INTEGRATION MODEL COMPARISON FOR COUPLED NAVIGATION USING AN IMU AND THE GPS THROUGH THE EKF
G. Baldo Carvalho Flight Test Division - GFT, EMBRAER - Empresa Brasileira de Aeronautica S.A. Av. dos Astronautas 2170, Putim, CEP 12227-901 S˜ ao Jos´e dos Campos-SP, Brazil E-mail:
[email protected] Under the support of Embraer S.A. and cooperation CNPq/DAAD. S. Theil Center of Applied Space Technology and Microgravity - ZARM, University of Bremen, Am Fallturm, D28359 Bremen, Germany E-mail:
[email protected] H. Koiti Kuga National Institute of Space Research - INPE, Division of Space Mechanics and Control - DMC, Av. dos Astronautas 1758, Jardim da Granja, CEP 12227-010 S˜ ao Jos´e dos Campos-SP, Brazil E-mail:
[email protected]
Abstract - In this work a comparison between 5 different implementation cases of a EKF (Extended Kalman Filter) for Spacecraf navigation system using the GPS (Global Positioning System) and the IMU (Inertial Measurement Unity) is presented. Among them, one is developped with a more realistic sensor simulation provided by EADS Space Transportation-Bremen, where the simulation data provided is related to the PHOENIX prototype. Keywords - Navigation, GPS, IMU, Extended Kalman Filter Resumo - Neste trabalho apresenta-se uma compara¸ca ˜o entre 5 casos diferentes de implementa¸ca ˜o de EKF (Extended Kalman Filter) aplicado a ` navega¸ca ˜o espacial com utiliza¸ca ˜o do sistema GPS (Global Positioning System) e uma IMU (Inertial Measurement Unity). Entre eles, 1 caso ´e desenvolvido utilizando-se de uma simula¸ca ˜o de sensores mais real´ıstica provida pela EADS Space Transportation-Bremen, onde os dados de simula¸ca ˜o s˜ ao referentes ao prot´ otipo PHOENIX. Palavras-chave - Naviga¸ca ˜o, GPS, IMU, Filtro Extendido de Kalman
IV Simp´ osio Brasileiro de Engenharia Inercial, INPE, S˜ ao Jos´e dos Campos, SP, 17 a 19/11/04
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1 Introduction The models here developed use different GPS measurement models: • Case 1: Position measurement in ECEF reference frame • Case 2: Pseudoranges measurement • Case 3: Pseudoranges measurement affected by a constant clock bias • Case 4: Pseudoranges measurement affected by a linear drifted clock bias • Case 5: Pseudoranges measurement affected by a linear drifted clock bias + EADS sensor simulation The inputs to the system are inertial measurement data sampled by the IMU. For comparison both, measurements and inputs, are simulated with embedded Gaussian noises ( [1], [8]). In case 5 unbiased features of the EADS sensor simulation are considered in order to see the behavior of the filter with non-Gaussian data. All cases here presented use trajectory simulation data, which was provided by EADS ( [10]) and that corresponds to a reentry vehicle trajectory. This trajectory data is used as base input data to flight simulation.
EADS Trajectory Data
IMU
-Body Acc. -Body to ECI Ang. Rate
GPS
-Spacecraft ECEF Pos. or -Pseudo-Ranges
Simulated GPS Sat. Positions
Figure 1: System simulation. The flight simulation is represented in the following figure. It gives the general situation about GPS satellites (in blue), Earth and the spacecraft trajectory. A zoom to the trajectory (in red) close to the Earth is shown. GPS and Nav 7
x 10
Z
1.5 1 0.5
Z
X 0
Y
−0.5
Entity
Sensor
Noise σ
Hz
Acc.
IMU
5 · 10−3 m/s2
100
Body
Ang. rate
IMU
5 · 10−8 rad/s
100
Body
Pos.
GPS
3m
10
ECEF
Frame
Table 1: Components of the navigation solution. To understand the system behavior, it is necessary to describe each sensor, the estimator and to attent to each of their particularities.
1.1 Inertial Measurement Unity - IMU The IMU ( [11]) senses, the three-dimensional acceleration and the angular rates related to the body frame and integrate its measurements to give a fast navigation solution. Although its advantage of high sample rate and of being a stand alone sensor, it has disadvantages with gyros drift and accelerometers bias, meaning poor long time accuracy. Hence, its fast navigation solution can not reach the precision requirements for long duration missions. Its measurements are used in the estimation in order to keep the track to the actual position, velocity and attitude.
1.2 Global Positioning System - GPS
System Simulation
2
The following table lists the sensors, their working rate and measuring reference frame for the simulation used in the comparison:
Although the GPS is not a stand alone sensor ( [5], [7]) , it can be thought as one, once it provides the spacecraft a navigation solution, independently of where it is around the Earth. Using incoming signals from the GPS satellites, the onboard GPS receiver can provide the position and velocity of the spacecraft, related to ECEF frame. Unlike the IMU, it can not provide a fast navigation solution because its low sample rate and that the spacecraft must have at least 4 GPS satellite signals. Although the advantage of its accuracy, of being a stand alone sensor and the long operation time stability, it has also associated inaccuracies due to the path of the satellite signals to the receiver. They suffer the influence of the atmosphere, multipath effects and delay in the receiver clock as well as its associated bias. Thus, the navigation solution provided by the GPS receiver can not reach high precision requirements. Its measurements are used in the estimation in order to keep the track to the actual position and velocity.
−1
1.3 Estimation and data fusion
−1.5 −2 2
1.5
−2 1
−1 0.5
0
0 −0.5
1 −1
7
2
x 10
7
x 10
X
Y
Figure 2: Trajectory scenario.
As already discussed, no sensor can reach high precision and timing requirements on the navigation solution when used alone. To improve the accuracy, an state estimator can be used, which is capable, based on system dynamics, of predicting a state condition in a given future time. However, the dynamic model may not correspond 100% to the actual system, hence diverging in long time missions. To keep the estimation in track, the sensors can be used to update the system model used by the estimator. Thus, accurate navigation solution can be provided through the integration between the system dynamic model and the data coming from the sensors to reach the specified requirements. In this work, the estimation is provided through an Extended Kalman Filter (EKF) estimator ( [3], [6]), through which data coming from sensors with different sampling rates can be used in an improved update process ( [12]).
IV Simp´ osio Brasileiro de Engenharia Inercial, INPE, S˜ ao Jos´e dos Campos, SP, 17 a 19/11/04
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2 System requirements
Inputs
u
System dynamics
In the estimator development, the navigation requirements for an real system must be established in order to test the capability of the estimator to provide the navigation solution within the specified requirements. As an example, the requirements here adopted were retrieved from the mission of the unmanned flying vehicle PHOENIX from the EADS ( [10]) program for demonstration of the navigating capabilities of future reusable reentry vehicles.
x
z
Sensors
w v
Measurements Noise
Dynamics Noise Inputs measurements Noise + +
+ +
z noisy
u noisy
Simulated System
States Propagation
x(-)
Observation model
+
h -
Linear F matrix
Covariance Propagation
Q matrix Propagation
P(-)
Linear H matrix
Kalman Gain
Figure 3: ASTRA-PHOENIX vehicle.
R matrix
The following table lists the requirements to be achieved by the estimator here developed:
P(+)
Covariances Update
States Update
z – h
x(+)
Entity Acc. Ang. rate Altit. Ground Pos. Vel. Att.
3σ 0.1 m/s2 0.2◦ /s 2m 10 m 0.2 m/s 0.3◦
Reference Body Body NED NED ECI ECI
Update Estimator
Figure 4: Simulation model description.
Table 2: Requirements list. To guarantee that the developed estimator has achieved the specified requirements, the estimation error must lie between a 3 σ range around zero: −3 σ < x ˆ−x
