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# Quaternion Basics

**Description:**The basics of quaternion usage in ROS.

**Keywords:**quaternion, rotation

**Tutorial Level:**INTERMEDIATE

Contents

## Components of a quaternion

ROS uses quaternions to track and apply rotations. A quaternion has 4 components (`x`,`y`,`z`,`w`). That's right, 'w' is last (but **beware**: some libraries like Eigen put `w` as the first number!). The commonly-used unit quaternion that yields no rotation about the x/y/z axes is (0,0,0,1):

(C++)

The magnitude of a quaternion should be one. If numerical errors cause a quaternion magnitude other than one, ROS will print warnings. To avoid these warnings, normalize the quaternion:

```
1 q.normalize();
```

ROS uses two quaternion datatypes: msg and 'tf.' To convert between them in C++, use the methods of tf2_geometry_msgs.

(C++)

(Python)

## Think in RPY then convert to quaternion

It's easy for humans 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) / (subsequent pitch about the Y-axis) / (subsequent yaw about the Z-axis), then convert to a quaternion:

(Python)

## 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++)

```
1 #include <tf2_geometry_msgs/tf2_geometry_msgs.h>
2
3 tf2::Quaternion q_orig, q_rot, q_new;
4
5 // Get the original orientation of 'commanded_pose'
6 tf2::convert(commanded_pose.pose.orientation , q_orig);
7
8 double r=3.14159, p=0, y=0; // Rotate the previous pose by 180* about X
9 q_rot.setRPY(r, p, y);
10
11 q_new = q_rot*q_orig; // Calculate the new orientation
12 q_new.normalize();
13
14 // Stuff the new rotation back into the pose. This requires conversion into a msg type
15 tf2::convert(q_new, commanded_pose.pose.orientation);
```

(Python)

## Inverting a quaternion

An easy way to invert a quaternion is to negate the w-component:

(Python)

```
1 q[3] = -q[3]
```

## 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**, to go from **q_1** to **q_2**:

```
1 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:

```
1 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:

(Python)

```
1 q1_inv[0] = prev_pose.pose.orientation.x
2 q1_inv[1] = prev_pose.pose.orientation.y
3 q1_inv[2] = prev_pose.pose.orientation.z
4 q1_inv[3] = -prev_pose.pose.orientation.w # Negate for inverse
5
6 q2[0] = current_pose.pose.orientation.x
7 q2[1] = current_pose.pose.orientation.y
8 q2[2] = current_pose.pose.orientation.z
9 q2[3] = current_pose.pose.orientation.w
10
11 qr = tf.transformations.quaternion_multiply(q2, q1_inv)
```

*-Dr. Andy Zelenak, University of Texas*