CAPITULO V OBLIGACIONES PARA EL TRASLADO DE BIENES
1. ANTES DEL INICIO DEL TRASLADO
The extended Kalman filter (EKF) uses the available sensors to determine the attitude of the satellite relative to the orbit-fixed reference frame and will supply estimates of the inertially referenced body rates and attitude quaternions. The current attitude of the satellite is required to point the satellite towards a reference attitude. The main difference between the EKF and the RKF is that the EKF contains the known non-linear dynamics. The inclusion of the non-linear model should increase the accuracy and bandwidth of the filter. The algorithm executes operations on matrices with7 × 7 elements, and the computational complexity is much higher than for the RKF.
The algorithm for implementing the EKF is shown in Figure 5.3. As with the RKF, the algorithm can be parsed into the time update and measurement update phases. The time update will occur every sample period,Tsand will execute the following steps:
1. The non-linear vector function,f xk/k, k, is the dynamic and kinematic equations seen in Equation 3.3.5 and Equation 3.3.16 for the derivative of the elements of the state vector:
ˆ
x(k) =hωˆTB/I(k) qˆT(k)iT (5.2.11)
Propagation of the non-linear model is performed using numeric integration.
ˆ
xk+1/k = ˆxk/k+
Z t=(k+1)Ts
t=kTs
f ˆxk/k, k·dt (5.2.12)
2. The perturbation covariance matrix,Pk, is dependent on the state perturbation vector, δxk, and is
defined asPk, Eδxk·δxTk . The propagation of the covariance matrix is updated as,
Pk+1/k= Φk+1/kPk/kΦTk+1/k+ Q, (5.2.13)
whereΦk+1/kis the discrete perturbation state matrix, andQis the covariance matrix of the system
noise. The discrete perturbation state matrix is approximated as,
Φk+1/k= 17×7+ F ˆxk+1/k, k Ts+ 0.5 (F ˆxk+1/k, k Ts2, (5.2.14) withF ˆxk+1/k, k , ∂x∂f x=ˆx k+1/k as defined in Appendix B.
The time update steps are followed by the measurement update steps below when a valid measurement becomes available.
3. The estimator feedback gain,Kk+1, is calculated by,
Kk+1= Pk+1/kHTk+1Hk+1Pk+1/kHk+1+ R −1
, (5.2.15)
withHk+1 the perturbation state output matrix, andR the measurement noise covariance matrix.
The output matrix isHk+1, ∂h∂x x=ˆx
k+1/k
as defined in Appendix B.
4. The innovation vector, ek+1, is the difference between the current sensor measurement vector,
vmeas,k+1, and the body-modelled estimate vector, ˆvbody,k+1/k. The body-modelled estimate vector is
determined by transforming the modelled orbit-referenced measurement vector, vmodel,k+1 to body
coordinates. This transformation is done by populating the DCM, ABO, with the estimated quaternion vector,q = [ˆˆ q1 q2ˆ q3ˆ q4]ˆ T. The innovation is calculated by
ek+1= vmeas,k+1− ˆvbody,k+1/k
= vmeas,k+1− ABO ˆqk+1/k vmodel,k+1.
(5.2.16)
5. The perturbation error is updated by the product between the estimator feedback gain and the innovation,
δ ˆxk+1= Kk+1ek+1. (5.2.17)
6. The state vector is updated by adding the perturbation state vector to the propagated state vector,
ˆ
xk+1/k+1= ˆxk+1/k+ δ ˆxk+1. (5.2.18) The estimated quaternion vector within the state vector is then normalised,
ˆ qk+1/k+1= qˆk+1/k+1 qk+1/k+1ˆ . (5.2.19)
7. Update the perturbation covariance matrix,
Pk+1/k+1=17×7− Kk+1Hk+1/k+1 Pk+1/k17×7− Kk+1Hk+1/k+1 T
Time Step? Propagate Non-Linear Model Propagate Perturbation Covariance Matrix Measurement Available? Update Filter Gain Determine Innovation Update State Perturbation Vector Update State Vector Update Perturbation Covariance Matrix yes no yes no 1 2 3 4 5 6 7
Figure 5.3 – EKF Algorithm
The EKF employs the fine sun sensor, the nadir sensor and magnetometer to supply the measured vector. Steps 3-7 are repeated for each sensor that has a valid measurement. The measurement is repeated with the sensor with the highest noise figure used first and the sensor with the lowest used last. A sensor with a high noise figure has a larger covariance and will produce an estimate with a large covariance (greater uncertainty). Thus by using the sensor with the lowest noise figure in the final iteration of the measurement update we can ensure that the resultant estimate with the smallest covariance is produced and therefore the best chance of being correct.
The EKF combines all the sensor measurements to obtain a single attitude estimate. The accuracy of the filter depends on the quality of the measurements. The noise of the CubeSense sensor presented in §3.2.5 is low and a vector direction error of well below1◦ should be obtainable. This performance will only be available in the sunlit part of the satellite’s orbit. The CubeSense sun and nadir sensor will not be available in eclipse and the estimated attitude accuracy will fall greatly during this fraction of the orbit (except if a star tracker is available, see §5.2.4). During this period, the magnetometer will be used, but will not provide the same attitude accuracy that the other sensors can provide. However, the requirement for attitude manoeuvres should be minimal during the period in eclipse.
The bandwidth (rate of convergence) of the filter depends on the accuracy of the theoretical model and the sensor measurement noise. If the theoretical model is a true reflection of the real system or the measurement noise is low, the states can converge quickly for a high bandwidth estimator and supply
accurate estimates to the control algorithm. A low bandwidth estimator will supply imprecise estimates, which will have dramatic effects when the controller controls a system quickly to a reference. The filter will start to diverge and become unstable if the mathematical model within the EKF is inaccurate and when the satellite performs a fast attitude manoeuvre. The TRIAD algorithm is a method that makes use of two vectors to obtain an instant, non-filtered attitude of the satellite. This method is not dependent on a dynamic model. The TRIAD method can be used to determine an accurate dynamic model during satellite operations. For example, the principal moment of inertia terms of the satellite can be determined if the satellite performs successive attitude rotations around each principal body axis. For a specific body axis, the input signal is retrieved from the reaction wheels and the attitude output from the TRIAD method. The attitude response around this axis has a direct correlation with the moment of inertia around that axis. The processing required during an iteration of the EKF is significant. Numerous operations on 7 × 7
matrices occur during each time step. The amount of floating-point computations will influence the power consumption of the OBC. The EKF must only be used when the satellite is rotating at low angular rates to have an acceptable convergence rate. The OBC must also have the capability of performing floating-point calculations on such matrices. In addition, the EKF should be able to produce accurate attitude and rate estimates, despite all these negative factors.