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Aas 09-123 Rigid Body in Er Tia Es Ti Ma Tion with Ap Pli Ca Tions to the Cap

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Aas 09-123 Rigid Body in Er Tia Es Ti Ma Tion with Ap Pli Ca Tions to the Cap AAS 09-123 RIGID BODY IN ER TIA ES TI MA TION WITH AP PLI CA TIONS TO THE CAP TURE OF A TUM BLING SAT EL LITE Dan iel Sheinfeld* and Ste phen Rock† A frame work for rigid-body in er tia es ti ma tion is pre sented which is gen eral and may be used for any rigid body un der go ing ei ther torque-free or non- torque-free motion. It is ap plied here to the case of a tum bling satel lite. In - cluded are a geo met ric in ter pre ta tion of the es ti ma tion prob lem which pro vides an in tu itive un der stand ing of when to ex pect good es ti ma tion re sults and sim u- la tion re sults to dem on strate the vi a bil ity of the method. INTRODUCTION A critical issue that must be addressed to enable on-orbit servicing of satellites is how to cap- ture and de-spin a satellite that may be tumbling. A satellite can lose attitude control and begin to tumble for any number of reasons: thruster failure, avionics failure, failed docking attempts, colli- sion with space debris, etc. In general, it cannot be assumed that the tumbling motion is benign. De-spinning is accomplished by applying forces to the tumbling satellite, and knowledge of the inertia properties of the satellite is necessary to enable the calculation of efficient and/or op- timal control forces and moments to accomplish this task (the satellite is assumed to be rigid). However in general, these properties may not be well known or they may be entirely unknown a priori so a means of producing estimates of them is required. Further, the problem is com- pounded by the fact that a damaged satellite is likely to be uncooperative or unable to cooperate (meaning that the satellite cannot transmit any information about itself including inertia proper- ties, state information, etc.). This paper presents a two-part inertia estimation procedure which is motivated by a two-step de-spin strategy. In the first step, the servicing satellite tracks the target and positions itself with respect to the target (but does not make contact). For example, the servicing satellite may posi- tion itself along the angular momentum vector of the target. In the second step, the servicing sa- tellite applies forces and moments to the target (e.g. by making contact with a manipulator) to accomplish the de-spin. The tracking task requires only that the motion of the tumbling satellite be estimated, which in turn requires knowledge only of the ratios of the moments and products of inertia and the direction of the constant angular momentum vector (not their absolute values). Generating estimates of these quantities is the first part of the inertia estimation procedure. The manipulation task requires knowledge of the scale factors on the moments and products of inertia * Ph.D. Can di date, De part ment of Aeronau tics and As tro nau tics, Stan ford Uni versity, Durand Build ing Room 250, 496 Lomita Mall, Stan ford, Cal i for nia 94305, USA. †Pro fes sor, De part ment of Aero nau tics and Astro nau tics, Stan ford Uni versity, Durand Build ing Room 250, 496 Lomita Mall, Stan ford, Cal i for nia 94305, USA. 343 and the total mass. Generating estimates of these scale factors is the second part of the procedure. Although it is applied to a tumbling satellite here, the procedure is generic and may be used to estimate the inertia properties of any rigid object undergoing either torque-free or non-torque-free motion. Note that the estimation of satellite inertia properties has been addressed by others in the lite- rature, however, the procedure presented here yields the results in a particularly convenient form for application to the two-step de-spin strategy.2,5 All of these techniques (including this one) are based on exploiting the conservation of (or change in) angular momentum. The remainder of the paper is organized as follows: The tracking step of the de-spin strategy is discussed in detail. The estimation procedure is then presented. Concise necessary conditions for a solution to exist are then provided. This is followed by a geometric interpretation of the inertia matrix estimation problem which provides an intuitive understanding of when good estimation results are expected. Simulation results are then presented to demonstrate the viability of the es- timation methods with a focus on the case of a tumbling satellite undergoing torque-free motion. Sensing Various sensor data are assumed to be available to the algorithms given below. Sensors which provide these data either exist or are under development. For part one of the procedure, sensors which can track features on the rigid body and provide the attitude rate and attitude of the rigid body are necessary. An example of this type of sensor is the system implemented by Zampato, et. al.6 This system tracks features on a rotating rigid body using a stereo camera pair and outputs attitude and attitude rate. Another example is the algorithm developed by Dubowsky and Lichter which produces a measurement of attitude from a 3D reconstruction of the body using range da- ta.1 If the second part of the procedure is used, measurements of all of the torques and forces be- ing applied to the object are also necessary. TRACKING In order to de-spin a tumbling satellite, forces and moments must be applied at a grapple point or some safe application point. However, torque-free motion can be quite complicated since the inertial orientation of a torque-free rigid body is not periodic in general. Figure 1. Inertial Path of a Point on a Torque-Free Rigid Body. 344 Figure 2. Path of a Point on a Torque-Free Rigid Body as Viewed by a Spinning Observer. Figure 1 shows an example of the path of a point on a torque-free rigid body as viewed by an inertial observer. Attempting to interact with the tumbling body while the point of interest is moving in a complicated manner is a difficult task. However, general torque-free motion has an element which is periodic, and this element can be exploited to simplify the motion of the tum- bling body relative to an observer. For example, Figure 2 shows the observed path of a point on the body in Figure 1 as viewed by an observer spinning about the angular momentum vector, h, at a particular constant rate (described below). Figure 3. Poinsot Construction. The periodic element can be understood using the standard Poinsot construction which is pre- sented in Figure 3.3 The tip of the angular velocity vector, Ȧ, is the contact point between the invariable plane and the inertia ellipsoid. As the body tumbles, the contact point traces out the polhode on the inertia ellipsoid. The polhode is a closed curve, so each point on the polhode (which is on the inertia ellipsoid) contacts the invariable plane repeatedly as the body tumbles. 345 The period of these repeated contacts is called the polhode period, p, and is given in several clas- sic texts as: III1** 2 3 p 4** K k 2 I3 I 2 *** M 2 E I 1 ʌ 2 du K k (1) ³ 2 2 0 1 k* sin u 2 2 I2 I 1 *** 2 E I3 M k 2 I3 I 2 *** M 2 E I 1 where I1, I2, I3 are the principal moments of inertia with I1 I2 I3, M is the magnitude of the an- gular momentum vector, E is the kinetic energy due to rotation, and K(k) is the complete elliptic 2 2 integral of the first kind. Equation (1) holds if M >2*E*I2. If M <2*E*I2, then I1 and I3 are swapped in Equation (1).4 The periodic element is seen by observing the body at intervals of one polhode period. The first observation is at time t. At time t, a single point, q, on the inertia ellipsoid is in contact with the invariable plane. After time t, the body continues to tumble and q moves away from the inva- riable plane. One polhode period later at time t + p, q is again in contact with the invariable plane, and since the center of the inertia ellipsoid and the invariable plane are fixed in space, the inertia ellipsoid is in its original configuration with respect to inertial space plus a pure rotation about h. The above implies that the net motion of the body after one polhode period is a pure rotation about h as viewed by an inertial observer. This implies that the orientation of the inertia ellipsoid with respect to h is the same at times t and t + p. Therefore, the motion (in inertial space) during the polhode period after time t + p is identical to the motion during the previous polhode period but it starts with an initial rotation about h. This is the periodic element of general torque-free motion. Figure 4. Invariable Plane. 346 The angle of rotation of the body after one polhode period, Į, is the angle swept out by the projection of Ȧ onto the invariable plane (this angle can be greater than 2ʌ in general). This is depicted in Figure 4 which shows the invariable plane at two times separated by one polhode pe- riod. At time t, the body is in a configuration such that point ct is the contact point between the inertia ellipsoid and the invariable plane and the vector Ȗt is the projection of Ȧ onto the invaria- ble plane.
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