- Email: [email protected]
Kalman Filter Analysis for Orbit Estimation using Pulsars for Interplanetary Missions
Optimization of Dynamical Systems 4th International Conference on in 4th International Conference on Advances Advances in Control Control and and February 1-5, 2016. NIT Tiruchirappalli, India 4th International Conference on in and 4th International Conference on Advances Advances in Control Control Optimization of Dynamical Systems Available onlineand at www.sciencedirect.com Optimization of Dynamical Systems Optimization of Dynamical Systems Optimization of Dynamical Systems February 1-5, 1-5, 2016. 2016. NIT NIT Tiruchirappalli, Tiruchirappalli, India India February February 1-5, 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India
ScienceDirect
Kalman Filter Analysis for 49-1 Orbit using Pulsars for IFAC-PapersOnLine (2016)Estimation 136–141 Interplanetary Missions using Kalman Filter Analysis for Orbit Estimation Pulsars for Kalman Kalman Filter Analysis for Orbit Estimation using Pulsars for Kalman Filter Filter Analysis Analysis for for Orbit Orbit Estimation Estimation using using Pulsars Pulsars for for Interplanetary Missions Interplanetary Missions Interplanetary Missions M Thameemunnisha*. M P. Ramachandran** Interplanetary Missions
M Thameemunnisha*. Thameemunnisha*. M M P. P. Ramachandran** Ramachandran** M M Thameemunnisha*. Thameemunnisha*. M P. Ramachandran** Ramachandran** M M P. *ISRO Satellite Centre, 560017 Bangalore, India *ISRO (Tel: 080-25084414; e-mail: [email protected]). Satellite Centre, Bangalore, 560017 Satellite Bangalore, *ISRO Satellite Centre, Bangalore, 560017 **India ISRO*ISRO Satellite Centre,Centre, Bangalore, 560017560017 India (e-mail: *ISRO Satellite Centre, Bangalore, 560017 (Tel: 080-25084414; e-mail: [email protected]). India (Tel: 080-25084414; e-mail: [email protected]). India (Tel: 080-25084414; e-mail: [email protected]). [email protected]) 080-25084414; e-mail: [email protected]). **India ISRO(Tel: Satellite Centre, Bangalore, Bangalore, 560017 India India (e-mail: (e-mail: ** ISRO Satellite Centre, 560017 ** Bangalore, ** ISRO ISRO Satellite Satellite Centre, Centre, Bangalore, 560017 560017 India India (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) [email protected]) Abstract: Autonomous Navigation in Interplanetary Missions is the main challenge due to lack of dense ground tracking networkNavigation measurements. This paper presents a novel for determining Abstract: Autonomous in Interplanetary Interplanetary Missions is the technique main challenge challenge due to to lackspacecraft of dense dense Abstract: Autonomous Navigation in Missions is due Abstract:and Autonomous Navigation inX-ray Interplanetary Missions is the the main main challenge due to lack lack of of of dense position velocity using celestial sources, such as pulsars. It portrays the formulation the Abstract: Autonomous Navigation in Interplanetary Missions is the main challenge due to lack of dense ground tracking network measurements. This paper presents a novel technique for determining spacecraft ground tracking measurements. This aa novel technique for spacecraft groundmeasurements tracking network network measurements. This paper paper presents presents novel technique for determining determining spacecraft pulsar and describes the development of a noise model for X-ray Navigation and brings ground tracking network measurements. This paper presents a novel technique for determining spacecraft position and and velocity velocity using using celestial celestial X-ray X-ray sources, sources, such such as as pulsars. pulsars. It It portrays portrays the the formulation formulation of of the the position position and using X-ray sources, as portrays the formulation of out themeasurements cumulative efforts in celestial this developing field. Insuch addition tomodel that,It also covers the blending position and velocity velocity using celestial X-ray sources, such as pulsars. pulsars. Ititfor portrays the formulation of the the pulsar and describes the development of a noise X-ray Navigation and brings pulsar measurements and describes development of aa noise model for brings pulsar measurements and the development of for X-ray X-ray Navigation and brings pulsar-derived measurements withdeveloping a the Kalman filter continuous determination of Navigation position andand pulsar and describes describes the development a noise noisetomodel model X-ray Navigation andvelocity brings out themeasurements cumulative efforts in this this field.forIn Inof addition that, it itforalso also covers the blending blending of the the out the cumulative efforts in developing field. addition to that, that, covers the of out the cumulative efforts in this developing field. In addition to it also covers the blending of of an Earth orbiting spacecraft. Several sampling analysis are presented to establish the expected out the cumulative efforts in this developing field. In addition to that, it also covers the blending of the the pulsar-derived measurements measurements with with aa Kalman Kalman filter filter for for continuous continuous determination determination of of position position and and velocity velocity pulsar-derived pulsar-derived measurements with aa Kalman filter for determination of position velocity performance measurements of pulsed X-ray signals. is theand first expected step and pulsar-derived measurements withobtained Kalmanfrom filtermodels for continuous continuous position and velocity of an an Earth Earth using orbiting spacecraft. Several sampling analysis aredetermination presented toofThis establish the of orbiting spacecraft. Several sampling analysis are presented to establish the expected of an Earth orbiting spacecraft. Several sampling analysis are presented to establish the expected future efforts shall make the orbit estimation more sophisticated. of an Earth using orbiting spacecraft. obtained Several from sampling analysis areX-ray presented to This establish performance measurements models of pulsed pulsed signals. is the the the first expected step and and performance using measurements obtained from models of X-ray signals. This is first step performance using measurements obtained from models of pulsed X-ray signals. This is the first step and performance using measurements obtained from models of pulsed X-ray signals. This is the first step and Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling future efforts shall make the orbit estimation more sophisticated. © 2016,efforts IFACshall (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. future make orbit more future efforts shall make the theFederation orbit estimation estimation more sophisticated. sophisticated. future efforts shall make the orbit estimation more sophisticated. Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling Keywords: Pulsar, Pulsar, Kalman Kalman Filter, Filter, Orbit Orbit Estimation, Estimation, GPS, GPS, Sampling Sampling Keywords: Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling both attitude and orbit. Man made constellation namely 1. INTRODUCTION Global Positioning (GPS)made is used in earth observing both attitude attitude and System orbit. Man Man constellation namely both and orbit. made constellation namely both attitude and orbit. Man made constellation namely INTRODUCTION spacecrafts wherein accurate knowledge is Orbit determination 1. (OD) is very important in interplanetary both and System orbit. Man constellation namely 1. Globalattitude Positioning (GPS)made is used used in of earthposition observing 1. INTRODUCTION INTRODUCTION Global Positioning System (GPS) is in earth observing 1. INTRODUCTION Global Positioning System (GPS) is used in earth observing essential. This constellation is at an altitude of 20,000 km and missions. Pulsar based orbit determination has gained more Global Positioning (GPS)knowledge is used in of earthposition observing whereinSystem accurate is Orbit determination determination (OD) (OD) is is very very important important in in interplanetary interplanetary spacecrafts spacecrafts knowledge of position is Orbit spacecrafts wherein accurate knowledge of position is Orbit (OD) is important in interplanetary is reliableThis inwherein aiding accurate manyis Low Earth Orbiting Satellite interest recently. These celestial sources has provide unique spacecrafts wherein accurate knowledge of position is Orbit determination determination (OD) is very very important in interplanetary essential. constellation at an altitude of 20,000 km and missions. Pulsar based orbit determination gained more essential. This constellation is at an altitude of 20,000 km and missions. Pulsar based orbit determination has gained more essential. This This constellation is at at an an altitude ofwith 20,000 km and missions. Pulsar based orbit determination determination has gained gained more Missions. This concept achieved is fructified the advent signals that can be detected by sensors placed onboard the essential. constellation is altitude of 20,000 km and missions. Pulsar based orbit has more is reliable in aiding many Low Earth Orbiting Satellite interest recently. These celestial sources provide unique is reliable in aiding many Low Earth Satellite interest These celestial sources provide unique is the reliable in clock. aidingThe many Lowdifference Earth Orbiting Orbiting Satellite interest recently. recently. TheseChester celestial sources provide unique of atomic marked iswith in using radio spacecraft, see Wood; and Butman. Detection and is reliable in aiding many Low Earth Orbiting Satellite interest recently. These celestial sources provide unique Missions. This concept achieved is fructified the advent signals that can be detected by sensors placed onboard the Missions. This concept achieved is fructified with the advent signals that can be detected by sensors placed onboard the Missions. This concept achieved is fructified with the advent signals that of canpulsar be detected detected byalong sensors placed onboard the Missions. wave broadcast of time and using a stable atomic clock. The processing signals with time of arrival This concept achieved is fructified with the advent signals that can be by sensors placed onboard the of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and frequency and deduced range enables to determine the information can be used to generate the range measurements of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and wave broadcast of time and using aa stable atomic clock. The processing of pulsar signals along with time of arrival wave broadcast of time and using atomic clock. The processing of pulsar signals along with time of arrival wave broadcast of time and using aa stable stable atomic clock. The processing of pulsar signals along with time of arrival position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. wave broadcast of time and using stable atomic clock. The processing of pulsar signals along with time of arrival frequency and and deduced deduced range range enables enables to to determine determine the the information can can be be used used to to generate generate the the range range measurements measurements frequency information frequency and from deduced range enables to determine the information can be used to generate the measurements by triangulation three satellites. Fourth satellite enables frequency deduced range enables tothe determine the information can used to generate the range range measurements position of and the orbiting orbiting satellite possessing GPS receiver receiver with respect respect tousing anbeinertial inertial frame. The aim of pulsars is to achieve autonomous and position of the satellite possessing the GPS with to an frame. position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. to accurately determine the time. Using dual frequencies and position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. by triangulation from three satellites. Fourth satellite enables absolute orbit determination in space faring missions. This by triangulation from three satellites. Fourth enables by triangulation from three satellites. Fourth satellite enables The aim of using pulsars is to achieve autonomous and resolving phase ambiguities haveUsing shown insatellite estimating the by triangulation from three satellites. Fourth satellite enables The aim of using pulsars is to achieve autonomous and to accurately determine the time. dual frequencies and The aim of using pulsars is to achieve autonomous and requires phase information available with the detector and to accurately determine the time. Using dual frequencies and The aim of using pulsars is to achieve autonomous and to accurately determine the time. Using dual frequencies and absolute orbit determination in space faring missions. This position accuracies close to centimetres. Pulsar based orbit to accurately determine the time. Using dual frequencies and absolute orbitway determination in space space faring missions. This resolving phase ambiguities have shown in estimating the absolute orbit in faring missions. This this is some to go. However the with next and intermediate ambiguities have in absolute orbit determination determination in space faring missions. This resolving phase phase ambiguities have shown in estimating estimating the the requires phase information available the detector and resolving determination closely resembles the shown GPS. Pulsar resolving phase ambiguities have shown in estimating the requires phase information available with the detector and position accuracies close to centimetres. based orbit requires phase information available with the detector and step could be relative orbit determination which is carried out position accuracies close to centimetres. Pulsar orbit requires phase information available with the detector and position accuracies close to centimetres. Pulsar based orbit this is is some some way way to to go. go. However However the the next next and and intermediate intermediate position accuracies close to centimetres. Pulsar based based orbit this determination closely resembles the GPS. this is some way to However the and currently. closely resembles this some to go. go. However the next next and isintermediate intermediate 1.2 Pulse emission and Detectorthe sizeGPS. determination closely resembles the GPS. step is could be way relative orbit determination which carried out out determination determination closely resembles the GPS. step could be relative orbit determination which is carried step could be relative orbit determination which is carried out step could be relative orbit determination which is carried out currently. Generally interplanetary missions need multiple stations currently. 1.2 Pulse emission and Detector size currently. 1.2 Pulse size currently. A neutron and star Detector is the result 1.2pulsar Pulseoremission emission and Detector sizeof a massive star that has 1.2 Pulse emission and Detector size tracking extended missions periods that prohibitively Generally for interplanetary need are multiple stations exhausted its nuclear fuel and undergone a core-collapse Generally interplanetary missions need multiple stations Generally interplanetary missions need multiple stations expensive. Orbiting around Lagrangian points is one of them A pulsar pulsar or or neutron neutron star star is is the the result result of of aa massive massive star star that that has has Generally interplanetary need are multiple stations A tracking for for extended missions periods that that prohibitively A pulsar neutron star is result of massive star has resulting a nuclear supernova explosion. newly tracking extended periods are prohibitively A pulsar or orinits neutron star fuel is the the result of aaYoung, massive star that thatborn has tracking for extended periods that are prohibitively as in ADITYA. It may also be noted that usually we have exhausted and undergone a core-collapse tracking for extended periods that are prohibitively exhausted its nuclear fuel and undergone a core-collapse expensive. Orbiting Orbiting around around Lagrangian Lagrangian points points is is one one of of them them neutron exhausted its nuclear fuel and undergone a core-collapse stars typically rotate with periods on the order of tens expensive. exhausted its nuclear fuel and undergone a core-collapse expensive. Orbiting around Lagrangian points is one of them tracking at least from Bangalore Indian Deep Space Network resulting in a supernova explosion. Young, newly born expensive. Orbiting around Lagrangian points is onewe of them in aa supernova explosion. Young, newly born as in ADITYA. It may also be noted that usually have resulting resulting in explosion. Young, newly born milliseconds, whilerotate older neutron stars through as in ADITYA. It mayavailable also be in noted that usually resulting in typically a supernova supernova explosion. Young, newlyenergy born as in It also noted that usually we have (IDSN). This is always the mission. The we orbithave for of neutron stars with periods on the order of tens as in ADITYA. ADITYA. It may may also be be Indian noted that usually we have neutron stars typically rotate with periods on the order of tens tracking at least from Bangalore Deep Space Network neutron stars typically rotate with periods on the order of tens tracking at least from Bangalore Indian Deep Space Network dissipation eventually slow down to periods on the order of neutron stars typically rotate with periods on the order of tens tracking athours leastuntil fromnext Bangalore Indian Deep Deep Space Network of next few OD is carried out is also available. milliseconds, while older neutron stars through energy tracking at least from Bangalore Indian Space Network of milliseconds, while older neutron stars through energy (IDSN). This is always available in the mission. The orbit for of milliseconds, while older neutron stars through energy (IDSN). This is always available in the mission. The orbit for several seconds. A unique aspect of this rotation is that the of milliseconds, while older neutron stars through energy (IDSN). This is always always available ininthe the mission. The orbit for dissipation This orbitThis is called reference orbit this paper. And relative eventually slow down to periods on the of (IDSN). is available in mission. The orbit for dissipation eventually slow down periods on the order order of next few hours until next OD is carried out is also available. dissipation eventually slow down to periods on of pulsations can beAextremely stableto and predictable. Pulsars next few hours until next OD is carried out is also available. dissipation eventually slow aspect down to periods on the theis order order of next few hours until next OD is carried out is also available. orbit estimated is to follow the reference orbit and arrest any several seconds. unique of this rotation that the next few hours until next OD is carried out is also available. several seconds. A unique aspect of this rotation is that the This orbit is called reference orbit in this paper. And relative several seconds. A unique aspect of this rotation is that the have been found to emit throughout the radio, infrared, This orbit is called reference orbit in this paper. And relative several seconds. A unique aspect of this rotation is that the This orbit is called reference orbit in this paper. And relative deviations that could happenthe especially it happens in Halo pulsations can can be be extremely extremely stable stable and and predictable. predictable. Pulsars Pulsars This is called orbit in thisasorbit paper. And relative orbit orbit estimated is to toreference follow reference and arrest any pulsations pulsations can be extremely stable and predictable. Pulsars visible (optical), X-ray, energies orbit estimated is follow the reference orbit and arrest any pulsations can beultraviolet, extremely stable and and gamma-ray predictable. Pulsars orbit estimated is to follow the reference orbit and arrest any orbits. have been found to emit throughout the radio, infrared, orbit estimated is to follow the reference orbit and arrest any have been found to emit throughout the radio, infrared, deviations that that could could happen happen especially especially as as it it happens happens in in Halo Halo of have been found to emit throughout the radio, infrared, the electromagnetic spectrum, see Sheikh and Becker. deviations have been found to emit throughout the radio, infrared, deviations that could happen especially as it happens in Halo visible (optical), (optical), ultraviolet, ultraviolet, X-ray, X-ray, and and gamma-ray gamma-ray energies energies deviations that could happen especially as it happens in Halo visible orbits. visible (optical), ultraviolet, X-ray, and gamma-ray energies orbits. However, detection within the X-ray allows for the visible (optical), ultraviolet, X-ray, and band gamma-ray energies 1.1 Closeness to GPS orbits. of the electromagnetic spectrum, see Sheikh and Becker. orbits. of the electromagnetic spectrum, see Sheikh and Becker. of the electromagnetic spectrum, see Sheikh and Becker. development of more compact detectors than other bands. of the electromagnetic spectrum, see band Sheikhallows and Becker. However, detection within the X-ray for the 1.1 to GPS 1.1 Closeness Closeness to Sun GPShave been used as navigational aids and However, However, detection detection within within the the X-ray X-ray band band allows allows for for the the Stars, Moon and 1.1 to However, detection within the X-ray band allows for the 1.1 Closeness Closeness to GPS GPS development of more compact detectors than other bands. The detector technology and material mass have seen development of more compact detectors than other bands. development of of more more compact compact detectors detectors than than other other bands. bands.more described in ancient texts all over the world. Today they are development Stars, Moon and Sun have been used as navigational aids and research and development activities. Stars, Moon and Sun been used as navigational and The detector technology and and material mass mass have have seen seen more more Stars,used Moon and Sunofhave have been used asstar navigational aids and The still with help sun all sensors and sensors toaids realize detector technology material Stars, Moon and Sun have been used as navigational aids and The detector technology and material described in ancient texts over the world. Today they are detector technology and material mass mass have have seen seen more more described research and development activities. described in in ancient ancient texts texts all all over over the the world. world. Today Today they they are are The research and development activities. described in ancient texts all over the world. Today they are research still used with help of sun sensors and star sensors to realize research and and development development activities. activities. still used with help of sun sensors and star sensors to realize still used with help of sun sensors and star sensors to realize still used ©with Copyright 2016help IFACof sun sensors and star sensors to realize 136 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 IFAC 136 Copyright © 2016 IFAC 136 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 136Control. Copyright © 2016 IFAC 136 10.1016/j.ifacol.2016.03.042
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
The detector size in the past have been huge that deter any possibility of having it onboard. The literature, as in Becker et al (2013), has three kinds of detector technologies: the glass pore, silicon drift and active antenna. The glass pore has a mass of 25kgs to 5 kg and can help to measure range accurately from 5km to 10km in space. In the case of active antenna assembly of 15kg with 250W, the accuracy can be 5kms. This development will surely reduce the total mass and power requirement and give 5kms and less accuracy.
d - Duty cycle of a pulse t - total observation time The accuracy of the range measurement, R, can be computed using the speed of light, c, and the pulse TOA accuracy, w, see Downs. 2.2 Noise sources The various noises in the pulsar measurements includes Source noise (steady and periodic), diffuse X-ray background noise, Cosmic Background Noise, Detector Noise, Local clock noise, Source shape uncertainty, pulse period uncertainty, source phase jitter, etc.
We shall concern here only about range measurement and not phase measurement.
The primary source of error in the navigation solution is the ability to measure the TOA of the pulse train from each source while contending with shot noise inherent in the faint signal, the diffuse X-ray background, cosmic ray events and detector back ground. The SNR of the measurement (and hence the accuracy of the TOA estimate) can be improved by using a larger detector, or increasing the observation time. However, physical limits of the spacecraft and mass considerations will limit the maximum possible collection area of the detector. Thus, the problem becomes one of extracting the most of photon impression from an observation in order to minimize the required detector area and observation time. This may require certain attitude rate and jitter control. Yet in interplanetary missions this is achievable.
2. PULSAR MEASUREMENTS AND NOISE 2.1 Range measurements To observe a source, an X-ray detector is initially aligned along the line of sight to the chosen source. This usually is along the celestial pole. Once photon events from this source are positively identified, components within the detector system record the time of arrival of each individual X-ray photon with respect to the system’s clock to high precision. During the total observation time of a specific source, a large number of photons will have each of their arrival times recorded. There are many approaches to arrive at the pulse time of arrival as discussed by Sheikh et. Al (2004). The pulse shapes are catalogued.
3. SIMULATIONS
Due to the unique, periodic, nature of the signal produced by these sources, pulse time-of-arrival (TOA) information, which can be deduced as range data can be used to update and then compute three-dimensional position and velocity solutions. As detector systems can be produced to monitor the whole sky, simultaneous observations of multiple source signals from different directions allow this concept to produce full 3-D solutions. Spacecraft that have accurate clocks onboard, can track these signals over time to maintain full dynamic trajectory solutions.
3.1 Noise and Range Simulation Various sources that affect the pulsar measurement and the time of arrival were identified. These had been modelled. The geostationary communication satellite GSAT-10 orbit was used. Any other trajectory could have been used. The extent measurement error would have been of the same order. This noise affects the range accuracy in pulsars. Also included is the timing measurement uncertainty. An algorithm using the modelled error sources was programmed. It has to be noted that the pulsars used for simulation are along the pitch axes of GSAT-10. The instantaneous noise (in km) is shown in the following plots (Fig. 1, Fig. 2, and Fig. 3). Simulated instantaneous noise along the three axes is given below(y axis is in km).
High accuracy measurements from these celestial sources can be utilized within the algorithms to produce improved spacecraft navigation solutions. The estimated accuracy of the arrival time measurement is an important aspect for navigation. It is important to determine the TOA with an accuracy that is determined by the magnitude of the Signalto-Noise-Ratio (SNR) of the measured source profile. In order to select the pulsars for autonomous navigation, the ranging accuracy of pulsars has to be analyzed based on their properties. Such a method for estimating accuracy is used in the present analysis that computes the SNR of a source, see Becker and John Hanson et. Al, based upon the known X-ray characteristics of the source, without requiring raw observation data. SNR = function (F, A, P, B, d, t) F - Total observed flux P - Pulsed fraction
137
(1)
A - Detector area B - Background radiation flux
Fig. 1. Simulated instantaneous noise along the X axis 137
IFAC ACODS 2016 138 M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141 February 1-5, 2016. NIT Tiruchirappalli, India
Fig. 5. Range accuracy sampled every 50 seconds
The instantaneous range accuracy can be about 3000 m. This is with respect to a certain pulsar observed. When observed for 6 seconds the range accuracy improves to 1300 m. And it is seen that the accuracy of the TOA estimate can be improved by collecting photons over an increased observation time and this helps in removing the noise thereby improving the range accuracy to 100 meter in 800 seconds. This is statistical phenomenon and TOA has gets refined. This will be the concern in the simulations of this paper.
Fig. 2. Simulated instantaneous noise along the Y axis
4. ORBIT DETERMINATION 4.1 GEO Orbit Model The technology demonstration of pulsar based orbit determination can be done by sending a detector piggy back on any Geostationary Earth Orbit (GEO) mission. The detector can be mounted on the pitch axis so as to get uninterrupted pulsar signals. With this thought in mind, further simulations were all carried out on a GEO orbit. To begin with, a GEO orbit propagator was modelled in Matlab. The orbit propagator includes the secular effects of J2 and J22, predominant perturbations at GEO orbit, see Chobotov.
Fig. 3. Simulated instantaneous noise along the Z axis
This simulation study was helpful in understanding the measurement noise. The photon collected is integrated over time and the pulse period and time of arrival is estimated. This is carried out next. The following plot in Fig. 4 and Fig. 5 shows range accuracy of a Pulsar which was simulated.
The second order theory for the secular perturbations has been used and the equations are as follows (see Vallado):
An interesting and important observation is that as observation increases the instantaneous noise gets diminished and range accuracies are improved. This was analysed using a typical pulsar from the catalogue.
Anomalistic mean motion 3 𝑛𝑛 = 𝑛𝑛0 1 + 𝐽𝐽2 2
1 − 𝑒𝑒 2 𝑝𝑝2
3 1 − sin2 𝑖𝑖 2
3 2 1 − 𝑒𝑒 2 𝐽𝐽 16 128 2 𝑝𝑝4 + 25 1 − 𝑒𝑒 2 − 15 + 30 − 96 1 − 𝑒𝑒 2 +
1 − 𝑒𝑒 2
− 90 1 − 𝑒𝑒 2 cos2 𝑖𝑖 + {105 + 144 1 − 𝑒𝑒 2 + 25 1 − 𝑒𝑒 2 } cos4 𝑖𝑖 −
45 𝐽𝐽 128 22
+ 35 cos 4 𝑖𝑖)
1 − 𝑒𝑒 2 2 𝑒𝑒 (3 − 30 cos2 𝑖𝑖 𝑝𝑝4
(2)
Fig. 4. Range accuracy sampled every 5 seconds Time derivative of right ascension of the ascending node 138
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
Ω=−
3 𝐽𝐽2 3 𝐽𝐽2 3 𝑒𝑒 2 5 5 𝑛𝑛 cos 𝑖𝑖 1 + 2 + − 2 1 − 𝑒𝑒 2 − − 𝑒𝑒 2 − 3 1 − 𝑒𝑒 2 sin2 𝑖𝑖 2 2 𝑝𝑝 2 𝑝𝑝 2 6 3 24 3 2 12 − 21 sin2 𝑖𝑖 35 𝐽𝐽22 cos 𝑖𝑖 𝑛𝑛 1 + 𝑒𝑒 + 2 14 8 𝑝𝑝4 0
and the state measurements in a covariance matrix, and then update the states by accounting for this error. Once the states are updated, they are propagated again according to the model until another measurement is taken, when the estimated states are updated once more. Kalman filtering loops through this process and tracks satellite motion as measurements are taken, requiring minimal data storage compared to other orbital estimators. The filter uses a measurement once, updates the covariance matrix calculations, and then discards that measurement.
(3) Time derivative of argument of perigee 𝜔𝜔 =
3 𝐽𝐽2 5 𝑛𝑛 2 − sin2 𝑖𝑖 1 2 2 𝑝𝑝 2 3 𝐽𝐽2 𝑒𝑒 2 + 2 + − 2 1 − 𝑒𝑒 2 2 2 𝑝𝑝 2 43 𝑒𝑒 2 − − − 3 1 − 𝑒𝑒 2 sin2 𝑖𝑖 24 48 5 𝐽𝐽22 2 − 𝑒𝑒 𝑛𝑛0 cos 4 𝑖𝑖 4 𝑝𝑝4 35 𝐽𝐽22 12 93 2 21 − 𝑛𝑛 − sin 𝑖𝑖 + sin4 𝑖𝑖 8 𝑝𝑝4 0 7 14 4 24 27 2 81 4 2 + 𝑒𝑒 − sin 𝑖𝑖 + sin 𝑖𝑖 14 4 16
Kalman estimation demonstrates a high degree of flexibility because Kalman filters can be “tuned” to rely more heavily on either the model or the measurements during the state update process. The Kalman filter algorithm includes two matrices designated as R and Q which account for measurement error and system modelling error, respectively. In a real system, true values for sensor noise error and modelling error will never be known, but by tuning the R and Q matrices to these sources of error as precisely as possible, the accuracy of the Kalman estimation can be optimized.
(4) where
4.4 Measurement Error Covariance for KF
ɳ0 : anomalistic mean motion where perturbations are neglected
Errors in the TOA estimate are created by errors in the measured photon arrival time (i.e. clock error), uncertainty in the true pulsar period and uncertainty in the Doppler shift. The error in the TOA estimate of a pulse train from a pulsar has two components, one describing the noise inherent in the signal itself and one describing the added shot noise due to all background photons, including the steady un-pulsed emission from the source, the diffuse X-ray background, cosmic rays and detector noise.
e : eccentricity i : orbital inclination
𝑝𝑝
139
− 𝑒𝑒
(5)
ae : equatorial radius of the earth The orbit propagator includes only gravitational effects of J2 and J22. Other perturbing forces like forces due to sun and moon has not been considered so far.
The covariance of the error in the TOA estimate is: (7)
4.2 State Transition Matrix
where
The state transition matrix maps deviation in the state vector from one time to another. In this case, deviations in the state are mapped from t0 to t. The state deviation vector at any time can be written in terms of x0, as follows (see Vallado):
t0 = time of arrival of the pulse ζs and ζb describe the impact of the pulse shape on the error for source noise and background noise, respectively. T = pulse period
(6)
A = collection area
Due to the limitation of the computing speed and memory of onboard computer, a simple method is applied for calculating the Transition matrix. Components for the calculation of Transition matrix used are the status of the satellite in time t k and time tk-1, gravity parameter μ, J2, Δt.
Δt = total observed time s = source flux b = background rate
4.3 Kalman Filtering
4.5 Linear Kalman Filter (KF) Algorithm
The design of a Kalman filter begins with the selection of the states to be estimated. For orbital estimation, the states are the six components comprising the satellite position and velocity vectors. A set of governing equations and guesses for the initial conditions of each state must be then chosen. Kalman filters take the initial conditions of the states, propagate them forward in time according to the assumed model, compare the propagated states to measurements, calculate and store the error between the state propagation
The blending of the spacecraft state dynamics and pulse ranging measurement can be implemented by using an extended Kalman filter technique. Towards this, a linear Kalman filter model was formulated as follows: This filter recursively incorporates pulse ranging measurements with an estimate of the orbit state. Time update: (8) 139
IFAC ACODS 2016 140 M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141 February 1-5, 2016. NIT Tiruchirappalli, India
Sampling rate: 10 minutes
(9) Measurement Update:
𝑖𝑖
𝑖𝑖
−
(10)
−
(11) (12) (13) 5. RESULTS
GEO orbit state vector was obtained using simulated pulsar measurement for over a period of 1 day with the noise characteristics considered from various sources as mentioned above and this data was used as measurement in the filter and convergence was observed for various sampling rate. The results from the filter are discussed below in Fig. 6 to Fig. 14.
Fig. 9. Error in Position in X axis sampled every 10 minutes
Here the paper discusses the range measurements sampled over different sampling times. Sampling rate: 1 second
Fig. 10. Error in Position in Y axis sampled every 10 minutes
Fig. 6. Error in Position in X axis sampled every 1 second
Fig. 11. Error in Position in Z axis sampled every 10 minutes
Sampling rate: 20 minutes
Fig. 7. Error in Position in Y axis sampled every 1 second
Fig. 12. Error in Position in X axis sampled every 20 minutes Fig. 8. Error in Position in Z axis sampled every 1 second 140
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
141
Relative orbit determination is autonomous onboard and this can help to arrest any violations in the case of a Halo orbit like ADITYA-1. In order to achieve better accuracy of the solution and faster convergence, and more importantly accounting the non-linear dynamics over extended sampling intervals the method of extended Kalman filtering is recommended for the autonomous navigation. However as a next step additional parameters will next be included in the state estimation within linear kalman filter. Also Unscented Kalman filter is suggested in literature and we will look through. Fig. 13. Error in Position in Y axis sampled every 20 minutes
The solar effects become predominant in the case of ADITYA-1 or in the case of any other interplanetary missions. Those perturbations need to be studied and modeled to be included in the orbit simulations purposes.
REFERENCES Becker, W. Mike Bernhardt, G. and Jessner, A. (2013). Autonomous Spacecraft Navigation with Pulsars. ACTA Futura 7, 11-28. Chester, T. J., and Butman S. A. (1981). Navigation Using Xray Pulsars. NASA Technical Reports N81-27129, 2225. Chobotov, A. Vladimir, (1996). Orbital Mechanics. American Institute of Aeronautics and Astronautics, Inc. Washington, DC. Downs, G. S. (1974). Interplanetary Navigation Using Pulsating Radio Sources. JPL Technical Report 32-1594. Jet Propulsion Laboratory, California. John Hanson, Suneel Sheikh, Paul Graven and John Collins. Noise Analysis for X-ray Navigation Systems. Sheikh, S. I. Pines, D. Ray, P. S. Wood, K. S. Lovellette, M. N. and Wolff, M. T. (2004). The use of X-ray Pulsars for Spacecraft Navigation. American Astronautical Society, 105-119. Vallado A. David. (2001). Fundamentals of Astrodynamics and Application. Kluwer Academic Publishers, Dordrecht/Boston/London. Wood, K. S. (1993). Navigation Studies Utilizing the NRL – 801 Experiment and the ARGOS Satellite. Small Satellite Technology and Applications III, Ed. B. J. Horais, SPIE Proceedings, 105-116.
Fig. 14. Error in Position in Z axis sampled every 20 minutes
We know from Fig. 5 that measurement noise becomes lesser with large integration times. That is from Fig. 5, we observe that when the integration time is 600 second, the measurement noise is about 150m. With this as input and as the dynamics become non-linear with sampling time, it is decided to analyse and demonstrate the Kalman filter performance for various sampling intervals. Hence Extended Kalman Filter or Unscented Kalman Filter shall be attemped for all future estimations.
6. CONCLUSIONS The orbit knowledge is about 250 m in about 200 seconds. This gives an input that integration of pulse can be accommodated when the measurement gets more refined as seen earlier. The celestial pulsed radiation could be used as navigation system for spacecraft. The pulsar characteristics and the various sources that affect the measurement of the time of arrival have been identified and modelled. Pulsar noise (in the range) characterization has been updated with current literature and simulated. A Kalman filter was formulated to estimate the orbit. The KF is first realized for verification purposes for a GEO type of orbit. This OD although carried out in GEO, pulsar based OD is invariant considering any futuristic application. In the sense, Pulse noise characteristics and deviations enable orbit estimation anywhere in space to the same accuracies. Therefore, this OD process can be used for MARS-2. This can however be test flown on a Geostationary mission. 141
ScienceDirect
Kalman Filter Analysis for 49-1 Orbit using Pulsars for IFAC-PapersOnLine (2016)Estimation 136–141 Interplanetary Missions using Kalman Filter Analysis for Orbit Estimation Pulsars for Kalman Kalman Filter Analysis for Orbit Estimation using Pulsars for Kalman Filter Filter Analysis Analysis for for Orbit Orbit Estimation Estimation using using Pulsars Pulsars for for Interplanetary Missions Interplanetary Missions Interplanetary Missions M Thameemunnisha*. M P. Ramachandran** Interplanetary Missions
M Thameemunnisha*. Thameemunnisha*. M M P. P. Ramachandran** Ramachandran** M M Thameemunnisha*. Thameemunnisha*. M P. Ramachandran** Ramachandran** M M P. *ISRO Satellite Centre, 560017 Bangalore, India *ISRO (Tel: 080-25084414; e-mail: [email protected]). Satellite Centre, Bangalore, 560017 Satellite Bangalore, *ISRO Satellite Centre, Bangalore, 560017 **India ISRO*ISRO Satellite Centre,Centre, Bangalore, 560017560017 India (e-mail: *ISRO Satellite Centre, Bangalore, 560017 (Tel: 080-25084414; e-mail: [email protected]). India (Tel: 080-25084414; e-mail: [email protected]). India (Tel: 080-25084414; e-mail: [email protected]). [email protected]) 080-25084414; e-mail: [email protected]). **India ISRO(Tel: Satellite Centre, Bangalore, Bangalore, 560017 India India (e-mail: (e-mail: ** ISRO Satellite Centre, 560017 ** Bangalore, ** ISRO ISRO Satellite Satellite Centre, Centre, Bangalore, 560017 560017 India India (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) [email protected]) Abstract: Autonomous Navigation in Interplanetary Missions is the main challenge due to lack of dense ground tracking networkNavigation measurements. This paper presents a novel for determining Abstract: Autonomous in Interplanetary Interplanetary Missions is the technique main challenge challenge due to to lackspacecraft of dense dense Abstract: Autonomous Navigation in Missions is due Abstract:and Autonomous Navigation inX-ray Interplanetary Missions is the the main main challenge due to lack lack of of of dense position velocity using celestial sources, such as pulsars. It portrays the formulation the Abstract: Autonomous Navigation in Interplanetary Missions is the main challenge due to lack of dense ground tracking network measurements. This paper presents a novel technique for determining spacecraft ground tracking measurements. This aa novel technique for spacecraft groundmeasurements tracking network network measurements. This paper paper presents presents novel technique for determining determining spacecraft pulsar and describes the development of a noise model for X-ray Navigation and brings ground tracking network measurements. This paper presents a novel technique for determining spacecraft position and and velocity velocity using using celestial celestial X-ray X-ray sources, sources, such such as as pulsars. pulsars. It It portrays portrays the the formulation formulation of of the the position position and using X-ray sources, as portrays the formulation of out themeasurements cumulative efforts in celestial this developing field. Insuch addition tomodel that,It also covers the blending position and velocity velocity using celestial X-ray sources, such as pulsars. pulsars. Ititfor portrays the formulation of the the pulsar and describes the development of a noise X-ray Navigation and brings pulsar measurements and describes development of aa noise model for brings pulsar measurements and the development of for X-ray X-ray Navigation and brings pulsar-derived measurements withdeveloping a the Kalman filter continuous determination of Navigation position andand pulsar and describes describes the development a noise noisetomodel model X-ray Navigation andvelocity brings out themeasurements cumulative efforts in this this field.forIn Inof addition that, it itforalso also covers the blending blending of the the out the cumulative efforts in developing field. addition to that, that, covers the of out the cumulative efforts in this developing field. In addition to it also covers the blending of of an Earth orbiting spacecraft. Several sampling analysis are presented to establish the expected out the cumulative efforts in this developing field. In addition to that, it also covers the blending of the the pulsar-derived measurements measurements with with aa Kalman Kalman filter filter for for continuous continuous determination determination of of position position and and velocity velocity pulsar-derived pulsar-derived measurements with aa Kalman filter for determination of position velocity performance measurements of pulsed X-ray signals. is theand first expected step and pulsar-derived measurements withobtained Kalmanfrom filtermodels for continuous continuous position and velocity of an an Earth Earth using orbiting spacecraft. Several sampling analysis aredetermination presented toofThis establish the of orbiting spacecraft. Several sampling analysis are presented to establish the expected of an Earth orbiting spacecraft. Several sampling analysis are presented to establish the expected future efforts shall make the orbit estimation more sophisticated. of an Earth using orbiting spacecraft. obtained Several from sampling analysis areX-ray presented to This establish performance measurements models of pulsed pulsed signals. is the the the first expected step and and performance using measurements obtained from models of X-ray signals. This is first step performance using measurements obtained from models of pulsed X-ray signals. This is the first step and performance using measurements obtained from models of pulsed X-ray signals. This is the first step and Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling future efforts shall make the orbit estimation more sophisticated. © 2016,efforts IFACshall (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. future make orbit more future efforts shall make the theFederation orbit estimation estimation more sophisticated. sophisticated. future efforts shall make the orbit estimation more sophisticated. Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling Keywords: Pulsar, Pulsar, Kalman Kalman Filter, Filter, Orbit Orbit Estimation, Estimation, GPS, GPS, Sampling Sampling Keywords: Keywords: Pulsar, Kalman Filter, Orbit Estimation, GPS, Sampling both attitude and orbit. Man made constellation namely 1. INTRODUCTION Global Positioning (GPS)made is used in earth observing both attitude attitude and System orbit. Man Man constellation namely both and orbit. made constellation namely both attitude and orbit. Man made constellation namely INTRODUCTION spacecrafts wherein accurate knowledge is Orbit determination 1. (OD) is very important in interplanetary both and System orbit. Man constellation namely 1. Globalattitude Positioning (GPS)made is used used in of earthposition observing 1. INTRODUCTION INTRODUCTION Global Positioning System (GPS) is in earth observing 1. INTRODUCTION Global Positioning System (GPS) is used in earth observing essential. This constellation is at an altitude of 20,000 km and missions. Pulsar based orbit determination has gained more Global Positioning (GPS)knowledge is used in of earthposition observing whereinSystem accurate is Orbit determination determination (OD) (OD) is is very very important important in in interplanetary interplanetary spacecrafts spacecrafts knowledge of position is Orbit spacecrafts wherein accurate knowledge of position is Orbit (OD) is important in interplanetary is reliableThis inwherein aiding accurate manyis Low Earth Orbiting Satellite interest recently. These celestial sources has provide unique spacecrafts wherein accurate knowledge of position is Orbit determination determination (OD) is very very important in interplanetary essential. constellation at an altitude of 20,000 km and missions. Pulsar based orbit determination gained more essential. This constellation is at an altitude of 20,000 km and missions. Pulsar based orbit determination has gained more essential. This This constellation is at at an an altitude ofwith 20,000 km and missions. Pulsar based orbit determination determination has gained gained more Missions. This concept achieved is fructified the advent signals that can be detected by sensors placed onboard the essential. constellation is altitude of 20,000 km and missions. Pulsar based orbit has more is reliable in aiding many Low Earth Orbiting Satellite interest recently. These celestial sources provide unique is reliable in aiding many Low Earth Satellite interest These celestial sources provide unique is the reliable in clock. aidingThe many Lowdifference Earth Orbiting Orbiting Satellite interest recently. recently. TheseChester celestial sources provide unique of atomic marked iswith in using radio spacecraft, see Wood; and Butman. Detection and is reliable in aiding many Low Earth Orbiting Satellite interest recently. These celestial sources provide unique Missions. This concept achieved is fructified the advent signals that can be detected by sensors placed onboard the Missions. This concept achieved is fructified with the advent signals that can be detected by sensors placed onboard the Missions. This concept achieved is fructified with the advent signals that of canpulsar be detected detected byalong sensors placed onboard the Missions. wave broadcast of time and using a stable atomic clock. The processing signals with time of arrival This concept achieved is fructified with the advent signals that can be by sensors placed onboard the of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and frequency and deduced range enables to determine the information can be used to generate the range measurements of the atomic clock. The marked difference is in using radio spacecraft, see Wood; Chester and Butman. Detection and wave broadcast of time and using aa stable atomic clock. The processing of pulsar signals along with time of arrival wave broadcast of time and using atomic clock. The processing of pulsar signals along with time of arrival wave broadcast of time and using aa stable stable atomic clock. The processing of pulsar signals along with time of arrival position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. wave broadcast of time and using stable atomic clock. The processing of pulsar signals along with time of arrival frequency and and deduced deduced range range enables enables to to determine determine the the information can can be be used used to to generate generate the the range range measurements measurements frequency information frequency and from deduced range enables to determine the information can be used to generate the measurements by triangulation three satellites. Fourth satellite enables frequency deduced range enables tothe determine the information can used to generate the range range measurements position of and the orbiting orbiting satellite possessing GPS receiver receiver with respect respect tousing anbeinertial inertial frame. The aim of pulsars is to achieve autonomous and position of the satellite possessing the GPS with to an frame. position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. to accurately determine the time. Using dual frequencies and position of the orbiting satellite possessing the GPS receiver with respect to an inertial frame. by triangulation from three satellites. Fourth satellite enables absolute orbit determination in space faring missions. This by triangulation from three satellites. Fourth enables by triangulation from three satellites. Fourth satellite enables The aim of using pulsars is to achieve autonomous and resolving phase ambiguities haveUsing shown insatellite estimating the by triangulation from three satellites. Fourth satellite enables The aim of using pulsars is to achieve autonomous and to accurately determine the time. dual frequencies and The aim of using pulsars is to achieve autonomous and requires phase information available with the detector and to accurately determine the time. Using dual frequencies and The aim of using pulsars is to achieve autonomous and to accurately determine the time. Using dual frequencies and absolute orbit determination in space faring missions. This position accuracies close to centimetres. Pulsar based orbit to accurately determine the time. Using dual frequencies and absolute orbitway determination in space space faring missions. This resolving phase ambiguities have shown in estimating the absolute orbit in faring missions. This this is some to go. However the with next and intermediate ambiguities have in absolute orbit determination determination in space faring missions. This resolving phase phase ambiguities have shown in estimating estimating the the requires phase information available the detector and resolving determination closely resembles the shown GPS. Pulsar resolving phase ambiguities have shown in estimating the requires phase information available with the detector and position accuracies close to centimetres. based orbit requires phase information available with the detector and step could be relative orbit determination which is carried out position accuracies close to centimetres. Pulsar orbit requires phase information available with the detector and position accuracies close to centimetres. Pulsar based orbit this is is some some way way to to go. go. However However the the next next and and intermediate intermediate position accuracies close to centimetres. Pulsar based based orbit this determination closely resembles the GPS. this is some way to However the and currently. closely resembles this some to go. go. However the next next and isintermediate intermediate 1.2 Pulse emission and Detectorthe sizeGPS. determination closely resembles the GPS. step is could be way relative orbit determination which carried out out determination determination closely resembles the GPS. step could be relative orbit determination which is carried step could be relative orbit determination which is carried out step could be relative orbit determination which is carried out currently. Generally interplanetary missions need multiple stations currently. 1.2 Pulse emission and Detector size currently. 1.2 Pulse size currently. A neutron and star Detector is the result 1.2pulsar Pulseoremission emission and Detector sizeof a massive star that has 1.2 Pulse emission and Detector size tracking extended missions periods that prohibitively Generally for interplanetary need are multiple stations exhausted its nuclear fuel and undergone a core-collapse Generally interplanetary missions need multiple stations Generally interplanetary missions need multiple stations expensive. Orbiting around Lagrangian points is one of them A pulsar pulsar or or neutron neutron star star is is the the result result of of aa massive massive star star that that has has Generally interplanetary need are multiple stations A tracking for for extended missions periods that that prohibitively A pulsar neutron star is result of massive star has resulting a nuclear supernova explosion. newly tracking extended periods are prohibitively A pulsar or orinits neutron star fuel is the the result of aaYoung, massive star that thatborn has tracking for extended periods that are prohibitively as in ADITYA. It may also be noted that usually we have exhausted and undergone a core-collapse tracking for extended periods that are prohibitively exhausted its nuclear fuel and undergone a core-collapse expensive. Orbiting Orbiting around around Lagrangian Lagrangian points points is is one one of of them them neutron exhausted its nuclear fuel and undergone a core-collapse stars typically rotate with periods on the order of tens expensive. exhausted its nuclear fuel and undergone a core-collapse expensive. Orbiting around Lagrangian points is one of them tracking at least from Bangalore Indian Deep Space Network resulting in a supernova explosion. Young, newly born expensive. Orbiting around Lagrangian points is onewe of them in aa supernova explosion. Young, newly born as in ADITYA. It may also be noted that usually have resulting resulting in explosion. Young, newly born milliseconds, whilerotate older neutron stars through as in ADITYA. It mayavailable also be in noted that usually resulting in typically a supernova supernova explosion. Young, newlyenergy born as in It also noted that usually we have (IDSN). This is always the mission. The we orbithave for of neutron stars with periods on the order of tens as in ADITYA. ADITYA. It may may also be be Indian noted that usually we have neutron stars typically rotate with periods on the order of tens tracking at least from Bangalore Deep Space Network neutron stars typically rotate with periods on the order of tens tracking at least from Bangalore Indian Deep Space Network dissipation eventually slow down to periods on the order of neutron stars typically rotate with periods on the order of tens tracking athours leastuntil fromnext Bangalore Indian Deep Deep Space Network of next few OD is carried out is also available. milliseconds, while older neutron stars through energy tracking at least from Bangalore Indian Space Network of milliseconds, while older neutron stars through energy (IDSN). This is always available in the mission. The orbit for of milliseconds, while older neutron stars through energy (IDSN). This is always available in the mission. The orbit for several seconds. A unique aspect of this rotation is that the of milliseconds, while older neutron stars through energy (IDSN). This is always always available ininthe the mission. The orbit for dissipation This orbitThis is called reference orbit this paper. And relative eventually slow down to periods on the of (IDSN). is available in mission. The orbit for dissipation eventually slow down periods on the order order of next few hours until next OD is carried out is also available. dissipation eventually slow down to periods on of pulsations can beAextremely stableto and predictable. Pulsars next few hours until next OD is carried out is also available. dissipation eventually slow aspect down to periods on the theis order order of next few hours until next OD is carried out is also available. orbit estimated is to follow the reference orbit and arrest any several seconds. unique of this rotation that the next few hours until next OD is carried out is also available. several seconds. A unique aspect of this rotation is that the This orbit is called reference orbit in this paper. And relative several seconds. A unique aspect of this rotation is that the have been found to emit throughout the radio, infrared, This orbit is called reference orbit in this paper. And relative several seconds. A unique aspect of this rotation is that the This orbit is called reference orbit in this paper. And relative deviations that could happenthe especially it happens in Halo pulsations can can be be extremely extremely stable stable and and predictable. predictable. Pulsars Pulsars This is called orbit in thisasorbit paper. And relative orbit orbit estimated is to toreference follow reference and arrest any pulsations pulsations can be extremely stable and predictable. Pulsars visible (optical), X-ray, energies orbit estimated is follow the reference orbit and arrest any pulsations can beultraviolet, extremely stable and and gamma-ray predictable. Pulsars orbit estimated is to follow the reference orbit and arrest any orbits. have been found to emit throughout the radio, infrared, orbit estimated is to follow the reference orbit and arrest any have been found to emit throughout the radio, infrared, deviations that that could could happen happen especially especially as as it it happens happens in in Halo Halo of have been found to emit throughout the radio, infrared, the electromagnetic spectrum, see Sheikh and Becker. deviations have been found to emit throughout the radio, infrared, deviations that could happen especially as it happens in Halo visible (optical), (optical), ultraviolet, ultraviolet, X-ray, X-ray, and and gamma-ray gamma-ray energies energies deviations that could happen especially as it happens in Halo visible orbits. visible (optical), ultraviolet, X-ray, and gamma-ray energies orbits. However, detection within the X-ray allows for the visible (optical), ultraviolet, X-ray, and band gamma-ray energies 1.1 Closeness to GPS orbits. of the electromagnetic spectrum, see Sheikh and Becker. orbits. of the electromagnetic spectrum, see Sheikh and Becker. of the electromagnetic spectrum, see Sheikh and Becker. development of more compact detectors than other bands. of the electromagnetic spectrum, see band Sheikhallows and Becker. However, detection within the X-ray for the 1.1 to GPS 1.1 Closeness Closeness to Sun GPShave been used as navigational aids and However, However, detection detection within within the the X-ray X-ray band band allows allows for for the the Stars, Moon and 1.1 to However, detection within the X-ray band allows for the 1.1 Closeness Closeness to GPS GPS development of more compact detectors than other bands. The detector technology and material mass have seen development of more compact detectors than other bands. development of of more more compact compact detectors detectors than than other other bands. bands.more described in ancient texts all over the world. Today they are development Stars, Moon and Sun have been used as navigational aids and research and development activities. Stars, Moon and Sun been used as navigational and The detector technology and and material mass mass have have seen seen more more Stars,used Moon and Sunofhave have been used asstar navigational aids and The still with help sun all sensors and sensors toaids realize detector technology material Stars, Moon and Sun have been used as navigational aids and The detector technology and material described in ancient texts over the world. Today they are detector technology and material mass mass have have seen seen more more described research and development activities. described in in ancient ancient texts texts all all over over the the world. world. Today Today they they are are The research and development activities. described in ancient texts all over the world. Today they are research still used with help of sun sensors and star sensors to realize research and and development development activities. activities. still used with help of sun sensors and star sensors to realize still used with help of sun sensors and star sensors to realize still used ©with Copyright 2016help IFACof sun sensors and star sensors to realize 136 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 IFAC 136 Copyright © 2016 IFAC 136 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 136Control. Copyright © 2016 IFAC 136 10.1016/j.ifacol.2016.03.042
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
The detector size in the past have been huge that deter any possibility of having it onboard. The literature, as in Becker et al (2013), has three kinds of detector technologies: the glass pore, silicon drift and active antenna. The glass pore has a mass of 25kgs to 5 kg and can help to measure range accurately from 5km to 10km in space. In the case of active antenna assembly of 15kg with 250W, the accuracy can be 5kms. This development will surely reduce the total mass and power requirement and give 5kms and less accuracy.
d - Duty cycle of a pulse t - total observation time The accuracy of the range measurement, R, can be computed using the speed of light, c, and the pulse TOA accuracy, w, see Downs. 2.2 Noise sources The various noises in the pulsar measurements includes Source noise (steady and periodic), diffuse X-ray background noise, Cosmic Background Noise, Detector Noise, Local clock noise, Source shape uncertainty, pulse period uncertainty, source phase jitter, etc.
We shall concern here only about range measurement and not phase measurement.
The primary source of error in the navigation solution is the ability to measure the TOA of the pulse train from each source while contending with shot noise inherent in the faint signal, the diffuse X-ray background, cosmic ray events and detector back ground. The SNR of the measurement (and hence the accuracy of the TOA estimate) can be improved by using a larger detector, or increasing the observation time. However, physical limits of the spacecraft and mass considerations will limit the maximum possible collection area of the detector. Thus, the problem becomes one of extracting the most of photon impression from an observation in order to minimize the required detector area and observation time. This may require certain attitude rate and jitter control. Yet in interplanetary missions this is achievable.
2. PULSAR MEASUREMENTS AND NOISE 2.1 Range measurements To observe a source, an X-ray detector is initially aligned along the line of sight to the chosen source. This usually is along the celestial pole. Once photon events from this source are positively identified, components within the detector system record the time of arrival of each individual X-ray photon with respect to the system’s clock to high precision. During the total observation time of a specific source, a large number of photons will have each of their arrival times recorded. There are many approaches to arrive at the pulse time of arrival as discussed by Sheikh et. Al (2004). The pulse shapes are catalogued.
3. SIMULATIONS
Due to the unique, periodic, nature of the signal produced by these sources, pulse time-of-arrival (TOA) information, which can be deduced as range data can be used to update and then compute three-dimensional position and velocity solutions. As detector systems can be produced to monitor the whole sky, simultaneous observations of multiple source signals from different directions allow this concept to produce full 3-D solutions. Spacecraft that have accurate clocks onboard, can track these signals over time to maintain full dynamic trajectory solutions.
3.1 Noise and Range Simulation Various sources that affect the pulsar measurement and the time of arrival were identified. These had been modelled. The geostationary communication satellite GSAT-10 orbit was used. Any other trajectory could have been used. The extent measurement error would have been of the same order. This noise affects the range accuracy in pulsars. Also included is the timing measurement uncertainty. An algorithm using the modelled error sources was programmed. It has to be noted that the pulsars used for simulation are along the pitch axes of GSAT-10. The instantaneous noise (in km) is shown in the following plots (Fig. 1, Fig. 2, and Fig. 3). Simulated instantaneous noise along the three axes is given below(y axis is in km).
High accuracy measurements from these celestial sources can be utilized within the algorithms to produce improved spacecraft navigation solutions. The estimated accuracy of the arrival time measurement is an important aspect for navigation. It is important to determine the TOA with an accuracy that is determined by the magnitude of the Signalto-Noise-Ratio (SNR) of the measured source profile. In order to select the pulsars for autonomous navigation, the ranging accuracy of pulsars has to be analyzed based on their properties. Such a method for estimating accuracy is used in the present analysis that computes the SNR of a source, see Becker and John Hanson et. Al, based upon the known X-ray characteristics of the source, without requiring raw observation data. SNR = function (F, A, P, B, d, t) F - Total observed flux P - Pulsed fraction
137
(1)
A - Detector area B - Background radiation flux
Fig. 1. Simulated instantaneous noise along the X axis 137
IFAC ACODS 2016 138 M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141 February 1-5, 2016. NIT Tiruchirappalli, India
Fig. 5. Range accuracy sampled every 50 seconds
The instantaneous range accuracy can be about 3000 m. This is with respect to a certain pulsar observed. When observed for 6 seconds the range accuracy improves to 1300 m. And it is seen that the accuracy of the TOA estimate can be improved by collecting photons over an increased observation time and this helps in removing the noise thereby improving the range accuracy to 100 meter in 800 seconds. This is statistical phenomenon and TOA has gets refined. This will be the concern in the simulations of this paper.
Fig. 2. Simulated instantaneous noise along the Y axis
4. ORBIT DETERMINATION 4.1 GEO Orbit Model The technology demonstration of pulsar based orbit determination can be done by sending a detector piggy back on any Geostationary Earth Orbit (GEO) mission. The detector can be mounted on the pitch axis so as to get uninterrupted pulsar signals. With this thought in mind, further simulations were all carried out on a GEO orbit. To begin with, a GEO orbit propagator was modelled in Matlab. The orbit propagator includes the secular effects of J2 and J22, predominant perturbations at GEO orbit, see Chobotov.
Fig. 3. Simulated instantaneous noise along the Z axis
This simulation study was helpful in understanding the measurement noise. The photon collected is integrated over time and the pulse period and time of arrival is estimated. This is carried out next. The following plot in Fig. 4 and Fig. 5 shows range accuracy of a Pulsar which was simulated.
The second order theory for the secular perturbations has been used and the equations are as follows (see Vallado):
An interesting and important observation is that as observation increases the instantaneous noise gets diminished and range accuracies are improved. This was analysed using a typical pulsar from the catalogue.
Anomalistic mean motion 3 𝑛𝑛 = 𝑛𝑛0 1 + 𝐽𝐽2 2
1 − 𝑒𝑒 2 𝑝𝑝2
3 1 − sin2 𝑖𝑖 2
3 2 1 − 𝑒𝑒 2 𝐽𝐽 16 128 2 𝑝𝑝4 + 25 1 − 𝑒𝑒 2 − 15 + 30 − 96 1 − 𝑒𝑒 2 +
1 − 𝑒𝑒 2
− 90 1 − 𝑒𝑒 2 cos2 𝑖𝑖 + {105 + 144 1 − 𝑒𝑒 2 + 25 1 − 𝑒𝑒 2 } cos4 𝑖𝑖 −
45 𝐽𝐽 128 22
+ 35 cos 4 𝑖𝑖)
1 − 𝑒𝑒 2 2 𝑒𝑒 (3 − 30 cos2 𝑖𝑖 𝑝𝑝4
(2)
Fig. 4. Range accuracy sampled every 5 seconds Time derivative of right ascension of the ascending node 138
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
Ω=−
3 𝐽𝐽2 3 𝐽𝐽2 3 𝑒𝑒 2 5 5 𝑛𝑛 cos 𝑖𝑖 1 + 2 + − 2 1 − 𝑒𝑒 2 − − 𝑒𝑒 2 − 3 1 − 𝑒𝑒 2 sin2 𝑖𝑖 2 2 𝑝𝑝 2 𝑝𝑝 2 6 3 24 3 2 12 − 21 sin2 𝑖𝑖 35 𝐽𝐽22 cos 𝑖𝑖 𝑛𝑛 1 + 𝑒𝑒 + 2 14 8 𝑝𝑝4 0
and the state measurements in a covariance matrix, and then update the states by accounting for this error. Once the states are updated, they are propagated again according to the model until another measurement is taken, when the estimated states are updated once more. Kalman filtering loops through this process and tracks satellite motion as measurements are taken, requiring minimal data storage compared to other orbital estimators. The filter uses a measurement once, updates the covariance matrix calculations, and then discards that measurement.
(3) Time derivative of argument of perigee 𝜔𝜔 =
3 𝐽𝐽2 5 𝑛𝑛 2 − sin2 𝑖𝑖 1 2 2 𝑝𝑝 2 3 𝐽𝐽2 𝑒𝑒 2 + 2 + − 2 1 − 𝑒𝑒 2 2 2 𝑝𝑝 2 43 𝑒𝑒 2 − − − 3 1 − 𝑒𝑒 2 sin2 𝑖𝑖 24 48 5 𝐽𝐽22 2 − 𝑒𝑒 𝑛𝑛0 cos 4 𝑖𝑖 4 𝑝𝑝4 35 𝐽𝐽22 12 93 2 21 − 𝑛𝑛 − sin 𝑖𝑖 + sin4 𝑖𝑖 8 𝑝𝑝4 0 7 14 4 24 27 2 81 4 2 + 𝑒𝑒 − sin 𝑖𝑖 + sin 𝑖𝑖 14 4 16
Kalman estimation demonstrates a high degree of flexibility because Kalman filters can be “tuned” to rely more heavily on either the model or the measurements during the state update process. The Kalman filter algorithm includes two matrices designated as R and Q which account for measurement error and system modelling error, respectively. In a real system, true values for sensor noise error and modelling error will never be known, but by tuning the R and Q matrices to these sources of error as precisely as possible, the accuracy of the Kalman estimation can be optimized.
(4) where
4.4 Measurement Error Covariance for KF
ɳ0 : anomalistic mean motion where perturbations are neglected
Errors in the TOA estimate are created by errors in the measured photon arrival time (i.e. clock error), uncertainty in the true pulsar period and uncertainty in the Doppler shift. The error in the TOA estimate of a pulse train from a pulsar has two components, one describing the noise inherent in the signal itself and one describing the added shot noise due to all background photons, including the steady un-pulsed emission from the source, the diffuse X-ray background, cosmic rays and detector noise.
e : eccentricity i : orbital inclination
𝑝𝑝
139
− 𝑒𝑒
(5)
ae : equatorial radius of the earth The orbit propagator includes only gravitational effects of J2 and J22. Other perturbing forces like forces due to sun and moon has not been considered so far.
The covariance of the error in the TOA estimate is: (7)
4.2 State Transition Matrix
where
The state transition matrix maps deviation in the state vector from one time to another. In this case, deviations in the state are mapped from t0 to t. The state deviation vector at any time can be written in terms of x0, as follows (see Vallado):
t0 = time of arrival of the pulse ζs and ζb describe the impact of the pulse shape on the error for source noise and background noise, respectively. T = pulse period
(6)
A = collection area
Due to the limitation of the computing speed and memory of onboard computer, a simple method is applied for calculating the Transition matrix. Components for the calculation of Transition matrix used are the status of the satellite in time t k and time tk-1, gravity parameter μ, J2, Δt.
Δt = total observed time s = source flux b = background rate
4.3 Kalman Filtering
4.5 Linear Kalman Filter (KF) Algorithm
The design of a Kalman filter begins with the selection of the states to be estimated. For orbital estimation, the states are the six components comprising the satellite position and velocity vectors. A set of governing equations and guesses for the initial conditions of each state must be then chosen. Kalman filters take the initial conditions of the states, propagate them forward in time according to the assumed model, compare the propagated states to measurements, calculate and store the error between the state propagation
The blending of the spacecraft state dynamics and pulse ranging measurement can be implemented by using an extended Kalman filter technique. Towards this, a linear Kalman filter model was formulated as follows: This filter recursively incorporates pulse ranging measurements with an estimate of the orbit state. Time update: (8) 139
IFAC ACODS 2016 140 M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141 February 1-5, 2016. NIT Tiruchirappalli, India
Sampling rate: 10 minutes
(9) Measurement Update:
𝑖𝑖
𝑖𝑖
−
(10)
−
(11) (12) (13) 5. RESULTS
GEO orbit state vector was obtained using simulated pulsar measurement for over a period of 1 day with the noise characteristics considered from various sources as mentioned above and this data was used as measurement in the filter and convergence was observed for various sampling rate. The results from the filter are discussed below in Fig. 6 to Fig. 14.
Fig. 9. Error in Position in X axis sampled every 10 minutes
Here the paper discusses the range measurements sampled over different sampling times. Sampling rate: 1 second
Fig. 10. Error in Position in Y axis sampled every 10 minutes
Fig. 6. Error in Position in X axis sampled every 1 second
Fig. 11. Error in Position in Z axis sampled every 10 minutes
Sampling rate: 20 minutes
Fig. 7. Error in Position in Y axis sampled every 1 second
Fig. 12. Error in Position in X axis sampled every 20 minutes Fig. 8. Error in Position in Z axis sampled every 1 second 140
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India M Thameemunnisha et al. / IFAC-PapersOnLine 49-1 (2016) 136–141
141
Relative orbit determination is autonomous onboard and this can help to arrest any violations in the case of a Halo orbit like ADITYA-1. In order to achieve better accuracy of the solution and faster convergence, and more importantly accounting the non-linear dynamics over extended sampling intervals the method of extended Kalman filtering is recommended for the autonomous navigation. However as a next step additional parameters will next be included in the state estimation within linear kalman filter. Also Unscented Kalman filter is suggested in literature and we will look through. Fig. 13. Error in Position in Y axis sampled every 20 minutes
The solar effects become predominant in the case of ADITYA-1 or in the case of any other interplanetary missions. Those perturbations need to be studied and modeled to be included in the orbit simulations purposes.
REFERENCES Becker, W. Mike Bernhardt, G. and Jessner, A. (2013). Autonomous Spacecraft Navigation with Pulsars. ACTA Futura 7, 11-28. Chester, T. J., and Butman S. A. (1981). Navigation Using Xray Pulsars. NASA Technical Reports N81-27129, 2225. Chobotov, A. Vladimir, (1996). Orbital Mechanics. American Institute of Aeronautics and Astronautics, Inc. Washington, DC. Downs, G. S. (1974). Interplanetary Navigation Using Pulsating Radio Sources. JPL Technical Report 32-1594. Jet Propulsion Laboratory, California. John Hanson, Suneel Sheikh, Paul Graven and John Collins. Noise Analysis for X-ray Navigation Systems. Sheikh, S. I. Pines, D. Ray, P. S. Wood, K. S. Lovellette, M. N. and Wolff, M. T. (2004). The use of X-ray Pulsars for Spacecraft Navigation. American Astronautical Society, 105-119. Vallado A. David. (2001). Fundamentals of Astrodynamics and Application. Kluwer Academic Publishers, Dordrecht/Boston/London. Wood, K. S. (1993). Navigation Studies Utilizing the NRL – 801 Experiment and the ARGOS Satellite. Small Satellite Technology and Applications III, Ed. B. J. Horais, SPIE Proceedings, 105-116.
Fig. 14. Error in Position in Z axis sampled every 20 minutes
We know from Fig. 5 that measurement noise becomes lesser with large integration times. That is from Fig. 5, we observe that when the integration time is 600 second, the measurement noise is about 150m. With this as input and as the dynamics become non-linear with sampling time, it is decided to analyse and demonstrate the Kalman filter performance for various sampling intervals. Hence Extended Kalman Filter or Unscented Kalman Filter shall be attemped for all future estimations.
6. CONCLUSIONS The orbit knowledge is about 250 m in about 200 seconds. This gives an input that integration of pulse can be accommodated when the measurement gets more refined as seen earlier. The celestial pulsed radiation could be used as navigation system for spacecraft. The pulsar characteristics and the various sources that affect the measurement of the time of arrival have been identified and modelled. Pulsar noise (in the range) characterization has been updated with current literature and simulated. A Kalman filter was formulated to estimate the orbit. The KF is first realized for verification purposes for a GEO type of orbit. This OD although carried out in GEO, pulsar based OD is invariant considering any futuristic application. In the sense, Pulse noise characteristics and deviations enable orbit estimation anywhere in space to the same accuracies. Therefore, this OD process can be used for MARS-2. This can however be test flown on a Geostationary mission. 141