Modelica.Mechanics.MultiBody.Frames

Functions to transform rotational frame quantities

Information

Package Frames contains type definitions and functions to transform rotational frame quantities. The basic idea is to hide the actual definition of an orientation in this package by providing essentially type Orientation together with functions operating on instances of this type.

Content

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

   Frames.Orientation R, R1, R2, R_rel, R_inv;
   Real[3,3]   T, 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.Package (Icon for standard packages).

Package Content

Name	Description
Orientation	Orientation object defining rotation from a frame 1 into a frame 2
orientationConstraint	Return residues of orientation constraints (shall be zero)
angularVelocity1	Return angular velocity resolved in frame 1 from orientation object
angularVelocity2	Return angular velocity resolved in frame 2 from orientation object
resolve1	Transform vector from frame 2 to frame 1
resolve2	Transform vector from frame 1 to frame 2
resolveRelative	Transform vector from frame 1 to frame 2 using absolute orientation objects of frame 1 and of 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 an angle along one frame axis
axesRotations	Return fixed rotation object to rotate in sequence around fixed angles along 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 fixed orientation object from n_x and n_y vectors
from_nxz	Return fixed orientation object from n_x and n_z vectors
from_T	Return orientation object R from transformation matrix T
from_T2	Return orientation object R from transformation matrix T and its derivative der(T)
from_T_inv	Return orientation object R from inverse transformation matrix T_inv
from_Q	Return orientation object R 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 R
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
axis	Return unit vector for x-, y-, or z-axis
Quaternions	Functions to transform rotational frame quantities based on quaternions (also called Euler parameters)
TransformationMatrices	Functions for transformation matrices

Modelica.Mechanics.MultiBody.Frames.Orientation

Orientation object defining rotation from a frame 1 into a frame 2

Information

This object describes the rotation from a frame 1 into a frame 2. An instance of this type should never be directly accessed but only with the access functions provided in package Modelica.Mechanics.MultiBody.Frames. As a consequence, it is not necessary to know the internal representation of this object as described in the next paragraphs.

"Orientation" is defined to be a record consisting of two elements: "Real T[3,3]", the transformation matrix to rotate frame 1 into frame 2 and "Real w[3]", the angular velocity of frame 2 with respect to frame 1, resolved in frame 2. Element "T" has the following interpretation:

   Orientation R;
   R.T = [ex, ey, ez];
       e.g., R.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 = R.T * v1

The inverse orientation R_inv.T = R.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 Frames.orientationConstraint.

R.w is the angular velocity of frame 2 with respect to frame 1, resolved in frame 2. Formally, R.w is defined as:
skew(R.w) = R.T*der(transpose(R.T)) with

             |   0   -w[3]  w[2] |
   skew(w) = |  w[3]   0   -w[1] |
             | -w[2]  w[1]     0 |

Extends from Modelica.Icons.Record (Icon for records).

Modelica.Mechanics.MultiBody.Frames.orientationConstraint

Return residues of orientation constraints (shall be zero)

Information

Syntax

residue = Frames.orientationConstraint(R);

Description

The function call Frames.orientationConstraint(R) returns the Real residue vector with 6 elements that describes the constraints between the 9 elements of the orientation matrix.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.angularVelocity1

Return angular velocity resolved in frame 1 from orientation object

Information

Syntax

w12_1 = Frames.angularVelocity1(R12);

Description

The function call Frames.angularVelocity1(R12) returns the the angular velocity w12_1 of frame 2 with respect to frame 1 resolved in frame 1, from the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.angularVelocity2

Return angular velocity resolved in frame 2 from orientation object

Information

Syntax

w12_2 = Frames.angularVelocity2(R12);

Description

The function call Frames.angularVelocity2(R12) returns the the angular velocity w12_2 of frame 2 with respect to frame 1 resolved in frame 2, from the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.resolve1

Transform vector from frame 2 to frame 1

Information

Syntax

v1 = Frames.resolve1(R12, v2);

Description

The function call Frames.resolve1(R12, v2) returns vector v resolved in frame 1 (= v1) from vector v resolved in frame 2 (= v2) using the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.resolve2

Transform vector from frame 1 to frame 2

Information

Syntax

v2 = Frames.resolve2(R12, v1);

Description

The function call Frames.resolve2(R12, v1) returns vector v resolved in frame 2 (= v2) from vector v resolved in frame 1 (= v1) using the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.resolveRelative

Transform vector from frame 1 to frame 2 using absolute orientation objects of frame 1 and of frame 2

Information

Syntax

v2 = Frames.resolveRelative(R1, R2, v1);

Description

The function call Frames.resolveRelative(R1,R2,v1) returns vector v resolved in frame 2 (= v1) from vector v resolved in frame 1 (= v1) given the orientation object R1 that describes the orientation to rotate frame 0 into frame 1 and orientation object R2 that describes the orientation to rotate frame 0 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.resolveDyade1

Transform second order tensor from frame 2 to frame 1

Information

Syntax

D1 = Frames.resolveDyade1(R12, D2);

Description

The function call Frames.Dyade1(R12, D2) returns the second order tensor D resolved in frame 1 (= D1) from its representation in frame 2 (= D2) using the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.resolveDyade2

Transform second order tensor from frame 1 to frame 2

Information

Syntax

D2 = Frames.resolveDyade2(R12, D1);

Description

The function call Frames.Dyade2(R12, D1) returns the second order tensor D resolved in frame 2 (= D2) from its representation in frame 1 (= D1) using the orientation object R12 that describes the orientation to rotate frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.nullRotation

Return orientation object that does not rotate a frame

Information

Syntax

R = Frames.nullRotation(R);

Description

The function call Frames.rnullRotation() returns an orientation matrix R 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).

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

Outputs

Modelica.Mechanics.MultiBody.Frames.inverseRotation

Return inverse orientation object

Information

Syntax

R_inv = Frames.inverseRotation(R);

Description

The function call Frames.inverseRotation(R) returns orientation object R_inv that describes the orientation to rotate from frame 2 to frame 1 from the orientation object R that describes the orientation to rotate from frame 1 into frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.relativeRotation

Return relative orientation object

Information

Syntax

R_rel = Frames.relativeRotation(R1,R2);

Description

The function call Frames.relativeRotation(R1,R2) returns orientation object R_rel that describes the orientation to rotate frame 1 to frame 2 from the orientation object R1 that describes the orientation to rotate from frame 0 to frame 1 and from the orientation object R2 that describes the orientation to rotate from frame 0 to frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.absoluteRotation

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

Information

Syntax

R2 = Frames.relativeRotation(R1,R_rel);

Description

The function call Frames.absoluteRotation(R1,R_rel) returns orientation object R2 hat describes the orientation frame 0 to frame 2 from the orientation object R1 that describes the orientation to rotate from frame 0 to frame 1 and from the relative orientation object R_rel that describes the orientation to rotate from frame 1 to frame 2.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.planarRotation

Return orientation object of a planar rotation

Information

Syntax

R = Frames.planarRotation(e, angle, der_angle);

Description

The function call Frames.planarRotation(e, angle, der_angle) returns orientation object R that describes the orientation to rotate in the plane along unit axis e from frame 1 into frame 2 with angle angle and derivative of angle der_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.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.planarRotationAngle

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

Information

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.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.axisRotation

Return rotation object to rotate around an angle along one frame axis

Information

Syntax

R = Frames.axisRotation(axis, angle, der_angle);

Description

The function call Frames.axisRotation(axis, angle, der_angle) returns orientation object R that describes the orientation to rotate along unit axis axis from frame 1 into frame 2 with angle angle and derivative of angle der_angle. For example, Frames.axisRotation(2, phi, der_phi) returns the same orientation object as with the call Frames.planarRotation({0,1,0}, phi, der_phi)

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.axesRotations

Return fixed rotation object to rotate in sequence around fixed angles along 3 axes

Information

Syntax

R = Frames.axesRotation(asequence, angles, der_angles);

Description

The function call Frames.axesRotation(sequence, angles, der_angles) returns orientation object R that describes the orientation to rotate along unit axis axis from frame 1 into frame 2 with angle angle and derivative of angle der_angle. For example, Frames.axisRotation(2, phi, der_phi) returns the same orientation object as with the call Frames.planarRotation({0,1,0}, phi, der_phi)

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.axesRotationsAngles

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

Information

A call to this function of the form

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

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

    R = 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 R may be in a singular configuration, i.e., there is an infinite number of angle values leading to the same R. 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

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.smallRotation

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

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_nxy

Return fixed orientation object from n_x and n_y vectors

Information

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 R to rotate from frame 1 to frame 2.

The function is robust in the sense that it returns always an orientation object R, 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.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_nxz

Return fixed orientation object from n_x and n_z vectors

Information

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 R to rotate from frame 1 to frame 2.

The function is robust in the sense that it returns always an orientation object R, 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.

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_T

Return orientation object R from transformation matrix T

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_T2

Return orientation object R from transformation matrix T and its derivative der(T)

Information

Computes the orientation object from a transformation matrix T and the derivative der(T) of the transformation matrix. Usually, it is more efficient to use function "from_T" instead, where the angular velocity has to be given as input argument. Only if this is not possible or too difficult to compute, use function from_T2(..).

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_T_inv

Return orientation object R from inverse transformation matrix T_inv

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.from_Q

Return orientation object R from quaternion orientation object Q

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.to_T

Return transformation matrix T from orientation object R

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.to_T_inv

Return inverse transformation matrix T_inv from orientation object R

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.to_Q

Return quaternion orientation object Q from orientation object R

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.to_vector

Map rotation object into vector

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.to_exy

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

Information

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

Inputs

Outputs

Modelica.Mechanics.MultiBody.Frames.axis

Return unit vector for x-, y-, or z-axis

Information

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

Inputs

Outputs