it is recalled that the lines of the magnetic field of the earth t have the general appearance shown by the lines b in fig1 in a plane that contains the poles . the use of a b dot relationship makes it possible to absorb the rotational energy presented by the satellite , after separating from the launcher or in the event of an incident , until the satellite becomes fixed relative to the lines of the earth &# 39 ; s magnetic field , which means that the satellite is then rotating at twice the angular frequency ω 0 of its orbit about the normal to the plane of its orbit . in order to avoid interactions between the magneto - couplers and the magnetometer , it is possible to take the measurements and to drive the magneto - couplers in alternation , for example . by way of example , the description below relates to a satellite whose general configuration is as shown in fig2 . the satellite comprises a casing 34 having solar generators 36 mounted thereon in a fixed orientation . an axis in the plane of the solar generators is written ys and the axis normal to the plane of the solar generators is written zs . the casing carries a three - axis magnetometer 38 and magneto - couplers ( not shown ) enabling moments to be applied to the satellite by interacting with the earth &# 39 ; s magnetic field . the satellite also carries a sun sensor 40 . it will be assumed that the satellite is to be placed on a low polar orbit that is heliosynchronous and that local time for its ascending node is close to noon - midnight , with the spin axis of the satellite being oriented towards the sun so as to obtain maximum illumination of its solar generators . the successive steps of a sequence for bringing the satellite into its permanent configuration on orbit are described below . 1 . a first step serves to reduce undesired speeds of rotation , to cause the satellite to spin , to bring the spin axis towards the normal to the plane of the orbit , and to damp nutation . to do this , relationship ( 2 ) is implemented as soon as the satellite separates from the launcher ( or on leaving a survival mode ). as mentioned above , by dissipating energy , using the relationship reduces the angular velocities due , for example , to angular momentum being transferred on separation . the bias { dot over ( b )} i introduced into the relationship ( 2 ) makes it possible to cause the angular velocities of the satellite to converge quickly on a constant angular velocity about the selected axis ( the axis zs for an appropriate value of { dot over ( b )} i ). establishing spin is performed in parallel as soon as the velocities are reduced . the angular velocity reached corresponds to the sum ω i + 2ω 0 , i . e . the sum of the set speed of rotation in satellite axes plus the speed of rotation of the magnetic field lines around the orbit . relationship ( 2 ) guarantees that the axis will finally be in alignment with the normal to the plane of the orbit without any risk of inverse capture : the stable converged state corresponds to the situation in which the satellite is turning at the highest possible angular velocity ( 2ω 0 + ω i , not − 2ω 0 + ω i ) about its spin axis . the progress of satellite attitude during its first orbit can be as shown in fig3 . in general , on injection at 10 a , the orientation and the angular velocities ( spin and nutation ) of the satellite are ill - defined . application of relationship ( 2 ) leads to a reduction in the stray angular velocities , up to state 10 b . progressively , the satellite is caused to turn about a selected spin axis at an angular frequency 2ω 0 ( positions 10 c to 10 d )+ ωi . in the final situation , in the daytime phase , the sun describes a circle of 60 ° angular radius about the aiming axis of the solar sensor 40 for a 10 : 00 am or a 2 : 00 pm orbit . under such circumstances , a single sensor suffices during the second step . if the angular radius exceeds 80 ° ( as is the case for an orbit for which local time of the node lies in the range 11 : 20 am to 12 : 40 pm ) it is no longer certain that the sun will be seen by the sensor whose aiming axis is orthogonal to the plane of the solar generators . an additional sensor then needs to be provided on one face of the body in order to know in which direction to apply the torque for erecting the spin axis . in the common case where the spin axis is not the axis of maximum angular inertia , the gain k in relationship ( 2 ) must be sufficient to ensure that the system remains stable . for this purpose , the control passband must be greater than the unstable pole of the free dynamics . to avoid nutation diverging , possibly as far as “ flat ” nutation , it will often be necessary to use a high value for k . 2 . the second step is constituted by “ righting ” or “ erecting ” the spin axis . it is controlled responsive to signals representing the angular difference ( in two directions that are generally orthogonal ) between the direction to the sun and zs ( normal to the plane of the solar generators ). [ 0050 ] fig4 is a diagram showing an acquisition mode that uses a solar sensor 40 having a wide angle α , providing a difference signal relative to the direction s to the sun in two directions so long as the satellite is not in the earth &# 39 ; s shadow . a second sensor , on a side face 44 , makes it possible to obtain the same kind of signal when the sun leaves this field , for orbits in the range 11 : 20 am to 12 : 40 pm . to reorient the spin axis , the magneto - couplers are energized to generate a torque perpendicular to the spin axis zs ( so as to avoid disturbing the modulus of the angular momentum vector ) and towards the sun . it is only possible to provide torque in the plane normal to the direction of the local magnetic field . torque is thus generated which lies simultaneously in the plane normal to the magnetic field { right arrow over ( b )} and normal to { right arrow over ( h )}. the direction in which action is applied is selected to be that which causes the spin axis to approach the direction of the sun , and the selected amplitude is an increasing function of the difference between the spin axis and the direction of the sun and of the ground ( e . g . proportional thereto ): c 1 − κ ( u sun . n b ) n b ( 4 ) where κ is a gain , u sun is the direction of the sun in the axes of the satellite , and n b is a direction vector of the intersection between the plane normal to b and the plane normal to h , i . e . the plane ( x sat , y sat ) where y sat is along the orbital normal in geocentric pointing . in all cases , erection takes place progressively for the usual values of spin rates and inertias , with erecting generally taking place over a single orbit . the torque used for “ righting ” or “ erecting ” the spin axis tends to excite nutation . relationship ( 2 ) ensures that nutation is damped , both in the daylight stage and during an eclipse , with damping being better during eclipse since there is then no excitation . 3 . instead of waiting until the step of reducing velocities and aligning the spin axis with the normal to the orbit has converged ( typically 2 to 3 orbits ) before starting use of the relationship for erecting the spin axis , it is possible to apply both relationships in parallel : if the sun is visible in the field of the solar sensor prior to convergence in the “ b spin ” stage , then axis erecting can already begin . thus , the total time required for acquisition is often equal to the time required for reducing angular velocities . the complete sequence thus takes place naturally without any transition logic . 4 . it is also possible to trigger the relationship erecting the spin axis at the same time as the step for reducing velocities and aligning the spin axis with the normal to the orbit . relationship ( 2 ) is applicable for bringing the satellite into survival mode and for keeping it there , under all circumstances . in the case shown in fig5 of survival on a heliosynchronous polar orbit close to 6 : 00 am - 6 : 00 pm , this relationship applied to the axis normal to the solar generators ensures that velocities are reduced , that dynamic behavior converges , that the sun is acquired , and that a stable survival state is achieved without any internal angular momentum vector and without a solar sensor . adjustment of the desired spin velocity ( 2ω 0 + ω i ) in pitching ( about the zs axis ) makes it possible to optimize performance depending on external disturbances . this survival mode is particularly robust and makes use only of the magnetic sensors and magnetics couples and of a control relationship that is extremely simple , without requiring any logic or any transition threshold . the solar panels are in a “ windmill ” configuration , i . e . they lie substantially in the plane of the orbit . for a heliosynchronous polar orbit close to noon - midnight ( as shown in fig3 ) or for equatorial orbits , the spin axis can no longer be aligned towards the sun . the spin axis aligns naturally with the normal to the orbit in application of relationship ( 2 ), while the sun is rather in the plane of the orbit . under such circumstances , the selected spin axis is one of the two axes in the plane of the solar generators . the converged state corresponds to a “ barbecue ” configuration in which the generators point towards the sun periodically ( with a minimum angle of incidence corresponding to the angle between the orbital plane and the direction to the sun ). nevertheless , if the power budget is unsatisfactory , and if it is desired to have a converged state similar to the “ windmill ” mode of a 6 : 00 am - 6 : 00 pm orbit , relationship ( 2 ) on its own is not sufficient and it is necessary to return to the above - mentioned erecting relationship . [ 0066 ] fig6 shows one possible theoretical structure for a control loop . this loop has a three - axis magnetometer 12 supplying output signals which are filtered at 14 to eliminate interference and transients . a computing unit 16 determines the time derivative of the measurement and multiplies it by a gain k . to obtain equal damping time constants about all three axes , the gain k for each axis is advantageously standardized by the moment of inertia about the axis in question . the term k ×{ dot over ( b )} i is subtracted at 18 . the control torque for application by the magneto - couplers is computed at 22 . the spin - imparting torque is computed using formula ( 2 ). it is equal to the vector product of the magnetic moment { right arrow over ( m )} generated by the magneto - couplers 26 multiplied by the actual magnetic field { right arrow over ( b )}. if the required torque corresponds to current that is so high that the magneto - couplers do not have a linear characteristic , then an additional computation unit can be provided to take account of saturation and limit the current which is applied to the magneto - couplers 26 .