rotmat
Convert quaternion to rotation matrix
Description
converts the quaternion, rotationMatrix
= rotmat(quat
,rotationType
)quat
, to an equivalent rotation matrix
representation.
Examples
Convert Quaternion to Rotation Matrix for Point Rotation
Define a quaternion for use in point rotation.
theta = 45; gamma = 30; quat = quaternion([0,theta,gamma],"eulerd","ZYX","point")
quat = quaternion
0.8924 + 0.23912i + 0.36964j + 0.099046k
Convert the quaternion to a rotation matrix.
rotationMatrix = rotmat(quat,"point")
rotationMatrix = 3×3
0.7071 -0.0000 0.7071
0.3536 0.8660 -0.3536
-0.6124 0.5000 0.6124
To verify the rotation matrix, directly create two rotation matrices corresponding to the rotations about the y- and x-axes. Multiply the rotation matrices and compare to the output of rotmat
.
theta = 45; gamma = 30; ry = [cosd(theta) 0 sind(theta) ; ... 0 1 0 ; ... -sind(theta) 0 cosd(theta)]; rx = [1 0 0 ; ... 0 cosd(gamma) -sind(gamma) ; ... 0 sind(gamma) cosd(gamma)]; rotationMatrixVerification = rx*ry
rotationMatrixVerification = 3×3
0.7071 0 0.7071
0.3536 0.8660 -0.3536
-0.6124 0.5000 0.6124
Convert Quaternion to Rotation Matrix for Frame Rotation
Define a quaternion for use in frame rotation.
theta = 45; gamma = 30; quat = quaternion([0,theta,gamma],"eulerd","ZYX","frame")
quat = quaternion
0.8924 + 0.23912i + 0.36964j - 0.099046k
Convert the quaternion to a rotation matrix.
rotationMatrix = rotmat(quat,"frame")
rotationMatrix = 3×3
0.7071 -0.0000 -0.7071
0.3536 0.8660 0.3536
0.6124 -0.5000 0.6124
To verify the rotation matrix, directly create two rotation matrices corresponding to the rotations about the y- and x-axes. Multiply the rotation matrices and compare to the output of rotmat
.
theta = 45; gamma = 30; ry = [cosd(theta) 0 -sind(theta) ; ... 0 1 0 ; ... sind(theta) 0 cosd(theta)]; rx = [1 0 0 ; ... 0 cosd(gamma) sind(gamma) ; ... 0 -sind(gamma) cosd(gamma)]; rotationMatrixVerification = rx*ry
rotationMatrixVerification = 3×3
0.7071 0 -0.7071
0.3536 0.8660 0.3536
0.6124 -0.5000 0.6124
Convert Quaternion Vector to Rotation Matrices
Create a 3-by-1 normalized quaternion vector.
qVec = normalize(quaternion(randn(3,4)));
Convert the quaternion array to rotation matrices. The pages of rotmatArray
correspond to the linear index of qVec
.
rotmatArray = rotmat(qVec,"frame");
Assume qVec
and rotmatArray
correspond to a sequence of rotations. Combine the quaternion rotations into a single representation, then apply the quaternion rotation to arbitrarily initialized Cartesian points.
loc = normalize(randn(1,3)); quat = prod(qVec); rotateframe(quat,loc)
ans = 1×3
0.9524 0.5297 0.9013
Combine the rotation matrices into a single representation, then apply the rotation matrix to the same initial Cartesian points. Verify the quaternion rotation and rotation matrix result in the same orientation.
totalRotMat = eye(3); for i = 1:size(rotmatArray,3) totalRotMat = rotmatArray(:,:,i)*totalRotMat; end totalRotMat*loc'
ans = 3×1
0.9524
0.5297
0.9013
Input Arguments
quat
— Quaternion to convert
quaternion
object | array of quaternion
objects
Quaternion to convert, specified as a quaternion
object or an array of quaternion
objects of any dimensionality.
rotationType
— Type or rotation
"frame"
| "point"
Type of rotation represented by the rotationMatrix
output, specified as "frame"
or
"point"
.
Data Types: char
| string
Output Arguments
rotationMatrix
— Rotation matrix representation
3-by-3 numeric matrix | 3-by-3-by-N numeric array
Rotation matrix representation, returned as a 3-by-3 numeric matrix or 3-by-3-by-N numeric array.
If
quat
is a scalar,rotationMatrix
is returned as a 3-by-3 matrix.If
quat
is non-scalar,rotationMatrix
is returned as a 3-by-3-by-N array, whererotationMatrix(:,:,i)
is the rotation matrix corresponding toquat(i)
.
The data type of the rotation matrix is the same as the underlying data
type of quat
.
Data Types: single
| double
Algorithms
Given a quaternion of the form
the equivalent rotation matrix for frame rotation is defined as
The equivalent rotation matrix for point rotation is the transpose of the frame rotation matrix:
References
[1] Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 2007.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
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