
An easy way to invert a quaternion is to negate the w-component:
q[3] = -q[3]
4 Relative rotations
Say you have two quaternions from the same frame, q_1 and q_2.
You want to find the relative rotation, q_r, that converts q_1 to q_2 in a following manner:
q_2 = q_r * q_1
You can solve for q_r similarly to solving a matrix equation.
Invert q_1 and right-multiply both sides. Again, the order of multiplication is important:
q_r = q_2 * q_1_inverse
Here’s an example to get the relative rotation from the previous robot pose to the current robot pose in python:
q1_inv[0] = prev_pose.pose.orientation.x
q1_inv[1] = prev_pose.pose.orientation.y
q1_inv[2] = prev_pose.pose.orientation.z
q1_inv[3] = -prev_pose.pose.orientation.w # Negate for inverse
q2[0] = current_pose.pose.orientation.x
q2[1] = current_pose.pose.orientation.y
q2[2] = current_pose.pose.orientation.z
q2[3] = current_pose.pose.orientation.w
qr = tf_transformations.quaternion_multiply(q2, q1_inv)