Orientation, Position, and Coordinate Systems

Quaternions, Euler angles, rotation matrices, and conversions

Represent orientation and rotation using the quaternion data type. Convert between quaternions and Euler angles, rotation matrices, and rotation vectors using the euler, rotmat, and rotvec functions.

To learn more about quaternion mathematics and how they are implemented in Sensor Fusion and Tracking Toolbox™, see Rotations, Orientation, and Quaternions. To learn more about conventions and coordinate systems in Sensor Fusion and Tracking Toolbox, see Orientation, Position, and Coordinate Convention.

Functions

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Initialization and Convenience Functions

`quaternion`	Create quaternion array
`ones`	Create quaternion array with real parts set to one and imaginary parts set to zero
`zeros`	Create quaternion array with all parts set to zero
`classUnderlying`	Class of parts within quaternion
`normalize`	Quaternion normalization
`randrot`	Uniformly distributed random rotations

Mathematics

`times, .*`	Element-wise quaternion multiplication
`mtimes, *`	Quaternion multiplication
`prod`	Product of quaternion array
`minus, -`	Quaternion subtraction
`uminus, -`	Quaternion unary minus
`conj`	Complex conjugate of quaternion
`ldivide, .\`	Element-wise quaternion left division
`rdivide, ./`	Element-wise quaternion right division
`mldivide, \`	Quaternion left division for scalars (Since R2024a)
`mrdivide, /`	Quaternion right division for scalars (Since R2024a)
`exp`	Exponential of quaternion array
`log`	Natural logarithm of quaternion array
`power, .^`	Element-wise quaternion power

Metrics and Interpolation

`dist`	Angular distance in radians
`norm`	Quaternion norm
`angvel`	Angular velocity from quaternion array
`meanrot`	Quaternion mean rotation
`slerp`	Spherical linear interpolation
`interp1`	Quaternion interpolation (table lookup) (Since R2025a)

Rotations

`rotateframe`	Quaternion frame rotation
`rotatepoint`	Quaternion point rotation
`so2`	SO(2) rotation (Since R2023b)
`so3`	SO(3) rotation (Since R2023b)

Array Manipulation

`ctranspose, '`	Complex conjugate transpose of quaternion array
`transpose, .'`	Transpose quaternion array

Representation Conversion

`rotmat`	Convert quaternion to rotation matrix
`rotvec`	Convert quaternion to rotation vector (radians)
`rotvecd`	Convert quaternion to rotation vector (degrees)
`parts`	Extract quaternion parts
`euler`	Convert quaternion to Euler angles (radians)
`eulerd`	Convert quaternion to Euler angles (degrees)
`compact`	Convert quaternion array to N-by-4 matrix

Coordinate Transformation

`transformMotion`	Compute motion quantities between two relatively fixed frames
`se2`	SE(2) homogeneous transformation (Since R2023b)
`se3`	SE(3) homogeneous transformation (Since R2023b)
`enu2lla`	Transform local east-north-up coordinates to geodetic coordinates
`ned2lla`	Transform local north-east-down coordinates to geodetic coordinates
`lla2ned`	Transform geodetic coordinates to local north-east-down coordinates
`lla2enu`	Transform geodetic coordinates to local east-north-up coordinates

Topics

Orientation, Position, and Coordinate Convention
Learn about toolbox conventions for spatial representation and coordinate systems.
Rotations, Orientation, and Quaternions
This example reviews concepts in three-dimensional rotations and how quaternions are used to describe orientation and rotations.
Lowpass Filter Orientation Using Quaternion SLERP
This example shows how to use spherical linear interpolation (SLERP) to create sequences of quaternions and lowpass filter noisy trajectories.