Development of On-board orbit Development of On-board orbit determination system for Low Earth Orbit determination system for Low Earth Orbit (LEO) satellite Using Global Navigation (LEO) satellite Using Global Navigation Satellite System (GNSS) ReceiverSatellite System (GNSS) Receiver

Presented by,

Mr. Sandip AghavMr. Sandip Aghav

Department of Electronic Science,Department of Electronic Science,

University of Pune,University of Pune,

PunePune

Introduction

Doppler Measurement

Orbit Determination Techniques

Ground Based

Laser Ranging

Sun sensor, star sensor

Space borne

GNSS Measurements

Classification of Orbit determination techniques

Problem Definition

A method is proposed to use onboard GPS Receiver stand-alone with a direct measurement of position, velocity and acceleration data for orbit determination instead of using differential technique and combined observational technique.

Use of Simplified force models for orbit determination and reduce the extra Burdon from hardware.

Application Target Area: Remote Sensing Satellites

Range: 500 Km to 1200 Km

Positional Accuracy: <50m and Velocity: 1m/sec

Disadvantages of Ground Based Orbit Determination Techniques

Common disadvantage: Data can be collected from satellite only when the satellite is in the

line of sight of the controlling Ground Station.

Methods Disadvantages

Doppler Shift Measurements •Tropospheric, ionospheric and multipath errors •Accuracy frequency dependent.

Triangulation Method •Large number of Ground stations and their maintenance

Laser Ranging Technique •weather conditions, •troposphere errors, •laser system drift, •station position errors, etc

Why GPS based position determination

Ground station is reduced of several operational burdens.

All time data collection is possible The cost of planning experimental observations

is substantially reduced. Scheduling the ground station operations and

data collection is easier and can be done in advance as needed.

Autonomous orbit determination possible

Need of Autonomous On-board satellite Navigation system

On-board collection of data reduces many errors in the orbit determination.

On Board real time orbit determination is possible.

Data processing can be done on-board.

On-board orbit correction is possible.

Concept of Autonomous Navigation System

Objectives of the proposed work

To design/simulate orbit determination algorithm to be used on-board for satellite navigation.

To design/simulate GPS data filtering technique to be placed on-board satellite.

To select a simplified satellite orbit models for on-board processing.

feasibility of Use of above mentioned software on-board a satellite to make the navigation autonomous.

Methodology

Orbit Integration Orbit Estimation

R-K method,R-K method, Cowell’s Method

Least Square, Kalman FilterKalman Filter

Flow chartSTART

ACQUIRE A PRIORI STATE AND COVARIANCE ESTIMATES AT t0

SET k=0, i.e Initialization

k=k+1ACQUIRE A MEMBER OF OBSERVATION VECTOR Yk

PROPAGATE STATE VECTOR TO tk, CALCULATE STATE

TRANSITION MATRIX Φ (tk, tk+1)

CALCULATE EXPEXTED MEASUREMENT Xk AND PARTIAL

DERIVATIVES OF Xk WITH RESPECT TO Xk-1(tk)

PROPAGATE STATE NOISE COVARIANCE MATRIX Q(tk, tk-1)

PROPAGATE ERROR COVARIANCE MATRIX Pk-1(tk)

CALCULATE GAIN MATRIX K

UPDATE X*k-1 TO BECOME kth STATE ESTIMATE

UPDATE ERROR COVARIANCE MATRIX Pk

PROPAGATE Xk(tk) TO ANY TIME OF THE INTREST

LAST OBSERVATION?

END

Y

N

Kalman Filter and

Orbit Estimation

Orbit Estimation Method

Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.

Uncertain: Model, Measurement, Perturbations, etc.

Kalman Filter: Orbit Determination

Kalman Filter Basics:

“An optimal recursive data processing algorithm”

An efficient recursive filter that estimates the state of a linear dynamic system from a series of noisy measurements.

Very well suited for Real Time Data Filtering.

Estimate the state and the covariance of the state at any time T, given observations, xT = x1, …, xT

Kalman FilterMathematical Background

H relates the state to the measurement z at step k.

R is the measurement noise covariance.

Kalman Filter: Non Linear System

State Vector Propagation/Update:

15

2

31

2

22

23 r

z

r

RJ

r

xx e

xx

yy

2

22

2353

2

31

r

z

r

RJ

r

zz e

Abovementioned equation of motion is numerically integrated using Runge-Kutta 4th order method.

Integration is taken over initial to final time. Results were tested for various time step.

Seed Orbital Elements

Six orbital elements semi major axis (a) eccentricity (e) inclination angle (i) longitude of ascending node (Ω)

argument of perigee ()

time of perigee passage ()

As a function of time ‘t’ from standard ground station.

From six orbital elements, ECEF coordinates of the satellites are calculated.

Position vector r(t) = x(t)i + y(t)j + z(t)k

Position measurements using on-board GPS receiver

Collects data from GPS receiver (RINEX format) as a function of time ‘tc’ Conversion of RINEX format data into position and velocity (ECEF

coordinates).

GPS receiver measurements are in Geodetic co-ordinate system. It needs to be converted in to geocentric coordinate system.

Again calculate of position, acceleration and velocity vectors by same method which is used for reference orbit calculation.

Use Extended Kalman filer algorithm to estimate the optimal state vector.

Error calculation and error minimization

Generate new corrected orbit

Simplified force model:

Pure Keplerian and Newtonian model of Satellite orbit is selected.

Gaussian nature with zero mean nose model is selected.

J2, J3, J4 Earth Gravity model is selected.

4th Order Runge-Kutta method is selected with fixed step size.

Kalman filter: Initial Calculations

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

-8.29E-07 4.91E-08 7.36E-07 0 0 0

4.91E-08 -1.01E-06 1.90E-07 0 0 0

7.36E-07 1.90E-07 1.84E-06 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0.01 0 0

0 0 0 0 0.01 0

0 0 0 0 0 0.01

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

1.027118 1.027118 1.027118 1.027118 1.027118 1.027118

Table 2: Initial Covariance matrix

1823.2 470.7 7066.7 -6.6 2.9 1.5

Table 1: Initial State Vector

Table 3: Propagated error Covariance matrix

Table 4: System Jacobian matrix

0 2 4 6 8 10

x 104

6828.94

6828.96

6828.98

6829

6829.02

6829.04

tsec

Sem

i-m

ajo

r axis

(a)

in K

m

Time Vs Semi-major axis

0 2 4 6 8 10

x 104

9.0154

9.0155

9.0156

9.0157

9.0158

9.0159

9.016

9.0161

9.0162x 10

-3

tsec

eccentr

icity(e

)

Time Vs eccentricity

0 2 4 6 8 10

x 104

28.474

28.474

28.474

28.474

28.474

28.474

28.474

28.474

28.474

28.474

tsec

Inclination A

ngle

(i)

in d

egre

es Time Vs Inclination Angle

0 2 4 6 8 10

x 104

35.9118

35.9118

35.9118

35.9118

35.9118

35.9118

35.9118

35.9118

35.9118

35.9118

tsec

Rig

ht

Assention o

f A

scendin

g N

ode(O

MG

) in

degre

es

Time Vs Right Assention of Ascending Node

0 2 4 6 8 10

x 104

-44.572

-44.571

-44.57

-44.569

-44.568

-44.567

-44.566

-44.565

tsec

Arg

um

ent

of

perigee(o

mg)

in d

egre

es

Time Vs Argument of perigee

0 2 4 6 8 10

x 104

0

20

40

60

80

100

120

140

160

180

tsec

Tru

e A

nom

oly

(v)

in d

egre

es

Time Vs True Anomoly

0 2 4 6 8 10

x 104

6824

6825

6826

6827

6828

6829

6830

tsec

Sem

i-m

ajo

r axis

(a)

in K

m

Time Vs Semi-major axis

0 2 4 6 8 10

x 104

7

7.5

8

8.5

9

9.5x 10

-3

tsec

eccentr

icity(e

)

Time Vs eccentricity

0 2 4 6 8 10

x 104

28.435

28.44

28.445

28.45

28.455

28.46

28.465

28.47

28.475

tsec

Inclination A

ngle

(i)

in d

egre

es Time Vs Inclination Angle

0 2 4 6 8 10

x 104

28

29

30

31

32

33

34

35

36

tsec

Rig

ht

Assention o

f A

scendin

g N

ode(O

MG

) in

degre

es

Time Vs Right Assention of Ascending Node

0 2 4 6 8 10

x 104

-60

-55

-50

-45

-40

-35

-30

tsec

Arg

um

ent

of

perigee(o

mg)

in d

egre

es

Time Vs Argument of perigee

0 2 4 6 8 10

x 104

0

20

40

60

80

100

120

140

160

180

tsec

Tru

e A

nom

oly

(v)

in d

egre

es

Time Vs True Anomoly

Fig:1: Orbital Elements with Pure Keplerian Equations

Fig:2: Orbital Elements with J2 Effect

Fig: Effect of Secular variation J2 ,J3, J4 on orbit geometry

-1-0.5

00.5

1

x 104

-1-0.5

0

0.51

x 104

-4000

-2000

0

2000

4000

-1-0.5

00.5

1

x 104

-1

-0.5

0

0.5

1

x 104

-4000

-2000

0

2000

4000

-1-0.5

00.5

1

x 104

-1

-0.5

0

0.5

1

x 104

-4000

-2000

0

2000

4000

x[Km]y[Km]

z[km

]

(a) Pure Keplerian

(b) J2

(a) J2,J3,J4

Conclusion Orbit Integration using Kepler’s and Newton’s Laws of

motion. GPS RINEX data file decoding. Extended Kalman Filter Representation Calculation of Jacobian Matrix for system equation. Calculation of Jacobian Matrix for system equation from

actual measurement (RINEX data file). Calculation of System Matrix. Calculation of initial Noise matrix and error covariance

matrix.

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Thank You