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Triad is one of the earliest and simplest solutions to the spacecraft attitude determination problem,<ref>{{cite journal|last=Black|first=Harold|title=A Passive System for Determining the Attitude of a Satellite|journal=AIAA Journal|month = July|year=1964|volume=2|issue=7|pages=1350–1351|doi=10.2514/3.2555|bibcode = 1964AIAAJ...2.1350. }}</ref><ref>{{cite journal|last=Black|first=Harold|title=Early Developments of Transit, the Navy Navigation Satellite System|journal=Journal of Guidance, Control and Dynamics|year=1990|month=July-Aug|volume=13|issue=4|pages=577–585|doi=10.2514/3.25373|bibcode = 1990JGCD...13..577B }}</ref> due to Harold Black. Black played a key role in the development of the guidance, navigation and control of the U.S. Navy's Transit satellite system at Johns Hopkins Applied Physics Laboratories. As evident from the literature, TRIAD represents the state of practice in spacecraft attitude determination, well before the advent of the [[Wahba's problem]] <ref>{{cite journal|last=Wahba|first=Grace|title=A Least Squares Estimate of Satellite Attitude, Problem 65.1|journal=SIAM Review|year=1966|month=July|pages=385–386}}</ref>  and its several optimal solutions. Given the knowledge of two vectors in the reference and body coordinates of a satellite, the TRIAD algorithm obtains the direction cosine matrix relating both frames. Covariance analysis for Black's classical solution was subsequently provided by Markley.<ref>{{cite journal|last=Markley|first=Landis|title=Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm|journal=The Journal of Astronautical Sciences|year=1993|month=April–June|volume=41|issue=2|pages=261–280|url=http://www.malcolmdshuster.com/FC_Markley_1993_J_FOAM_JAS_MDSscan.pdf|accessdate=April 18, 2012}}</ref>
Fixed-blade knives are as easy to describe as they sound. The blade is fixed and does not fold. The benefits to a knife of this form is that they offer the most effective durability and are incredibly straightforward knives to clean. They are also the ideal fit for when much more “heavy-duty” operate is expected of a hunting knife, although they are not devoid of their downsides either.<br><br>Although you've already read my review of the top 5 survival knives, I've also compiled a huge tabular guide that compares all the top survival blades in the market at the present. This guide will help you make an informed choice of which knife to choose based on your needs and help you determine the best survival knife for you. Blade shape is very important when choosing a knife, you need to match the intended use with the best blade shape, for a skinning knife I prefer a drop point blade – my Gerber pocket knife is a drop point and despite being a small blade it is quick and easy to use – it has proven to be a great knife for skinning deer. 12.5 420HC Steel 6. In case you have any queries with regards to wherever as well as how you can employ [http://Thebestpocketknifereviews.com/gerber-knives-review/ Gerber Knives], you are able to contact us on our own website. 6 11.6 $$$ 4.4 Introduction to the Best Hunting Knife Knife Sharpening Information and Theory<br><br>The SOG Aura Hunting Fixed Blade uses a high impact and challenging glass-reinforced polymer handle that is overmolded with a much more pliable material for non-slip comfort. The 7CR13 blade is five inches and you can unscrew the carbide sharpener from the back of the handle for fast, in the field fixes. Similarly, a spear point knife can be applied as a hunting weapon–either by itself or lashed to a pole to produce a longer attain spear. I preserve the allen wrench (which came with my knife) in my knife sheath pocket at all occasions. This enables me to eliminate the scales and lash the full tang blade almost seamlessly onto a employees as a spear-point. Buck 110 Folding Hunter – 9/ten vs. Boker Folding Lockback – 7/ten Individual Preference and Gut Feeling Categories of Knives<br><br>The Buck 110 Folding Hunter is a timeless American classic that measures four 7/8 inches closed and weighs a hefty 7.2 ounces. It options an elegant, semi-hollow ground, three three/four inch clip point blade made from 420HC (high carbon) stainless steel hardened to 57-59 Rockwell. It also options higher top quality brass bolsters with rounded edges and increasingly uncommon Macassar Ebony wood slabs. Lockback Hunter on the other hand, measures five 1/4″ closed and weighs 5.six ounces. Plus, it [http://sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%5B%5Biframe+%2F%2Fwww.youtube.com%2Fembed%2FNt9fxWq5qg0+height%3D%22360%22+width%3D%22640%22%5D%5DDown+behind+enemy+traces%3F+Left+to+fend+for+your+self%3F++Should+you+have+just+about+any+%5Bhttp%3A%2F%2Fwww.Twitpic.com%2Ftag%2Fconcerns+concerns%5D+concerning+where+and+the+best+way+to+make+use+of+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fgerber-knives-review%2F+Thebestpocketknifereviews.com%5D%2C+you+can+call+us+with+the+site.+These+are+the+%5Bhttp%3A%2F%2FSearch.un.org%2Fsearch%3Fie%3Dutf8%26site%3Dun_org%26output%3Dxml_no_dtd%26client%3DUN_Website_en%26num%3D10%26lr%3Dlang_en%26proxystylesheet%3DUN_Website_en%26oe%3Dutf8%26q%3Dscenarios%26Submit%3DGo+scenarios%5D+that+%5Bhttp%3A%2F%2FBrowse.Deviantart.com%2F%3Fqh%3D%26section%3D%26global%3D1%26q%3Dimpressed+impressed%5D+the+LMF+II.+Former+army+man%2C+Jeff+Freeman+led+the+cost+to+engineer+this+fearless+new+10%22+survival+knife.+And+we+field-tested+with+the+troops.%0D%0A%0D%0ANow+lets+get+again+to+business+and+assessment+the+Paraframe+I.+The+Paraframe+I+has+eight+totally+different+knives+within+the+sequence.carry+the+serrated+for+those+further+robust+slicing+jobs.+The+Paraframe+I+relies+on+a+minimal+body+lock+design.+It%27s+splendidly+simple%2C+open+body%28skeleton%29+knife+with+a+serrated+locking+blade+that+is+super+light-weight%2C+it+cleans+simply+and+opens+effortlessly.+The+Paraframe+is+constructed+with+a+easy+chrome+steel+handle+and+a+pocket+clip+to+secure+the+blade+to+your+pocket.+The+knife+has+an+general+size+of+seven.01+inches+with+a+blade+size+of+three.01+inches.+The+Paraframe+is+a+great+on+a+regular+basis+folder.%0D%0A%0D%0AIf+you+happen+to+want+an+enormous%2C+beefy+combat+knife+that+may+lower+and+slice+with+the+best+of+them%2C+the+Gerber+DMF+is+one+I+like+to+recommend.+After+utilizing+the+knife+for+30+days%2C+I%27ve+come+to+the+conclusion+that+it+is+a+well+made%2C+sturdy+knife+that+may+take+heavy+abuse.+With+an+S30V+blade%2C+G-10+handle+scales+that+give+your+hand+plenty+of+grip%2C+and+a+no+nonsense+attitude%2C+the+DMF+will+get+an+A+grade.+The+Gerber+Steadfast+is+sold+complete+with+a+Cordura+sheath+that+includes+a+MOLLE+attachment+system+on+the+again+in+addition+to+a+large+belt+loop%2C+a+handy+and+a+leg+strap+with+quick+launch+buckle+is+also+supplied.+The+black+is+lined+with+a+plastic+protecting+liner+that+keeps+the+sheath+in+shape+and+prevents+the+sting+from+damaging+it.+Opening+up+a+can+with+the+knife%0D%0A%0D%0AErgonomics+for+this+folder+are+outstanding%21+The+deal+with+is+coated+with+a+grippy+rubbery+material+that%2C+together+with+the+striped+pattern+within+the+handle%2C+jimping%2C+and+a+deep+finger+choile%2C+provide+the+consumer+a+secure+grip+regardless+of+how+slippery+the+situations+get.+Throughout+the+handle+is+a+lockback+mechanism+supporting+the+folding+blade.+This+design+is+usually+stronger+than+a+linerlock+mechanism%2C+making+this+Gerber+Bear+Grylls+product+one+of+the+better+decisions+for+a+survival+folder.%0D%0A%0D%0AThe+sheath+is+reversible+and+allows+you+to+carry+the+knife+with+the+tip+pointing+up%2C+down%2C+or+horizontally.+The+sheath+additionally+has+a+belt+loop+that+is+adjustable+and+may+be+eliminated+and+laced+to+tactical+and+other+gear.+The+sheath+has+good+retention+and+seats+with+an+audible+click%2C+however+does+not+require+the+push+of+a+button+to+draw+the+knife+%28Massive+plus+IMHO%29.+Chances+are+you%27ll+check+your+order+status+by+clicking+the+Order+Standing+tab+in+the+LA+Police+Gear+Customer+Service+Center.+Please+be+sure+to+enter+the+zip+code+you+entered+in+your+billing+address.+You+probably+have+registered+for+an+account+with+LA+Police+Gear%2C+you+possibly+can+view+your+entire+order+history+by+clicking+My+Account.+Can+I+buy+from+LA+Police+Gear+if+I%27m+not+within+the+navy%2C+legislation+enforcement%2C+and+many+others.%3F%0D%0A%0D%0AThis+knife+is+a+good+software+to+keep+in+your+tool+field+or+tool+belt.+It+is+a+good+working+knife+since+it+might+probably+assist+with+the+tools+like+cleaning+off+the+underside+of+lawn+mower+deck.+Gerber+has+made+it+a+degree+to+use+scallop+serrations+since+it%27s+not+onerous+to+re+sharpen+the+knife.+For+those+who+dislike+difficult+sharpening+procedures+this+is+the+right+knife+that+permits+its+proprietor+to+make+use+of+it+over+and+over+again.+This+knife+has+a+slim+deal+with+which+has+two+lashing+holes+in+it+when+there%27s+a+have+to+affix+a+spear.%0D%0A%0D%0ANot+all+knife+brands+would+not+offer+a+special+provide+for+their+customers.+Thankfully%2C+if+you+happen+to%27re+prepared+to+purchase+a+gerber+knife+then+we+now+have+a+excellent+news+for+you.+All+the+gerber+knives+are+backed+by+full+lifetime+guarantee+%28+25+years+outside+of+the+country+%29.+This+give+a+guarantee+that+the+knife+will+final+for+lifetime.+This+is+likely+one+of+the+very+unique+guarantee+that+no+other+firm+would+possibly+offer.+The+Silver+Trident+is+the+most+advanced+Gerber+Knife+you+Blade+ownCould+Gerber%27s+superior+recognition+for+high-quality+makes+this+Gerber%27s+many+innovative+knife+and+one+that+may+actually+final+you+generations.+The+Gerber+LMF+II+survival+knife+has+a+really+wide+blade+that%27s+3%2F16s+of+an+inch+assume%2C+4.eighty+four+inches+lengthy.+Stainless+Steel+Mounted+Blade+Searching+Knife+made+by+Gerber%0D%0A%0D%0AThe+Kershaw+1780CB+Cutting+knife%2C+known+additionally+because+the+Kershaw+Rake%2C+is+a+composite+folding+knife+that+features+wear-resistant+D2+steel+on+its+cutting+edge+and+Sandvik+14C28N+metal+on+the+spine.+This+pretty+priced+knife+has+high+opinions+and+ratings+on-line.+Amazon+customers+were+impressed+with+many+options+of+the+Fast+Draw.+One+buyer+wrote+that+it+holds+the+edge+effectively+and+sharpens+fairly+easily.+One+other+wrote+that+the+design+was+very+lightweight+and+compact%2C+and+that+the+deal+with+matches+snug+in+your+hand.+Others+famous+that+the+pocket+clip+was+a+great+function.+General%2C+Amazon+reviews+wrote+that+at+lower+than+%2430%2C+the+Quick+Draw+was+a+stable+possibility+in+the+event+you%27re+searching+for+an+everyday+carry+pocket+knife.+Gerber+brand+on+machete%2Fsaw+combo+Priorities+of+survival+information&create=Create capabilities] a, Saber Ground, three 1/8″ Clip Point blade created from either 440C stainless steel with Rockwell Hardness of 58 or Cronidur 30 stainless steel with Rockwell Hardness of 58 and a classic Lockback locking mechanism. Excellent choice for hunters who also want a knife for everyday use<br><br>Whatever knife you pick, it will have to be kept really sharp actually sharp adequate to shave the hair off of your arm. Most hunting knives, which are designed for cutting meat, have a cutting edge sharpened at about a 20-degree angle. This is the angle [http://En.wikipilipinas.org/index.php/Gerber_Infantry_Knife_Reviews utilized] by Randall, for instance. I use a Buck knife sharpening kit, which includes medium and fine Arkansas stones and knife sharpening oil. Equivalent kits are obtainable from other sources the brand doesn't matter. Comply with the directions that are supplied with the sharpening set for correct use.<br><br>On the more give, the Trailing Point cutter type integrates a extended, concave, contour along the duration of the spinal column which areas the notion of the cutter over the spinal column therefore boosting the span of the stretch however lowering the quantity of management the person possesses over the recommendation of the cutter. When skinning this style of creature, I learn that I normally just make use of the initial 1/2″ in order to 3/4″ of the cutter back through the suggestion in order to separate this layer as well as therefore, each the Clip Point and also the Drop Point cutter forms are essentially outstanding choices for this kind of skinning<br><br>Gerber is recognized for quality at an economical value. The Moment continues that tradition and will locate favor with practically any hunter. It's very sturdy, with excellent ergonomics and a wonderfully grippy, textured rubber handle. The only region where this drop point is lacking is in out-of-the-box keenness, so you'll want to sharpen it ahead of making use of. The nylon sheath is rigid and [http://Dallaskoreans.com/index.php?document_srl=360432&mid=Test durable]. All in all, this is an excellent value—a knife that will serve its owner
 
==Summary==
We consider the linearly independent reference vectors <math>\vec{R}_{1}</math> and <math>\vec{R}_2 </math>. Let <math> \vec{r}_1, \vec{r}_2</math> be the corresponding measured directions of the reference unit vectors as resolved in a body fixed frame of reference. Then they are related by the equations,
 
{{NumBlk|:|
<math>
\vec{R}_i = A \vec{r}_i
</math>
|{{EquationRef|1}}}}
 
for <math> i = 1,2 </math>, where <math> A </math> is a rotation matrix (sometimes also known as a proper [[orthogonal matrix]], i.e., <math>A^{T}A = I, det(A) = +1 </math>). <math> A </math> transforms vectors in the body fixed frame into the frame of the reference vectors. Among other properties, rotational matrices preserve the length of the vector they operate on. Note that the direction cosine matrix <math> A </math> also transforms the cross product vector, written as,
 
{{NumBlk|:|
<math>
\vec{R}_1 \times \vec{R}_2 = A \left( \vec{r}_1 \times \vec{r}_2 \right)
</math>
|{{EquationRef|2}}}}
 
Triad proposes an estimate of the direction cosine matrix <math> A </math> as a solution to the linear system equations given by
 
{{NumBlk|:|
<math>
\left[ \vec{R}_1 ~\vdots~ \vec{R}_2 ~\vdots~ \left(\vec{R}_1 \times \vec{R}_2 \right) \right] = A \left[ \vec{r}_1 ~\vdots~ \vec{r}_2 ~\vdots~ \left( \vec{r}_1 \times \vec{r}_2 \right) \right]
</math>
|{{EquationRef|3}}}}
 
where <math> \vdots </math> have been used to separate different column vectors.  
 
The solution presented above works well in the noise-free case. However, in practice, <math> \vec{r}_1, \vec{r}_2 </math> are noisy and the orthogonality condition of the attitude matrix (or the direction cosine matrix) is not preserved by the above procedure. Triad incorporates the following elegant procedure to redress this problem. To this end, we define unit vectors
 
{{NumBlk|:|
<math>
\hat{S} = \frac{\vec{R}_1}{|| \vec{R}_1||}
</math>
|{{EquationRef|4}}}}
{{NumBlk|:|
<math>
\hat{s} = \frac{\vec{r}_1}{||\vec{r}_1||}
</math>
|{{EquationRef|5}}}}
 
and
 
{{NumBlk|:|
<math>
\hat{M} = \frac{\vec{R}_1 \times \vec{R}_2}{|| \vec{R}_1 \times \vec{R}_2 ||}
</math>
|{{EquationRef|6}}}}
{{NumBlk|:|
<math>
\hat{m} = \frac{\vec{r}_1 \times \vec{r}_2 }{||\vec{r}_1 \times \vec{r}_2 ||}
</math>
|{{EquationRef|7}}}}
 
to be used in place of the first two columns of ({{EquationNote|3}}). Their cross product is used as the third column in the linear system of equations obtaining a proper orthogonal matrix for the spacecraft attitude given by
 
{{NumBlk|:|
<math>
\left[ \hat{S} ~ \vdots ~ \hat{M} ~\vdots~ \hat{S} \times \hat{M} \right] = A \left[ \hat{s} ~\vdots~ \hat{m} ~\vdots~ \hat{s} \times \hat{m} \right]
</math>
|{{EquationRef|8}}}}
 
While the normalizations of Equations ({{EquationNote|4}}) - ({{EquationNote|7}}) are not necessary, they have been carried out to achieve a computational advantage in solving the linear system of equations in ({{EquationNote|8}}). Thus an estimate of the spacecraft attitude is given by the proper orthogonal matrix as
{{NumBlk|:|
<math>
\hat{A} = \left[ \hat{S} ~ \vdots ~ \hat{M} ~\vdots~ \hat{S} \times \hat{M} \right] \left[ \hat{s} ~\vdots~ \hat{m} ~\vdots~ \hat{s} \times \hat{m} \right]^T.
</math>
|{{EquationRef|9}}}}
Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose. Equation({{EquationNote|8}}) shows that the matrices used for computing attitude are each composed of an orthogonal triad of basis vectors. "TRIAD" derives its name from this observation.
 
==Triad Attitude Matrix and Handed-ness of Measurements==
It is of consequence to note that the Triad method always produces a proper orthogonal matrix irrespective of the handedness of the reference and body vectors employed in the estimation process. This can be shown as follows.
Let us re-write  Eq. ({{EquationNote|8}}) in a matrix form given by
 
{{NumBlk|:|
<math>
\Gamma = A \Delta
</math>
|{{EquationRef|10}}}}
 
where  <math>
\Gamma := \left[ \hat{S} ~ \vdots ~ \hat{M} ~\vdots~ \hat{S} \times \hat{M} \right]
</math>
and  
<math>
\Delta = \left[ \hat{s} ~\vdots~ \hat{m} ~\vdots~ \hat{s} \times \hat{m} \right].
</math>
Note that if the columns of <math> \Gamma </math> form a left handed triad, then the columns of <math> \Delta </math> are also left handed because of the one-one correspondence between the vectors. This is because of the simple fact that, in Euclidean geometry, the angle between any two vectors remains invariant to coordinate transformations. Therefore, the determinant <math> det \left( \Gamma \right) </math> is <math> 1 </math> or <math> -1 </math> depending on whether its columns are right or left - handed respectively (similarly, <math> \Delta  = \pm 1</math> ). Taking determinant on both sides of the relation in Eq. ({{EquationNote|10}}), we conclude that  
{{NumBlk|:|
<math>
det \left( A \right) = 1.
</math>
|{{EquationRef|11}}}}
This is quite useful in practical applications since the analyst is always guaranteed a proper orthogonal matrix irrespective of the nature of the reference and measured vector quantities.
 
==Applications==
Triad was used as an attitude determination technique to process the telemetry data from the Transit satellite system (used by the U.S. Navy for navigation). It is interesting to note that the principles of the Transit system gave rise to what we now know as the GPS satellite constellation. In an application problem, the reference vectors are usually known directions (e.g., stars, Earth magnetic field, gravity vector, etc.). Body fixed vectors are the measured directions as observed by an on-board sensor (e.g., star tracker, magnetometer, etc.). With advances in micro-electronics, attitude determination algorithms such as Triad have found their place in a variety of devices (e.g., smart phones, cars, tablets, UAVs etc.) with a broad impact on the modern society.
 
==See also==
*[[Attitude dynamics and control|Attitude Dynamics and Control]]
*[[Orientation (geometry)|Orientation (Geometry)]]
* [[Wahba's problem]]
 
== References ==
{{Reflist}}
 
[[Category:Spacecraft attitude control]]
[[Category:Rotation in three dimensions]]

Latest revision as of 09:56, 4 November 2014

Fixed-blade knives are as easy to describe as they sound. The blade is fixed and does not fold. The benefits to a knife of this form is that they offer the most effective durability and are incredibly straightforward knives to clean. They are also the ideal fit for when much more “heavy-duty” operate is expected of a hunting knife, although they are not devoid of their downsides either.

Although you've already read my review of the top 5 survival knives, I've also compiled a huge tabular guide that compares all the top survival blades in the market at the present. This guide will help you make an informed choice of which knife to choose based on your needs and help you determine the best survival knife for you. Blade shape is very important when choosing a knife, you need to match the intended use with the best blade shape, for a skinning knife I prefer a drop point blade – my Gerber pocket knife is a drop point and despite being a small blade it is quick and easy to use – it has proven to be a great knife for skinning deer. 12.5 420HC Steel 6. In case you have any queries with regards to wherever as well as how you can employ Gerber Knives, you are able to contact us on our own website. 6 11.6 $$$ 4.4 Introduction to the Best Hunting Knife Knife Sharpening Information and Theory

The SOG Aura Hunting Fixed Blade uses a high impact and challenging glass-reinforced polymer handle that is overmolded with a much more pliable material for non-slip comfort. The 7CR13 blade is five inches and you can unscrew the carbide sharpener from the back of the handle for fast, in the field fixes. Similarly, a spear point knife can be applied as a hunting weapon–either by itself or lashed to a pole to produce a longer attain spear. I preserve the allen wrench (which came with my knife) in my knife sheath pocket at all occasions. This enables me to eliminate the scales and lash the full tang blade almost seamlessly onto a employees as a spear-point. Buck 110 Folding Hunter – 9/ten vs. Boker Folding Lockback – 7/ten Individual Preference and Gut Feeling Categories of Knives

The Buck 110 Folding Hunter is a timeless American classic that measures four 7/8 inches closed and weighs a hefty 7.2 ounces. It options an elegant, semi-hollow ground, three three/four inch clip point blade made from 420HC (high carbon) stainless steel hardened to 57-59 Rockwell. It also options higher top quality brass bolsters with rounded edges and increasingly uncommon Macassar Ebony wood slabs. Lockback Hunter on the other hand, measures five 1/4″ closed and weighs 5.six ounces. Plus, it capabilities a, Saber Ground, three 1/8″ Clip Point blade created from either 440C stainless steel with Rockwell Hardness of 58 or Cronidur 30 stainless steel with Rockwell Hardness of 58 and a classic Lockback locking mechanism. Excellent choice for hunters who also want a knife for everyday use

Whatever knife you pick, it will have to be kept really sharp actually sharp adequate to shave the hair off of your arm. Most hunting knives, which are designed for cutting meat, have a cutting edge sharpened at about a 20-degree angle. This is the angle utilized by Randall, for instance. I use a Buck knife sharpening kit, which includes medium and fine Arkansas stones and knife sharpening oil. Equivalent kits are obtainable from other sources the brand doesn't matter. Comply with the directions that are supplied with the sharpening set for correct use.

On the more give, the Trailing Point cutter type integrates a extended, concave, contour along the duration of the spinal column which areas the notion of the cutter over the spinal column therefore boosting the span of the stretch however lowering the quantity of management the person possesses over the recommendation of the cutter. When skinning this style of creature, I learn that I normally just make use of the initial 1/2″ in order to 3/4″ of the cutter back through the suggestion in order to separate this layer as well as therefore, each the Clip Point and also the Drop Point cutter forms are essentially outstanding choices for this kind of skinning

Gerber is recognized for quality at an economical value. The Moment continues that tradition and will locate favor with practically any hunter. It's very sturdy, with excellent ergonomics and a wonderfully grippy, textured rubber handle. The only region where this drop point is lacking is in out-of-the-box keenness, so you'll want to sharpen it ahead of making use of. The nylon sheath is rigid and durable. All in all, this is an excellent value—a knife that will serve its owner