Quaternions

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

Package Contents

	Orientation Orientation type defining rotation from a frame 1 into a frame 2 with quaternions {p1,p2,p3,p0}
	der_Orientation (1/s) 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

Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

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.