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/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.

Extends from Modelica.Icons.Library (Icon for library).

Package Content

NameDescription
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
Modelica.Mechanics.MultiBody.Frames.Quaternions.orientationConstraint orientationConstraint Return residues of orientation constraints (shall be zero)
Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity1 angularVelocity1 Compute angular velocity resolved in frame 1 from quaternion orientation object and its derivative
Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity2 angularVelocity2 Compute angular velocity resolved in frame 2 from quaternions orientation object and its derivative
Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve1 resolve1 Transform vector from frame 2 to frame 1
Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve2 resolve2 Transform vector from frame 1 to frame 2
Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve1 multipleResolve1 Transform several vectors from frame 2 to frame 1
Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve2 multipleResolve2 Transform several vectors from frame 1 to frame 2
Modelica.Mechanics.MultiBody.Frames.Quaternions.nullRotation nullRotation Return quaternions orientation object that does not rotate a frame
Modelica.Mechanics.MultiBody.Frames.Quaternions.inverseRotation inverseRotation Return inverse quaternions orientation object
Modelica.Mechanics.MultiBody.Frames.Quaternions.relativeRotation relativeRotation Return relative quaternions orientation object
Modelica.Mechanics.MultiBody.Frames.Quaternions.absoluteRotation absoluteRotation Return absolute quaternions orientation object from another absolute and a relative quaternions orientation object
Modelica.Mechanics.MultiBody.Frames.Quaternions.planarRotation planarRotation Return quaternions orientation object of a planar rotation
Modelica.Mechanics.MultiBody.Frames.Quaternions.smallRotation smallRotation Return rotation angles valid for a small rotation
Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T from_T Return quaternions orientation object Q from transformation matrix T
Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T_inv from_T_inv Return quaternions orientation object Q from inverse transformation matrix T_inv
Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T to_T Return transformation matrix T from quaternion orientation object Q
Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T_inv to_T_inv Return inverse transformation matrix T_inv from quaternion orientation object Q

Types and constants

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

    extends Internal.QuaternionBase;

    encapsulated function equalityConstraint 
    "Return the constraint residues to express that two frames have the same quaternion orientation"

      import Modelica;
      import Modelica.Mechanics.MultiBody.Frames.Quaternions;
      extends Modelica.Icons.Function;
      input Quaternions.Orientation Q1 
      "Quaternions orientation object to rotate frame 0 into frame 1";
      input Quaternions.Orientation Q2 
      "Quaternions orientation object to rotate frame 0 into frame 2";
      output Real residue[3] 
      "The half of the rotation angles around x-, y-, and z-axis of frame 1 to rotate frame 1 into frame 2 for a small rotation (shall be zero)";
    algorithm 
      residue := [Q1[4], Q1[3], -Q1[2], -Q1[1]; -Q1[3], Q1[4], Q1[1], -Q1[2];
         Q1[2], -Q1[1], Q1[4], -Q1[3]]*Q2;
    end equalityConstraint;

  end Orientation;

  type der_Orientation = Real[4] (each unit="1/s") 
  "First time derivative of Quaternions.Orientation";


Modelica.Mechanics.MultiBody.Frames.Quaternions.orientationConstraint Modelica.Mechanics.MultiBody.Frames.Quaternions.orientationConstraint

Return residues of orientation constraints (shall be zero)

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2

Outputs

TypeNameDescription
Realresidue[1]Residue constraint (shall be zero)

Modelica definition

function orientationConstraint 
  "Return residues of orientation constraints (shall be zero)"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Real residue[1] "Residue constraint (shall be zero)";
algorithm 
  residue := {Q*Q - 1};
end orientationConstraint;

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity1 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 a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
der_Orientationder_Q Derivative of Q [1/s]

Outputs

TypeNameDescription
AngularVelocityw[3]Angular velocity resolved in frame 1 [rad/s]

Modelica definition

function angularVelocity1 
  "Compute angular velocity resolved in frame 1 from quaternion orientation object and its derivative"

  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input der_Orientation der_Q "Derivative of Q";
  output Modelica.SIunits.AngularVelocity w[3] 
    "Angular velocity resolved in frame 1";
algorithm 
  w := 2*([Q[4], -Q[3], Q[2], -Q[1]; Q[3], Q[4], -Q[1], -Q[2]; -Q[2], Q[1],
     Q[4], -Q[3]]*der_Q);
end angularVelocity1;

Modelica.Mechanics.MultiBody.Frames.Quaternions.angularVelocity2 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 a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
der_Orientationder_Q Derivative of Q [1/s]

Outputs

TypeNameDescription
AngularVelocityw[3]Angular velocity of frame 2 with respect to frame 1 resolved in frame 2 [rad/s]

Modelica definition

function angularVelocity2 
  "Compute angular velocity resolved in frame 2 from quaternions orientation object and its derivative"

  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input der_Orientation der_Q "Derivative of Q";
  output Modelica.SIunits.AngularVelocity w[3] 
    "Angular velocity of frame 2 with respect to frame 1 resolved in frame 2";
algorithm 
  w := 2*([Q[4], Q[3], -Q[2], -Q[1]; -Q[3], Q[4], Q[1], -Q[2]; Q[2], -Q[1],
     Q[4], -Q[3]]*der_Q);
end angularVelocity2;

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve1 Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve1

Transform vector from frame 2 to frame 1

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
Realv2[3] Vector in frame 2

Outputs

TypeNameDescription
Realv1[3]Vector in frame 1

Modelica definition

function resolve1 "Transform vector from frame 2 to frame 1"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input Real v2[3] "Vector in frame 2";
  output Real v1[3] "Vector in frame 1";
algorithm 
  v1 := 2*((Q[4]*Q[4] - 0.5)*v2 + (Q[1:3]*v2)*Q[1:3] + Q[4]*cross(Q[1:3],
    v2));
end resolve1;

Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve2 Modelica.Mechanics.MultiBody.Frames.Quaternions.resolve2

Transform vector from frame 1 to frame 2

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
Realv1[3] Vector in frame 1

Outputs

TypeNameDescription
Realv2[3]Vector in frame 2

Modelica definition

function resolve2 "Transform vector from frame 1 to frame 2"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input Real v1[3] "Vector in frame 1";
  output Real v2[3] "Vector in frame 2";
algorithm 
  v2 := 2*((Q[4]*Q[4] - 0.5)*v1 + (Q[1:3]*v1)*Q[1:3] - Q[4]*cross(Q[1:3],
    v1));
end resolve2;

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve1 Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve1

Transform several vectors from frame 2 to frame 1

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
Realv2[3, :] Vectors in frame 2

Outputs

TypeNameDescription
Realv1[3, size(v2, 2)]Vectors in frame 1

Modelica definition

function multipleResolve1 
  "Transform several vectors from frame 2 to frame 1"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input Real v2[3, :] "Vectors in frame 2";
  output Real v1[3, size(v2, 2)] "Vectors in frame 1";
algorithm 
  v1 := ((2*Q[4]*Q[4] - 1)*identity(3) + 2*([Q[1:3]]*transpose([Q[1:3]]) +
    Q[4]*skew(Q[1:3])))*v2;
end multipleResolve1;

Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve2 Modelica.Mechanics.MultiBody.Frames.Quaternions.multipleResolve2

Transform several vectors from frame 1 to frame 2

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2
Realv1[3, :] Vectors in frame 1

Outputs

TypeNameDescription
Realv2[3, size(v1, 2)]Vectors in frame 2

Modelica definition

function multipleResolve2 
  "Transform several vectors from frame 1 to frame 2"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  input Real v1[3, :] "Vectors in frame 1";
  output Real v2[3, size(v1, 2)] "Vectors in frame 2";
algorithm 
  v2 := ((2*Q[4]*Q[4] - 1)*identity(3) + 2*([Q[1:3]]*transpose([Q[1:3]]) -
    Q[4]*skew(Q[1:3])))*v1;
end multipleResolve2;

Modelica.Mechanics.MultiBody.Frames.Quaternions.nullRotation Modelica.Mechanics.MultiBody.Frames.Quaternions.nullRotation

Return quaternions orientation object that does not rotate a frame

Information

Extends from Modelica.Icons.Function (Icon for a function).

Outputs

TypeNameDescription
OrientationQQuaternions orientation object to rotate frame 1 into frame 2

Modelica definition

function nullRotation 
  "Return quaternions orientation object that does not rotate a frame"

  extends Modelica.Icons.Function;
  output Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
algorithm 
  Q := {0,0,0,1};
end nullRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.inverseRotation Modelica.Mechanics.MultiBody.Frames.Quaternions.inverseRotation

Return inverse quaternions orientation object

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2

Outputs

TypeNameDescription
OrientationQ_invQuaternions orientation object to rotate frame 2 into frame 1

Modelica definition

function inverseRotation 
  "Return inverse quaternions orientation object"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Quaternions.Orientation Q_inv 
    "Quaternions orientation object to rotate frame 2 into frame 1";
algorithm 
  Q_inv := {-Q[1],-Q[2],-Q[3],Q[4]};
end inverseRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.relativeRotation Modelica.Mechanics.MultiBody.Frames.Quaternions.relativeRotation

Return relative quaternions orientation object

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ1 Quaternions orientation object to rotate frame 0 into frame 1
OrientationQ2 Quaternions orientation object to rotate frame 0 into frame 2

Outputs

TypeNameDescription
OrientationQ_relQuaternions orientation object to rotate frame 1 into frame 2

Modelica definition

function relativeRotation 
  "Return relative quaternions orientation object"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q1 
    "Quaternions orientation object to rotate frame 0 into frame 1";
  input Quaternions.Orientation Q2 
    "Quaternions orientation object to rotate frame 0 into frame 2";
  output Quaternions.Orientation Q_rel 
    "Quaternions orientation object to rotate frame 1 into frame 2";
algorithm 
  Q_rel := [Q1[4], Q1[3], -Q1[2], -Q1[1]; -Q1[3], Q1[4], Q1[1], -Q1[2]; Q1[
    2], -Q1[1], Q1[4], -Q1[3]; Q1[1], Q1[2], Q1[3], Q1[4]]*Q2;
end relativeRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.absoluteRotation 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 a function).

Inputs

TypeNameDefaultDescription
OrientationQ1 Quaternions orientation object to rotate frame 0 into frame 1
OrientationQ_rel Quaternions orientation object to rotate frame 1 into frame 2

Outputs

TypeNameDescription
OrientationQ2Quaternions orientation object to rotate frame 0 into frame 2

Modelica definition

function absoluteRotation 
  "Return absolute quaternions orientation object from another absolute and a relative quaternions orientation object"

  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q1 
    "Quaternions orientation object to rotate frame 0 into frame 1";
  input Quaternions.Orientation Q_rel 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Quaternions.Orientation Q2 
    "Quaternions orientation object to rotate frame 0 into frame 2";
algorithm 
  Q2 := [Q_rel[4], Q_rel[3], -Q_rel[2], Q_rel[1]; -Q_rel[3], Q_rel[4],
    Q_rel[1], Q_rel[2]; Q_rel[2], -Q_rel[1], Q_rel[4], Q_rel[3]; -Q_rel[1],
     -Q_rel[2], -Q_rel[3], Q_rel[4]]*Q1;
end absoluteRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.planarRotation Modelica.Mechanics.MultiBody.Frames.Quaternions.planarRotation

Return quaternions orientation object of a planar rotation

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

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

Outputs

TypeNameDescription
OrientationQQuaternions orientation object to rotate frame 1 into frame 2 along axis e

Modelica definition

function planarRotation 
  "Return quaternions orientation object of a planar rotation"
  import Modelica.Math;
  extends Modelica.Icons.Function;
  input Real e[3](each final unit="1") 
    "Normalized axis of rotation (must have length=1)";
  input Modelica.SIunits.Angle angle 
    "Rotation angle to rotate frame 1 into frame 2 along axis e";
  output Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2 along axis e";
algorithm 
  Q := vector([e*Math.sin(angle/2); Math.cos(angle/2)]);
end planarRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.smallRotation Modelica.Mechanics.MultiBody.Frames.Quaternions.smallRotation

Return rotation angles valid for a small rotation

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2

Outputs

TypeNameDescription
Anglephi[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 definition

function smallRotation 
  "Return rotation angles valid for a small rotation"
  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Modelica.SIunits.Angle 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";
algorithm 
  phi := 2*{Q[1],Q[2],Q[3]};
end smallRotation;

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T

Return quaternions orientation object Q from transformation matrix T

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

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

Outputs

TypeNameDescription
OrientationQQuaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)

Modelica definition

function from_T 
  "Return quaternions orientation object Q from transformation matrix T"

  extends Modelica.Icons.Function;
  input Real T[3, 3] 
    "Transformation matrix to transform vector from frame 1 to frame 2 (v2=T*v1)";
  input Quaternions.Orientation Q_guess=nullRotation() 
    "Guess value for Q (there are 2 solutions; the one close to Q_guess is used";
  output Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)";
protected 
  Real paux;
  Real paux4;
  Real c1;
  Real c2;
  Real c3;
  Real c4;
  constant Real p4limit=0.1;
  constant Real c4limit=4*p4limit*p4limit;
algorithm 
  /*
   Note, for quaternions, Q and -Q have the same transformation matrix.
   Calculation of quaternions from transformation matrix T:
   It is guaranteed that c1>=0, c2>=0, c3>=0, c4>=0 and
   that not all of them can be zero at the same time
   (e.g. if 3 of them are zero, the 4th variable is 1).
   Since the sqrt(..) has to be performed on one of these variables,
   it is applied on a variable which is far enough from zero.
   This guarantees that the sqrt(..) is never taken near zero
   and therefore the derivative of sqrt(..) can never be infinity.
   There is an ambiguity for quaternions, since Q and -Q
   lead to the same transformation matrix. The ambiguity
   is resolved here by selecting the Q that is closer to
   the input argument Q_guess.
*/
  c1 := 1 + T[1, 1] - T[2, 2] - T[3, 3];
  c2 := 1 + T[2, 2] - T[1, 1] - T[3, 3];
  c3 := 1 + T[3, 3] - T[1, 1] - T[2, 2];
  c4 := 1 + T[1, 1] + T[2, 2] + T[3, 3];

  if c4 > c4limit or (c4 > c1 and c4 > c2 and c4 > c3) then
    paux := sqrt(c4)/2;
    paux4 := 4*paux;
    Q := {(T[2, 3] - T[3, 2])/paux4,(T[3, 1] - T[1, 3])/paux4,(T[1, 2] - T[
      2, 1])/paux4,paux};

  elseif c1 > c2 and c1 > c3 and c1 > c4 then
    paux := sqrt(c1)/2;
    paux4 := 4*paux;
    Q := {paux,(T[1, 2] + T[2, 1])/paux4,(T[1, 3] + T[3, 1])/paux4,(T[2, 3]
       - T[3, 2])/paux4};

  elseif c2 > c1 and c2 > c3 and c2 > c4 then
    paux := sqrt(c2)/2;
    paux4 := 4*paux;
    Q := {(T[1, 2] + T[2, 1])/paux4,paux,(T[2, 3] + T[3, 2])/paux4,(T[3, 1]
       - T[1, 3])/paux4};

  else
    paux := sqrt(c3)/2;
    paux4 := 4*paux;
    Q := {(T[1, 3] + T[3, 1])/paux4,(T[2, 3] + T[3, 2])/paux4,paux,(T[1, 2]
       - T[2, 1])/paux4};
  end if;

  if Q*Q_guess < 0 then
    Q := -Q;
  end if;
end from_T;

Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T_inv Modelica.Mechanics.MultiBody.Frames.Quaternions.from_T_inv

Return quaternions orientation object Q from inverse transformation matrix T_inv

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

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

Outputs

TypeNameDescription
OrientationQQuaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)

Modelica definition

function from_T_inv 
  "Return quaternions orientation object Q from inverse transformation matrix T_inv"

  extends Modelica.Icons.Function;
  input Real T_inv[3, 3] 
    "Inverse transformation matrix to transform vector from frame 2 to frame 1 (v1=T_inv*v2)";
  input Quaternions.Orientation Q_guess=nullRotation() 
    "Guess value for output Q (there are 2 solutions; the one closer to Q_guess is used";
  output Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2 (Q and -Q have same transformation matrix)";
algorithm 
  Q := from_T(transpose(T_inv), Q_guess);
end from_T_inv;

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T

Return transformation matrix T from quaternion orientation object Q

Information

Extends from Modelica.Icons.Function (Icon for a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2

Outputs

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

Modelica definition

function to_T 
  "Return transformation matrix T from quaternion orientation object Q"

  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Real T[3, 3] 
    "Transformation matrix to transform vector from frame 1 to frame 2 (v2=T*v1)";
algorithm 
  /*
  T := (2*Q[4]*Q[4] - 1)*identity(3) + 2*([Q[1:3]]*transpose([Q[1:3]]) - Q[4]*
    skew(Q[1:3]));
*/
  T := [2*(Q[1]*Q[1] + Q[4]*Q[4]) - 1, 2*(Q[1]*Q[2] + Q[3]*Q[4]), 2*(Q[1]*Q[
    3] - Q[2]*Q[4]); 2*(Q[2]*Q[1] - Q[3]*Q[4]), 2*(Q[2]*Q[2] + Q[4]*Q[4])
     - 1, 2*(Q[2]*Q[3] + Q[1]*Q[4]); 2*(Q[3]*Q[1] + Q[2]*Q[4]), 2*(Q[3]*Q[2]
     - Q[1]*Q[4]), 2*(Q[3]*Q[3] + Q[4]*Q[4]) - 1];
end to_T;

Modelica.Mechanics.MultiBody.Frames.Quaternions.to_T_inv 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 a function).

Inputs

TypeNameDefaultDescription
OrientationQ Quaternions orientation object to rotate frame 1 into frame 2

Outputs

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

Modelica definition

function to_T_inv 
  "Return inverse transformation matrix T_inv from quaternion orientation object Q"

  extends Modelica.Icons.Function;
  input Quaternions.Orientation Q 
    "Quaternions orientation object to rotate frame 1 into frame 2";
  output Real T_inv[3, 3] 
    "Transformation matrix to transform vector from frame 2 to frame 1 (v1=T*v2)";
algorithm 
  /*
  T_inv := (2*Q[4]*Q[4] - 1)*identity(3) + 2*([Q[1:3]]*transpose([Q[1:3]]) + Q[
    4]*skew(Q[1:3]));
*/
  T_inv := [2*(Q[1]*Q[1] + Q[4]*Q[4]) - 1, 2*(Q[2]*Q[1] - Q[3]*Q[4]), 2*(Q[
    3]*Q[1] + Q[2]*Q[4]); 2*(Q[1]*Q[2] + Q[3]*Q[4]), 2*(Q[2]*Q[2] + Q[4]*Q[
    4]) - 1, 2*(Q[3]*Q[2] - Q[1]*Q[4]); 2*(Q[1]*Q[3] - Q[2]*Q[4]), 2*(Q[2]*
    Q[3] + Q[1]*Q[4]), 2*(Q[3]*Q[3] + Q[4]*Q[4]) - 1];
end to_T_inv;

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