A Sensor Driven Trade Study for Autonomous Navigation Capabilities

Transcription

1 A Sensor Driven Trade Study for Autonomous Navigation Capabilities Sebastián Muñoz and E. Glenn Lightsey The University of Texas at Austin, Austin, TX, Traditionally, most interplanetary exploration missions have relied heavily on navigation aids from Earth based resources. As new missions are considered, management of Earth Based Resources such as Deep Space Network tracking becomes more complex. On-board autonomous navigation capability adds reliability and robustness to these missions and mitigates the dependency on Earth based resources. This paper presents a sensor trade study to determine which combination of sensors is best suited to accomplish autonomous navigation for interplanetary missions. To perform this sensor trade study a sensor database has been created that is used with a sensor fusion navigation algorithm to determine which combination of sensors is best suited for a cislunar test case. The sensors / measurements that are considered in this sensor database are accelerometers, gyroscopes, sun sensors, star trackers, GPS receivers, Pulsar range observations, DSN 1-way range observations, and centroid and apparent diameter observations. I. Introduction and Motivation As new interplanetary exploration missions are considered, there is a growing need to increase the autonomy of the on board navigation capabilities. Given the fact that currently there are only three Deep Space Network (DSN) tracking stations, managing their resources to successfully aid in a spacecraft s navigation will become more complex, demanding, and expensive as the number of missions increases while the Earth based resources stay the same. 1 Apart from reducing the dependency on Earth based tracking resources, there are multiple reasons for wanting to increase the degree of autonomy of the spacecraft s navigation such as adding reliability and robustness to the system. If these missions include human exploration missions, then a completely autonomous navigation system would address safety concerns for the on board crew and allow the mission to return safely to Earth in case of any communication failures with Earth based resources. If on the other hand these missions include robotic spacecraft visiting asteroids or planets where during parts of their mission trajectories they might not be able to communicate with Earth, then adding an autonomous navigation capability greatly increases the missions robustness and reliability and decreases its operational risks. There have been multiple sensors that have been identified to increase the level of autonomy of the on board navigation system such as centroid and apparent diameter measurements of planetary bodies, angular measurements between surface features or horizon of a planetary body and reference stars, and x- ray pulsar based measurements. 2, 3 Apart from these sensors, other sensors that have been traditionally used for on-board navigation purposes include Inertial Measurement Units (IMUs), star trackers, sun sensors, and Global Positioning System (GPS) receivers. For completeness and to be able to compare against current navigation system architectures, the DSN 1-way ranging tracking measurement and GPS measurements are also included in the sensor database created for this study. This trade study compares these different sensor types and their ability to increase the spacecraft s autonomous navigation capability. It also combines the sensors using a sensor fusion algorithm in a navigation filter to determine which combinations of sensors increase the reliability of the autonomous navigation capabilities. To perform these sensor trade studies, one test case scenario was chosen of human exploration Graduate Research Assistant, Department of Aerospace Engineering and Engineering Mechanics, and AIAA Student Member. Professor, Department of Aerospace Engineering and Engineering Mechanics, and AIAA Associate Fellow. 1 of 16

2 mission in a lunar return trajectory. The following section discuss the navigation filter implementation chosen to perform these trade studies as well as all of the sensor models that are included in the sensor database. Although not all of them are used to perform the sensor trade studies, all of them are presented for completeness. Following the system description and an explanation of the test case scenario, results of the sensor trade study analysis are presented. II. Navigation Filter The navigation filter chosen to perform this trade study is a Multiplicative Extended Kalman Filter (MEKF). The reason this filter was chosen is because it allows the capability of estimating translational states, attitude states, and sensor parameters in one filter. One advantage of this filter structure is that it can be designed in a modular way that allows different sensor combinations to be characterized without having to change the design of the filter. For the MEKF, the state vector, x will be defined as x sc x = q (1) where x sc contains all of the spacecraft states to be estimated except for the spacecraft s attitude quaternion q which has been stated explicitly as it requires a different treatment as will be shown. The spacecraft state vector can include the spacecraft s position (r sc ), velocity (r sc ), and its angular velocity (ω); however these will be determined based on the particular test mission being studied. Additionally, the to be state vector to be estimated in the filter can include other parameters such as sensors biases which will be represented [ ] T by x p. The equations of motion of the state vector ẋ = x T sc q T ẋ T p are defined by x p ẋ sc = f (x) + w sc (2) [ ] q = 1 ω q (3) 2 ẋ p = g (x) + w p (4) whose non-quaternion dynamics are affected by process noise represented by zero-mean white noise processes such that the i-th component of each process noise vector behaves according to w sci N (, σsc 2 ) i (5) w pi N (, σp 2 ) i (6) unless otherwise specified. It should be noted that if the state vector includes the velocity of the spacecraft, the velocity vector is not affected by process noise. One of the special characteristics of the MEKF, is that instead of estimating the spacecraft s attitude quaternion, a small angle deviation (δα) is estimated which is then used to update the spacecraft s attitude estimation. 4, 5 For this reason, the state deviation model for the MEKF can be defines as δx = δx sc δα δx p where the deviations for the non-quaternion states are defined as (7) δx sc = x sc ˆx sc (8) δx p = x p ˆx p (9) such that they are the difference between the true and estimated states. The small angle deviation, δα is then defined by the following property [ ] δ q = q ˆ q δα (1) 1 2 of 16

3 where this approximation is valid to first-order if the deviation quaternion (δ q) from the true attitude quaternion ( q) to the estimated attitude quaternion (ˆ q) is small. The state deviation is governed by δẋ = F (ˆx) δx + w (11) where F (ˆx) is the Jacobian of the state vector evaluated at the current estimate of the state, and w is the vector containing all the zero-mean white noise processes (w sc, w p ) whose corresponding process noise covariance is given by Q such that [ ] E w (t) w (τ) T = Qδ (t τ) (12) Note that because the small angle deviation is included in the state deviation model rather than the full attitude quaternion deviation, the dimension of the state deviation vector is (n 1) 1 if the full state vector is of dimension n 1. For all the different sensors in the navigation filter, the measurement model at time t k is given by y k = h (x) + v k (13) where h (x) is the nonlinear measurement model, and v k is a zero-mean white noise process with corresponding measurement covariance given by R k such that E [ v k v T i ] = {R k,, i = k i k (14) It should be noted that it is assumed that at any time t k, the measurement noise is uncorrelated with the process noise at any other time such that E [ w (t) vk T ] = (15) For the MEKF, given some initial conditions, ˆx (t o ) = ˆx o, and an initial error covariance, P (t o ) = P o, the equations of motion are propagated forward until a measurement becomes available at time t k. Since one of the motivating factors behind choosing the MEKF implementation was for it to be modular such that multiple combinations of sensors can be studied, every time a new sensor measurement is available, the state is updated. If two measurements from different sensors become available at the same time, then they are accumulated as a single measurement to be update the state. Although it is possible to update the state individually with each sensor measurement assuming a zero time step in between updates and as as long as measurement errors are uncorrelated them, for numerical reasons it was decided to process them as a single measurement. 6 Each time a sensor is processed at time t k, the Kalman Gain, K k, and the state-observation matrix, H k, are evaluated at the current estimate, ˆx 6 k, and computed to be:5, K k = P k H [ k Hk P ] 1 k HT k + R k (16) [ H k = h x sc h α ] h x p x=ˆx k where P k is the error covariance at time t k before the update. After these matrices are computed, the state and covariance can be updated as given by (17) P + k = [I K kh k ] P k (18) δˆx + k = K [ )] k yk h k (ˆx k (19) ˆx + sc k = ˆx sc k + δˆx + sc k (2) Ξ (ˆ q ] ) [ˆq k = 4k I [ˆq ] (21) ˆq T ˆ q + k = ˆ q k Ξ (ˆ q k ) δ ˆα + k (22) ˆ q + k = ˆ q + k ˆ q + k (23) ˆx + p k = ˆx p k + δˆx + p k (24) 3 of 16

4 where as a precaution the updated quaternion has been renormalized, although it should already be unity norm to first-order. 5, 7, 8 After updating the state estimate and error covariance, the state estimate and error covariance can be propagated propagated forward to time t k+1 when a new measurement will be available by integrating The procedure for the MEKF is summarized in Figure 1. ˆx sc = f (ˆx) [ ] (25) ˆ q = 1 ˆω 2 ˆ q (26) ˆx p = g (ˆx) (27) Ṗ = FP + PF T + Q (28) Figure 1: Navigation Filter Flow Diagram The dynamic models used for the spacecraft translational equations of motion for the different trade studies can include two-body motion, with non-spherical central body perturbations, third body perturbations, and atmospheric drag. The inclusion of which models to use is based on the trade study being performed. For the rotational dynamics, we assume that the spacecraft is a rigid body with fixed mass properties, so the equations of motion are given by Euler s equations. The equations of motion for all of these models can be referenced in Vallado 9 and Crassidis. 5 III. Sensor Database Given that the navigation filter was designed with modularity in mind, a common sensor structure was designed to be able to create a sensor database. This database contains a common sensor structure where each sensor has four main components as shown in Figure 2. The Type of Sensor field defines what type of sensor each sensor is which allows multiple kinds of the same sensor with different performance or operating specifications within the database. The Status of Sensor field defines whether the sensor is active or not in a particular simulation segment which allows different combinations of sensors to be tried out or be active during different segment runs. The Frequency field for each sensor specifies the output frequency of the measurement for that particular sensor when it is active during a simulation segment. Finally the Sensor Specs field is a general structure which contains the performance specifications for each sensor and has a different structure depending on which type of sensor is defined in the database. These last three are implemented in such a way that if a simulation run includes several segments, then the sensor can have different parameters for each segment. Following are the descriptions of all the measurement models for the different sensors that are incorporated into the database. 4 of 16

5 Figure 2: Sensor Structure III.A. Inertial Measurement Unit Traditionally one of the primary sensors used for navigation purposes is the Inertial Measurement Unit (IMU). An IMU is used to sense the spacecraft s maneuvers as well as any non-conservative forces or external torques that affect the spacecraft s trajectory and orientation. In most cases, the IMU is used to dead-reckon in between navigation updates which means that instead of being used as a measurement in the navigation filter it is used directly to propagate the internal spacecraft s equations of motion. The IMU provides two types of measurements, one coming from the accelerometer and one from the gyroscope. Only for these two measurements in the database, if active they can be specified to be used as measurements in the navigation filter or directly to propagate the internal spacecraft s equations of motion. III.A.1. Accelerometers The accelerometer is one of the two sensors included in the Inertial Measurement Unit (IMU) and it is used to measure the non-conservative forces (i.e: drag, solar radiation pressure, thrusting maneuvers) a spacecraft might experience. Its measurement model is given by y k = (S a + S a ) T A A T A Ba B + b a + n a (29) where S a is the scale factor matrix, S a is the scale factor error matrix, T A A is the transformation matrix representing the misalignment and non-orthogonal error rotation matrix of the accelerometer frame (A), T A B is rotation matrix from the body frame (B) to the accelerometer frame (A), a B is the actual acceleration measured by the accelerometer in the body frame, b a is the accelerometer bias, and finally n a w N (, σ 2 n a I ) is a zero-mean white noise error that affects the measurement. Given that an accelerometer can only measure the non-conservative acceleration on a spacecraft, if we assume that the accelerometer is not located at the center of mass of the spacecraft, then a B = a B sc,nc + ω ω r B a/sc + ω rb IMU/sc (3) where a B sc,nc is the non-conservative acceleration of spacecraft at the center of mass, r B a/sc is relative location of the accelerometer with respect to the center of mass of the spacecraft expressed in the body frame, ω is the angular velocity of the spacecraft, and ω is the angular acceleration of the spacecraft. III.A.2. Gyroscopes The gyroscope is the other sensor making up the IMU and it measures the angular velocity of the spacecraft directly. As with the accelerometer, the gyroscope measurement model is very similar and is given by y k = (S g + S g ) T G G T G Bω + b g + n g (31) where S g is the gyro scale factor matrix, S g is the gyro scale factor error matrix, T G G is the transformation matrix representing the misalignment and non-orthogonal error rotation matrix of the gyroscope frame (G), 5 of 16

6 T G B is rotation matrix from the body frame (B) to the gyroscope frame (G), ω is the angular velocity of the spacecraft, b g is the gyroscope bias, and finally n g N (, σngi ) 2 is a zero-mean white noise error affecting the measurement. In the case of the gyroscope, the bias drifts with time and can be modeled as a first-order Gauss-Markov process given by ḃ g = 1 τ g b g + n bg (32) with variance (σ 2 bg ) and where τ g is the correlation time and n bg process. 1 ( ) N, 2σ2 bg τ g is a zero-mean white noise III.B. Sun Sensors As it name implies, a sun sensor is used to measure the direction of the sun as seen from the spacecraft. This measurement is mostly used to estimate the spacecraft s attitude to a sub-degree accuracy but it also provides some information on the spacecraft s position. The sun sensor measurement model is illustrated in Figure 3. The direction of the sun is given as azimuth and elevation measurements such that the measurement model Figure 3: Sun Sensor Measurement Model is given by [ ] [ ] Az n y k = + az El n el (33) where a simplifying assumption has been made that the measurement noise is uncorrelated so that n az N (, σ 2 az) and nel N (, σ 2 el) are zero-mean white noise error affecting the measurement, and and where T SS B Az = tan 1 ( u SS sun/ss,y u SS sun/ss,x El = sin 1 ( u SS sun/ss,z ) ) (34) (35) u SS sun/ss = TSS B T ( q) u sun/ss (36) is the rotation matrix between the body frame and the sun sensor frame, T ( q) is the rotation matrix between the inertial frame and the body frame, and u sun/ss is the unit-vector direction of the sun relative to the spacecraft. III.C. Star Trackers One of the most reliable and accurate sensors for attitude estimation (arc second accuracy) on a spacecraft are star trackers. In most cases star trackers include internal processing algorithms that compare star unit vector observations (u B ) against known star inertial unit vectors (u I ) to generate an observed quaternion 6 of 16

7 q obs with a corresponding measurement covariance P αα. Figure 4 shows the internal measurements of the star tracker which are given by the QUEST measurement model 11, z k as z k = u B i + n i (37) where u B i is the unit vector observation in the body frame to the i-th star and n i is a zero-mean white noise corrupting the measurement. It is assumed that the internal processing of the star tracker uses the QUEST algorithm 11 to generate an observed quaternion q obs. To process this observed quaternion as a measurement in the MEKF, it is assumed that a small angle deviation (δα obs ) from the current estimate quaternion (ˆ q ) can be obtained such that where δ q obs (δα obs ) = q obs (ˆ q ) 1 (38) δα obs 2δq obs (39) is valid if a small angle deviation is assumed from the current best estimate. Based on this, the measurement model for the sensor database can now be constructed to be y k = δα + n st = δα obs (4) where n st is a zero-mean white noise corrupting the measurement which has a corresponding error covariance matrix given by [ n 1 [ P αα = I 3 3 ( u B ) ( ) i u B T ] ] 1 i (41) σ 2 i=1 i which is the the QUEST error covariance matrix and where σ i is the standard deviation of the measurement 7, 11 noise n i for each unit vector observation. Figure 4: Star Tracker Measurement Model III.D. Centroid and Apparent Diameter Like the sun sensor and star tracker measurements, another optical measurement that is included in the sensor database is that of the direction of a planet or moon centroid and its apparent size (diameter) in the sensor s field of view. This type of measurement is useful in determining the relative range of the spacecraft to a celestial body if its size is known. 2 An illustration of this type of measurement is shown in Figure 5. As with the sun sensor, the direction of the planet is given by azimuth and elevation measurements of the centroid of the celestial body, and the apparent size is given by the apparent diameter of the celestial body. The measurement model is then given by Az n az y k = El + n el (42) γ n γ 7 of 16

8 Figure 5: Centroid and Apparent Diameter Measurement Model where again the simplifying assumption has been made that the measurements are uncorrelated and n az N (, σaz) 2, nel N (, σel) 2, and nγ N (, σγ) 2 are zero-mean white noise errors affecting the measurement. In this case the Azimuth, Elevation and Apparent Diameter are given by Az = tan 1 ( u CAM p/sc,y u CAM p/sc,x ( ) El = sin 1 u CAM p/sc,z γ = 2 sin 1 ( Rp r p/sc where u CAM p/sc is the unit vector direction to the centroid of the planet or moon in the camera frame, R p is the planet or moon s radius, and r p/sc is the relative distance between the planet/moon and the spacecraft. The unit vector direction to the centroid of the planet/moon in the camera frame is given by u CAM p/sc ) ) (43) (44) (45) = T CAM B T ( q) u p/sc (46) u p/sc = r p r r p/sc (47) r p/sc = r p r (48) where T CAM B is rotation matrix between the body frame (B) and the camera frame (CAM), and r p is the position of the planet/moon. III.E. X-Ray Pulsar Navigation One of the most promising measurement for autonomous navigation in interplanetary space is that of x-ray pulsar based ranging measurements as pulsar emit very accurate timing signals which can be used to navigate in space. For this measurement it is assumed that the spacecraft has an x-ray detector on board that can track variable celestial x-ray pulsars. 3 The measurement model is given by 3 y k = c (t SSBi t sci ) + b c + n pls (49) where c is the speed of light, t SSBi is the estimated time of arrival of the pulse from the i-th pulsar at the solar system barycenter, t sci is the time of arrival of the pulse of i-th pulsar at the spacecraft, b c is the satellite clock bias, and n pi N ( ), σp 2 i is a zero-mean white noise error affecting the measurement. Given that the spacecraft has an almanac with the unit vector directions to the available pulsars, for a given pulsar the measurement can be related to the spacecraft position by c (t SSB t sc ) = u T i r sc/ssb (5) r sc/ssb = r E + r (51) where u i is the unit vector direction to the i-th x-ray pulsar, and r E is the position of the Earth. The concept for this measurement is shown in Figure 6. It should be noted that if there is any celestial body in between the spacecraft and the x-ray pulsar the measurement will not be available. 8 of 16

9 Figure 6: Pulsar Based Measurement Model III.F. Global Positioning System The Global Position System (GPS) measurement is simulated as a pseudorange measurement between the spacecraft and each of the GPS satellite and is given by 12 y k = ρ GP Si + b c + n GP Si (52) where ρ GP Si is the range from the i-th GPS satellite to the spacecraft, b c is again the satellite s clock bias, and n GP Si N ( ), σgps 2 i is a a zero-mean white noise error affecting the measurement of the i-th GPS satellite. The range from the i-th GPS satellite to the spacecraft is given by as shown in Figure 7. ρ GP Si = r r GP Si (53) Figure 7: GPS Measurement Model III.G. Deep Space Network For completeness and to be able to compare current navigation architectures with the sensors provided in this database, a DSN 1-way ranging measurement model is also included in the database. In this case the measurement model is given by 1 y k = ρ GSi + b c + n GSi (54) where ρ GSi is the range from the one of the three DSN tracking ground stations to the satellite as shown in Figure 8, b c is the satellite clock bias, and n GSi N ( ), σgs 2 i is a zero-mean white noise error affecting the measurement from the i-th DSN ground station. The range is given by ρ = r r dsn (55) where r GSi is the inertial position of the DSN tracking station. It should be noted that no measurement is available if the spacecraft is not within line of sight of the tracking station. 9 of 16

10 Figure 8: DSN 1-Way Ranging Measurement Model IV. Simulation Scenario - Spacecraft in Lunar Return Trajectory To perform the trade studies presented in this paper and illustrate the autonomous navigation capabilities of different combination of sensors for an interplanetary mission a test case scenario where a human exploration mission is in a return trajectory from the Moon was chosen. To make this comparable to other interplanetary deep-space missions it was decided that GPS measurements would not be available to the spacecraft. IV.A. Navigation Requirements To further motivate these trade studies, it is a assumed that the human crewed vehicle on the lunar return trajectory to Earth has a communication failure with Earth occurs that occurs 3 hours prior to Entry Interface (EI). This prevents the vehicle from obtaining any navigation updates through DSN and hence only on-board sensors will be available to meet the navigation requirements. For this scenario, the highest priority is to try to get the astronauts safely back to Earth and land which implies that the flight path angle error requirement at Entry Interface (EI) to guarantee the crew s safety. IV.B. Initial Conditions φ e.1 deg (56) For all of the results presented, it is assumed that the spacecraft can be modeled as a rigid cylinder with constant mass whose parameters are given in Table 1. The spacecraft s x-axis is assumed to be along the long axis of the rigid cylinder. Table 1: Spacecraft Mass Properties Mass Height Radius 16,7 kg 8 m 2.5 m The true initial position and velocity of the spacecraft were taken from a reference lunar return trajectory provided by NASA JSC for the Orion spacecraft. The true initial attitude was chosen so that body frame was coincident with the inertial frame at the the start of the scenario. The initial true angular velocity was chosen randomly with a magnitude of.5 deg/s. A summary of the initial conditions is given in Table 2. If not specified in this table then they are assumed to be zero initially. IV.C. Sensor Specifications The sensors included on the spacecraft for navigation purposes in the simulation scenario are a Gyroscope, two Star Trackers, two Centroid & Apparent Diameter Measurements, three X-Ray Pulsar Range Measurements, and for comparison purposes the three DSN ground station 1-way range measurements are also included. The specifications used in this test case scenario are given in Tables 3-7. As part of the simulation the 1 of 16

11 Table 2: Spacecraft Initial Conditions when Communication Failure Occurs Position Velocity Attitude Angular Velocity (km) (km/s) (deg/s) r o = v o = q o = ω o = bias of the gyroscope is being estimated while the gyroscope is not used as a measurement but instead it is used to propagate the internal state of the filter which means that the angular velocity is not estimated. Note that if a parameter is not specified in the Sensor Specification Tables, it is assumed to zero or identity matrix if appropriate. Table 3: Gyro Specifications Parameter Bias (1σ) Bias Time Constant Noise (1σ) Value.5 deg/hr 1 hr.5 deg/hr Table 4: Star Tracker Specifications Parameter Star Tracker 1 Star Tracker 2 Field of View 22 deg 22 deg Observation Noise (1σ) 8 arcsec 8 arcsec Boresight Vector u B st1 = u B st1 = 1 1 V. Analysis & Results Two different trade studies were performed on the simulation scenario described above. Both of these trade studies looked at what sensor combinations would be better suited to meet the Flight Path Angle Error at Entry Interface Navigation Requirement. Under both of these trade studies the star tracker was used to estimate the attitude of the spacecraft while the gyro was used internally to propagate the filter s equations of motion directly. V.A. Sensor Combinations at Fixed Frequencies The first trade study performed looked at all possible autonomous sensor combinations with the available sensors where all the measurements were available at the same frequency. In this trade study, these combinations were then compared with one where only measurements from the Deep Space Network were available and one where all of the measurements were available to see which combinations perform better. Table 8 shows the summary of the results of all of these simulation runs. Figure 9-1 show the filter performance during one of the simulated runs as a representative example. In all cases, the filter behaved very similarly, being capable of estimating both the position and velocity of the spacecraft. What changed from run to run as expected was how much the error covariance estimate would shrink based on the available sensor combinations. 11 of 16

12 Table 5: Centroid & Apparent Diameter Specifications Parameter Field of View Azimuth Noise (1σ) Elevation Noise (1σ) Apparent Diameter Noise (1σ) Value 5 deg 8 arcsec 8 arcsec 8 arcsec Table 6: X-Ray Pulsar Specifications 3 Parameter Pulsar 1 Pulsar 2 Pulsar 3 Name B B B Right Ascension deg deg deg Declination deg deg deg Range Noise (1σ) 19 m 325 m 344 m Table 7: DSN 1-Way Range Specifications Parameter Station 1 Madrid Goldstone Name Canberra B B Latitude deg 4.41 deg deg Longitude deg deg deg Range Noise (1σ) 1 m 1 m 1 m Table 8: Sensor Combination Trade Study for Meeting FPA Navigation Requirement Simulation Sensor Measurement EI Autonomous Meets Nav Run Combination Frequency FPA Error (deg) Requirement 1 CAD 1 / Hr.659 Yes Yes 2 Pulsar 1 / Hr.154 Yes Yes 3 CAD, Pulsar 1 / Hr.144 Yes Yes 4 DSN 1 / Hr.33 No Yes 5 CAD, Pulsar, DSN 1 / Hr No Yes 12 of 16

13 5 Position Error Radial (km) Tangential (km) Normal (km) Time from EI (sec) Figure 9: Position Error Estimates before Entry Interface for the Fifth simulation run with all of the measurements available. Radial Vel (km/s) Tangential Vel (km/s) Normal Vel (km/s).2 Velocity Error Time from EI (sec) Figure 1: Velocity Error Estimates before Entry Interface for the Fifth simulation run with all of the measurements available. V.B. Sensor Combinations at Different Frequencies The second trade study performed was similar to the first but in this case in only focused on the autonomous sensor combinations and instead of having the measurements processed at the same frequency, changing the frequency of the sensors was investigated. In the first case, both sensors were sampled at the same frequency, and in the following 2 cases one sensor was sampled 6 times faster than the other to see if there would be any advantage over choosing one over the other. Table 9 shows the results of this trade study. From this table it is clear that if the Centroid & Apparent Diameter measurements are sampled in between the Pulsar measurements, the navigation performance is better. Figure 11 show the filter performance on estimating the position of spacecraft of this scenario as representative example. 13 of 16

14 Table 9: Sensor Combination Trade Study for Meeting FPA Navigation Requirement with Different Frequencies Simulation Sensor Measurement EI Autonomous Meets Nav Run Combination Frequencies FPA Error Requirement 1 CAD, Pulsar 1 / Hr, 1 / Hr.144 Yes Yes 2 CAD, Pulsar 1 / Hr, 6 / Hr.48 Yes Yes 3 CAD, Pulsar 6 / Hr, 1 / Hr.28 Yes Yes Figure 11: Position Error Estimates before Entry Interface for the Third simulation run with Centroid & Apparent Diameter and Pulsar. V.C. Attitude Estimation & Gyro Bias During both of the trade studies the star tracker was generating measurements at a rate of once per minute and the filter was capable of estimating both the attitude of the spacecraft and the gyro bias. Figures 12 and 13 show a representative run where the star tracker was now generating measurements at a rate of once per second and we can see that the filter does a good job of estimating both the attitude of the spacecraft as well as the bias of the gyroscope. VI. Conclusions From the results presented in this paper we can conclude that in trade studies performed, the navigation filter was able to show that better navigation performance was obtained when combining multiple autonomous sensors. It also showed that adding these sensors will improve the performance of the system even if ground based resources such as DSN are used. Also, it showed that changing the frequencies of different sensor combinations can lead to better results such as was the case of the Third simulation run in the second trade study. It was also shown that the attitude of the spacecraft and gyro bias were well estimated by the filter. Acknowledgements The authors wish to thank John A. Christian and Chris D Souza of the NASA Johnson Space Center. 14 of 16

15 5 x 1 4 Attitude Error Roll (deg) Pitch (deg) Yaw (deg) 5 x x Time from EI (sec) Figure 12: Attitude Error Estimates before Entry Interface with Star Tracker running at 1 Hz. b gx (deg/s) b gy (deg/s) b gz (deg/s) 2 x 1 6 Gyro Bias Error x x Time from EI (sec) Figure 13: Gyro Bias Error Estimates before Entry Interface with Star Tracker running at 1 Hz References 1 Thornton, C.L. and Border, J.S., Radiometric Tracking Techniques for Deep-Space Navigation, Jet Propulsion Laboratory, 2. 2 Christian,J.A. and Lightsey, E.G., A Review of Options for Autonomous Cislunar Navigation, AIAA Guidance, Navigation and Control Conference and Exhibit, No. AIAA , Honolulu, HI, Aug Sheikh, S., The Use of Variable Celestial X-Ray Sources for Spacecraft Navigation, Ph.D. thesis, University of Maryland, Markley, F. and Mortari, D., Quaternion Attitude Estimation Using Vector Observations, The Journal of the Astronautical Sciences, Vol. 48, No. 2/3, April-September 2, pp Crassidis, J.L. and Junkins, J.L., Optimal Estimation of Dynamical Systems, CRC Press LLC, Brown, R.G. and Hwang, P.Y.C., Introduction to Random Signals and Applied Kalman Filtering, John Wiley & Sons, 3rd ed., of 16

16 7 Markley, F.Landis, Attitude Error Representations for Kalman Filtering, Journal of Guidance, Control, and Dynamics, Vol. 26, No. 2, March-April Zanetti, R., Advanced Navigation Algorithms for Precision Landing, Ph.D. thesis, The University of Texas at Austin, December Vallado, D., Fundamentals of Astrodynamics and Applications, Springer, 3rd ed., Gelb, A., editor, Applied Optimal Estimation, The MIT press, Massachusetts Institute of Technology, Cambridge, MA, Shuster, M. and Oh, S., Three-Axis Attitude Determination from Vector Observations, Journal of Guidance and Control, Vol. 4, No. 1, January-February 1981, pp Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, Ganga-Jamuna Press, of 16