Quaternion estimator algorithm - Revision history

OAbot: Open access bot: hdl updated in citation with #oabot.

2024-07-22T03:24:45Z

Open access bot: hdl updated in citation with #oabot.

← Previous revision		Revision as of 03:24, 22 July 2024
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	* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of Guidance and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.\|doi=10.2514/3.19717\|bibcode=1981JGCD....4...70S}}		* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of Guidance and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.\|doi=10.2514/3.19717\|bibcode=1981JGCD....4...70S}}
	* {{cite journal\|title=Fast linear quaternion attitude estimator using vector observations\|year=2017\|journal=IEEE Transactions on Automation Science and Engineering\|volume=15\|pages=307–319\|issue=1\|publisher=IEEE\|last1=Wu\|first1=Jin\|last2=Zhou\|first2=Zebo\|last3=Gao\|first3=Bin\|last4=Li\|first4=Rui\|last5=Cheng\|first5=Yuhua\|last6=Fourati\|first6=Hassen\|doi=10.1109/TASE.2017.2699221\|s2cid=3455346\|url=https://hal.inria.fr/hal-01513263/file/bare_jrnl.pdf}}		* {{cite journal\|title=Fast linear quaternion attitude estimator using vector observations\|year=2017\|journal=IEEE Transactions on Automation Science and Engineering\|volume=15\|pages=307–319\|issue=1\|publisher=IEEE\|last1=Wu\|first1=Jin\|last2=Zhou\|first2=Zebo\|last3=Gao\|first3=Bin\|last4=Li\|first4=Rui\|last5=Cheng\|first5=Yuhua\|last6=Fourati\|first6=Hassen\|doi=10.1109/TASE.2017.2699221\|s2cid=3455346\|url=https://hal.inria.fr/hal-01513263/file/bare_jrnl.pdf}}
	* {{cite journal\|title=A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements\|year=2008\|journal=IEEE Transactions on Instrumentation and Measurement\|volume=57\|pages=638–650\|issue=3\|publisher=IEEE\|last1=Yun\|first1=Xiaoping\|last2=Bachmann\|first2=Eric R\|last3=McGhee\|first3=Robert B\|doi=10.1109/TIM.2007.911646\|s2cid=15571138}}		* {{cite journal\|title=A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements\|year=2008\|journal=IEEE Transactions on Instrumentation and Measurement\|volume=57\|pages=638–650\|issue=3\|publisher=IEEE\|last1=Yun\|first1=Xiaoping\|last2=Bachmann\|first2=Eric R\|last3=McGhee\|first3=Robert B\|doi=10.1109/TIM.2007.911646\|s2cid=15571138\|hdl=10945/46081\|hdl-access=free}}

	== External links ==		== External links ==

97.113.187.176: Adding the word "conjugate" to clarify for people who are not as familiar with the * notation in the domain of conjugates.

2023-08-11T21:49:36Z

Adding the word "conjugate" to clarify for people who are not as familiar with the * notation in the domain of conjugates.

← Previous revision		Revision as of 21:49, 11 August 2023
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	</math>		</math>

	that gives the quaternion representation of the optimal rotation as		that gives the conjugate quaternion representation of the optimal rotation as

	:<math> \mathbf{q}^* = \frac{1}{\sqrt{\gamma^2 + \left\| \mathbf{x} \right\|^2}} (\mathbf{x}, \gamma)^\top </math>		:<math> \mathbf{q}^* = \frac{1}{\sqrt{\gamma^2 + \left\| \mathbf{x} \right\|^2}} (\mathbf{x}, \gamma)^\top </math>

SunDawn: Importing Wikidata short description: "Algorithm to solve Wahba's problem"

2022-07-02T04:36:42Z

Importing Wikidata short description: "Algorithm to solve Wahba's problem"

← Previous revision		Revision as of 04:36, 2 July 2022
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			{{Short description\|Algorithm to solve Wahba's problem}}
	The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a [[quadratic form]], using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.		The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a [[quadratic form]], using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.

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2022-03-05T03:19:34Z

Alter: journal. Add: url, s2cid, bibcode, doi. | Use this bot. Report bugs. | Suggested by Headbomb | Linked from Wikipedia:WikiProject_Academic_Journals/Journals_cited_by_Wikipedia/Sandbox | #UCB_webform_linked 300/387

← Previous revision		Revision as of 03:19, 5 March 2022
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	== Sources ==		== Sources ==
	* {{cite journal\|title=Survey of nonlinear attitude estimation methods\|year=2007\|journal=Journal of ~~guidance~~, ~~control~~, and ~~dynamics~~\|volume=30\|pages=12–28\|issue=1\|last1=Crassidis\|first1=John L\|last2=Markley\|first2=F Landis\|last3=Cheng\|first3=Yang}}		* {{cite journal\|title=Survey of nonlinear attitude estimation methods\|year=2007\|journal=Journal of Guidance, Control, and Dynamics\|volume=30\|pages=12–28\|issue=1\|last1=Crassidis\|first1=John L\|last2=Markley\|first2=F Landis\|last3=Cheng\|first3=Yang\|doi=10.2514/1.22452\|bibcode=2007JGCD...30...12C}}
	* {{cite journal\|title=Quaternion attitude estimation using vector observations\|year=2000\|journal=The Journal of the Astronautical Sciences\|volume=48\|pages=359–380\|issue=2\|publisher=Springer\|last1=Markley\|first1=F Landis\|last2=Mortari\|first2=Daniele}}		* {{cite journal\|title=Quaternion attitude estimation using vector observations\|year=2000\|journal=The Journal of the Astronautical Sciences\|volume=48\|pages=359–380\|issue=2\|publisher=Springer\|last1=Markley\|first1=F Landis\|last2=Mortari\|first2=Daniele\|doi=10.1007/BF03546284\|bibcode=2000JAnSc..48..359M}}
	* {{cite journal\|title=Attitude-determination filtering via extended quaternion estimation\|year=2000\|journal=Journal of Guidance, Control, and Dynamics\|volume=23\|pages=206–214\|issue=2\|last1=Psiaki\|first1=Mark L}}		* {{cite journal\|title=Attitude-determination filtering via extended quaternion estimation\|year=2000\|journal=Journal of Guidance, Control, and Dynamics\|volume=23\|pages=206–214\|issue=2\|last1=Psiaki\|first1=Mark L\|doi=10.2514/2.4540\|bibcode=2000JGCD...23..206P}}
	* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of ~~guidance~~ and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.}}		* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of Guidance and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.\|doi=10.2514/3.19717\|bibcode=1981JGCD....4...70S}}
	* {{cite journal\|title=Fast linear quaternion attitude estimator using vector observations\|year=2017\|journal=IEEE Transactions on Automation Science and Engineering\|volume=15\|pages=307–319\|issue=1\|publisher=IEEE\|last1=Wu\|first1=Jin\|last2=Zhou\|first2=Zebo\|last3=Gao\|first3=Bin\|last4=Li\|first4=Rui\|last5=Cheng\|first5=Yuhua\|last6=Fourati\|first6=Hassen}}		* {{cite journal\|title=Fast linear quaternion attitude estimator using vector observations\|year=2017\|journal=IEEE Transactions on Automation Science and Engineering\|volume=15\|pages=307–319\|issue=1\|publisher=IEEE\|last1=Wu\|first1=Jin\|last2=Zhou\|first2=Zebo\|last3=Gao\|first3=Bin\|last4=Li\|first4=Rui\|last5=Cheng\|first5=Yuhua\|last6=Fourati\|first6=Hassen\|doi=10.1109/TASE.2017.2699221\|s2cid=3455346\|url=https://hal.inria.fr/hal-01513263/file/bare_jrnl.pdf}}
	* {{cite journal\|title=A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements\|year=2008\|journal=IEEE Transactions on ~~instrumentation~~ and ~~measurement~~\|volume=57\|pages=638–650\|issue=3\|publisher=IEEE\|last1=Yun\|first1=Xiaoping\|last2=Bachmann\|first2=Eric R\|last3=McGhee\|first3=Robert B}}		* {{cite journal\|title=A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements\|year=2008\|journal=IEEE Transactions on Instrumentation and Measurement\|volume=57\|pages=638–650\|issue=3\|publisher=IEEE\|last1=Yun\|first1=Xiaoping\|last2=Bachmann\|first2=Eric R\|last3=McGhee\|first3=Robert B\|doi=10.1109/TIM.2007.911646\|s2cid=15571138}}

	== External links ==		== External links ==

Tino: Rename vector Q to v for consistency (to have all vectors denoted with lowercase letters, and all matrices denoted with uppercase)

2022-02-28T21:34:19Z

Rename vector Q to v for consistency (to have all vectors denoted with lowercase letters, and all matrices denoted with uppercase)

← Previous revision		Revision as of 21:34, 28 February 2022
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	where <math>\mathbf{B} = \textstyle \sum_i a_i \mathbf{w}_i \mathbf{v}_i^\top</math> is known as the attitude profile matrix.		where <math>\mathbf{B} = \textstyle \sum_i a_i \mathbf{w}_i \mathbf{v}_i^\top</math> is known as the attitude profile matrix.

	In order to reduce the number of variables, the problem can be reformulated by parametrising the rotation as a unit [[quaternion]] <math>\mathbf{q} = \left( ~~Q_1~~, ~~Q_2~~, ~~Q_3~~, q\right)</math> with vector part <math>\mathbf{Q} = \left( ~~Q_1~~, ~~Q_2~~, ~~Q_3~~ \right)</math> and scalar part <math>q</math>, representing the rotation of angle <math>\theta = 2 \cos^{-1} q</math> around an axis whose direction is described by the vector <math>\textstyle \frac{1}{\sin \frac{\theta}{2}} \mathbf{Q}</math>, subject to the unity constraint <math>\mathbf{q}^\top \mathbf{q} = 1</math>. It is now possible to express <math>\mathbf{A}</math> in terms of the quaternion parametrisation as		In order to reduce the number of variables, the problem can be reformulated by parametrising the rotation as a unit [[quaternion]] <math>\mathbf{q} = \left( v_1, v_2, v_3, q\right)</math> with vector part <math>\mathbf{v} = \left( v_1, v_2, v_3 \right)</math> and scalar part <math>q</math>, representing the rotation of angle <math>\theta = 2 \cos^{-1} q</math> around an axis whose direction is described by the vector <math>\textstyle \frac{1}{\sin \frac{\theta}{2}} \mathbf{v}</math>, subject to the unity constraint <math>\mathbf{q}^\top \mathbf{q} = 1</math>. It is now possible to express <math>\mathbf{A}</math> in terms of the quaternion parametrisation as

	:<math> \mathbf{A} = \left( q^2 - \mathbf{Q} \cdot \mathbf{Q} \right) \mathbf{I} + 2 \mathbf{Q}\mathbf{Q}^\top + 2 q \mathbf{Q}_\times </math>		:<math> \mathbf{A} = \left( q^2 - \mathbf{v} \cdot \mathbf{v} \right) \mathbf{I} + 2 \mathbf{v}\mathbf{v}^\top + 2 q \mathbf{V}_\times </math>

	where <math>\mathbf{Q}_\times</math> is the [[skew-symmetric matrix]]		where <math>\mathbf{V}_\times</math> is the [[skew-symmetric matrix]]

	:<math> \mathbf{Q}_\times =		:<math> \mathbf{V}_\times =
	\begin{pmatrix}		\begin{pmatrix}
	0 & ~~Q_3~~ & -~~Q_2~~ \\		0 & v_3 & -v_2 \\
	-~~Q_3~~ & 0 & ~~Q_1~~ \\		-v_3 & 0 & v_1 \\
	~~Q_2~~ & -~~Q_1~~ & 0 \\		v_2 & -v_1 & 0 \\
	\end{pmatrix}		\end{pmatrix}
	</math>.		</math>.
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	</math>		</math>

	where <math>\mathbf{y} = \textstyle \frac{1}{q} \mathbf{Q}</math> is the [[Rodrigues' rotation formula\|Rodrigues vector]]. Substituting <math>\mathbf{y}</math> in the second equation with the first, it is possible to derive an expression of the [[characteristic polynomial\|characteristic equation]]		where <math>\mathbf{y} = \textstyle \frac{1}{q} \mathbf{v}</math> is the [[Rodrigues' rotation formula\|Rodrigues vector]]. Substituting <math>\mathbf{y}</math> in the second equation with the first, it is possible to derive an expression of the [[characteristic polynomial\|characteristic equation]]

	:<math> \lambda = \sigma + \mathbf{z}^\top \left( (\lambda + \sigma) \mathbf{I} - \mathbf{S} \right)^{-1} \mathbf{z} </math>.		:<math> \lambda = \sigma + \mathbf{z}^\top \left( (\lambda + \sigma) \mathbf{I} - \mathbf{S} \right)^{-1} \mathbf{z} </math>.

Tino: fix typo: lambda is the (scalar valued) maximum value of the loss, not the (matrix valued) argument of the maximum

2022-02-28T21:24:19Z

fix typo: lambda is the (scalar valued) maximum value of the loss, not the (matrix valued) argument of the maximum

← Previous revision		Revision as of 21:24, 28 February 2022
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	:<math> \lambda = \sigma + \mathbf{z}^\top \left( (\lambda + \sigma) \mathbf{I} - \mathbf{S} \right)^{-1} \mathbf{z} </math>.		:<math> \lambda = \sigma + \mathbf{z}^\top \left( (\lambda + \sigma) \mathbf{I} - \mathbf{S} \right)^{-1} \mathbf{z} </math>.

	Since <math>\lambda_{\text{max}} = ~~\arg~~\max g\left(\mathbf{A}\right)</math>, it follows that <math>\lambda_{\text{max}} = 1 - ~~\arg~~\min l\left(\mathbf{A}\right)</math> and therefore <math>\lambda_{\text{max}} \approx 1</math> for an optimal solution (when the loss <math>l</math> is small). This permits to construct the optimal quaternion <math>\mathbf{q}^*</math> by replacing <math>\lambda_{\text{max}}</math> in the Rodrigues vector <math>\mathbf{y}</math>		Since <math>\lambda_{\text{max}} = \max g\left(\mathbf{A}\right)</math>, it follows that <math>\lambda_{\text{max}} = 1 - \min l\left(\mathbf{A}\right)</math> and therefore <math>\lambda_{\text{max}} \approx 1</math> for an optimal solution (when the loss <math>l</math> is small). This permits to construct the optimal quaternion <math>\mathbf{q}^*</math> by replacing <math>\lambda_{\text{max}}</math> in the Rodrigues vector <math>\mathbf{y}</math>

	:<math> \mathbf{q}^* = \frac{1}{\sqrt{1 + \left\| \mathbf{y}_{\lambda_{\text{max}}} \right\|^2}} (\mathbf{y}, 1)^\top </math>.		:<math> \mathbf{q}^* = \frac{1}{\sqrt{1 + \left\| \mathbf{y}_{\lambda_{\text{max}}} \right\|^2}} (\mathbf{y}, 1)^\top </math>.

Tino: fix bilinear form -> quadratic form, brief comparison with other methods, more applications

2022-02-27T21:04:53Z

fix bilinear form -> quadratic form, brief comparison with other methods, more applications

← Previous revision		Revision as of 21:04, 27 February 2022
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		⚫	The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a [[quadratic form]], using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.
	{{one source\|date=February 2022}}

			The algorithm was introduced by [[Malcolm D. Shuster]] in 1981, while working at [[Computer Sciences Corporation]].<ref name="shuster">Shuster and Oh (1981)</ref> While being in principle less robust than other methods such as Davenport's q method or [[singular value decomposition]], the algorithm is significantly faster and reliable in practical applications,<ref name="markley_and_mortari">Markley and Mortari (2000)</ref><ref>Crassidis (2007)</ref> and it is used for [[flight dynamics\|attitude determination]] problem in fields such as [[robotics]] and [[avionics]].<ref>Psiaki (2000)</ref><ref>Wu et al. (2017)</ref><ref>Xiaoping et al. (2008)</ref>
⚫	The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a ~~bilinear~~ form, using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.

	The algorithm has applications to the [[flight dynamics\|attitude determination]] problem in fields such as [[avionics]], and it was introduced by [[Malcolm D. Shuster]] in 1981, while working at [[Computer Sciences Corporation]].<ref name="shuster">Shuster and Oh (1981)</ref>

	== Formulation of the problem ==		== Formulation of the problem ==
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	</math>.		</math>.

	Substituting <math>\mathbf{A}</math> with the quaternion representation and simplifying the resulting expression, the gain function can be written as a [[~~bilinear~~ form]] in <math>\mathbf{q}</math>		Substituting <math>\mathbf{A}</math> with the quaternion representation and simplifying the resulting expression, the gain function can be written as a [[quadratic form]] in <math>\mathbf{q}</math>

	:<math> g(\mathbf{q}) = \mathbf{q}^\top \mathbf{K} \mathbf{q} </math>		:<math> g(\mathbf{q}) = \mathbf{q}^\top \mathbf{K} \mathbf{q} </math>
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	</math>		</math>

	This ~~bilinear~~ form can be optimised under the unity constraint by adding a [[Lagrange multiplier]] <math>-\lambda \mathbf{q}^\top \mathbf{q}</math>, obtaining an unconstrained gain function		This quadratic form can be optimised under the unity constraint by adding a [[Lagrange multiplier]] <math>-\lambda \mathbf{q}^\top \mathbf{q}</math>, obtaining an unconstrained gain function

	:<math> \hat{g} \left( \mathbf{q} \right) = \mathbf{q}^\top \mathbf{K} \mathbf{q} - \lambda \mathbf{q}^\top \mathbf{q} </math>		:<math> \hat{g} \left( \mathbf{q} \right) = \mathbf{q}^\top \mathbf{K} \mathbf{q} - \lambda \mathbf{q}^\top \mathbf{q} </math>
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	:<math>\mathbf{K} \mathbf{q} = \lambda \mathbf{q}</math>.		:<math>\mathbf{K} \mathbf{q} = \lambda \mathbf{q}</math>.

	This implies that the optimal rotation is parametrised by the quaternion <math>\mathbf{q}^*</math> that is the [[eigenvector]] associated to the largest [[eigenvalue]] <math>\lambda_{\text{max}}</math> of <math>\mathbf{K}</math>.<ref name="shuster"/>		This implies that the optimal rotation is parametrised by the quaternion <math>\mathbf{q}^*</math> that is the [[eigenvector]] associated to the largest [[eigenvalue]] <math>\lambda_{\text{max}}</math> of <math>\mathbf{K}</math>.<ref name="shuster"/><ref name="markley_and_mortari"/>

	== Solution of the characteristic equation ==		== Solution of the characteristic equation ==
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	</math>		</math>

	whose root can be efficiently approximated with the [[Newton–Raphson method]], taking 1 as initial guess of the solution in order to converge to the highest eigenvalue (using the fact, shown above, that <math>\lambda_{\text{max}} \approx 1</math> when the quaternion is close to the optimal solution).<ref name="shuster"/>		whose root can be efficiently approximated with the [[Newton–Raphson method]], taking 1 as initial guess of the solution in order to converge to the highest eigenvalue (using the fact, shown above, that <math>\lambda_{\text{max}} \approx 1</math> when the quaternion is close to the optimal solution).<ref name="shuster"/><ref name="markley_and_mortari"/>

	== See also ==		== See also ==
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	== Sources ==		== Sources ==
			* {{cite journal\|title=Survey of nonlinear attitude estimation methods\|year=2007\|journal=Journal of guidance, control, and dynamics\|volume=30\|pages=12–28\|issue=1\|last1=Crassidis\|first1=John L\|last2=Markley\|first2=F Landis\|last3=Cheng\|first3=Yang}}
			* {{cite journal\|title=Quaternion attitude estimation using vector observations\|year=2000\|journal=The Journal of the Astronautical Sciences\|volume=48\|pages=359–380\|issue=2\|publisher=Springer\|last1=Markley\|first1=F Landis\|last2=Mortari\|first2=Daniele}}
			* {{cite journal\|title=Attitude-determination filtering via extended quaternion estimation\|year=2000\|journal=Journal of Guidance, Control, and Dynamics\|volume=23\|pages=206–214\|issue=2\|last1=Psiaki\|first1=Mark L}}
	* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of guidance and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.}}		* {{cite journal\|title=Three-axis attitude determination from vector observations\|year=1981\|journal=Journal of guidance and Control\|volume=4\|pages=70–77\|issue=1\|last1=Shuster\|first1=M.D.\|last2=Oh\|first2=S.D.}}
			* {{cite journal\|title=Fast linear quaternion attitude estimator using vector observations\|year=2017\|journal=IEEE Transactions on Automation Science and Engineering\|volume=15\|pages=307–319\|issue=1\|publisher=IEEE\|last1=Wu\|first1=Jin\|last2=Zhou\|first2=Zebo\|last3=Gao\|first3=Bin\|last4=Li\|first4=Rui\|last5=Cheng\|first5=Yuhua\|last6=Fourati\|first6=Hassen}}
			* {{cite journal\|title=A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements\|year=2008\|journal=IEEE Transactions on instrumentation and measurement\|volume=57\|pages=638–650\|issue=3\|publisher=IEEE\|last1=Yun\|first1=Xiaoping\|last2=Bachmann\|first2=Eric R\|last3=McGhee\|first3=Robert B}}

	== External links ==		== External links ==

John B123: Added tags to the page using Page Curation (one source)

2022-02-27T17:08:54Z

Added tags to the page using Page Curation (one source)

← Previous revision		Revision as of 17:08, 27 February 2022
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			{{one source\|date=February 2022}}

	The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a bilinear form, using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.		The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a bilinear form, using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.

Tino: title

2022-02-26T17:53:48Z

title

← Previous revision		Revision as of 17:53, 26 February 2022
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	The '''quaternion ~~estimation~~ algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a bilinear form, using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.		The '''quaternion estimator algorithm''' ('''QUEST''') is an [[algorithm]] designed to solve [[Wahba's problem]], that consists of finding a [[rotation matrix]] between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a bilinear form, using the [[Cayley–Hamilton theorem]] and the [[Newton's method\|Newton–Raphson method]] to efficiently solve the eigenvalue problem and construct a [[numerical stability\|numerically stable]] representation of the solution.

	The algorithm has applications to the [[flight dynamics\|attitude determination]] problem in fields such as [[avionics]], and it was introduced by [[Malcolm D. Shuster]] in 1981, while working at [[Computer Sciences Corporation]].<ref name="shuster">Shuster and Oh (1981)</ref>		The algorithm has applications to the [[flight dynamics\|attitude determination]] problem in fields such as [[avionics]], and it was introduced by [[Malcolm D. Shuster]] in 1981, while working at [[Computer Sciences Corporation]].<ref name="shuster">Shuster and Oh (1981)</ref>

Tino: Tino moved page Quaternion estimation algorithm to Quaternion estimator algorithm: accurate name

2022-02-26T17:53:21Z

Tino moved page Quaternion estimation algorithm to Quaternion estimator algorithm: accurate name

← Previous revision	Revision as of 17:53, 26 February 2022
(No difference)

← Previous revision		Revision as of 03:19, 5 March 2022
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	== Sources ==		== Sources ==
	* {{cite journal\|title=Survey of nonlinear attitude estimation methods\|year=2007\|journal=Journal of ~~guidance~~, ~~control~~, and ~~dynamics~~\|volume=30\|pages=12–28\|issue=1\|last1=Crassidis\|first1=John L\|last2=Markley\|first2=F Landis\|last3=Cheng\|first3=Yang}}		* {{cite journal\|title=Survey of nonlinear attitude estimation methods\|year=2007\|journal=Journal of Guidance, Control, and Dynamics\|volume=30\|pages=12–28\|issue=1\|last1=Crassidis\|first1=John L\|last2=Markley\|first2=F Landis\|last3=Cheng\|first3=Yang\|doi=10.2514/1.22452\|bibcode=2007JGCD...30...12C}}
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	== External links ==		== External links ==