Modelica.Mechanics.MultiBody.Frames.TransformationMatrices

Functions for transformation matrices

Information

Package Frames.TransformationMatrices contains type definitions and functions to transform rotational frame quantities using transformation matrices.

Content

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

   Orientation T, T1, T2, T_rel, T_inv;
   Real[3]     v1, v2, w1, w2, n_x, n_y, n_z, e, e_x, 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 a transformation matrix
der_Orientation	New type defining the first time derivative of Orientation
orientationConstraint	Return residues of orientation constraints (shall be zero)
angularVelocity1	Return angular velocity resolved in frame 1 from orientation object and its derivative
angularVelocity2	Return angular velocity resolved in frame 2 from 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
resolveDyade1	Transform second order tensor from frame 2 to frame 1
resolveDyade2	Transform second order tensor from frame 1 to frame 2
nullRotation	Return orientation object that does not rotate a frame
inverseRotation	Return inverse orientation object
relativeRotation	Return relative orientation object
absoluteRotation	Return absolute orientation object from another absolute and a relative orientation object
planarRotation	Return orientation object of a planar rotation
planarRotationAngle	Return angle of a planar rotation, given the rotation axis and the representations of a vector in frame 1 and frame 2
axisRotation	Return rotation object to rotate around one frame axis
axesRotations	Return rotation object to rotate in sequence around 3 axes
axesRotationsAngles	Return the 3 angles to rotate in sequence around 3 axes to construct the given orientation object
smallRotation	Return rotation angles valid for a small rotation and optionally residues that should be zero
from_nxy	Return orientation object from n_x and n_y vectors
from_nxz	Return orientation object from n_x and n_z vectors
from_T	Return orientation object R from transformation matrix T
from_T_inv	Return orientation object R from inverse transformation matrix T_inv
from_Q	Return orientation object T from quaternion orientation object Q
to_T	Return transformation matrix T from orientation object R
to_T_inv	Return inverse transformation matrix T_inv from orientation object R
to_Q	Return quaternion orientation object Q from orientation object T
to_vector	Map rotation object into vector
to_exy	Map rotation object into e_x and e_y vectors of frame 2, resolved in frame 1

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.Orientation

Orientation type defining rotation from a frame 1 into a frame 2 with a transformation matrix

Information

This type describes the rotation from a frame 1 into a frame 2. An instance R of type Orientation has the following interpretation:

   T = [ex, ey, ez];
       e.g., T = [1,0,0; 0,1,0; 0,0,1]

where e_x,e_y,e_z are unit vectors in the direction of the x-axis, y-axis, and z-axis of frame 1, resolved in frame 2, respectively. Therefore, if v₁ is vector v resolved in frame 1 and v₂ is vector v resolved in frame 2, the following relationship holds:

    v2 = T * v1

The inverse orientation T_inv = T^T describes the rotation from frame 2 into frame 1.

Since the orientation is described by 9 variables, there are 6 constraints between these variables. These constraints are defined in function TransformationMatrices.orientationConstraint.

Note, that in the MultiBody library the rotation object is never directly accessed but only with the access functions provided in package TransformationMatrices. As a consequence, other implementations of Rotation can be defined by adapting this package correspondingly.

Extends from Internal.TransformationMatrix.

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.der_Orientation

New type defining the first time derivative of Orientation

Parameters

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.orientationConstraint

Return residues of orientation constraints (shall be zero)

Information

Syntax

residue = TransformationMatrices.orientationConstraint(T);

Description

This function returns the Real residue vector with 6 elements that describes the constraints between the 9 elements of the transformation matrix T.

See also

Frames.orientationConstraint, Quaternions.orientationConstraint.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.angularVelocity1

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

Information

Syntax

w = TransformationMatrices.angularVelocity1(T, der_T);

Description

This function returns the angular velocity w of frame 2 with respect to frame 1, resolved in frame 1, from the transformation matrix T that describes the orientation to rotate frame 1 into frame 2 and from its first time derivative der_T:

w = vec( der(transpose(T)) * T ).

See also

Frames.angularVelocity1, Quaternions.angularVelocity1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.angularVelocity2

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

Information

Syntax

w = TransformationMatrices.angularVelocity2(T, der_T);

Description

This function returns the angular velocity w of frame 2 with respect to frame 1, resolved in frame 2, from the transformation matrix T that describes the orientation to rotate frame 1 into frame 2 and from its first time derivative der_T:

w = vec( T * der(transpose(T)) ).

See also

Frames.angularVelocity2, Quaternions.angularVelocity2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.resolve1

Transform vector from frame 2 to frame 1

Information

Syntax

v1 = TransformationMatrices.resolve1(T, v2);

Description

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

See also

resolve2, Frames.resolve1, Quaternions.resolve1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.resolve2

Transform vector from frame 1 to frame 2

Information

Syntax

v2 = TransformationMatrices.resolve2(T, v1);

Description

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

See also

resolve1, Frames.resolve2, Quaternions.resolve2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.multipleResolve1

Transform several vectors from frame 2 to frame 1

Information

Syntax

v1 = TransformationMatrices.multipleResolve1(T, v2);

Description

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

See also

Quaternions.multipleResolve1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.multipleResolve2

Transform several vectors from frame 1 to frame 2

Information

Syntax

v2 = TransformationMatrices.multipleResolve2(T, v1);

Description

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

See also

Quaternions.multipleResolve2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.resolveDyade1

Transform second order tensor from frame 2 to frame 1

Information

Syntax

D1 = TransformationMatrices.resolveDyade1(T, D2);

Description

This function returns the second order tensor D resolved in frame 1 (= D1) from its representation in frame 2 (= D2) using the transformation matrix T that describes the orientation to rotate frame 1 into frame 2.

See also

Frames.resolveDyade1.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.resolveDyade2

Transform second order tensor from frame 1 to frame 2

Information

Syntax

D2 = TransformationMatrices.resolveDyade2(T, D1);

Description

This function returns the second order tensor D resolved in frame 2 (= D2) from its representation in frame 1 (= D1) using the transformation matrix T that describes the orientation to rotate frame 1 into frame 2.

See also

Frames.resolveDyade2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.nullRotation

Return orientation object that does not rotate a frame

Information

Syntax

T = TransformationMatrices.nullRotation();

Description

This function returns a transformation matrix T 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, Quaternions.nullRotation.

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

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.inverseRotation

Return inverse orientation object

Information

Syntax

T_inv = TransformationMatrices.inverseRotation(T);

Description

This function returns transformation matrix T_inv that describes the orientation to rotate from frame 2 to frame 1 from the transformation matrix T that describes the orientation to rotate from frame 1 into frame 2.

See also

Frames.inverseRotation, Quaternions.inverseRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.relativeRotation

Return relative orientation object

Information

Syntax

T_rel = TransformationMatrices.relativeRotation(T1, T2);

Description

This function returns transformation matrix T_rel that describes the orientation to rotate frame 1 to frame 2 from the transformation matrix T1 that describes the orientation to rotate from frame 0 to frame 1 and from the transformation matrix T2 that describes the orientation to rotate from frame 0 to frame 2.

See also

Frames.relativeRotation, Quaternions.relativeRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.absoluteRotation

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

Information

Syntax

T2 = TransformationMatrices.absoluteRotation(T1, T_rel);

Description

This function returns transformation matrix T2 that describes the orientation frame 0 to frame 2 from the transformation matrix T1 that describes the orientation to rotate from frame 0 to frame 1 and from the relative transformation matrix T2_rel that describes the orientation to rotate from frame 1 to frame 2.

See also

Frames.absoluteRotation, Quaternions.absoluteRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.planarRotation

Return orientation object of a planar rotation

Information

Syntax

T = TransformationMatrices.planarRotation(e, angle);

Description

This function returns transformation matrix T 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

planarRotationAngle, Frames.planarRotation, Quaternions.planarRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.planarRotationAngle

Return angle of a planar rotation, given the rotation axis and the representations of a vector in frame 1 and frame 2

Information

Syntax

angle = TransformationMatrices.planarRotationAngle(e, v1, v2);

Description

A call to this function of the form

  Real[3]                e, v1, v2;
  Modelica.SIunits.Angle angle;
equation
  angle = planarRotationAngle(e, v1, v2);

computes the rotation angle "angle" of a planar rotation along unit vector e, rotating frame 1 into frame 2, given the coordinate representations of a vector "v" in frame 1 (v1) and in frame 2 (v2). Therefore, the result of this function fulfills the following equation:

v2 = resolve2(planarRotation(e,angle), v1)

The rotation angle is returned in the range

-p <= angle <= p

This function makes the following assumptions on the input arguments

• Vector e has length 1, i.e., length(e) = 1
• Vector "v" is not parallel to e, i.e., length(cross(e,v1)) ≠ 0

The function does not check the above assumptions. If these assumptions are violated, a wrong result will be returned and/or a division by zero will occur.

See also

planarRotation, Frames.planarRotationAngle.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.axisRotation

Return rotation object to rotate around one frame axis

Information

Syntax

T = TransformationMatrices.axisRotation(axis, angle);

Description

This function returns transformation matrix T that describes the orientation to rotate along unit axis axis from frame 1 into frame 2 with angle angle. For example, TransformationMatrices.axisRotation(2, phi) returns the same orientation object as with the call TransformationMatrices.planarRotation({0,1,0}, phi)

See also

planarRotation, Frames.axisRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.axesRotations

Return rotation object to rotate in sequence around 3 axes

Information

Syntax

T = TransformationMatrices.axesRotations(sequence, angles);

Description

This function returns transformation matrix T that describes the orientation defined by three elementary rotations in a given sequence, each rotated by angles.

See also

Frames.axesRotations.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.axesRotationsAngles

Return the 3 angles to rotate in sequence around 3 axes to construct the given orientation object

Information

Syntax

angles = TransformationMatrices.axesRotationsAngles(T, sequence, guessAngle1);

Description

A call to this function of the form

  TransformationMatrices.Orientation T;
  parameter Integer sequence[3] = {1,2,3};
  Modelica.SIunits.Angle angles[3];
equation
  angle = axesRotationAngles(T, sequence);

computes the rotation angles "angles[1:3]" to rotate frame 1 into frame 2 along axes sequence[1:3], given the transformation matrix T from frame 1 to frame 2. Therefore, the result of this function fulfills the following equation:

T = axesRotation(sequence, angles)

The rotation angles are returned in the range

-p <= angles[i] <= p

There are two solutions for "angles[1]" in this range. Via the third argument guessAngle1 (default = 0) the returned solution is selected such that |angles[1] - guessAngle1| is minimal. The orientation object T may be in a singular configuration, i.e., there is an infinite number of angle values leading to the same T. The returned solution is selected by setting angles[1] = guessAngle1. Then angles[2] and angles[3] can be uniquely determined in the above range.

Note, that input argument sequence has the restriction that only values 1,2,3 can be used and that sequence[1] ≠ sequence[2] and sequence[2] ≠ sequence[3]. Often used values are:

sequence = {1,2,3}  // Cardan angle sequence
         = {3,1,3}  // Euler angle sequence
         = {3,2,1}  // Tait-Bryan angle sequence

See also

Frames.axesRotationsAngles.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.smallRotation

Return rotation angles valid for a small rotation and optionally residues that should be zero

Information

Syntax

phi = TransformationMatrices.smallRotation(T, withResidues);

Description

This function returns rotation angles valid for a small rotation of x-y-z sequence (i.e. {1,2,3}). Optionally, residues are returned as well if withResidues=true.

See also

Frames.smallRotation, Quaternions.smallRotation.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.from_nxy

Return orientation object from n_x and n_y vectors

Information

Syntax

T = TransformationMatrices.from_nxy(n_x, n_y);

Description

It is assumed that the two input vectors n_x and n_y are resolved in frame 1 and are directed along the x and y axis of frame 2 (i.e., n_x and n_y are orthogonal to each other). The function returns the orientation object T to rotate from frame 1 to frame 2.

The function is robust in the sense that it returns always an orientation object T, even if n_y is not orthogonal to n_x. This is performed in the following way:
If n_x and n_y are not orthogonal to each other, first a unit vector e_y is determined that is orthogonal to n_x and is lying in the plane spanned by n_x and n_y. If n_x and n_y are parallel or nearly parallel to each other, a vector e_y is selected arbitrarily such that e_x and e_y are orthogonal to each other.

See also

Frames.from_nxy.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.from_nxz

Return orientation object from n_x and n_z vectors

Information

Syntax

T = TransformationMatrices.from_nxz(n_x, n_z);

Description

It is assumed that the two input vectors n_x and n_z are resolved in frame 1 and are directed along the x and z axis of frame 2 (i.e., n_x and n_z are orthogonal to each other). The function returns the orientation object T to rotate from frame 1 to frame 2.

The function is robust in the sense that it returns always an orientation object T, even if n_z is not orthogonal to n_x. This is performed in the following way:
If n_x and n_z are not orthogonal to each other, first a unit vector e_z is determined that is orthogonal to n_x and is lying in the plane spanned by n_x and n_z. If n_x and n_z are parallel or nearly parallel to each other, a vector e_z is selected arbitrarily such that n_x and e_z are orthogonal to each other.

See also

Frames.from_nxz.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.from_T

Return orientation object R from transformation matrix T

Information

Syntax

R = TransformationMatrices.from_T(T);

Description

This function returns a transformation matrix R which is equal to a real input matrix T.

See also

from_T_inv, to_T, Frames.from_T.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.from_T_inv

Return orientation object R from inverse transformation matrix T_inv

Information

Syntax

R = TransformationMatrices.from_T_inv(T_inv);

Description

This function returns a transformation matrix R which is inverse to real input matrix T_inv.

See also

to_T_inv, from_T, Frames.from_T_inv.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.from_Q

Return orientation object T from quaternion orientation object Q

Information

Syntax

T = TransformationMatrices.from_Q(Q);

Description

This function returns a transformation matrix T computed from a quaternion object Q

See also

to_Q. Frames.from_Q.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.to_T

Return transformation matrix T from orientation object R

Information

Syntax

T = TransformationMatrices.to_T(R);

Description

This function returns a real matrix T which is equal to a transformation matrix R.

See also

from_T, to_T_inv, Frames.to_T, Quaternions.to_T.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.to_T_inv

Return inverse transformation matrix T_inv from orientation object R

Information

Syntax

T_inv = TransformationMatrices.to_T_inv(R);

Description

This function returns a real matrix T_inv which is inverse to a transformation matrix R.

See also

from_T_inv, to_T, Frames.to_T_inv, Quaternions.to_T_inv.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.to_Q

Return quaternion orientation object Q from orientation object T

Information

Syntax

Q = TransformationMatrices.to_Q(T, Q_guess);

Description

This function returns a quaternion object Q computed from a transformation matrix T and depending on the initial guess Q_guess.

See also

from_Q Frames.to_Q.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.to_vector

Map rotation object into vector

Information

Syntax

vec = TransformationMatrices.to_vector(T);

Description

This function returns a vector vec which contains elements of a transformation matrix T.

See also

to_T, Frames.to_vector.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.to_exy

Map rotation object into e_x and e_y vectors of frame 2, resolved in frame 1

Information

Syntax

exy = TransformationMatrices.to_exy(T);

Description

This function returns unit vectors e_x and e_y which define axes of frame 2 resolved in frame 1, provided T is a transformation matrix to rotate frame 1 into  2.

See also

from_nxy, from_nxz, Frames.to_exy.

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

Inputs

Outputs