User Manual 4.0 Kinematics - Historique des versions

Admin : /* Differentiate a quaternion */

2018-03-20T16:34:26Z

Differentiate a quaternion

← Version précédente		Version du 20 mars 2018 à 16:34
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	As a time-dependent quaternion can be represented by the <code>OrientationFunction</code> interface, the derivative of the quaternion can be computed using the <code>derivative()</code> method. As this method is implemented by the user, he can choose to use a numerical differentiation method or to compute the exact derivative, when possible.		As a time-dependent quaternion can be represented by the <code>OrientationFunction</code> interface, the derivative of the quaternion can be computed using the <code>derivative()</code> method. As this method is implemented by the user, he can choose to use a numerical differentiation method or to compute the exact derivative, when possible.

	The derivative of a quaternion can be also computed using the kinematics equation: <math>\dot{Q}_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.		The derivative of a quaternion can be also computed using the kinematics equation: <math>\dot{Q}_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.<br>
	This equation is implemented by the following method in the kinematics toolkit:		This equation is implemented by the following method in the kinematics toolkit:

Admin le 20 février 2018 à 16:40

2018-02-20T16:40:51Z

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			__NOTOC__

	== Introduction ==		== Introduction ==
	=== Scope ===		=== Scope ===

Admin : /* Javadoc */

2018-02-20T16:00:56Z

Javadoc

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	The kinematics objects are available in the package fr.cnes.sirius.patrius.attitudes.kinematics.		The kinematics objects are available in the package fr.cnes.sirius.patrius.attitudes.kinematics.

	\|=~~(% colspan~~="3" %)Library~~\|=(% colspan~~="6" %)Javadoc		{\| class="wikitable"
	\|~~(% colspan="3" %)~~Orekit\|~~(% colspan="6" %)~~[{{~~CurrentPathJavaDoc~~}}/fr/cnes/sirius/patrius/attitudes/kinematics/package-summary.html Package fr.cnes.sirius.patrius.attitudes.kinematics]		\|-
			! scope="col"\| Library
			! scope="col"\| Javadoc
			\|-
			\|Orekit
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/attitudes/kinematics/package-summary.html Package fr.cnes.sirius.patrius.attitudes.kinematics]
			\|}

	=== Links ===		=== Links ===

Admin : /* Package Overview */

2018-02-20T15:57:42Z

Package Overview

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	=== Package Overview ===		=== Package Overview ===
	The following diagram represents the main classes of the <~~syntaxhighlight lang="java"~~>fr.cnes.sirius.patrius.attitudes.kinematics</~~syntaxhighlight~~> ~~package:~~		The following diagram represents the main classes of the package :
			<code>fr.cnes.sirius.patrius.attitudes.kinematics</code>

	[[File:kinematics.~~bmp~~\|center]]		[[File:kinematics.png\|center]]

	== Features Description ==		== Features Description ==

Admin : /* Classes */

2018-02-20T15:56:36Z

Classes

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	=== Classes ===		=== Classes ===
	\|=~~(% colspan~~="3" %)Class~~\|=(% colspan~~="6" %)Summary~~\|=(% colspan~~="1" %)Javadoc
	\|~~(% colspan="3" %)~~'''AbstractOrientationFunction'''\|~~(% colspan="6" %)~~This class represents an orientation function.\|[{{~~CurrentPathJavaDoc~~}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractOrientationFunction.html ...]		{\| class="wikitable"
	\|~~(% colspan="3" %)~~'''AbstractVector3DFunction'''\|~~(% colspan="6" %)~~This class represents a spin function, or a spin n-th derivative function.\|[{{~~CurrentPathJavaDoc~~}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractVector3DFunction.html ...]		\|-
			! scope="col"\| Class
			! scope="col"\| Summary
			! scope="col"\| Javadoc
			\|-
			\|'''AbstractOrientationFunction'''
			\|This class represents an orientation function.
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractOrientationFunction.html ...]
			\|-
			\|'''AbstractVector3DFunction'''
			\|This class represents a spin function, or a spin n-th derivative function.
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractVector3DFunction.html ...]
			\|}

	[[Category:User_Manual_4.0_Attitude]]		[[Category:User_Manual_4.0_Attitude]]

Admin : /* Interfaces */

2018-02-20T15:55:06Z

Interfaces

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	== Contents ==		== Contents ==
	=== Interfaces ===		=== Interfaces ===
	\|=~~(% colspan~~="3" %)Interface~~\|=(% colspan~~="6" %)Summary~~\|=(% colspan~~="1" %)Javadoc		{\| class="wikitable"
	\|~~(% colspan="3" %)~~'''OrientationFunction'''\|~~(% colspan="6" %)~~This interface represents a generic univariate orientation function.\|[{{~~CurrentPathJavaDoc~~}}/fr/cnes/sirius/patrius/attitudes/kinematics/OrientationFunction.html ...]		\|-
	\|~~(% colspan="3" %)~~'''Vector3DFunction'''\|~~(% colspan="6" %)~~This interface represents a generic univariate 3-D vector function.\|[{{~~CurrentPathJavaDoc~~}}/fr/cnes/sirius/patrius/attitudes/kinematics/Vector3DFunction.html ...]		! scope="col"\| Interface
			! scope="col"\| Summary
			! scope="col"\| Javadoc
			\|-
			\|'''OrientationFunction'''
			\|This interface represents a generic univariate orientation function.
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/attitudes/kinematics/OrientationFunction.html ...]
			\|-
			\|'''Vector3DFunction'''
			\|This interface represents a generic univariate 3-D vector function.
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/attitudes/kinematics/Vector3DFunction.html ...]
			\|}

	=== Classes ===		=== Classes ===

Admin le 16 février 2018 à 11:01

2018-02-16T11:01:36Z

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	\|(% colspan="3" %)'''AbstractOrientationFunction'''\|(% colspan="6" %)This class represents an orientation function.\|[{{CurrentPathJavaDoc}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractOrientationFunction.html ...]		\|(% colspan="3" %)'''AbstractOrientationFunction'''\|(% colspan="6" %)This class represents an orientation function.\|[{{CurrentPathJavaDoc}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractOrientationFunction.html ...]
	\|(% colspan="3" %)'''AbstractVector3DFunction'''\|(% colspan="6" %)This class represents a spin function, or a spin n-th derivative function.\|[{{CurrentPathJavaDoc}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractVector3DFunction.html ...]		\|(% colspan="3" %)'''AbstractVector3DFunction'''\|(% colspan="6" %)This class represents a spin function, or a spin n-th derivative function.\|[{{CurrentPathJavaDoc}}/fr/cnes/sirius/patrius/attitudes/kinematics/AbstractVector3DFunction.html ...]

			[[Category:User_Manual_4.0_Attitude]]

Admin : Admin a déplacé la page Kinematics vers User Manual 4.0 Kinematics sans laisser de redirection

2018-02-16T11:01:15Z

Admin a déplacé la page Kinematics vers User Manual 4.0 Kinematics sans laisser de redirection

← Version précédente	Version du 16 février 2018 à 11:01
(Aucune différence)

Admin le 16 février 2018 à 10:59

2018-02-16T10:59:03Z

Admin le 16 février 2018 à 10:52

2018-02-16T10:52:29Z

← Version précédente		Version du 16 février 2018 à 10:52
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	=== Package Overview ===		=== Package Overview ===
	The following diagram represents the main classes of the <syntaxhighlight lang="java">fr.cnes.sirius.patrius.attitudes		The following diagram represents the main classes of the <syntaxhighlight lang="java">fr.cnes.sirius.patrius.attitudes.kinematics</syntaxhighlight> package:
	.kinematics</syntaxhighlight> package:
	[[File:kinematics.bmp\|center]]		[[File:kinematics.bmp\|center]]

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	* The quaternion differentiation’s formula <math>\.Q_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.		* The quaternion differentiation’s formula <math>\.Q_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.
	This formula connects time derivative of the quaternion q(t) with the vector of angular velocity W(t).		This formula connects time derivative of the quaternion q(t) with the vector of angular velocity W(t).

	* When the quaternion is represented by a set of differentiable functions (Fourier series or polynomials), a direct analytic computation can be performed to get its derivative;		* When the quaternion is represented by a set of differentiable functions (Fourier series or polynomials), a direct analytic computation can be performed to get its derivative;
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	The derivative of a quaternion can be also computed using the kinematics equation: <math>\.Q_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.		The derivative of a quaternion can be also computed using the kinematics equation: <math>\.Q_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.
	This equation is implemented by the following method in the kinematics toolkit:		This equation is implemented by the following method in the kinematics toolkit:
	<~~code~~>public static Quaternion differentiateQuaternion(final Quaternion q, final Vector3D spin)</~~code~~>
			<syntaxhighlight lang="java">public static Quaternion differentiateQuaternion(final Quaternion q, final Vector3D spin)</syntaxhighlight>

	=== Integrate a quaternion ===		=== Integrate a quaternion ===
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	* Estimation: two rotations corresponding to two different dates are used to compute the spin (the velocity is supposed to be constant during the time interval between the two rotations).		* Estimation: two rotations corresponding to two different dates are used to compute the spin (the velocity is supposed to be constant during the time interval between the two rotations).
	This function is implemented by the following method in the kinematics toolkit:		This function is implemented by the following method in the kinematics toolkit:
	<~~code~~>public static Vector3D estimateSpin(final Rotation start, final Rotation end, final double dt)</~~code~~>.
			<syntaxhighlight lang="java">public static Vector3D estimateSpin(final Rotation start, final Rotation end, final double dt)</syntaxhighlight>.

	* <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\.Q_{S/R}</math>, when knowing the derivative of the quaternion.		* <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\.Q_{S/R}</math>, when knowing the derivative of the quaternion.
	This equation is implemented by the following kinematics toolkit method:		This equation is implemented by the following kinematics toolkit method:
	<~~code~~>public static Vector3D computeSpin(final Quaternion q, final Quaternion qd)</~~code~~>.
			<syntaxhighlight lang="java">public static Vector3D computeSpin(final Quaternion q, final Quaternion qd)</syntaxhighlight>.

	* Using the derivative of the Euler or Cardan angles:		* Using the derivative of the Euler or Cardan angles:
	** Euler (not to be used when the angles are small):		** Euler (not to be used when the angles are small):
	<math>\Omega_{S/R}^{[S]} = \left[ \begin{array}{c} \dot{\psi}\sin(\theta)\sin(\phi) + \dot{\theta}\cos(\phi) \\ \dot{\psi}\sin(\theta)\cos(\phi)- \dot{\theta}\sin(\phi) \\ \dot{\psi}\cos(\theta) + \dot{\phi} \end{array} \right]</math>		<math>\Omega_{S/R}^{[S]} = \left[ \begin{array}{c} \dot{\psi}\sin(\theta)\sin(\phi) + \dot{\theta}\cos(\phi) \\ \dot{\psi}\sin(\theta)\cos(\phi)- \dot{\theta}\sin(\phi) \\ \dot{\psi}\cos(\theta) + \dot{\phi} \end{array} \right]</math>
	** Cardan (only to be used when the angles are small):		** Cardan (only to be used when the angles are small):
	<math>\Omega_{S/R}^{[S]} = \left[ \begin{array}{c}-\dot{\psi}\sin(\theta)+\dot{\phi} \\ \dot{\psi}\cos(\theta)\sin(\phi) + \dot{\theta}\cos(\phi) \\ \dot{\psi}\cos(\theta)\cos(\phi) - \dot{\theta}\sin(\phi) \end{array} \right]</math>		<math>\Omega_{S/R}^{[S]} = \left[ \begin{array}{c}-\dot{\psi}\sin(\theta)+\dot{\phi} \\ \dot{\psi}\cos(\theta)\sin(\phi) + \dot{\theta}\cos(\phi) \\ \dot{\psi}\cos(\theta)\cos(\phi) - \dot{\theta}\sin(\phi) \end{array} \right]</math>

	These formula are implemented by the following kinematics toolkit method:		These formula are implemented by the following kinematics toolkit method:
	<~~code~~>public static Vector3D computeSpin(final double[] ang, final double[] angd, final RotationOrder order)</~~code~~>.
			<syntaxhighlight lang="java">public static Vector3D computeSpin(final double[] ang, final double[] angd, final RotationOrder order)</syntaxhighlight>.

	* Analytic computation: if the spin is represented by a function, its instantaneous value must be obtained by direct analytic computation (the <code>IVector3DFunction</code> classes have been created for this purpose)		* Analytic computation: if the spin is represented by a function, its instantaneous value must be obtained by direct analytic computation (the <code>IVector3DFunction</code> classes have been created for this purpose)

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	A quaternion can be differentiated using various methods:		A quaternion can be differentiated using various methods:

	* The quaternion differentiation’s formula <math>\~~.Q_~~{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.		* The quaternion differentiation’s formula <math>\dot{Q}_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.
	This formula connects time derivative of the quaternion q(t) with the vector of angular velocity W(t).		This formula connects time derivative of the quaternion q(t) with the vector of angular velocity W(t).

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	* return the function representing the derivative of the quaternion components, by the <code>derivative()</code> method. A numerical differentiation method is used to compute the derivatives, nevertheless the classes inheriting the <code>AbstractOrientationFunction</code> can override this method and return the analytical function representing the derivatives, when possible.		* return the function representing the derivative of the quaternion components, by the <code>derivative()</code> method. A numerical differentiation method is used to compute the derivatives, nevertheless the classes inheriting the <code>AbstractOrientationFunction</code> can override this method and return the analytical function representing the derivatives, when possible.

	* compute the function representing the spin using the <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\~~.Q_~~{S/R}</math> formula, by the <code>computeSpinFunction()</code> method. The derivatives of the quaternion are computed thanks to the <code>derivative()</code> method.		* compute the function representing the spin using the <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\dot{Q}_{S/R}</math> formula, by the <code>computeSpinFunction()</code> method. The derivatives of the quaternion are computed thanks to the <code>derivative()</code> method.

	* estimate the function representing the spin using two rotations corresponding to two different dates (the velocity is supposed to be constant during the time interval between the two rotations), by the <code>estimateRateFunction(dt)</code> method.		* estimate the function representing the spin using two rotations corresponding to two different dates (the velocity is supposed to be constant during the time interval between the two rotations), by the <code>estimateRateFunction(dt)</code> method.
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	As a time-dependent quaternion can be represented by the <code>OrientationFunction</code> interface, the derivative of the quaternion can be computed using the <code>derivative()</code> method. As this method is implemented by the user, he can choose to use a numerical differentiation method or to compute the exact derivative, when possible.		As a time-dependent quaternion can be represented by the <code>OrientationFunction</code> interface, the derivative of the quaternion can be computed using the <code>derivative()</code> method. As this method is implemented by the user, he can choose to use a numerical differentiation method or to compute the exact derivative, when possible.

	The derivative of a quaternion can be also computed using the kinematics equation: <math>\~~.Q_~~{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.		The derivative of a quaternion can be also computed using the kinematics equation: <math>\dot{Q}_{S/R} = \frac{1}{2}Q_{S/R}\Omega_{S/R}^{[S]}</math>.
	This equation is implemented by the following method in the kinematics toolkit:		This equation is implemented by the following method in the kinematics toolkit:

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	<syntaxhighlight lang="java">public static Vector3D estimateSpin(final Rotation start, final Rotation end, final double dt)</syntaxhighlight>.		<syntaxhighlight lang="java">public static Vector3D estimateSpin(final Rotation start, final Rotation end, final double dt)</syntaxhighlight>.

	* <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\~~.Q_~~{S/R}</math>, when knowing the derivative of the quaternion.		* <math>\Omega_{S/R}^{[S]} = 2Q_{S/R}\dot{Q}_{S/R}</math>, when knowing the derivative of the quaternion.
	This equation is implemented by the following kinematics toolkit method:		This equation is implemented by the following kinematics toolkit method:

User Manual 4.0 Kinematics - Historique des versions

Admin : /* Differentiate a quaternion */

Admin le 20 février 2018 à 16:40

Admin : /* Javadoc */

Admin : /* Package Overview */

Admin : /* Classes */

Admin : /* Interfaces */