## Coordinate Transformations in Robotics

In robotics applications, many different coordinate systems can be used to define where
robots, sensors, and other objects are located. In general, the location of an object in 3-D space
can be specified by position and orientation values. There are multiple possible representations
for these values, some of which are specific to certain applications. Translation and rotation are
alternative terms for position and orientation. Robotics System Toolbox™ supports representations that are commonly used in robotics and allows you to
convert between them. You can transform between coordinate systems when you apply these
representations to 3-D points. These supported representations are detailed below with brief
explanations of their usage and numeric equivalent in MATLAB^{®}. Each representation has an abbreviation for its name. This is used in the naming of
arguments and conversion functions that are supported in this toolbox.

At the end of this section, you can find out about the conversion functions that we offer to convert between these representations.

Robotics System Toolbox assumes that positions and orientations are defined in a right-handed Cartesian coordinate system.

### Axis-Angle

**Abbreviation: axang**

A rotation in 3-D space described by a scalar rotation around a fixed axis defined by a vector.

**Numeric Representation:** 1-by-3 unit vector and a scalar
angle combined as a 1-by-4 vector

For example, a rotation of `pi/2 `

radians around the
*y*-axis would be:

axang = [0 1 0 pi/2]

### Euler Angles

**Abbreviation: eul**

Euler angles are three angles that describe the orientation of a rigid body. Each angle is a
scalar rotation around a given coordinate frame axis. The Robotics System Toolbox supports two rotation orders. The `'ZYZ'`

axis order is commonly
used for robotics applications. We also support the `'ZYX'`

axis order which is
also denoted as “Roll Pitch Yaw (rpy).” Knowing which axis order you use is
important for apply the rotation to points and in converting to other representations.

**Numeric Representation:** 1-by-3 vector of scalar
angles

For example, a rotation around the *y* -axis of pi would be expressed
as:

eul = [0 pi 0]

*Note:* The axis order is not stored in the transformation, so you must
be aware of what rotation order is to be applied.

### Homogeneous Transformation Matrix

**Abbreviation: tform**

A homogeneous transformation matrix combines a translation and rotation into one matrix.

**Numeric Representation:** 4-by-4 matrix

For example, a rotation of angle α around the *y* -axis and a translation
of 4 units along the *y* -axis would be expressed as:

tform = cos α 0 sin α 0 0 1 0 4 -sin α 0 cos α 0 0 0 0 1

You should **pre-multiply** your transformation matrix with
your homogeneous coordinates, which are represented as a matrix of row vectors
(*n*-by-4 matrix of points). Utilize the transpose (`'`

) to
rotate your points for matrix multiplication. For example:

points = rand(100,4); tformPoints = (tform*points')';

### Quaternion

**Abbreviation: quat**

A quaternion is a four-element vector with a scalar rotation and 3-element vector.
Quaternions are advantageous because they avoid singularity issues that are inherent in other
representations. The first element, *w*, is a scalar to normalize the vector
with the three other values, *[x y z]* defining the axis of rotation.

**Numeric Representation:** 1-by-4 vector

For example, a rotation of `pi/2 `

around the *y* -axis
would be expressed as:

quat = [0.7071 0 0.7071 0]

### Rotation Matrix

**Abbreviation: rotm**

A rotation matrix describes a rotation in 3-D space. It is a square, orthonormal matrix with a determinant of 1.

**Numeric Representation:** 3-by-3 matrix

For example, a rotation of **α** degrees around the
*x*-axis would be:

rotm = 1 0 0 0 cos α -sin α 0 sin α cos α

You should **pre-multiply** your rotation matrix with your
coordinates, which are represented as a matrix of row vectors (*n*-by-3 matrix
of points). Utilize the transpose (`'`

) to rotate your points for matrix
multiplication. For example:

points = rand(100,3); rotPoints = (rotm*points')';

### Translation Vector

**Abbreviation: trvec**

A translation vector is represented in 3-D Euclidean space as Cartesian coordinates. It only involves coordinate translation applied equally to all points. There is no rotation involved.

**Numeric Representation:** 1-by-3 vector

For example, a translation by 3 units along the *x* -axis and 2.5 units
along the *z* -axis would be expressed as:

trvec = [3 0 2.5]

### Conversion Functions and Transformations

Robotics System Toolbox provides conversion functions for the previously mentioned transformation representations. Not all conversions are supported by a dedicated function. Below is a table showing which conversions are supported (in blue). The abbreviations for the rotation and translation representations are shown as well.

Converting To | Axis-Angle (`axang` ) | Euler Angles (`eul` ) | Numeric Quaternion (`quat` ) | Quaternion Object (`quaternion` ) | Rotation Matrix (`rotm` ) | Transformation Matrix (`tform` ) | Translation Vector (`trvec` ) | |

Converting From | ||||||||

Axis-Angle (`axang` ) | — | — | `axang2quat` | `quaternion` | `axang2rotm` | `axang2tform` | — | |

Euler Angles (`eul` ) | — | — | `eul2quat` | `quaternion` | `eul2rotm` | `eul2tform` | ||

Numeric Quaternion
(`quat` ) | `quat2axang` | `quat2eul` | — | `quaternion` | `quat2rotm` | `quat2tform` | ||

Quaternion Object
(`quaternion` ) | — | `euler` | `compact` | — | — | |||

Rotation Matrix (`rotm` ) | `rotm2axang` | `rotm2eul` | `rotm2quat` | `quaternion` | — | `rotm2tform` | ||

Transformation Matrix
(`tform` ) | `tform2axang` | `tform2eul` | `tform2quat` | — | `tform2rotm` | — | `tform2trvec` | |

Translation Vector
(`trvec` ) | — | — | — | — | — | `trvec2tform` | — |

The names of all the conversion functions follow a standard format. They follow the form
`alpha2beta`

where `alpha`

is the abbreviation for what you
are converting from and `beta`

is what you are converting to as an abbreviation.
For example, converting from Euler angles to quaternion would be
`eul2quat`

.

All the functions expect valid inputs. If you specify invalid inputs, the outputs will be undefined.

There are other conversion functions for converting between radians and degrees, Cartesian and homogeneous coordinates, and for calculating wrapped angle differences. For a full list of conversions, see Coordinate Transformations.