Modelica.Mechanics.MultiBody.Frames.Quaternions

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

Information

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/type*	*Description*
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.

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
Orientation	Orientation type defining rotation from a frame 1 into a frame 2 with quaternions {p1,p2,p3,p0}
der_Orientation	First time derivative of Quaternions.Orientation
orientationConstraint	Return residues of orientation constraints (shall be zero)
angularVelocity1	Compute angular velocity resolved in frame 1 from quaternion 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

Modelica.Mechanics.MultiBody.Frames.Quaternions.Orientation

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

Information

This type describes the rotation to rotate a frame 1 into a frame 2 using quaternions (also called Euler parameters) according to the following definition:

   Quaternions.Orientation Q;
   Real  n[3];
   Real  phi(unit="rad");
   Q = [ n*sin(phi/2)
           cos(phi/2) ]

where "n" is the axis of rotation to rotate frame 1 into frame 2 and "phi" is the rotation angle for this rotation. Vector "n" is either resolved in frame 1 or in frame 2 (the result is the same since the coordinates of "n" with respect to frame 1 are identical to its coordinates with respect to frame 2).

The term "quaternions" is preferred over the historically more reasonable "Euler parameters" in order to not get confused with Modelica "parameters".

Extends from Internal.QuaternionBase.

Modelica.Mechanics.MultiBody.Frames.Quaternions.der_Orientation

First time derivative of Quaternions.Orientation

Parameters

Name	Description
unit

Modelica.Mechanics.MultiBody.Frames.Quaternions.orientationConstraint

Return residues of orientation constraints (shall be zero)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
residue[1]	Residue constraint (shall be zero)

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity1

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

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
der_Q	Derivative of Q [1/s]

Outputs

Name	Description
w[3]	Angular velocity resolved in frame 1 [rad/s]

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity2

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

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
der_Q	Derivative of Q [1/s]

Outputs

Name	Description
w[3]	Angular velocity of frame 2 with respect to frame 1 resolved in frame 2 [rad/s]

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve1

Transform vector from frame 2 to frame 1

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
v2[3]	Vector in frame 2

Outputs

Name	Description
v1[3]	Vector in frame 1

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve2

Transform vector from frame 1 to frame 2

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
v1[3]	Vector in frame 1

Outputs

Name	Description
v2[3]	Vector in frame 2

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve1

Transform several vectors from frame 2 to frame 1

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
v2[3, :]	Vectors in frame 2

Outputs

Name	Description
v1[3, size(v2, 2)]	Vectors in frame 1

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve2

Transform several vectors from frame 1 to frame 2

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2
v1[3, :]	Vectors in frame 1

Outputs

Name	Description
v2[3, size(v1, 2)]	Vectors in frame 2

Modelica.Mechanics.MultiBody.Frames.Quaternions.nullRotation

Return quaternion orientation object that does not rotate a frame

Information

Extends from Modelica.Icons.Function (Icon for functions).

Outputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Modelica.Mechanics.MultiBody.Frames.Quaternions.inverseRotation

Return inverse quaternions orientation object

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
Q_inv	Quaternions orientation object to rotate frame 2 into frame 1

Modelica.Mechanics.MultiBody.Frames.Quaternions.relativeRotation

Return relative quaternions orientation object

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q1	Quaternions orientation object to rotate frame 0 into frame 1
Q2	Quaternions orientation object to rotate frame 0 into frame 2

Outputs

Name	Description
Q_rel	Quaternions orientation object to rotate frame 1 into frame 2

Modelica.Mechanics.MultiBody.Frames.Quaternions.absoluteRotation

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

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q1	Quaternions orientation object to rotate frame 0 into frame 1
Q_rel	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
Q2	Quaternions orientation object to rotate frame 0 into frame 2

Modelica.Mechanics.MultiBody.Frames.Quaternions.planarRotation

Return quaternion orientation object of a planar rotation

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
e[3]	Normalized axis of rotation (must have length=1) [1]
angle	Rotation angle to rotate frame 1 into frame 2 along axis e [rad]

Outputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2 along axis e

Modelica.Mechanics.MultiBody.Frames.Quaternions.smallRotation

Return rotation angles valid for a small rotation

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
phi[3]	The rotation angles around x-, y-, and z-axis of frame 1 to rotate frame 1 into frame 2 for a small relative rotation [rad]

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T

Return quaternion orientation object Q from transformation matrix T

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
T[3, 3]	Transformation matrix to transform vector from frame 1 to frame 2 (v2=T*v1)
Q_guess	Guess value for Q (there are 2 solutions; the one close to Q_guess is used

Outputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T_inv

Return quaternion orientation object Q from inverse transformation matrix T_inv

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
T_inv[3, 3]	Inverse transformation matrix to transform vector from frame 2 to frame 1 (v1=T_inv*v2)
Q_guess	Guess value for output Q (there are 2 solutions; the one closer to Q_guess is used

Outputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T

Return transformation matrix T from quaternion orientation object Q

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
T[3, 3]	Transformation matrix to transform vector from frame 1 to frame 2 (v2=T*v1)

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T_inv

Return inverse transformation matrix T_inv from quaternion orientation object Q

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Q	Quaternions orientation object to rotate frame 1 into frame 2

Outputs

Name	Description
T_inv[3, 3]	Transformation matrix to transform vector from frame 2 to frame 1 (v1=T*v2)

Automatically generated Tue Apr 05 09:36:40 2016.