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;

Extends from Modelica.Icons.FunctionsPackage (Icon for packages containing functions).

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 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

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

Modelica.Mechanics.MultiBody.Frames.Quaternions.orientationConstraint

Return residues of orientation constraints (shall be zero)

Information

Syntax

residue = Quaternions.orientationConstraint(Q);

Description

This function returns the Real residue vector with 1 element only that describes the constraints between the 4 elements of the quaternions orientation Q.

See also

Frames.orientationConstraint, TransformationMatrices.orientationConstraint.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity1

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

Information

Syntax

w = Quaternions.angularVelocity1(Q, der_Q);

Description

This function returns the angular velocity w of frame 2 with respect to frame 1, resolved in frame 1, from the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2 and from its first time derivative der_Q.

See also

Frames.angularVelocity1, TransformationMatrices.angularVelocity1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity2

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

Information

Syntax

w = Quaternions.angularVelocity2(Q, der_Q);

Description

This function returns the angular velocity w of frame 2 with respect to frame 1, resolved in frame 2, from the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2 and from its first time derivative der_Q.

See also

Frames.angularVelocity2, TransformationMatrices.angularVelocity2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve1

Transform vector from frame 2 to frame 1

Information

Syntax

v1 = Quaternions.resolve1(Q, v2);

Description

This function returns vector v resolved in frame 1 (=v1) from vector v resolved in frame 2 (=v2) using the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2.

See also

resolve2, Frames.resolve1, TransformationMatrices.resolve1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve2

Transform vector from frame 1 to frame 2

Information

Syntax

v2 = Quaternions.resolve2(Q, v1);

Description

This function returns vector v resolved in frame 2 (=v2) from vector v resolved in frame 1 (=v1) using the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2.

See also

resolve1, Frames.resolve2, TransformationMatrices.resolve2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve1

Transform several vectors from frame 2 to frame 1

Information

Syntax

v1 = Quaternions.multipleResolve1(Q, v2);

Description

This function returns vectors v resolved in frame 1 (=v1) from vectors v resolved in frame 2 (=v2) using the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2.

See also

TransformationMatrices.multipleResolve1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve2

Transform several vectors from frame 1 to frame 2

Information

Syntax

v2 = Quaternions.multipleResolve2(Q, v1);

Description

This function returns vectors v resolved in frame 2 (=v2) from vectors v resolved in frame 1 (=v1) using the quaternions orientation Q that describes the orientation to rotate frame 1 into frame 2.

See also

TransformationMatrices.multipleResolve2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.nullRotation

Return quaternion orientation object that does not rotate a frame

Information

Syntax

Q = Quaternions.nullRotation();

Description

This function returns a quaternions orientation Q describing the orientation object to rotate frame 1 into frame 2, if frame 1 and frame 2 are identical. (= transformation matrix is identity matrix and angular velocity is zero).

See also

Frames.nullRotation, TransformationMatrices.nullRotation.

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

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.inverseRotation

Return inverse quaternions orientation object

Information

Syntax

Q_inv = Quaternions.inverseRotation(Q);

Description

This function returns quaternions orientation Q_inv that describes the orientation to rotate from frame 2 to frame 1 from the quaternions orientation Q that describes the orientation to rotate from frame 1 into frame 2.

See also

Frames.inverseRotation, TransformationMatrices.inverseRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.relativeRotation

Return relative quaternions orientation object

Information

Syntax

Q_rel = Quaternions.relativeRotation(Q1, Q2);

Description

This function returns quaternions orientation Q_rel that describes the orientation to rotate frame 1 to frame 2 from the quaternions orientation Q1 that describes the orientation to rotate from frame 0 to frame 1 and from the quaternions orientation Q2 that describes the orientation to rotate from frame 0 to frame 2.

See also

Frames.relativeRotation, TransformationMatrices.relativeRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.absoluteRotation

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

Information

Syntax

Q2 = Quaternions.absoluteRotation(Q1, Q_rel);

Description

This function returns quaternions orientation Q2 that describes the orientation frame 0 to frame 2 from the quaternions orientation Q1 that describes the orientation to rotate from frame 0 to frame 1 and from the relative quaternions orientation Q_rel that describes the orientation to rotate from frame 1 to frame 2.

See also

Frames.absoluteRotation, TransformationMatrices.absoluteRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.planarRotation

Return quaternion orientation object of a planar rotation

Information

Syntax

Q = Quaternions.planarRotation(e, angle);

Description

This function returns quaternions orientation Q that describes the orientation to rotate in the plane along unit axis e from frame 1 into frame 2 with angle angle. Note, "e" must be a unit vector. However, this is not checked in this function and the function will return a wrong result, if length(e) is not one.

See also

Frames.planarRotation, TransformationMatrices.planarRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.smallRotation

Return rotation angles valid for a small rotation

Information

Syntax

phi = Quaternions.smallRotation(Q);

Description

This function returns rotation angles valid for a small rotation of x-y-z sequence (i.e. {1,2,3}).

See also

Frames.smallRotation, TransformationMatrices.smallRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T

Return quaternion orientation object Q from transformation matrix T

Information

Syntax

Q = Quaternions.from_T(T, Q_guess);

Description

This function returns a quaternions orientation Q. computed from a transformation matrix T and depending on the initial guess Q_guess. Generally, the transformation matrix T can be gained using a function from the TransformationMatrices package.

See also

to_T. from_T_inv, Frames.from_T, TransformationMatrices.from_T.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T_inv

Return quaternion orientation object Q from inverse transformation matrix T_inv

Information

Syntax

Q = Quaternions.from_T_inv(T_inv, Q_guess);

Description

This function returns a quaternions orientation Q. computed from an inverse transformation matrix T_inv and depending on the initial guess Q_guess. Generally, the transformation matrix T_inv can be gained using a function from the TransformationMatrices package, e.g. using T_inv = inverseRotation(T).

See also

from_T, Frames.from_T_inv, TransformationMatrices.from_T_inv.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T

Return transformation matrix T from quaternion orientation object Q

Information

Syntax

T = Quaternions.to_T(Q);

Description

This function returns a real matrix T computed from a quaternions orientation Q. The matrix T is considered to be an object transformation matrix.

See also

from_T, to_T_inv, Frames.to_T, TransformationMatrices.to_T.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T_inv

Return inverse transformation matrix T_inv from quaternion orientation object Q

Information

Syntax

T_inv = Quaternions.to_T_inv(Q);

Description

This function returns a real matrix T_inv computed from a quaternions orientation Q. The matrix T is considered to be an inverse transformation matrix.

See also

to_T, from_T_inv, Frames.to_T_inv, TransformationMatrices.to_T_inv.

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

Inputs

Outputs