Angular Velocity Bounds via Light Curve Glint Duration

Transcription

1 Angular Velocity Bouns via Light Curve Glint Duration Joanna C. Hinks an John L. Crassiis The time uration of specular glint events within a light curve is relate to the angular velocity of the reflecting space object. Upper an lower bouns on the magnitue of the observable part of angular velocity can be erive with some reasonable assumptions. The propose metho is inepenent of object shape or signal perioicity, an only loosely epenent on the choice of mathematical reflection moel. This technique may be valuable in etecting when the overall spin rate of an object has change, or in etermining that a spacecraft that nominally has a fixe attitue is unergoing an attitue maneuver. I. Introuction A major research topic in recent years has been space situational awareness (SSA, which is concerne with the ientification an tracking of all space objects (SOs in orbit aroun Earth. This task faces many challenges, one of the greatest of which is inaequate ata ue to the limite number of available sensors. Consequently, many research efforts focus on extracting as much information as possible from the ata. Of particular interest are techniques which etermine or constrain the SO s attitue or angular velocity. These quantities may help in such tasks as SO ientification or anomaly etection. One commonly available observation that epens strongly on attitue an angular velocity is the history of total reflecte sunlight, or light curve. Light curves have historically been use to investigate SOs in two ways: as part of a shape-moel-base approach, or using moel-inepenent techniques that focus preominantly on signal characteristics. Moelbase approaches require a geometric moel of the shape of the reflecting object, as in Refs Filters or other estimators can calculate the expecte brightness sunlight reflecte from a particular shape an compare those values to the observe light curve. Quantities such as attitue, angular rate, an surface material properties can be estimate. In the absence of reliable or sufficiently etaile moels, however, moel-base approaches may perform very poorly. Signal-base approaches, in contrast, analyze the behavior of the observe light curve without making assumptions about the shape of the SO. Thus they may be very useful in cases where the shape is not well-known a priori. Moel inepenence comes at a cost, however: a significant reuction in the types or precision of quantities that can be estimate. One example of a signal-base approach is the use of simultaneous observations at multiple wavelengths to estimate surface materials. 5, 6 One large family of signal-base algorithms exploits frequency content to spin rate, an in some cases, spin axis. These algorithms inclue, but are not limite to, Fourier analysis, 7 9 wavelet analysis, 5, 6 10, 11 an approaches base on feature perioicity (especially specular glint perioicity. While such methos perform well for tumbling or spin-stabilize objects, they cannot be applie to signals that o not exhibit perioic behavior such as three-axis-stabilize, geostationary objects. Furthermore, shape symmetries may cause the observe frequencies to be greater than the physical frequencies. This paper presents proposes an algorithm that oes not rely on mulispectral ata or on frequency content. It analyzes the time uration (with of iniviual specular glint events, an relates that value to SO angular rate. Reference 1 previously stuie the angular extent of specular glints (as observe via uration or their spatial range of visiblity on Earth. However, that source emphasize the relationship of glint size to facet angle an surface properties, an i not rigorously evelop the connection to angular rate. Postoctoral Assistant, Department of Mechanical & Aerospace Engineering, University at Buffalo, State University of New York, Amherst, NY, jchinks@buffalo.eu. Member AIAA. CUBRC Professor in Space Situational Awareness, Department of Mechanical & Aerospace Engineering, University at Buffalo, State University of New York, Amherst, NY, johnc@buffalo.eu. Fellow AIAA. 1 of 11

2 The major weakness of this paper s algorithm is that it is imprecise; rather than giving a specific estimate of angular rate it merely provies upper an lower bouns on the observable component of that quantity. Even such limite information may prove valuable, however. One coul etect unexpecte attitue motion, ientify changes in overall angular rate, or obtain a rough angular rate estimate that can be use to initialize a filter. The remainer of this paper is organize as follows: Section Iefines the problem geometry an erives the relationship between angular velocity an a quantity herein calle glint rate. In Section III, the relationship between glint rate an the light curve signal (specifically glint uration is evelope. Section IV aresses the role playe by uncertainties in real applications an suggests some implementation guielines. The algorithms are applie to three ifferent ata sets in Section V, an the results are iscusse. Section VI raws some conclusions. II. Glint Geometry an Rate A. Scenario Configuration an Definitions The goal of this paper is to erive bouns on the angular velocity ω, which escribes the rotation of the space object s boy-fixe B frame relative to the inertial I frame. At any given time, the two frames are relate by tje attitue matrix A true, such that the representations of any vector v in the I an B frames are relate by v B = A true v I. It is common to istinguish between synoic angular rate an siereal angular rate. The esire ω is the siereal angular rate; it is inepenent of the observer an escribes only the physical relative motion of the two frames. The synoic angular rate is the name given to the apparent angular rate etermine from the perioicity of a light curve; it aitionally epens on the observer location an the relative motion of the Sun, SO, an observer. This paper s techniques are inepenent of the particular shape of the reflecting space object. They o, however, require that the glint conitions are met: Namely, that the object inclues some sufficiently large flat surface with surface normal vector u n off of which sunlight reflects in a mirrorlike fashion. The object (or at least the currently glinting portion of it is assume to be a rigi boy, an the surface normal vector is assume to be constant in the boy-fixe B frame (written as u B n. The inertial positions of the SO, Sun, an observer are given by the vectors R SO, R sun an R obs, respectively. These vectors, an their erivatives, are assume to be known an available. For the present application, it is most convenient to work in a coorinate system centere on the SO. In such a system, the relative positions of the Sun an the observer are given by r sun = R sun R SO an r obs = R obs R SO. The corresponing line of sight (LOS unit vectors are u obs an u sun. Glints occur when any of the SO s facet normal vectors u n gets very close to the specular irection. The specular irection is aligne with the unit half-vector u h, which bisects the angle between u obs an u sun : u h = u obs + u sun u obs + u sun (1 Another name for u h is the phase angle bisector, because this vector bisects the Sun-SO-observer phase angle. Figure 1 illustrates the surface basis vectors an LOS vectors that enter the light curve glint calculations. Closeness to the glint irection (where u n an u h are aligne is escribe by the angle θ between these two unit vectors. For this paper s purposes, a full glint is efine as one for which θ reaches 0. Practically speaking, most light curve glints are at best mostly full. Recall that the angle θ is relate to the ot prouct between the two vectors by cos θ = u h u n ( B. Glint Rate The uration of a glint event is etermine by how quickly the angle θ approaches a value near zero an then increases. This glint rate, θ, epens on the rates of change of both u n an u h in the inertial frame. Note that the glint rate θ is not to be confuse with the frequency at which glints occur in a perioic light curve. The rate of change of u n is relate to the angular velocity of the B frame relative to the I frame, whereas the rate of change of u h epens on the relative motion of the Sun, SO, an observer. An equation that of 11

3 Figure 1. Reflection Geometry escribes the contributions from each of these sources can be erive as follows. Unless otherwise specifie, erivatives are taken with respect to the inertial (I frame. Start by applying the prouct rule to the ot prouct (u h u n : t (u h u n = ( I t u h u n + u h ( I t u n Equation (3 inclues two unit vector erivatives. When taking the erivative of a unit vector, which may rotate but not stretch, it is necessary to explicitly inclue the normalization. Suppose an arbitrary unit vector u is efine as: u = v (4 v The erivative of u is then given by: ( I uu T t u = v t v (5 Applying Eq. (5 to the unit vector u h as efine in Eq. (1 gives: ( I t u uh u T ( I h = h I u obs + u sun t u obs + t u sun The erivatives of u obs an u sun are in turn given by ( ( I t u uobs u T obs = obs I r obs t r uobs u T obs = obs (V obs V SO r obs (7a ( ( I t u usun u T sun = sun I r sun t r usun u T sun = sun (V sun V SO r sun (7b Returning to Eq. (3, the erivative of u n in the I frame is foun by applying the transport theorem. The boy-frame erivative of the facet normal vector is zero, so only the cross prouct term remains: Substituting Eq. (8 into Eq. (3 yiels: B t u n = t u n + ω u n = ω u n (8 I ( I t (u h u n = t u h ( I = t u h u n + u h (ω u n (3 (6 u n + ω (u n u h (9 3 of 11

4 The secon form of the equation uses an equivalent form of the scalar triple prouct. Also, note that t u h coul be written in terms of known position an velocity vectors using Eqs. (6-(7. It has been left in its original form here for conciseness. Next, ecompose the facet normal u n into a component parallel to u h an a component orthogonal to u h : u n = cos θu h + sin θu p (10 In Eq. (10, the vector u p is a unit vector that lies in the irection of the projection of u n onto the plane orthogonal to u h. For purposes of this algorithm, it is a mathematical construct that oes not nee to be explicitly compute. Substitute Eq. (10 into Eq. (9 an simplify. The simplification exploits the fact that the erivative of a unit vector is orthogonal to the vector itself: u u = 0. I ( I t (u h u n = t u h (cos θu h + sin θu p + ω [(cos θu h + sin θu p u h ] [( I ] = sin θ t u h u p + sin θ [ω (u p u h ] (11 An alternative form of this erivative can be foun by ifferentiating Eq. (: I t (u h u n = t (cos θ = θ sin θ (1 Set Eqs. (11 an (1 equal an cancel the common sin θ term to obtain ( I θ = t u h u p ω (u p u h (13 Finally, rearrange Eq. (13 to isolate the term with ω: ( I θ + t u h u p = ω (u h u p (14 Equation (14 woul be most useful if the attitue (an thus u p were known. In the absence of attitue knowlege, however, it is still possible to use the form of the equation to boun the magnitue of the angular velocity. The unit vectors u h an u p are orthogonal, so u h u p = 1. Let ω x ω (u h u p be the component of ω orthogonal to both u h an u p. (It is also orthogonal to u n. This is the only component of ω that has( an effect on glint rate θ, so it can also be thought of as the observable component of ω. Also, t u I h t u h u p t u h. The upper an lower bouns on ω x are thus given by: ω x max = θ + t u h (15a θ t ω x min = u h if θ > t u h 0 if θ t u (15b h The erivative norm t u h can be calculate using Eqs. (6-(7. The glint rate θ is estimate from the glint uration, as escribe in Section III. III. Angular Velocity from Glint Duration In the absence of attitue control, the time history of the angle θ (t near a glint event is well-approximate by an hyperbola. (Even if attitue control is present, this is likely a goo approximation in many cases. The angle θ (t between u h an u n tens to ecrease at a relatively constant rate β until it nears the glint time t g, transition smoothly through a minimum value of θ g at t g, an then increase at the same constant rate β. For a given set of surface properties, the angular with of glints is relatively constant. It can be specifie by 4 of 11

5 the ege angle θ e, such that glints occur for times such that θ (t θ e. The times corresponing to the glint eges are t e1 an t e, an the overall glint with is t = t e t e1. In cases without goo attitue knowlege, the minimum angle θ g is not known accurately. The true θ hyperbola for a given glint is one of a family of hyperbolae parameterize by the minimum angle θ g, as illustrate in Fig.. For now, assume that θ e an θ g are known. This assumption will be aresse in Figure. Family of hyperbolae for given θ e an t Section IV. The hyperbolic shape of θ (t makes it possible to erive an expression for the erivative θ in terms of θ e, θ g, an the glint with t. The hyperbola θ(t is escribe by or, in a more convenient form, θ (t θ g (t t g (θ g /β = 1 (16 θ (t = θ g + β (t t g (17 One can erive an expression for the asymptotic slope β as a function of θ e, θ g, an the glint with t by substituting the point (t e, θ e into Eq. (17 an solving for β : θ e = θ g + β (t e t g (18 ( θ e θ g β = (t e t g (19 Assuming that t e1 t g = t e t g = 0.5 t, the hyperbola is escribe by θ (t = θg + 4 ( θe θg t (t t g (0 The angular velocity bouns of Eq. (15 require the glint rate θ at some given time t. One cannot use the time of maximum glint, t g, for this purpose, because θ goes to zero an Eq. (14 has a potential singularity (in the case where u n = u h, so that u p is unefine. Instea, the glint ege times t e1 an t e, which are relatively easy to measure in a ata set, are use. The glint rate θ shoul have equal magnitues at the two eges. Therefore, taking the erivative of Eq. (0 an solving for θ: θ θ = 8 ( θe θg t (t t g (1 5 of 11

6 Evaluating Eq. ( at (t e, θ e gives θ = 4 ( θe θg θ t (t t g ( θ (t e = 4 ( θe θg θ e t (t e t g = 4 ( θe θg θ e t = ( θe θg θ e t t = θ e t θ g θ e t For a given θ e, θ g, an measure t, Eq. (3 gives the value of θ that can be substitute into the angular velocity boun equations. (3 IV. Implementation Issues While Eqs. (15 an (3 for calculating angular velocity bouns from glint uration are simple, there are several ifficulties that arise in implementation. First, it may be ifficult to accurately measure the with of a glint event in a real ata set with irregularly space observations, gaps, noise, etc. especially if the glint is a shallow one. Secon, the ege angle θ e epens on the surface material properties of the reflecting object. Such properties may be known poorly, if at all. Finally, the glint angle θ g (the smallest angle between u n an u h for a given glint event cannot be etermine accurately without knowing something about the object shape an attitue. Generally speaking, that information is not available in cases where the angular velocity is completely unknown. A. Incorporating Uncertainty One simple approach to the aforementione issues is to incorporate all the existing sources of uncertainty irectly into the angular velocity bouns. The form of Eq. (14 is such that only the component of ω orthogonal to both u h an u n is at all observable from glint rate, an only upper an lower bouns on that component are possible without attitue knowlege. Suppose that the quantities t, θ e, an θ g are also known only to lie within some range: t min t t max, θ e,min θ e θ e,max, an θ g,min θ g θ g,max. Then the bouning values can be use in Eqs. (15 an (3 as necessary to create the most conservative upper an lower bouns. First, one must take this approach with Eq. (3 to fin minimum an maximum values of θ (t e : θ min (t e = θ e,min t max θ g,max θ e,min t max (4a θ max (t e = θ e,max t min θ g,min θ e,max t min = θ e,max t min The final simplification in Eq. (4b recognizes that the minimum possible value of θ g is zero; this occurs for the case of a full glint. When applying Eq. (4a, one must take care to use a consistent pair of values for θ e,min an θ g,max : For a given glint, θ e > θ g because θ g is by efinition the minimum θ achieve uring the glint. In fact, θ g must be sufficiently less than θ e in orer for any glint to be visible in the ata. Once θ min (t e an θ max (t e have been calculate, these values can be use within the conservative version of Eqs. (15a an (15b: ω x max = θ max + t u h (5a θ min t ω x min = u h if θ min > t u h 0 if θ min (5b t u h (4b 6 of 11

7 B. Selecting θ e an θ g The uncertainty associate with t is largely etermine by the quality of available ata, an it is not ifficult to select reasonable bouns for this quantity. Even with the conservative bouns escribe in Section A, it is necessary to have a reasonable starting point for the angles θ e an θ g if such values are not provie. As escribe in the previous subsection, the choice of θ e an θ g cannot be completely inepenent. The following iscussion is not intene to be the final wor on this subject, but it shoul provie some useful guielines. Glints occur when one of an object s facet normal vectors u n gets close enough to the specular irection u h that bright specular reflection ominates iffuse reflection. The range of angles within which this occurs is known as the specular lobe, an the size of the specular lobe epens on the surface material an surface finish associate with that facet. The more shiny an mirror-like a facet is, the smaller the size of the specular lobe, an vice versa. The threshol angle for specular reflection is efine in this ocument as the glint ege angle θ e. If one knows the reflecting object to be very shiny, a small value of θ e can be use, an a large value can be selecte if the reflecting object is less mirror-like. In practice, it may be necessary to use a relatively small θ e,min an a relatively large θ e,max to account for uncertainty (an in such cases, the angular velocity bouns will not be tight. If many eep, istinct glints are observe in the ata, it is likely that θ e is small. Conversely, the presence of only shallower, less-istinct glints suggests a larger value of θ e. For a given BRDF moel, one can use simulations to empirically etermine the typical values of θ e for a particular set of surface parameters. One such moel is the Ashikhmin-Shirley BRDF moel. 13 Note that although this moel allows for anisotropic reflection, this paper s iscussion assumes isotropy. This assumption is enforce by setting the parameters n u an n v equal; the resulting single parameter is herein calle n uv. In Ashikhmin s moel, the size of the specular lobe is controlle primarily by a single expression in the numerator: (u h u n nuv. Uner the assumption of a relatively shiny facet with n uv = 1000, the angle θ e has been foun in simulations to be roughly raians ( This value is fairly consistent over multiple simulations an observe glints, varying by roughly 1 at most. Rather than running new simulations an eriving an empirical θ e for each possible n uv, a scaling proceure is recommene. One woul expect the glint threshol to occur for roughly the same value of (u h u n nuv, regarless of the specific n uv. This expression can be rewritten as (cos θ nuv. Let θ e0 = raians an n uv0 = 1000 be reference values. Then (cos θ e nuv = (cos θ e0 nuv0 cos θ e = (cos θ e0 n uv0 nuv θ e = cos 1 [ (cos θ e0 n uv0 nuv ] (6 Aitional simulations for several ifferent values of n uv such as 500 or 100 have inicate that this scaling proceure is fairly successful at preicting the values of θ at the glint eges. The observe errors have been similar to the level of variation seen in the simulations themselves. It is further conceivable that a similar scaling proceure might be erive for a ifferent BRDF moel, provie one coul isolate the terms responsible for the size of the specular lobe. Once upper an lower bouns for θ e have been selecte, one can consier θ g. As previously mentione, the minimum possible value of θ g is zero; this occurs uring a full glint, when u n an u h align perfectly. For purposes of Eq. (4a, the maximum value is require this value correspons to a very shallow glint. In orer for any istinct glint to be visible in a light curve, θ e must excee θ g by some minimum, non-negligible amount. In simulations using the Ashikhmin-Shirley BRDF moel, the shallowest istinctly recognizable glints for a given n uv were sought. For the n uv = 1000 case, such shallow glints were foun to have a glint angle of roughly θ g = raians (6.066, or about 1.5 less than θ e. As θ e becomes larger, however, the ifference between θ e an θ g require to prouce a clear glint also becomes larger. One reasonably successful approach is to reprise the scaling proceure introuce for θ e : [ ] θ g = cos 1 (cos θ g0 n uv0 nuv (7 where θ g0 = raians an n uv0 = 1000 can be use as reference values. 7 of 11

8 V. Results The glint uration techniques escribe in the preceing sections were applie in several contexts: to simulate light curves, ata from a laboratory experiment, an ata from the Galaxy 15 communications satellite. A. Results from Simulate Data A number of truth moel simulations were performe in orer to etermine reasonable rules of thumb for selecting θ e an θ g. This subsection escribes the results of one representative simulation in etail. For the simulate scenario, the reflecting object is a shiny rectangular prism (with BRDF parameters R spec = 0.7, R iff = 0.3, an n uv = The object s orbit is near, but not exactly, geostationary, an it tumbles with an average angular rate of raians/secon. One istinct glint was observe in the light curve uring the first 10 minutes of simulate time, as shown in Fig. 3. The glint uration t Figure 3. Simulate light curve with glint was roughly 6 secons. Assuming that the parameter n uv = 1000 is fairly well-known for this case, raians was selecte as the glint ege angle θ e (with no uncertainty, an raians was selecte as the maximum glint angle θ g,max. The minimum an maximum glint withs were chosen as t min = 58 secons an t max = 66 secons. Application of Eqs. (4a an (4b gives the minimum an maximum glint rates θ min = ra/sec an θ max = ra/sec. These values are in turn use to evaluate Eqs. (5a an (5b. The magnitue of t u h, the inertial erivative of u h, was foun to be approximately , so the bouns on the observable component of ω are ω x min = ra/sec an ω x max = ra/sec. For comparison, the true (simulate value of the observable component of ω at the glint eges was foun to be ra/sec, which lies well within the compute bouns. In this particular example, the overall angular velocity magnitue also falls within the bouns, but that is not always the case. The angular velocity results are summarize in Table 1. Table 1. Summary of results for simulate light curve t 6 sec ω x min ra/sec ω x max ra/sec ω x true ra/sec ω true ra/sec 8 of 11

9 B. Results from Experimental Data A series of laboratory experiments were conucte to create synthetic light curves for some well-characterize test objects. In each experiment, a light source was focuse on the test object, which rotate on a turntable. Other sources of light an reflection were eliminate. A series of images was collecte by a camera, an the light curve was constructe by calculating the total brightness in the image pixels at each time step. The size an shape of the test objects, an locations of the object, camera, an light source, were foun by measuring. One of the test objects which exhibite istinct specular glints was a flat, relatively shiny piece of sheet metal. The corresponing light curve is shown in Fig. 4. As the light source, object, an camera were all Figure 4. Experimental light curve with multiple glints at fixe locations, the glint angle u h was constant an its erivative was zero in the inertial frame. The average glint uration over the ata set was t = 9.5 samples. (The camera sampling interval was use for the time units. It was possible to calculate the glint ege angles θ e from the known configuration an object rotation angles: On average they were about θ e = raians (3.8. Using the scaling proceure in Eq. (6, this correspons to a parameter of n uv = For this n uv, Eq. (7 suggests θ g,max = raians (19.4. The actual average θ g for this experiment was 0.01 raians (11.51, so the estimate θ g,max is quite conservative. A glint uration uncertainty of ± samples was assume. The glint rate bouns were foun from Eq. (4 to be θ (t e ra/sample. As t u h = 0 for this case, the bouns on the observable component of angular velocity were exactly the bouns on θ: ω x (t e ra/sample. The true magnitue of the observable component of ω was ω x = ra/sample, an the magnitue of the overall rotation rate was ω = ra/sample. Table summarizes these results. Table. Summary of results for experimental light curve t 9.5 samp ω x min ra/samp ω x max ra/samp ω x true ra/samp ω true ra/samp C. Results from Galaxy 15 Data The glint uration algorithms were also applie to real light curve ata from the Galaxy 15 communications satellite. Galaxy 15 is nominally geostationary an nair-pointing, but no etaile shape moel or attitue 9 of 11

10 history were ever provie, so it was not possible to compute true values of the angle θ or relate quantities. The nominal angular velocity for the nair-pointing configuration was compute. Surface reflection parameters were not given. The light curve exhibite a ominant eep V shape of increase brightness ue to the relative motion of the Sun, but it was not possible to clearly ientify a specular glint within this region. In aition, however, there were five smaller istinct glints, as inicate in Fig. 5. The average uration of these glints was about secons, an the corresponing bouns were compute. The results presente here are the averages over the five glint events. Figure 5. Galaxy 15 light curve with small glints The angular velocity bouns were compute uner several ifferent assumptions for the reflection parameter n uv. Large values of n uv (more mirror-like reflection give conservative lower boun estimates for ω x, an smaller values of n uv give conservative upper boun estimates. Assuming n uv = 1000, the lower boun on the observable part of ω is ω x min = ra/sec. Of course, this is also a lower boun on the magnitue of the overall vector ω. For an assumption of n uv = 100, the compute upper boun was ω x max = ra/sec. In comparison, the magnitue of the nominal angular velocity for the geostationary, nair-pointing configuration is ω nom = ra/sec. These results are summarize in Table 3. Table 3. Summary of results for Galaxy 15 light curve t sec ω x min ra/sec ω x max ra/sec ω nom ra/sec Note that the nominal angular velocity magnitue is about half that of the conservative lower boun calculate from the glint withs. There are several possible explanations for this iscrepancy. The most obvious conclusion is that the observe glints are cause by some form of attitue maneuvers - such as perioic solar panel rotations intene to keep the solar panels in an approximately sun-pointing configuration. 14, 15 Before accepting that hypothesis, it is wise to look at previous analyses of the Galaxy 15 light curve. Reference 15, in particular, examines these faint glints in epth, an conclues that the shift in glint epoch time from ay to ay is consistent with glints from components mounte on the main bus, rather than the solar panels. As shown here, the uration of the glints strongly suggests that these components are moving. Alternatively, the glinting components may not conform to the theoretical glint with because they fail to satisfy the flat facet assumption an are in fact cylinrical or more complex in shape. This case inicates the power of the propose analysis: While it cannot etermine angular velocity in 10 of 11

11 any exact sense, measurements of glint with can be use to ientify non-nominal behavior or changes in behavior, even when etails like attitue an shape are poorly known. VI. Conclusions In this paper, a technique was evelope that constrains the magnitue of SO angular velocity base on the time uration of specular glint events. The algorithm erives a set of equations that relate the observable part of the siereal angular rate to the erivative of the angle between the phase angle bisector an the normal vector of the reflecting facet. An aitional erivation relates this erivative to the observe glint with. Uncertainties in the form of upper an lower bouns on measure or assume quantities can be easily incorporate. This paper s metho is applie to ata from three sources: simulations, laboratory experiments, an a real satellite. The compute bouns successfully brackete the observable component of angular velocity in all cases for which the truth was available for comparison, although in many cases the bouns were not very tight. However, even such limite rate information may be useful in etecting anomalous rate behavior, or in selecting a reasonable initial rate estimate for a Kalman filter. References 1 Jah, M. an Maler, R., Satellite Characterization: Angles an Light Curve Data Fusion for Spacecraft State an Parameter Estimation, Proceeings of the Avance Maui Optical an Space Surveillance Technologies Conference, Vol. 49, Sept. 007, Maui, HI. Linares, R., Shoemaker, M., Walker, A., Mehta, P. M., Palmer, D. M., Thompson, D. C., Koller, J., an Crassiis, J. L., Photometric Data from Non-Resolve Objects for Space Object Characterization an Improve Atmospheric Moeling, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 013, Maui, HI. 3 Linares, R., Jah, M. K., Crassiis, J. L., an Nebelecky, C. K., Space Object Shape Characterization an Tracking Using Light Curve an Angles Data, J. of Guiance, Control, an Dynamics, Vol. 37, No. 1, 014, pp Wetterer, C. J., Hunt, B., Kervin, P., an Jah, M., Comparison of Unscente Kalman Filter an Unscente Schmit Kalman Filter in Preicting Attitue an Associate Uncertainty of a Geosynchronous Satellite, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 014, Maui, HI. 5 Alcala, C. M. an Brown, J. H., Space Object Characterization Using Time-Frequency Analysis of Multi-Spectral Measurements from the Magalena Rige Observatory, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 009, Maui, HI. 6 Papushev, P., Karavaev, Y., an Mishina, M., Investigations of the Evolution of Optical Characteristics an Dynamics of Proper Rotation of Uncontrolle Geostationary Artificial Satellites, Av. in Space Research, Vol. 43, No. 9, Hall, D., Africano, J., Archambeault, D., Birge, B., Witte, D., an Kervin, P., AMOS Observations of NASA s IMAGE Satellite, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 006, Maui, HI. 8 Binz, C. R., Davis, M. A., Kelm, B. E., an Moore, C. I., Optical Survey of the Tumble Rates of Retire GEO Satellites, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 014, Maui, HI. 9 Cognion, R. L., Rotation Rates of Inactive Satellites Near Geosynchronous Orbit, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 014, Maui, HI. 10 Wallace, B., Somers, P., an Scott, R., Determination of Spin Axis Orientation of Geosynchronous Objects Using Space- Base Sensors: An Initial Feasibility Investigation, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 010, Maui, HI. 11 Somers, P., Cylinrical RSO Signatures, Spin Axis Orientation an Rotation Perio Determination, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 011, Maui, HI. 1 Vrba, F. J., DiVittorio, M. E., Hinsley, R. B., Schmitt, H. R., Armstrong, J. T., Shanklan, P. D., Hutter, D. J., an Benson, J. A., A Survey of Geosynchronous Satellite Glints, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 009, Maui, HI. 13 Ashikhmin, M. an Shirley, P., An Anisotropic Phong Light Reflection Moel, Tech. Rep. UUCS , University of Utah, Salt Lake City, UT, Hall, D., AMOS Galaxy 15 Satellite Observations an Analysis, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 011, Maui, HI. 15 Hall, D. an Kervin, P., Analysis of Faint Glints from Stabilize GEO Satellites, Avance Maui Optical an Space Surveillance Technologies Conference, Sept. 013, Maui, HI. 11 of 11