Method for attitude determination of spacecraft using a direction vector and a total momentum measurement

The present invention relates to a method for determining the attitude of a spacecraft based on two vector determinations, wherein sensor data and spin data are determined as output variables for position determination.

From the prior art methods are known, with which a spacecraft, equipped with an earth sensor (= direction measurement) and swirl wheels, which produce a swirl target value, can be stabilized around all 3 axes (eg Whecon principle).

All of these methods utilize the dynamic coupling between rolls and yaw caused by the swirl set point to either directly through appropriately selected actuator control (jets or wheels) or by using an observer to determine the missing attitude information (typically the yaw attitude) To stabilize spacecraft.

Reference is made by way of example to EP 0 785 132. There, on the one hand, the Whecon principle is explained, on the other hand, a method for determining the attitude of a spacecraft is described, in which a decoupling of the individual moments takes place, which act on the spacecraft. Decoupling terms are therefore determined there in order to achieve a decoupling of the roll-yaw coupling terms. However, this represents a relatively high computational effort.

From US 6,282,467 a system and a method for determining the position of a spin-stabilized spacecraft in an inertial coordinate system is known. The following steps are carried out: determination of a spin vector direction in the inertial coordinate system on the basis of measured data, in particular sensor data; Determining a spin vector direction in a body-fixed coordinate system based on known inertial data; Determination of a direction vector from uniaxial sensor data; Determination of a reference direction vector using an ephemeris model; Determining the position of the spacecraft from said four vectors and propagating the location using sensor data. This method is not applicable to three-axis stabilized spacecraft and moreover requires a relatively large number of known inputs and sensor data.

The object of the present invention is therefore to provide a simplified and improved method for determining the position of a spacecraft, in particular a three-axis-stabilized spacecraft. This object is solved by the features of claims 1 and 5.

The invention relates to a method for determining the attitude of a spacecraft based on the determination of direction vectors and spin vectors, wherein sensor data and spin data are determined as output variables for determining the position, in which the first vector determination is a measurement of a direction vector in a body-fixed coordinate system with the aid of a sensor, the second vector determination is the determination of a reference direction vector in a reference coordinate system on the basis of the orbital position of the spacecraft and an orbit model, the third vector determination determines the total spin vector of the spacecraft in the body-fixed coordinate system, is determined as the fourth vector determination of the reference total spin vector of the spacecraft in a reference coordinate system by temporally propagating known initial values ​​of the total spin of the spacecraft or by temporally tracking a reference model for the total spin and Based on the four vectors, a determination of the position of the spacecraft takes place.

This method is simpler than the above-mentioned method of the prior art, since the use of the knowledge of an initial value of the total twist on the one hand, a costly determination of Entkopplungstermen is unnecessary and the individual determination steps to determine the situation are simplified, as will be explained below , In particular, in principle only a single uniaxial vector measurement with the aid of a sensor is required, namely for measuring a direction vector in a body-fixed coordinate system.

The total spin can be determined, for example, as a deviation from a reference total spin. The reference total spin can be determined on the one hand by temporal propagation of known initial values ​​of the total spin. Therefore, a measurement or estimation of the total spin at a certain starting time is required. However, it is also possible to determine the reference total spin by temporally tracking a reference model for the total spin. This basically corresponds to the method of using an observer, as it is known in principle from the prior art. In principle, the reference direction vector can also be determined-analogously to the reference total spin-either by temporally propagating known initial values ​​of the direction vector or by temporally tracking a reference model for the direction vector.

A specific embodiment of the method according to the invention provides that, as an intermediate step for determining the position, first the rotational speed or a part of the rotational speed between a body-fixed coordinate system and the reference coordinate system is determined. It is provided that at least from the direction vector, the reference direction vector and a temporally propagated from known initial values ​​or by means of a reference model temporally tracked total spin initially the longitudinal rate component of the rotational speed or of the rotational velocity vector in the direction of the direction vector between the body-fixed coordinate system and the reference coordinate system is determined. The total spin can be used in a suitable representation, for example in an orbit coordinate system. With the help of the longitudinal rate component, the total spin vector is determined in the body-fixed coordinate system. On the basis of the direction vector, the reference direction vector, the total spin vector and the reference total spin vector, a transformation matrix is ​​determined which describes the deviations of the actual position of the body-fixed coordinate system of the spacecraft from the reference coordinate system representing the target position of the body-fixed coordinate system, ie a transformation matrix of the deviations of the actual position of the body-fixed coordinate system of the spacecraft from the reference coordinate system determined. The reference coordinate system thus represents the desired position of the body-fixed coordinate system. For this process step, in turn, the overall twist is used. It is therefore necessary for the entire orientation only the knowledge of the direction vector and the total spin, which greatly simplifies the process. Moreover, in the last method step, it is not the total twist required in direction and magnitude that suffices for the direction of the total spin alone.

To determine the total spin vector in the body-fixed coordinate system, at least the direction vector and the reference direction vector are additionally used in particular.

The overall spin is essentially determined by whether the spacecraft has spin wheels whose wheel spin constitutes an essential component of the total spin. The method described so far can also be carried out on a spacecraft with twisting wheels without an explicit measurement of the wheel spin. It can then be the rotational speed or of rotational speed components may additionally be determined on the basis of estimates or measurements for wheel spin. In particular, the determination of the longitudinal rate component and the determination of the total spin vector can thus additionally be carried out on the basis of the determined spin vector of spiral wheels of the spacecraft.

The present method can be carried out particularly advantageous if the deviations of the actual position from the desired position are small. This is especially true when the angular deviations φ<i>_BR</i>: in the transformation matrix T^BR= I - φ<i>_BR</i> smaller than 0.1 rad.

Another object of the present invention is an arrangement for determining the attitude of a spacecraft, comprising the following components:<ul id="ul0001" list-style="dash" compact="compact"><li>a device for measuring a direction vector in a body-fixed coordinate system,</li><li>a device for determining the orbital position of the spacecraft,</li><li>an arrangement for determining a reference direction vector in a reference coordinate system based on the orbital position of the spacecraft and an orbit model,</li><li>a device for determining a total spin vector in the body-fixed coordinate system,</li><li>an arrangement for determining a reference total spin vector in the reference coordinate system and, wherein the reference total spin vector is determined by temporally propagating known initial values ​​of the total spin of the spacecraft or by temporally tracking a reference model for the total spin, and</li><li>a device for determining the position of the spacecraft on the basis of the four vectors.</li></ul>

The arrangements mentioned can consist of one or more functional components. Such an arrangement is used in particular for carrying out the above-mentioned method. There are also the advantages and possibilities already mentioned.

A further development of the arrangement according to the invention additionally comprises: a device for determining the longitudinal rate component of the rotational speed or of the rotational speed vector in the direction of the direction vector between the body-fixed coordinate system and the reference coordinate system.

In addition, the arrangement according to the invention can have a device for measuring or estimating the swirl vector of swirl wheels of the spacecraft.

A specific embodiment of the present invention will be described below with reference to FIG.

It shows:<dl id="dl0001" compact="compact"><dt>Fig. 1:</dt><dd>Block diagram of the process sequence according to the invention</dd></dl>

The following relationships are used to derive the method<ul id="ul0002" list-style="bullet" compact="compact"><li>Rotational speed between body-fixed and reference coordinate system<maths id="math0001"><math display="block"><mfenced><mn>1</mn></mfenced><mi>   </mi><msub><mi>ω</mi><mi mathvariant="italic">BR</mi></msub><mo>=</mo><mo>-</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>+</mo><msub><mi mathvariant="italic">ce</mi><mi>B</mi></msub><mo>+</mo><msup><mi>T</mi><mi mathvariant="italic">BR</mi></msup><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub></math><img file="EP1514799B1_D0001.tif" /></maths> With<dl id="dl0002" compact="compact"><dt>e_B = T^BR e_R:</dt><dd>Unit vector (EV) of the direction measurement in the body-fixed system (ie direction vector, obtained from direction vector measurement)</dd><dt>e_R:</dt><dd>associated EV in the reference coordinate system with the additional property that the corresponding direction in the inertial system is time-variable (ie reference direction vector obtained from reference direction vector measurement)</dd><dt>T^BR= I - φ<i>_BR</i>:</dt><dd>to be determined transformation matrix small deviations between body-fixed and reference coordinate system, also corresponds to the representation of the deviation of the total spin from the reference total spin</dd><dt>φ_BR:</dt><dd>Vector of small angles describing the deviation between body-fixed and reference system</dd><dt>"~":</dt><dd>Cross product operation</dd><dt>c = e_B^Tω_BR:</dt><dd>to be determined rotational speed component in the direction of e_B, hereinafter referred to as "longitudinal rate"</dd></dl></li><li>Spin vector in the body-fixed coordinate system<maths id="math0002"><math display="block"><mtable><mtr><mtd><msub><mrow><mfenced><mn>2</mn></mfenced><mi> H</mi></mrow><mi>B</mi></msub></mtd><mtd columnalign="left"><mo>=</mo><msub><mi mathvariant="normal">J</mi><mi mathvariant="normal">s</mi></msub><mrow><mo>(</mo><msub><mi>ω</mi><mi>BR</mi></msub><mo>+</mo><msup><mi>T</mi><mi>BR</mi></msup><mo>⁢</mo><msub><mi>ω</mi><mi>RI</mi></msub><mo>)</mo><mo>+</mo><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">w</mi></msub></mrow></mtd><mtd><mi mathvariant="normal"> </mi></mtd><mtd><mi mathvariant="normal"> </mi></mtd></mtr><mtr><mtd><mi mathvariant="normal"> </mi></mtd><mtd columnalign="left"><mo>≈</mo><msub><mi mathvariant="italic">J</mi><mi mathvariant="italic">s</mi></msub><mrow><mo>[</mo><mo>-</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>+</mo><msub><mi mathvariant="italic">ce</mi><mi>B</mi></msub><mo>+</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>-</mo><msub><mover><mi>φ</mi><mo></mo></mover><mi mathvariant="italic">BR</mi></msub></mrow></mtd><mtd><msub><mrow><mrow><mo>(</mo><msub><mi>ω</mi><mi mathvariant="italic">RI</mi></msub><mo>+</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub></mrow><munder><mrow><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mrow><mo>)</mo></mrow><mrow><mo>]</mo></mrow><mo>+</mo><msub><mi mathvariant="italic">J</mi><mi mathvariant="italic">s</mi></msub><mo>⁢</mo><mi>ω</mi></mrow><mo>}</mo></munder></mrow><mi mathvariant="italic">RI</mi></msub><mo>+</mo><msub><mi mathvariant="italic">H</mi><mi mathvariant="italic">w</mi></msub></mtd><mtd><mi mathvariant="normal"> </mi></mtd></mtr><mtr><mtd><mi mathvariant="normal"> </mi></mtd><mtd columnalign="left"><mo>≈</mo><msub><mi mathvariant="italic">J</mi><mi mathvariant="italic">s</mi></msub><mo>⁢</mo><mfenced><mo>-</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>+</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>+</mo><msub><mi mathvariant="italic">ce</mi><mi>B</mi></msub></mfenced></mtd><mtd><mo>+</mo><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">s</mi></msub></mtd><mtd><mi> there </mi><msub><mi mathvariant="normal">φ</mi><mi>BR</mi></msub><mo>⁢</mo><mi>small </mi></mtd></mtr></mtable></math><img file="EP1514799B1_D0002.tif" /></maths> With<dl id="dl0003" compact="compact"><dt>J_s:</dt><dd>Inertia momentum matrix of spacecraft</dd><dt>ω_RI:</dt><dd>Rotation speed of the reference system compared to the inertial system</dd><dt>hw:</dt><dd>swirl content stored in the swirl wheels (= wheel spin)</dd></dl>First, the rotational speed component c is calculated. Since the angle between measuring and reference vectors is independent of the selected coordinate system, the following applies<maths id="math0003"><math display="block"><mrow><mo mathvariant="normal">(</mo><mn mathvariant="normal">3</mn><mo mathvariant="normal">)</mo><mi mathvariant="normal">   </mi><msup><msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">B</mi></msub><mi mathvariant="normal">T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">B</mi></msub><mo mathvariant="normal">=</mo><msup><msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">R</mi></msub><mi mathvariant="normal">T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">R</mi></msub></mrow><mo mathvariant="normal">=</mo><msup><msub><mi mathvariant="normal">e</mi><mn>0</mn></msub><mi mathvariant="normal">T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">H</mi><mn>0</mn></msub></math><img file="EP1514799B1_D0003.tif" /></maths>in which <maths id="math0004"><math display="block"><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">R</mi></msub><mo mathvariant="normal">=</mo><msup><mi mathvariant="normal">T</mi><mrow><mi mathvariant="normal">R</mi><mo>⁢</mo><mn mathvariant="normal">0</mn></mrow></msup><mo>⁢</mo><msub><mi mathvariant="normal">H</mi><mn mathvariant="normal">0</mn></msub></math><img file="EP1514799B1_D0004.tif" /></maths>With<dl id="dl0004" compact="compact"><dt>T^R0:</dt><dd>Transformation matrix between reference and orbit system; It describes the desired position of the spacecraft and is usually given by bias values, which can also be time-varying.</dd><dt>H₀:</dt><dd>as assumed assumed spin of the spacecraft in the orbit system</dd><dt>e₀:</dt><dd>Measuring direction in the orbit system, known from orbit data</dd></dl>From (3) follows with (1) and (2), 3rd line immediately<maths id="math0005"><math display="block"><mfenced><mn>4</mn></mfenced><mi mathvariant="italic">   c</mi><mo>=</mo><mfrac><mrow><msubsup><mi>e</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>H</mi><mn>0</mn></msub><mo>-</mo><msubsup><mi>e</mi><mi>B</mi><mi>T</mi></msubsup><mo>⁢</mo><mfenced><msub><mi>H</mi><mi>s</mi></msub><mo>-</mo><msub><mi>J</mi><mi>s</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>+</mo><msub><mi>J</mi><mi>s</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub></mfenced></mrow><mrow><msubsup><mi>e</mi><mi>B</mi><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>J</mi><mi>s</mi></msub><mo>⁢</mo><msub><mi>e</mi><mi>B</mi></msub></mrow></mfrac></math><img file="EP1514799B1_D0005.tif" /></maths>Since the term J_s (ω_RI + <i>é_Rė_R</i>) by the choice of Raddralls small compared to h_w stays, is the term <i>J_s</i>φ<i>_BR</i> (ω<i>_RI</i> + <i>é_Rė_R</i>) of second order small, whereby h_B practically independent of φ_BR and thus T^BR becomes. Except for the determination of h_B By way of c according to equation (2), (4), it is alternatively possible to use an observer or a Kalman filter such that the vector h reduced by the term with the unknown c as the measured variable h_B used. the equations are:<maths id="math0006"><math display="block"><mtable><mtr><mtd><mrow><mo mathvariant="normal">(</mo><mn mathvariant="normal">2</mn><mo>⁢</mo><mi mathvariant="normal">a</mi><mo mathvariant="normal">)</mo><mi>   </mi><mo>(</mo><mi mathvariant="italic">aa</mi><mo>)</mo><msub><mi>H</mi><mi>m</mi></msub><mo>=</mo><msub><mi>J</mi><mi>s</mi></msub><mrow><mo>(</mo><mo>-</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>B</mi></msub><mo>+</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>⁢</mo><msub><mover><mi>e</mi><mo></mo></mover><mi>R</mi></msub><mo>)</mo><msub><mi>H</mi><mi>s</mi></msub></mrow></mrow></mtd></mtr><mtr><mtd><mfenced><mi mathvariant="italic">bb</mi></mfenced><mo>⁢</mo><msub><mover><mi>ω</mi><mo>^</mo></mover><mi mathvariant="italic">BI</mi></msub><mo>=</mo><msubsup><mi>J</mi><mi>s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mo>⁢</mo><mfenced><mi>z</mi><mo>-</mo><msub><mi>H</mi><mi>w</mi></msub></mfenced></mtd></mtr><mtr><mtd><mfenced><mi mathvariant="italic">cc</mi></mfenced><mo>⁢</mo><mover><mi mathvariant="italic">z</mi><mo mathvariant="italic"></mo></mover><mo>+</mo><msub><mover><mover><mi>ω</mi><mo>^</mo></mover><mo></mo></mover><mi mathvariant="italic">BI</mi></msub><mo>⋅</mo><mi>z</mi><mo>=</mo><mi>K</mi><mo>⁢</mo><mfenced><msub><mi>H</mi><mi>m</mi></msub><mo>-</mo><mi>z</mi></mfenced></mtd></mtr></mtable></math><img file="EP1514799B1_D0006.tif" /></maths> With<dl id="dl0005" compact="compact"><dt>z:</dt><dd>Estimation vector for h_B</dd><dt>K:</dt><dd>Gain matrix, e.g. B. "Kalman" reinforcement</dd></dl>In both cases, the searched matrix T^BR from the vector pairs e_B/ e_n and h_B/H_n or z / h_R be determined with one of the well-known methods. There are thus position and speed deviations with respect to the reference system, with which the spacecraft can be aligned in a known manner by connecting these two shares to the actuators with respect to the reference system. To explain the practical procedure, consider an earth-oriented spacecraft on a circular path. For orbit twist h₀ The simple relationship applies here <maths id="math0007"><math display="block"><mfenced><mn>5</mn></mfenced><mi> </mi><msub><mover><mi>H</mi><mo></mo></mover><mn>0</mn></msub><mo>=</mo><mo>-</mo><msub><mover><mi>ω</mi><mo></mo></mover><mn>0</mn></msub><mo>⁢</mo><msub><mi>H</mi><mn>0</mn></msub><mo>+</mo><msubsup><mi>t</mi><mi>s</mi><mfenced><mn>0</mn></mfenced></msubsup></math><img file="EP1514799B1_D0007.tif" /></maths>With<maths id="math0008"><math display="block"><msub><mi>ω</mi><mn>0</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi mathvariant="italic">w</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math><img file="EP1514799B1_D0008.tif" /></maths><maths id="math0009"><math display="inline"><msubsup><mi>t</mi><mi>s</mi><mfenced><mn>0</mn></mfenced></msubsup></math><img file="EP1514799B1_D0009.tif" /></maths>: External moments on the spacecraft, expressed in the orbit system</li></ul>

As already mentioned, there are two approaches for determining h₀(t) to:<ul id="ul0003" list-style="none"><li>a) Propagation according to equation (5): For this purpose, in sections optical 3-axis reference, z. B. from a sun sensor needed to the unknown proportions in t_s⁽⁰⁾ (eg solar disturbances) during these phases and h₀ initialize to an initial value at the end of these phases</li><li>b) Tracking a reference model for h₀: This will be charged additionally<maths id="math0010"><math display="block"><mtable><mtr><mtd><mo>(</mo><mn>6</mn><mo>)</mo></mtd><mtd columnalign="left"><msub><mtable><mtr><mtd><mtext> </mtext><mover><mtable><mtr><mtd><mtext> </mtext><mover><mi>H</mi><mo>^</mo></mover></mtd></mtr></mtable><mo></mo></mover></mtd></mtr></mtable><mn>01</mn></msub><mo>=</mo><msub><mi>w</mi><mn>0</mn></msub><mo>⁢</mo><msub><mover><mi>H</mi><mo>^</mo></mover><mn>03</mn></msub><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mo>⁢</mo><mfenced><msub><mi>H</mi><mn>03</mn></msub><mo>-</mo><msub><mover><mi>H</mi><mo>^</mo></mover><mn>03</mn></msub></mfenced><mo>+</mo><msub><mover><mi>t</mi><mo>^</mo></mover><mn>1</mn></msub><mtext> </mtext></mtd></mtr><mtr><mtd><mi mathvariant="normal"> </mi></mtd><mtd columnalign="left"><msub><mtable><mtr><mtd><mover><mtable><mtr><mtd><mover><mi>H</mi><mo>^</mo></mover></mtd></mtr></mtable><mo></mo></mover></mtd></mtr></mtable><mn>03</mn></msub><mo>=</mo><mo>+</mo><msub><mi>w</mi><mn>0</mn></msub><mo>⁢</mo><msub><mover><mi>H</mi><mo>^</mo></mover><mn>01</mn></msub><mo>+</mo><msub><mi>K</mi><mn>3</mn></msub><mo>⁢</mo><mfenced><msub><mi>H</mi><mn>03</mn></msub><mo>-</mo><msub><mover><mi>H</mi><mo>^</mo></mover><mn>03</mn></msub></mfenced><mo>+</mo><msub><mover><mi>t</mi><mo>^</mo></mover><mn>3</mn></msub></mtd></mtr></mtable><mtext> </mtext></math><img file="EP1514799B1_D0010.tif" /></maths><dl id="dl0006" compact="compact"><dt>H₀₃ = e_B^T<i>H_B</i>:</dt><dd>there e₀ = e₃ at earth reference</dd><dt><i>H_oi</i></dt><dd>Estimate for h_0i</dd><dt>K_i:</dt><dd>Switch-on values, eg K₁ = 3w₀, K₃ = 4 | w₀|</dd><dt><i>t_i</i>:</dt><dd>Estimates for t_si⁽⁰⁾</dd></dl></li></ul>

Equation (6) ensures that for t_si⁽⁰⁾ = <i>t_i</i> the estimates <i>H</i>_0i against <i>H</i>_0i converge; for the given values ​​of K_i This is the case after an orbit. The missing 2nd component of<i>H</i>₀ is calculated according to<maths id="math0011"><math display="block"><msub><mover><mi>H</mi><mo>^</mo></mover><mn>02</mn></msub><mo>=</mo><msup><mfenced><msubsup><mi>H</mi><mi>B</mi><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>H</mi><mi>B</mi></msub><mo>-</mo><msup><mfenced><msubsup><mi>e</mi><mi>B</mi><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>H</mi><mi>B</mi></msub></mfenced><mn>2</mn></msup><mo>-</mo><msup><msub><mover><mi>H</mi><mo>^</mo></mover><mn>01</mn></msub><mn>2</mn></msup></mfenced><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msub><mi mathvariant="italic">signw</mi><mn>0</mn></msub></math><img file="EP1514799B1_D0011.tif" /></maths> ie the nominal wheel spin h_w is chosen such that the 2nd component of its transformation in the orbit system has the same sign as the orbitate w₀ Has.

The essential advantage of the method described lies in the fact that, irrespective of the time-variant desired orientation with respect to the orbit system (for example in so-called yaw steering), all computation steps remain unchanged since only the matrix T^R0 is selected accordingly. In the case of the methods mentioned at the outset, extensive calculations for compensation terms or even time-variable dynamics for the observer are accepted. Furthermore, various types of sensors (eg earth sensor, magnetometer) can be used as sensors.

These advantages make it possible to use the method in all classes of spacecraft (LEO, MEO, GEO).

1 shows as a specific embodiment a block diagram of components that can be used to implement the method described here:

From a known orbital position of the spacecraft (for example by temporal propagation of initial position and temporal position change values ​​of the spacecraft, ideally taking into account external disturbances, by tracking a train model or by spacecraft internal or external position measurements such as GPS), which is determined by means of a web position device 11, is using a stored orbit model in a first Orbitmodell device 1, the reference unit vector e<i>_O</i> determined and a means for longitudinal rate determination 2 for the determination of c according to equation (4) supplied. However, this device 2 is also the means of the known, stored transformation matrix T<i>^RO</i> by a transformation device 3, transformed into the reference system unit vector e<i>_R</i>, as well as its time derivative, obtained via the differentiator 4, respectively. The unit vector e<i>_R</i> is further fed into the module for swirl determination 5, as well as in the module for determining the position 6.

With the aid of a sensor 9 (for example, an earth sensor, a magnetometer) shown in Fig. 1, the unit vector e<i>_B</i> determined and the modules for longitudinal rate determination 2, for spin determination 5, and for determining the position 6, respectively. The two first-mentioned modules are also the difference in 7, derived time derivation of e<i>_B</i> fed.

From measurement signals of a measuring device 10 of swirl wheels, not shown in Fig. 1 for reasons of simplicity, the swirl vector of the wheels, h<i>_w</i>, won. This is fed into the modules for longitudinal rate determination 2 and for swirl determination 5. The longitudinal rate c determined in module 2 is fed to the swirl determination module 5, the result of which, h<i>_B</i>, in turn, in the module for determining the position 6, as well as in a reference model module for the Orbitdrall 8, is fed. This reference model module 8 is also supplied with estimates i for external disturbance torques.

The result vector of the reference model module 8, <i>H</i>₀, is fed back into the longitudinal rate determination module 2, and transformed into the reference system via the module 3 shown again in FIG. 1 below for reasons of clarity and the result, namely the total spin vector h<i>_R</i> in the reference coordinate system, fed to the position determination module 6. This module 6 finally determines according to the above-mentioned method from the two vector pairs e supplied to it<i>_B</i>, e<i>_R</i>, H<i>_B</i>, H<i>_R</i> the satellite matrix describing transformation matrix T<i>^BR</i>, This transformation matrix describes the transformation between body-fixed coordinate system and reference coordinate system. Likewise, this transformation matrix describes the deviation of the total spin from the reference total spin.