AS3010: Introduction to Space Technology

Transcription

1 AS3010: Introduction to Space Technology L E C T U R E 22 Part B, Lecture April, 2017 C O N T E N T S Attitude stabilization passive and active. Actuators for three axis or active stabilization. Recall that we have looked at the stability of rigid body rotation in the last class. The main result was that while rotations about major and minor axes are stable, rotation about intermediate axis is unstable. The Explorer case: Now, recall that in one of the introductory lectures, we discussed the case of Explorer I that was spin stabilized by giving a rotation about its minor axis. A few hours after the deployment, Explorer I started tumbling and finally started to rotate about its major axis. That is, it went to a flat spin. From our analysis, we saw that the rotation about minor axis is supposed to be stable. Then, why did the spin about minor axis, in the case of Explorer I, become unstable? There was a key assumption in all the analysis that we did that the body is rigid. However, no body is perfectly rigid, and especially not Explorer I which had four flexible antennas as shown in the figure below. A flexible body vibrates, and the vibrational modes dissipate energy in form of heat. Therefore, the kinetic energy is no more conserved if the body is not rigid. However, the angular momentum is conserved. h = I max ω major = I min ω minor For a given h, the satellite will have different angular velocities depending on whether it is rotating about major or minor axis, and can be obtained as ω major = h, ω minor = h I max I min

2 Now, we can compute the rotational kinetic energy of the satellite with angular momentum h if it were to rotate about its major axis as follows. T major = 1 2 I maxω 2 major ( h ) 2 = 1 2 I max = 1 2 h 2 I max I max Similarly, the kinetic energy of the satellite, if it were to rotate about the minor axis can be obtained as And, we see that T minor = 1 2 h 2 I min T minor > T major In case of energy dissipation, that is for flexible satellites, the satellite will have the tendency to go to the lower energy state which is the rotation about the major axis. This is just a heuristic explanation for why the spin of Explorer I about minor axis became unstable. The take away is if you want to spin stabilize your satellite, then spin it about its major axis. In other words, spin stabilized satellites should look like discs rather than pencils! Attitude stabilization In the presence of disturbance torques, something should be done to maintain or stabilize attitude or orientation of the satellite. The attitude stabilization in satellites can be classified as passive and active. The active stabilization is also called 3-axis stabilization. One of the primary means of passive stabilization is spin stabilization. We have already looked at how a stable spin stabilization can be achieved by providing a rotation about major axis. Sometimes it may not be good to spin the entire satellite one may not like the antenna to spin along with the satellite. In such a case, one could employ dual spin stabilization in which the upper part of the satellite will rotate to stabilize the attitude while the lower part having antenna will not rotate or will rotate at much lower angular speeds. Such a dual spin stabilized satellite is also called a Gyrostat. Another passive attitude stabilization technique is called gravity gradient stabilization. Since the gravitational field of Earth is not uniform (it varies inversely with the square of distance from the center of Earth 1 ) a satellite that is long enough will experience a r2 torque because of the gravity gradient. 2

3 The reason for the gravity gradient torque to exist is because, in a non-uniform gravitational field, the center of mass (CM) of the satellite will be different from its center of gravity (CG). The gravitational force acts through the CG while the centrifugal force acts through CM. If the CM and CG of the satellite are different (which will be as the gravitational field is not uniform), the forces through CM and CG will form a moment couple that will cause the satellite to rotate. This is illustrated in the figure below. Exercise Consider a satellite in the form of a dumbbell as shown in the figure below. Show that the restoring torque acting on the satellite is T = 3 GMmd 2 θ 2 r 3 Assume small θ and use the fact that angular velocity of the satellite is ω = GM r 3. Practically to achieve a gravity gradient based stabilization, the satellites are designed as having long booms as shown in the figure below. 3

4 Note, from the exercise given above, that, the restoring torque due to gravity gradient does not have damping. Therefore, artificial dampers (called libration dampers) are employed in a gravity gradient stabilized satellite to drive the oscillations to zero. Attitude control Active attitude stabilization involves using actuators to actively control the attitude of the satellite. To control the attitude or orientation of a satellite, we need to produce a torque. We will have a quick look at some of the attitude control methods commonly used in satellites. Thrusters/Gas jets: A force not aligned with the center of mass (CM) of a body produces a torque on that body this is the easiest way to produce torque. It is this principle that makes thrusters or gas jets useful as attitude control devices in satellites and spacecraft. A thruster or a gas jet works on the principle of momentum conservation via mass expulsion. A fluid at some velocity is exhausted to produce a reaction thrust which, as not passing through the CM, causes a torque. The mass expelled can be cold gas (compressed and stored), or hot gas (that exhausted after a combustion). Since the thrust requirements are small for attitude control, electric propulsion can also be used in satellites in which ample power is available. A couple of thrusters can be used to produce a torque about one axis as shown in the figure below. Thrusters that produce rotations about each of the 3 principal axes are sufficient to control the orientation of a satellite. On spin stabilized satellites, thrusters are used to (i) control the spin of the satellite, and ii) orient the spin axis. Thruster positioning for control of spin angular velocity is shown in the figure below. 4

5 To change the direction of spin axis, consider a single thruster mounted on the circumference of a spinning satellite, as shown in the figure below, at a distance R from the spin axis. A body fixed axis system for this scenario is shown below. If the thruster produces a force F, then the moment about axis 2 is M = F R Now, due to gyroscopic effect, the satellite will precess about axis 1. Note that since the body-fixed axis is rotating about axis 3, the axis of precession axis 1 is also rotating with respect to an inertial frame. So, appropriate thruster firing strategies need to be employed to re-orient the spin axis along the desired direction. Reaction wheels: Some satellites use reaction wheels, one mounted along each axis, for their attitude control. A reaction wheel arrangement is shown in the figure below. Whenever a torque acts on a satellite imparting it an angular velocity, the reaction wheels will rotate to cancel out the satellite s rotation. Thus reactions wheel work on the principle of conservation of angular momentum. For axis 1, this can be represented as I 1 ω 1 + I R1 ω R1 = 0 5

6 Here I 1 is the total moment of inertia of the satellite about axis 1 (this includes the moment of inertia of the reaction wheel also). I R1 is the moment of inertia about axis 1 of the reaction wheel R 1 alone. The angular velocity ω R1 of the rotor is with respect to the spacecraft/satellite. A disadvantage of using reaction wheels for attitude control is that, if a constant torque is being acted on the satellite, then after some time interval, the motor rotating the wheel will reach its peak rotational speed leading to the reaction wheels getting saturated. Thus these reaction wheels need periodic de-saturation. A reaction wheel is de-saturated by firing thrusters to create a moment in the opposite direction and bringing the rotation of reaction wheels to zero during that time. Let M be the moment generated by firing the thruster. If M is constant (assuming that the thruster produce constant thrust), and the thrusters are fired for a time T, then the net change in angular momentum (due to firing of thruster) is given by h = T 0 Mdt = MT Since the reaction wheel has an angular momentum I R1 ω R1, the change in angular momentum if it is brought to rest (with respect to the satellite) is h = I R1 ω R1 Therefore, the time for which the thruster needs to be fired to bring its rotation of reaction wheel to zero without imparting an angular velocity to the satellite is given by T = I R 1 ω R1 M Control moment gyro: Control moment gyro is another attitude control mechanism. It typically consist of gimbaled momentum wheels. These wheels, rotate with a constant angular speed of ω R and thus have an angular momentum I R ω R. The gimbals are controlled by motors. Thus, using these motors, the direction of the spin axis of a momentum wheel can be changed. A change in the direction of spin axis implies a change in the momentum vector. A change in the momentum vector causes a torque (reaction gyroscopic torque) and a resultant rotation. This is explained in the figure below. 6

7 For simplicity, we consider only one momentum wheel with moment of inertia I R with an angular velocity of ω R about axis 1 as shown in the figure. Thus, we have an angular momentum along axis 1 given by h = I R ω R î. Now, if the gimbaled momentum wheel is rotated about axis 2 by a small angle θ, then the previous and the current angular momentum vectors will be on the plane 1-3 (as illustrated in the figure), and the change in angular momentum can be easily seen to be h sin θ ˆk. As the change in angular momentum is about axis 3, a rotation will occur about axis 3. Magnetic torquer: A magnetic torquer works on the principle that a current carrying coil creates a magnetic dipole which, when placed in a magnetic field, experiences a moment perpendicular both to the direction of the dipole as well as the direction of the external magnetic field 1. Let D be the dipole created by the current carrying coil, and let B be local magnetic field of Earth. Then, the coil experiences a moment given by M = D B Using three such current carrying coils, the orientation of the satellite about any axis can be controlled. Yo-yo mechanism Yo-yo mechanism is a simple and elegant way to completely de-spin a spacecraft. Sometimes, it becomes necessary to spin a spacecraft to high angular rates to provide it an angular momentum. This angular momentum will allow the spacecraft to retain the desired direction of motion while imparting a V even in the presence of small thrust misalignments. The spin is unnecessary after the firing of the rocket motor, and it is a waste of fuel to fire thrusters to de-spin it. A cheap, simple, and elegant alternative is to use a yo-yo mechanism to completely de-spin the satellite. A yo-yo mechanism consists of two small masses tied at the end of a string and wound around the satellite as shown below. The masses are initially clamped and kept to prevent it from flying away due to centrifugal force when the spacecraft is put to spin. When the vehicle needs a de-spin, the masses are un-clamped causing it to unwind as shown in the figure below. field. 1 Because of this, a torque cannot be produced if the magnetic torquer is along the external magnetic 7

8 L When it has completely unwound, the strings are cut-off and if the length of string is appropriately chosen, the masses will fly away carrying all the angular momentum rendering the spacecraft to zero spin. Let us do a quick analysis to find out the length of the string required to completely de-spin the satellite. The initial angular momentum (due to satellite and the masses) is h initial = Iω 0 + 2mR 2 ω 0 The final angular momentum is (as the satellite has zero spin and therefore zero angular momentum) h final = mv (2L + 2R) By conservation of angular momentum, we have Iω 0 + 2mR 2 ω 0 = mv (2L + 2R) To solve for L, we need to know V. Toward this, we employ the conservation of kinetic energy. T initial = 1 2 Iω2 0 + mr 2 ω 2 0 and T final = mv 2 Solving, we get I L = R + 2m + R2 What is remarkable about this is that the length of the string for the yo-yo mechanism to completely de-spin the satellite from an initial angular velocity ω 0 is independent of ω 0. Therefore, a yo-yo mechanism can be used to effectively de-spin a spacecraft even in the presence of burn-out uncertainties of the thrusters that have put the vehicle into a spin. 8