WOLFRAM SYSTEM MODELER

Quaternions

Functions to transform rotational frame quantities based on quaternions (also called Euler parameters)

Package Contents

Orientation

Orientation type defining rotation from a frame 1 into a frame 2 with quaternions {p1,p2,p3,p0}

der_Orientation (1/s)

First time derivative of Quaternions.Orientation

orientationConstraint

Return residues of orientation constraints (shall be zero)

angularVelocity1

Compute angular velocity resolved in frame 1 from quaternions orientation object and its derivative

angularVelocity2

Compute angular velocity resolved in frame 2 from quaternions orientation object and its derivative

resolve1

Transform vector from frame 2 to frame 1

resolve2

Transform vector from frame 1 to frame 2

multipleResolve1

Transform several vectors from frame 2 to frame 1

multipleResolve2

Transform several vectors from frame 1 to frame 2

nullRotation

Return quaternion orientation object that does not rotate a frame

inverseRotation

Return inverse quaternions orientation object

relativeRotation

Return relative quaternions orientation object

absoluteRotation

Return absolute quaternions orientation object from another absolute and a relative quaternions orientation object

planarRotation

Return quaternion orientation object of a planar rotation

smallRotation

Return rotation angles valid for a small rotation

from_T

Return quaternion orientation object Q from transformation matrix T

from_T_inv

Return quaternion orientation object Q from inverse transformation matrix T_inv

to_T

Return transformation matrix T from quaternion orientation object Q

to_T_inv

Return inverse transformation matrix T_inv from quaternion orientation object Q

Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

Package Frames.Quaternions contains type definitions and functions to transform rotational frame quantities with quaternions. Functions of this package are currently only utilized in MultiBody.Parts.Body components, when quaternions shall be used as parts of the body states. Some functions are also used in a new Modelica package for B-Spline interpolation that is able to interpolate paths consisting of position vectors and orientation objects.

Content

In the table below an example is given for every function definition. The used variables have the following declaration:

Quaternions.Orientation Q, Q1, Q2, Q_rel, Q_inv;
Real[3,3]   T, T_inv;
Real[3]     v1, v2, w1, w2, n_x, n_y, n_z, res_ori, phi;
Real[6]     res_equal;
Real        L, angle;
Function/typeDescription
Orientation Q; New type defining a quaternion object that describes
the rotation of frame 1 into frame 2.
der_Orientation der_Q; New type defining the first time derivative of Frames.Quaternions.Orientation.
res_ori = orientationConstraint(Q); Return the constraints between the variables of a quaternion object
(shall be zero).
w1 = angularVelocity1(Q, der_Q); Return angular velocity resolved in frame 1 from quaternion object Q
and its derivative der_Q.
w2 = angularVelocity2(Q, der_Q); Return angular velocity resolved in frame 2 from quaternion object Q
and its derivative der_Q.
v1 = resolve1(Q,v2); Transform vector v2 from frame 2 to frame 1.
v2 = resolve2(Q,v1); Transform vector v1 from frame 1 to frame 2.
[v1,w1] = multipleResolve1(Q, [v2,w2]); Transform several vectors from frame 2 to frame 1.
[v2,w2] = multipleResolve2(Q, [v1,w1]); Transform several vectors from frame 1 to frame 2.
Q = nullRotation() Return quaternion object R that does not rotate a frame.
Q_inv = inverseRotation(Q); Return inverse quaternion object.
Q_rel = relativeRotation(Q1,Q2); Return relative quaternion object from two absolute quaternion objects.
Q2 = absoluteRotation(Q1,Q_rel); Return absolute quaternion object from another absolute
and a relative quaternion object.
Q = planarRotation(e, angle); Return quaternion object of a planar rotation.
phi = smallRotation(Q); Return rotation angles phi valid for a small rotation.
Q = from_T(T); Return quaternion object Q from transformation matrix T.
Q = from_T_inv(T_inv); Return quaternion object Q from inverse transformation matrix T_inv.
T = to_T(Q); Return transformation matrix T from quaternion object Q.
T_inv = to_T_inv(Q); Return inverse transformation matrix T_inv from quaternion object Q.

Wolfram Language

In[1]:=
SystemModel["Modelica.Mechanics.MultiBody.Frames.Quaternions"]
Out[1]:=