package

LieGroups.SO3.Quat

SO(3) quaternion parameterization — q = {w, x, y, z}

Rotation convention

A unit quaternion is {w,x,y,z}, with Hamilton multiplication. A body-to-world quaternion rotates body vectors into world coordinates with rotate. Keep quaternion sign ambiguity separate from the represented rotation.

Example

For Real q[4] and Real v[3], this algorithm fragment rotates the x axis by 90 degrees about positive z:

q := LieGroups.SO3.Quat.exp_map({0,0,1.5707963267948966});
v := LieGroups.SO3.Quat.rotate(q, {1,0,0});

The result is approximately {0,1,0}. exp_map accepts a rotation vector in radians; log_map returns the local rotation vector. Group operations expect valid rotations; use normalize where a caller must explicitly restore quaternion norm.

Members

Name	Kind	Description
normalize	function	Normalize quaternion to unit length
product	function	Hamilton quaternion product q * p
inverse	function	Quaternion inverse (conjugate for unit quaternions)
exp_map	function	Exponential map: so(3) rotation vector -> unit quaternion
log_map	function	Logarithmic map: unit quaternion -> so(3) rotation vector
to_DCM	function	Convert unit quaternion to 3x3 rotation matrix (DCM)
from_DCM	function	Convert 3x3 rotation matrix to unit quaternion (Shepperd's method)
rotate	function	Rotate a 3-vector by a quaternion: v' = q @ v
kinematics	function	Quaternion kinematics: q_dot = 0.5 * q * [0, omega]
adjoint	function	Adjoint Ad_X: for SO(3), Ad_q = R (the rotation matrix)
small_adjoint	function	Small adjoint ad_xi: the 3x3 skew-symmetric matrix [v]x
wedge	function	Wedge map: so(3) vector -> 3x3 skew-symmetric matrix (xi^wedge)
vee	function	Vee map: 3x3 skew-symmetric matrix -> so(3) vector (S^vee)
left_jacobian	function	Left Jacobian J_l(v) of SO(3)
left_jacobian_inv	function	Inverse left Jacobian J_l^{-1}(v) of SO(3)
right_jacobian	function	Right Jacobian J_r(v) of SO(3)
right_jacobian_inv	function	Inverse right Jacobian J_r^{-1}(v) of SO(3)
left_jacobian_exact	function	Left Jacobian J_l(v) of SO(3), closed form down to the exp_map branch radius
left_jacobian_inv_exact	function	Inverse left Jacobian J_l^-1(v) of SO(3), closed form down to 1e-4 rad
right_jacobian_exact	function	Right Jacobian J_r(v) of SO(3), closed form down to the exp_map branch radius
right_jacobian_inv_exact	function	Inverse right Jacobian J_r^-1(v) of SO(3), closed form down to 1e-4 rad
exp_map_jacobian	function	Right-trivialized derivative of exp_map with respect to its rotation vector
log_map_jacobian	function	Right-trivialized derivative of log_map with respect to its quaternion
product_left_factor_jacobian	function	Right-trivialized derivative of product with respect to its left factor
product_right_factor_jacobian	function	Right-trivialized derivative of product with respect to its right factor
inverse_jacobian	function	Right-trivialized derivative of inverse with respect to its quaternion
rotate_rotation_jacobian	function	Right-trivialized derivative of rotate with respect to its quaternion
rotate_vector_jacobian	function	Derivative of rotate with respect to the vector being rotated

Modelica declarations and implementation

Show parameters, interfaces and source

Parameters, inputs, outputs, defaults and equations from the source file.

package Quat "SO(3) quaternion parameterization — q = {w, x, y, z}"
  annotation(Documentation(info = "<html>
    <h4>Rotation convention</h4>
<p>A unit quaternion is {w,x,y,z}, with Hamilton multiplication. A body-to-world
quaternion rotates body vectors into world coordinates with
<a href=\"modelica://LieGroups.SO3.Quat.rotate\">rotate</a>. Keep quaternion
sign ambiguity separate from the represented rotation.</p>
<h4>Example</h4>
<p>For <code>Real q[4]</code> and <code>Real v[3]</code>, this algorithm fragment
rotates the x axis by 90 degrees about positive z:</p>
<pre>q := LieGroups.SO3.Quat.exp_map({0,0,1.5707963267948966});
v := LieGroups.SO3.Quat.rotate(q, {1,0,0});</pre>
<p>The result is approximately {0,1,0}.
<a href=\"modelica://LieGroups.SO3.Quat.exp_map\">exp_map</a> accepts a rotation
vector in radians; <a href=\"modelica://LieGroups.SO3.Quat.log_map\">log_map</a>
returns the local rotation vector. Group operations expect valid rotations;
use <a href=\"modelica://LieGroups.SO3.Quat.normalize\">normalize</a> where a
caller must explicitly restore quaternion norm.</p>
    </html>"));
end Quat;