Innovative Gyroless Attitude Estimation Method

Innovative Gyroless Attitude Estimation Method Paul Dermarkar¹

Astrium Satellites, Toulouse, 31402, France,

A key aspect in Attitude and Orbital Control Systems (AOCS) is the ability to estimate accurately a spacecraft’s attitude and angular rate. While a classic gyrometer and star-tracker hybridization features high performances, gyroless solutions using only star-trackers as attitude sensors are often chosen to reduce the mission’s cost, when possible. Such an estimation is improved using the knowledge of the torque commanded by the spacecraft’s reaction wheels. The torque estimate is however imperfect, mostly due to friction occurring on the wheels’ shafts. This paper presents the research undertaken to improve this knowledge by adding a secondary torque estimator to an existing gyroless estimator. An additional filter is developed for that matter, tuned, and the global AOCS performance is tested in diverse operating conditions to assess the new method’s benefits. While converged state performances are not improved, this solution is shown to make the estimator more robust to friction torque spikes occurring on some reaction wheels, and to improve settling performances after spacecraft manoeuvers.

The numerical values of the results displayed in this paper are voluntarily modified or removed for confidentiality reasons.

The reader should therefore not pay attention to their magnitude.

Nomenclature

AKE = Absolute Knowledge Error AOCS = Attitude and Orbit Control System APE = Absolute Pointing Error

CAP = Custom Accurate Pointing (Imaging phase) CE = Circular Error

DSE = Dynamic Stellar Estimator

ECRV = Exponentially Correlated Random Variable ERS = Earth Remote Sensor

GSE = Gyro Stellar Estimator IAE = Inertial Attitude Estimation ImP = Imaging Phase

MAN = MANoeuvre

NM = Normal Mode

OH = Optical Head

RWCM = Reaction Wheel Cluster Management

RW = Reaction Wheel

S/C = SpaceCraft

STR = Star TRacker

TSE = Tacho Stellar Estimator

WC = Worst Case

CE99.7 = 99.7-th centile of circular error distribution CE86.5 = 86.5-th centile of circular error distribution

, meas OH





est, est



real, real



guid, guid





  = attitude and angular rate error (around specified direction)

meas, meas

 



Tguid = guidance (feedforward) control torque

Tctrl = control torque

, , cmd cmd real

T T = torque command expected and realized by AOCS control loop

TRW = torque command distributed to the cluster of actuators

, , f f est

T T = real reaction wheel friction torque and estimated one

Text = external torques applied onto the spacecraft

fRW = reaction wheel gain factor (or efficiency error) 1. Introduction

pacecraft attitude control is based mostly upon three subsystems, displayed in Figure 1:

1 Ecole Centrale Paris (MAE) / KTH, Stockholm (Aerospace Engineering) student, paul.dermarkar@student.ecp.fr,

S

KTH, Royal Institute of Technology – Aerospace Engineering 2

 Sensors which provide spacecraft attitude (and possibly angular rate) measurement. The attitude measurement of the Earth Observation Satellite (ERS), whose simulator is used throughout this document, is provided by star-trackers.

 A controller which computes the torque to be applied by actuators on the spacecraft to control its attitude.

 Actuators, which apply the previously computed torque onto the spacecraft. ERS generates this torque using a cluster of four reaction wheels (RW).

The sensor measurements are not used directly by the controller in practice due to measurement noises and robustness issues. They are thus usually combined to an estimator, which provides an improved spacecraft attitude and angular rate knowledge.

The Astrosat250 family estimation function is based upon two types of estimators:

 The GSE (Gyro-Stellar Estimator) which hybridizes gyrometer and star-tracker measurements.

 The DSE (Dynamic- Stellar Estimator)

which only uses star-tracker measurements, hybridizing them using the spacecraft dynamics knowledge. A simplified control loop featuring the DSE is represented in Figure 1.

The second solution makes it possible to reduce the satellite’s cost by removing the need for expensive space qualified gyrometers. The DSE however provides lower performance than the GSE as its only measurements suffers from several drawbacks:

 STR high frequency noise transmission.

 STR delay due to the stellar acquisition process which results in the corresponding attitude measurement.

 Measurement losses which may occur, especially if all optical heads are lost simultaneously.

To tackle this issue, the DSE hybridizes STR attitude measurements to a predicted (propagated) spacecraft attitude based on the spacecraft dynamics knowledge since the previous estimation time step. This prediction is also imperfect as it relies on parameters and quantities which are not perfectly known or unknown to the on-board software and which may evolve over time, typically:

 RW inertia (especially around its axis of rotation).

 RW axis of rotation direction with respect to spacecraft body frame.

 RW efficiency, which is modeled by a gain factor 1 f_RW(assumed constant for a given RW), which makes the torque realized T_RW by a RW different from the commanded one T_cmd when f_RW is non-zero according to relation:

(1 )

RW RW cmd

T   f T

 Spacecraft inertia.

 Spacecraft applied torque, to which the two main contributors are:

o External torques which are un-estimated in Astrosat250.

o Reaction wheel actuated torque, which is imperfectly realized due to RW efficiency and friction torque applied on the RW. The latter friction torque is estimated in a RW Cluster Management block (see Figure 2), referred to as RWCM, and subtracted to the torque command to try and cancel the friction based on this estimation.

This paper deals with the study of an improvement of the DSE called TSE (Torque Secondary Estimator), which is also a gyroless estimator, but includes an estimator of the commanded torque T_{cmd real,} . This estimate replaces the usual DSE input

Tcmd to better perform the spacecraft dynamics propagation. A similar estimation is already done in the RWCM to estimate the friction torque applied on each RW, but it has a low responsiveness, therefore penalizing the spacecraft performance, especially when unsteady, quickly varying friction occurs (manoeuver endpoint, friction spikes). The new estimator’s benefit arises from a different handling of the tachometer measurements. The TSE is shown from this analysis to mostly feature a different friction estimation method than the one performed in the RWCM. This filter providing fair static performance, one can expect the converged state performance to be little improved by using the TSE. It may even be worsened by transmitting measurement noise to a larger extent.

However, the bandwidth of the RWCM friction estimator is low, and one can therefore expect the TSE to provide better performance for high frequency phenomena, especially in the following operating conditions (see §2.1.5):

 RW friction spikes, or “oil drop”.

Estimator Controller

Dynamics

Measure Actuator

est

, cmd real real T





T

Figure 1. Global AOCS control loop with DSE

 Bang-bang manoeuver phases which induce high RW angular rate variations, and therefore high viscous friction variation.

The TSE may therefore improve dynamic performance through a “stiffer” filter tuning. This tuning and the performance of the so-tuned filter are detailed in the Results section.

2. Methods

2.1 Simulation hypothesis 2.1.1 Reference frames

Two reference frames are considered and used to perform the simulations detailed hereafter:

 The Inertial reference frame R_i.

 The AOCS reference frame Rv.

The detailed characteristics of these frames is given in Appendix 1.

2.1.2 Simulator architecture

The Normal Mode Simulink phase B ERS simulator (featuring a DSE) used throughout the study follows the logical architecture and splitting presented in Figure 2. This simulator is implemented in a Simulink environment, enabling access to a library of existing classical S/C Dynamics functions developed by Astrium. Appendix 2 describes the function and main characteristics of each block in the simulator.

2.1.3 Simulation requirements

Simulations are performed in this study to perform two tasks:

 TSE filter tuning

 TSE performance assessment w.r.t. DSE

In both cases, the simulations must be representative of a range of different, typical operating conditions in order to draw statistical conclusions from them. In this study, it is done by disturbing S/C parameter values as detailed in the next paragraph, and by running simulations under different operating conditions detailed in §2.1.5.

2.1.4 Statistical parameter disturbance

The attitude estimation and control is performed in the flight software assuming nominal S/C characteristics. An incorrect knowledge of these characteristics would therefore damage both TSE and DSE performances. The simulations are performed by disturbing the S/C model parameters in reasonable ranges. The disturbed parameters and the ranges of variation used during the simulation campaigns are given in Table 1.

STR Function

Attitude Estimator (DSE or TSE)

RW Cluster Management

cmd ctrl guid

T  T T

, cmd

RW f est

T T T

S/C Dynamics

RW Model

,

cmd real RW f

T T T

CTRL

meas, meas

 

real, real

 

meas

est

Tguid

Tctrl

Tcmd

, cmd real

T Text

guid, guid

 



STR Model

est

, meas OH



IAE

TRW

Guidance Function

Figure 2. AOCS control loop for DSE or TSE

KTH, Royal Institute of Technology – Aerospace Engineering 4

Parameter Model disturbance range and comments

Satellite

Inertia matrix

Main inertias randomly disturbed in a ±5% range (uniform distribution).

Products of inertia error I_ij randomly chosen in a ±1.5% range (uniform distribution) of corresponding main inertia:

1.5 1.5 100 ,100

ij k k

I  I I 

    (i,j,k different axis index)

RW

Main Inertia Main inertia randomly disturbed for each RW in a ±1% range (uniform distribution).

Axes

misalignment

Wheels are placed in a symmetrical configuration, parameterized by the azimuth





Efficiency Randomly chosen for each RW, in a ± 9% range (uniform distribution).

Table 1. Statistical model parameters for simulation campaigns [6]

2.1.5 Simulated operating conditions

The performance of each filter is tested under different operating conditions detailed hereafter.

Converged state (CAP)

Converged state refers to an ImP during which:

 Each subsystem operates in a nominal way.

 The guidance function sets each guidance command (feedforward torque and S/C angular rate) to zero.

 Any filter (especially RWCM’s friction estimator and TSE) has reached steady state.

Friction spike (CAP)

A friction spike (or oil drop) is simulated during a converged state phase (see previous paragraph). During this phase, an additional typical friction profile represented in Figure 3 is applied on RW1.

The friction spike characteristics analysed throughout the document have the typical characteristics below:

, f spike

T = 50 mNm

, up spike

t = t_{down spike,} = 15 s

Note that the analysed time interval contains the spike duration

(30 seconds) as well as the 100 seconds following it to account for the estimator settling capability.

STR loss (CAP)

STR loss (blinding of all OHs) is simulated by setting simultaneously to zero the STR measurement validity flags (DSE function inputs) for a chosen duration. STR loss is also initiated during a converged state phase.

Post-manoeuver phase (Early CAP)

A satellite manoeuver was simulated. The main characteristics of the manoeuver tested were:

,1

Tf

(mN.m)

, up spike

t

Time (s)

, f spike

T

, down spike

t

Figure 3. Friction spike profile

 Bang-bang torque profiles, that is using the maximum allocated RW torque during the manoeuver. In case the required RW angular momentum of a given RW exceeds the allocated one (user defined value), a bang-stop-bang manoeuver is performed (see profile sketched in Figure 4). This leaves the RW angular momentum constant in-between the two “bang”

phases. The manoeuver duration is thus minimized with respect to the allocated RW torque angular momentum.

 User defined manoeuver amplitude and direction (rotation eigenvector) in AOCS reference frame R_v.

The estimator performance is analysed right after such a manoeuver, as the global satellite performance depends on its ability to initiate ImP quickly after the MAN phase. For ERS for instance, the ImP starts 30 to 40 s after the beginning of the CAP.

2.2 TSE implementation

The TSE plugin is added as an input of the DSE in the Simulink model according to the scheme in Figure 5. This new output is either used or not using as switch block to enable DSE and TSE performance comparison using the same simulator.

This plugin is made up of four Kalman filters (one filter for each RW) running in parallel. The detail of one of these Kalman filters is given in Figure 6.

Time (s)

Time (s)

Tcmd (mNm)

S/C angular rate (rad/s)

Tmax

(max. allocated torque)

Tmax



(ceiling due to max allocated RW angular rate)

stop/bang

stop/bang bang/ stop

Figure 4. Bang-stop-bang manoeuver profile (along manoeuver axis)

bang/ stop

Kalman gain computation

propagation update

Figure 6. TSE Kalman filter (RW1)

, 1 cmd k

T _

, meas k

 _{est k,}

, est k

TSE

plugin DSE

, 1 cmd k

T _

, meas k



, est k



, meas k



, , 1

cmd est k

T _

TSE

, est k



DSE

Figure 5. Comparison scheme DSE vs. TSE

KTH, Royal Institute of Technology – Aerospace Engineering 6

2.3 Performance indicators

To assess the DSE/TSE relative performance, as well as to select a suitable tuning of the TSE, performance indicators (or cost functions) are required. These indicators are briefly presented and motivated hereafter.

2.3.1 Circular Error (CE)

The main indicator considered is the S/C angular rate Circular Error (CE), that is the norm of angular rate or attitude absolute error projected on the roll-pitch plane in the AOCS frame:

2 2

2 2

roll pitch

roll pitch

CE CE



 



    



   

 ⁽¹⁾

Where the attitude is expressed in Euler angle form (suitable when very low angles are involved, as it is the case throughout the paper). The CE analysed throughout the paper to asses the estimator performance is the estimation error (i.e. AKE), which is the difference between actual S/C attitude or angular rate and the estimated one.

The yaw direction (along the camera axis) is not considered since roll/pitch error in the same order of magnitude would have much higher influence on the picture sharpness (see sketch in Figure 7). For a given angular rate error in yaw ε, the ground scrolling error would be at most

εRframe with R_frame the maximum frame radius on ground, in the order of

45 km for ERS. For the same angular rate error in roll (assuming no pitch error), the ground scrolling error would be at most



ERS. Considering only the image sharpness factor, the yaw accuracy requirement is therefore approximately h² R²_frame R_frame≈ 17 times lower than pitch/roll requirement.

Figure 8 displays the roll/pitch angular rate measurement error over a sample of simulated points. The norm of each point (distance to the origin which is indicated by a blue cross) corresponds to the CE.

One can define statistical values for such a set of simulated points: CEi (i ranging from 0 to 100) is the value such that i % of the simulated points have a CE lower than CEi (i.e. i-th centile). On Figure 8 are displayed two of these values, namely CE86.5 and CE100 (which is the maximum CE over the observed sample).

2.3.2 Estimator performance indicators

While attitude error deals with the ability to perform ImP on the desired Earth surface and is not usually an issue considering state of the art performance, angular rate is considered more critical as it will directly influence the sharpness

of the resulting picture. The angular rate CE is therefore monitored in priority when assessing system performance during the filter tuning process in §3.1 and when analysing TSE benefits relative to DSE in §3.2. The attitude is also monitored for validation matters, but is not displayed in this paper for brevity reasons. The following indicators are considered:

 CE99.7 which corresponds to the maximum circular error, having rejected irrelevant values from a performance analysis point of view.

 CE86.5 which corresponds to two standard deviations



 The overall CE cumulative distribution function (obtained by sorting out the simulated values)

Max yaw ground scrolling error ∝ Rframe

Max roll/pitch ground scrolling error ∝ (h²+Rframe²)^1/2

ε

ε

ε

h

Rframe

Figure 7. Compared ImP influence of angular rate error along either yaw or pitch/roll

90 100d^S

dS(total settling distance)



Figure 9. CE and remaining maximum CE with respect to time during MAN-CAP phase

CE100 CE86.5

Figure 8. Measurement CE sample corresponding CE90 and 100.

 The CE over time (and especially the remaining maximum CE over time) during post-manoeuver phases. It is computed by scanning over time the values of the signal (error) from current time to the end of a selected time interval (typically until the signal is settled), and by computing the maximum value of this signal over the scanned values (see Figure 9).

This value indicates at each time instant between which values the signal will be constrained until steady state is reached, and makes it possible to assess the settling performance of the estimator (AKE).

 From the previously introduced remaining maximum graph, the time constant τ of the system can be computed, making it possible to quantity its settling speed by a scalar value. It can be computed as the time to achieve a chosen fraction of the settling from initial time to convergence, for instance 90% of the steady state value (settling time 10% away of converged state, τ90% in Figure 9).

2.4 Filter tuning

The TSE performance depends on the RW friction modeling, described as an ECRV by two tuning parameters σ_f and τ_f. The first subsection proposes a method to select suitable values of these two parameters, while the tuning retained is justified and given in the second subsection.

2.4.1 Tuning process

A simulation based tuning of the TSE using the modified NM ERS simulator (see §2.2) is performed.

A trade-off between the filter performance during converged state and friction spike is expected. These two operating conditions are therefore tested during each simulation.

A set of randomly disturbed model parameter values is generated according to §2.1.4. Simulations are then performed using different (σ_f,τf) combination, and under each of the generated disturbed model parameter values. The range of the σ_f and τ_f intervals are chosen as follows:

 τf must be high enough with respect to AOCS period to perform a valid ECRV modeling [4], and in the order of magnitude of typical time constants such as the 30 s friction spike duration ;

 σf must be chosen as a compromise between the residual RW friction ΔT_f during converged state on the one hand, and dynamic phases (friction spike, manoeuver) on the other hand.

These intervals are then refined using a process of trial and error on the TSE simulated performance. The size of each of these sets is then chosen to find a trade-off between tuning accuracy and computational cost.

The simulation results can then be post-processed to observe the performance indicators introduced in §2.3 and select an optimal tuning with respect to these criteria.

2.4.2 Computational cost

The computational cost of the simulation campaign is proportional to the product of the size of the σ_f and τ_f sets introduced previously, and of the amount of disturbed model parameter values simulated at each tuning point (which deal with the ability to draw statistical tendencies from the results and to ensure robust tuning). In order to limit the computation cost, the τf and σ_f selections are performed separately:

 The τf selection is chosen to be performed at first by scanning both τf and σ_f sets, using a limited amount of 10 generated disturbed models for each tuning point. The reasons for performing the τ_f selection independently is based on the a- posteriori observation that the optimal τ_f value is similar for any σf value (see Figure 10 and Table 2).

 The σf selection is performed by scanning only the σ_f sets (having already selected τf). This enables to increase the amount of simulations performed for each tuning point, keeping reasonable computational cost. 100 disturbed models are thus generated.

3. Results

3.1 Tuning selection 3.1.1 τ_f selection

For each (σ_f,τf) combination, the worst case measurement CE86.5 among the 10 generated disturbed models is displayed in Figure 10. Only simulation results during a friction spike are considered for now as the τ selection must mostly enable proper

Figure 10. Worst case CE86.5 with respect to σf and τ. Black asterisks indicate simulated tuning points

KTH, Royal Institute of Technology – Aerospace Engineering 8

among the simulated ones for each simulated σ_f value taken separately, as performed in Table 2. Either the 10 or 15 s value could be optimal depending on the σf value. The greatest value (τf = 15 seconds) is retained in order to ensure correct ECRV approximation. This value is used to further tune the filter (that is to select the σ_f value). Note that this value matches well with the friction spike duration of 30 seconds, while the RWCM’s friction estimator time constant used to be 2π/ω_f = 100 s, therefore being unable to estimate quick friction variations.

σf value [mNm] 5 10 15 20 25 30 35 40 45

Optimal τ_f value [s] (minimum

worst case CE86.5) 10 10 15 10 15 15 15 15 15

Table 2. Optimal simulated τf value with respect to simulated σf value 3.1.2 σ_f selection

Figure 11 shows the CE86.5 distribution over a set of 100 disturbed models obtained using either the DSE or the TSE under the selected σf set, both in converged state and during a friction spike. As expected, increasing the σ_f value on the considered range damages the TSE performance in converged state, but improves the TSE performance during a friction spike.

Such a graph makes it possible to choose an optimal σ_f value achieving a good trade-off between robustness to friction spikes and system performance in regular converged state. One can especially select a σ_f value for which TSE and DSE converged state performance are similar, and for which the TSE significantly improves the performance during a friction spike relative to the DSE.

This σ_f selection obviously depends on the mission requirements and hypothesis.

One can however suggest a σ_f value of 25 mNm:

 Friction spike’s CE86.5 tend to level-off as σf is increased above this value, as the noise transmitted

onto the friction estimate by the filter starts to be in the order of magnitude of the friction spike itself. The performance over each quartile is significantly improved (13 to 16 µrad/s over each quartile).

 Converged state’s CE86.5 performance is reasonably altered (less than 2.5 µrad/s over each quartile of the ).

3.2 Tuned estimator performance analysis

This section analyses the performance of the TSE tuned as suggested in §3.1.2 (τ_f = 15 s, σ_f = 25 mNm). The simulation results considered are the ones obtained when tuning the filter (σ_f selection), therefore providing the results of 100 simulations for the selected tuning.

3.2.1 Friction spike monitoring

State estimation monitoring

The simulation results under the worst case operating condition (with respect to the CE99.7 criterion) during a spike event are displayed in Figure 12. Note that the TSE friction estimation is still disturbed after the friction spike has occurred, motivating to account for this settling when analyzing the friction spike performance, as suggested in §2.1.5 and done hereafter. The RW angular rate measurement noise due to tachometer quantization (or sampling) is efficiently smoothed by the filter. Differentiating this state in order to estimate the RW realized torque is however shown to result in an extremely noisy estimate, motivating the retained estimation method.

Figure 11. Simulated CE86.5 “box plot” distribution wrt σf (100 simulations per sigma value)

Figure 12. RW1 states estimation using TSE filter during converged state and friction spike

(worst case operating condition)

Converged state Friction spike phase

The RW angle measurement noise is transmitted by the filter to a greater extent by the TSE.

S/C angular rate estimation monitoring

Figure 13 displays the S/C angular rate CE cumulative distribution function merging 100 simulations performed during the friction spike phase using either DSE or TSE, allowing comparison between the two estimation methods.

3.2.2 Converged state monitoring

State estimation monitoring

The TSE state estimates during converged state are also displayed in Figure 12 (before the friction spike occurs).

The TSE doesn’t improve the friction estimation performance whatsoever during such a phase, as it adds some model noise to enable friction spike estimation.

S/C angular rate estimation monitoring

Figure 14 displays the S/C angular CE cumulative distribution function merging 100 simulations performed during converged state.

3.2.3 STR loss monitoring - S/C attitude and angular rate estimation A 60-seconds STR-loss phase followed by a 15-seconds retrieval phase is monitored. TSE and DSE provide almost identical performance during the STR-loss phase. Their performances are also extremely close during the STR-retrieval phase, though the angular rate CE is slightly improved using the TSE, taking advantage of its better responsiveness.

3.2.4 MAN and early CAP monitoring

State and S/C angular rate estimation monitoring The TSE states under to the worst case operating condition (with respect to the CE99.7 criterion during spike) during MAN is displayed in Figure 15. This manoeuver has the characteristics given in §2.1.5 and amplitude of 35° around the roll axis. Due to allocated RW angular momentum limitations and initial RW angular momentums, this manoeuver is such that:

 Maximum allocated RW angular momentum is reached (by two RW), therefore requiring a bang-stop-bang manoeuver to be performed.

 The two other RW’s angular momentums cross the zero value, therefore causing the RWCM to activate the zero- crossing mode detailed in Appendix 1.1.5.

An important dynamic error is featured in the RWCM friction estimation causing high (but somehow constant) friction estimation error when the RW angular rate varies in a linear

Figure 13. S/C angular rate CE cumulative distribution function during friction spike phase

bang/stop stop/bang zero-crossing

stop/bang

Figure 14. S/C angular rate CE cumulative distribution function during converged state

Figure 15. RW1 friction estimation using TSE filter during MAN (worst case operating condition)

zero-crossing Bang

zero-crossing

Bang Stop

KTH, Royal Institute of Technology – Aerospace Engineering 10

fashion [5]. Once this error is settled, the TSE fails to compensate for this error (see Figure 15). The TSE however impacts this estimation during transition phases such as bang/stop, stop/bang, zero-crossing or early CAP phases. This impact is not clearly beneficial during the MAN phase itself (see Figure 16). The stop-bang and zero-crossing phases are actually worsened by the TSE as it settles faster towards the friction estimation dynamic error pointed out previously. Only the bang- stop and early CAP phases are improved as the settling is accelerated towards converged state error.

As justified in §2.3.2, the early CAP phase is however the most critical from an ImP point of view and is the only one further analysed in this paper. It is shown in Figure 15 that friction estimation benefits from using the TSE during this phase.

S/C angular rate estimation monitoring

Figure 17 displays the remaining maximum estimation CE over time for the S/C angular rate over 100 simulations performed straight after the previously described MAN. This makes it possible to assess the settling of DSE and TSE at the beginning of a CAP phase. The TSE is especially shown to converge faster towards a converged estimation error than the DSE, with a settling time (1% away of settled CE value) decreased by 36% for the worst case scenario (roughly 40% on average).

4. Discussion

4.1 Increased tachometer resolution

In brief, the TSE principle consists in relying more on the tachometer measurements than using the DSE. One can therefore expect the filter performance to benefit from an increase in the tachometer resolution (reducing the quantization step), therefore improving the measurement’s accuracy. Furthermore, this tachometer resolution is already accounted for in the measurement model used by the TSE Kalman filter.

A tachometer resolution increase is simulated under the previously selected TSE tuning and under constant DSE tuning. Figure 13 and Figure 17 are then reproduced, displaying one curve per tachometer resolution on Figure 18 and Figure 19. Tested tachometer resolutions are resp. 8, 16, 32, 64 and 128 pulses per revolution, and 10 simulations are performed at each resolution.

Both friction spike and early CAP settling performances are shown to be improved by increasing the tachometer resolution using the TSE, while the DSE, tuned to filter out the initial quantization noise does not benefit from this increase whatsoever. The TSE improvement levels off between roughly 32 and 64 pulses per revolution, with a CE86.5 during friction spike improved up to 66%, and a settling time decreased by 60%. Note that the converged state performance does not benefit from a higher resolution.

The convergence occurs towards a non-zero error despite a perfect RW angle knowledge during friction spike. This is mostly due to the model disturbance introduced in §2.1.4 and to the unestimated external torques applied onto the S/C. It is also due to the limited friction estimation bandwidth, which results from the AOCS frequency which is also used by the TSE estimator. A potential improvement would be to run the estimator at a (much) higher frequency, provided technical feasibility.

Figure 17. Settling phase using either TSE or DSE over 100 simulations

Figure 19. Tachometer resolution increase influence on early CAP settling

Increasing resolution

Figure 18. Tachometer resolution increase influence on friction spike phase

Increasingresolution

In addition, the remaining control loop does not entirely “trust” the TSE information, which is weighted by controller gains.

This issue is discussed and dealt with in the next paragraph.

4.2 Alternative and global filter tuning

Though the previously introduced tuning method has the advantage to be performance based, therefore offering optimal and robust performance in the tested operating conditions, it has two main drawbacks. The first one is its high computational cost, which increases complexity, and especially the process of trial and error. The resulting tuning also depends on the remaining control loop estimator’s tunings: with lower DSE Kalman gains, therefore relying more on the S/C Dynamics knowledge (and therefore on the TSE’s output), the quantization noise transmission throughout the TSE would be higher for a given TSE plugin tuning. This is also true for the controller and RWCM’s friction estimator tunings.

A way to perform such a tuning independently from the control loop is to analyse the TSE Kalman filter (Figure 6) independently and to analyse:

 The noise transmission from measured RW angle to the estimated friction torque, the ratio of the two quantities being equal to the corresponding Kalman gain.

 The friction estimation settling time (1% away of settled value) in response to a step input on the measured RW angle.

Displaying these two adversary criteria with respect to

the τf and σ_f sets introduced previously makes it possible to find a trade-off between converged state performance (first criterion) and filter responsiveness (second criterion). Such a graph is displayed in Figure 20, and one can especially find the previously selected tuning by:

 Selecting an 0.1 Kalman gain value, which keeps the transmitted noise in the order of magnitude of the RWCM’s one (that is worsens reasonably converged state performance).

 Rejecting τ_f values lower than 15 s (due to ECRV considerations and simulation results in section 3.1.1).

 Minimizing friction estimation settling time.

5. Conclusion

By taking advantage of a quicker response time, the TSE not only improves the expected robustness to friction spikes (CE86.5 improved by 31%), but also improves the angular rate estimation settling (estimator’s settling time reduced by roughly 40%). It however worsens converged state performance by approximately 6% (CE86.5).

Depending on the mission’s requirements, the TSE may be relevant or not: if a low amount of high quality frames are desired, one should keep the converged state performance as high as possible and keep using the DSE. If a higher frame amount of frame is desired, the TSE would make it possible to reduce the allocated CAP duration per frame, and therefore increase the daily amount of acquired frames.

This estimator is also even more efficient under low external torque perturbations, as it only estimates the torque realized by the RW. It could therefore suit upcoming inter-planetary missions and GEO missions (especially the GO-3S mission, which also has imaging requirements).

Though not reaching GSE like performance, especially in converged state, improvements discussed in the previous section such as increased controller computation frequency and RW tachometer resolution could make the TSE a capable and easy- to-implement estimator.

Rejected τf values Performance

based tuning

Figure 20. Friction estimation settling time and friction estimation Kalman gain (noise transmission) with respect

to σf and τf

KTH, Royal Institute of Technology – Aerospace Engineering 12

Appendix 1. Reference Frames

1 Inertial reference frame

The inertial reference frame Ri is J2000 reference frame, defined by the mean celestial reference frame in 2000 January, the 1^st, at noon (date defined by universal time UT1).

This reference frame is thus defined by:

 O: Earth reference ellipsoid centre

 Z: direction between Earth ellipsoid centre and the mean pole

 X: mean vernal direction on 2000/01/01 at 12h00 UT1

 Y: completes the right handed (X, Y, Z) trihedral

 J2000 time origin: 2000/01/01 at 12h00 UT1

2 Inertial reference frame

The AOCS reference frame (or S/C frame) that shall be controlled, at any instant, by the AOCS corresponds to the instrument line of sight reference frame R_v. Its definition can be described as follows:

 Z_v: instrument line of sight axis, oriented in Earth center direction during the GAP phases (data transfer phase),

 X_v: orthogonal to the images lines (orthogonal to the detector) oriented in direction of spacecraft in orbit velocity,

 Y_v: parallel with the images lines (parallel with the detector), completes the right handed trihedral.

Appendix 2. Simulator Architecture and Components Clarification 1 Flight software (FSW)

1.1 STR function

STRs provide S/C attitude measurement with respect to the inertial frame. In ERS, three STRs are used simultaneously to ensure robustness to STR loss in case optical heads are blinded (by the sun or the Earth for instance) leaving at least one STR operational, as well as to improve attitude estimation accuracy when at least two OH are available. An OH indeed features more accurate attitude measurements along its transverse axis than along its longitudinal axis (considerations similar to the ones in §2.3.1 and Figure 7). The measurements provided by each OH are merged in the STR function, resulting in an absolute attitude measurement (quaternion) from the inertial frame to the S/C, taking advantage of the accuracy of each OH along their transverse directions.

Note that the DSE features an STR-loss mode in case all optical heads are blurred or non-operational. In that case, only the S/C dynamics prediction detailed in the next subsection is used to perform attitude estimation (Kalman gains set to zero).

1.2 Gyroless attitude estimator (DSE)

This subsection is a brief summary of the complete Gyroless Attitude Estimation Function Design [2] to which the reader should refer to for further information.

The DSE is made of a Kalman filter, which hybridizes the measured quaternion from the STR function with the predicted quaternion and angular rate based on the S/C dynamics using constant Kalman gains. This S/C dynamics prediction is based on the fundamental angular momentum equation:

_

Rv Ri

tot tot

sat tot ext

Ri Rv

dH dH

H T

dt  dt     (A1)

where

 

 

   

 

 

v i

x R _ R

sat y

z

is the angular rate from the inertial reference frame to the AOCS frame

Htot is the total angular momentum, with H_tot  I ^{Rv Ri}_sat^_ H_cluster^Rv (I is the satellite inertia matrix and H_cluster^Rv the RW cluster angular momentum in the spacecraft frame)

T_ext is the total external torque

Equation (A1) is simplified assuming no external torque is applied on the S/C that isT_ext 0. In the particular case of the DSE, one assumes that the commanded torque to the RW is perfectly realized thanks to the friction pre- compensation occurring in the RW Cluster Management (see §1.5).

Rv cluster

Rv

dH

dt which represents the torques applied by the RW cluster on the spacecraft is therefore simplified as equal to the commanded torque T_cmd. Note that this constitutes one of the main difference between DSE and TSE, as the TSE computes this angular rate variation based on the RW measurements.

Under this assumption, equation (A1) can be simplified and inverted to result in the S/C angular rate equation:

 

 

_ 1 _ _

Rv Ri Rv Rv Ri Rv Ri Rv

sat I^ Tcmd sat I sat Hcluster

        (A2)

where H_cluster^Rv is the total RW angular momentum measured by the RW tachometers, and all other right hand side terms are known or estimated by the filter. Equation (A2) can therefore be discretized as is (first order Euler) to perform angular rate prediction in the DSE Kalman filter.

The attitude quaternion prediction can be performed directly from the previous angular rate and attitude estimation using Edwards method. This equation as well as the discrete form of equation (A2) which is available in [2] are not recalled here as only the estimation method is useful to know in this paper. The attitude and angular rate innovation computation and updates resulting in their estimates are also detailed in this reference document.

1.3 Guidance function

This function gives a commanded quaternion which is compared to the estimated quaternion in the controller.

It also gives a feed forward commanded torqueT_guid, which is added to the control torque T_ctrl from the controller in order to improve the system responsiveness during MAN.

1.4 Controller

A Proportional Derivative controller is used to compute the control torqueT_ctrl. It uses proportional control on both the attitude (quaternion) and angular rate error (difference between estimated value from DSE and reference value from the guidance function):

_ _ _ _

( ^{Rv Ri Rv Ri}) ( ^{Rv Ri Rv Ri})

ctrl e est ref v est ref

T K





where K_e is the 3x1 proportional gain

Kv is the 3x1 derivative gain

_ Rv Ri

 is the eigenvector of rotation of the S/C from Ri to Rv is the element-by-element product (i.e. array multiplication) 1.5 RW cluster management function

This function distributes the commanded torque T_cmd among the cluster of RW (four in ERS), resulting in a 4- components vectorTcmd cluster_, . T_cmd is computed by adding control torqueT_ctrl, guidance feed forward torqueT_guid, and subtracting estimated friction torque T_{fric est,} :

,

cmd ctrl guid fric est

T T T T (A4)

The RW friction torque estimator is located there, and is based on the RW tachometer.

To avoid friction estimation delays occurring when the RW angular momentum value crosses zero, the RWCM’s friction estimator estimates the friction torque absolute valueT_{f est,} and this torque’s sign separately. This torque sign is estimated in different ways depending on the estimated RW angular momentum valueH_est:

 Above a given angular momentum threshold, the friction sign is estimated opposed to the one of H_est.

KTH, Royal Institute of Technology – Aerospace Engineering 14

 Below this threshold, the friction sign is either estimated opposed to the one of the torque command, or zero if this torque command’s absolute value is lower than a chosen threshold.

This particular zero crossing handling to estimate RW friction is not performed by the TSE.

2 Models

2.1 STR noise and disturbances model

Three kinds of STR noise alter the STR measurement and are taken into account in the simulator:

 Temporal noise or Noise Equivalent Angle (NEA) due to the STR electronics.

 Pixel noise (PIX) due to non-uniform pixel response.

 Field Of View noise (FOV) due to optical distortion.

No thermo elastic noise have been taken into account.

2.2 RW model

The modelled RW account for:

 A 25 ms actuation delay.

 The friction torque T_fric which is modelled as the sum of a constant dry torque (which only depends on the RW spin direction) and a viscous torque proportional to the RW angular rate.

 Misalignment errors which are defined by the angle  (different for each wheel) between the actual RW main inertia axis and the expected one (see Figure 21).

 The RW efficiency f_RW, which is modeled by a gain factor

1 f_RW(different for each of the four RWs) such that the net torque received by the RW cluster is:

, (1 )

tot cluster

fric RW RW

Rv

dH T f T

dt    (A5)

 The RW torque saturation and angular momentum saturation, due to the chosen motor maximum torque and power.

2.3 RW tachometer model

The tachometer model gives RWs angle and angular rate measurements at the AOCS frequency of 16Hz. The angular measurement is modeled accounting for the resolution of the tachometer (amount of pulses, or measurable angular positions per RW rotation).

The angular rate measurement is then computed performing a first order derivation of the angle measurement between two consecutive time steps.

2.4 External torques model

Four different external torque contributions are considered:

 Gravity gradient torque.

 Aerodynamic torque (due to aerodynamic drag as ERS is in low earth orbit).

 Magnetic residual torque (contribution from the S/C structure and electronic components).

 Solar wind torque.

These torque computations require S/C geometry information, which are extracted from ERS’ FACT model. They are computed under chosen operating conditions, which depend on numerous factors such as the satellites orbit description, considered time and position on the orbit, seasonal configuration etc. which are added to the simulator.

Note however that little attention is brought to these latter parameters, as the study mostly requires performing simulations under typical, realistic operating conditions.

2.5 S/C dynamics model

Basic rigid dynamics are considered (no flexible modes are modeled), performing discrete integration of equation (A1).

RW expected main inertia RW 

RW actual main inertia

Figure 21. RW misalignment sketch

Acknowledgments

I would like to thank first of all Christophe RABEJAC for the confidence he expressed by hiring me for this 6 - months internship, as well as for his permission to issue this report despite strict confidentiality constraints.

My deep appreciation goes to my Astrium supervisor Romain SEPTIER for his endless scientific support, explanations regarding tools and habits of the department, thorough proof reading and the time he allocated for all this. I also would like to express my sincere gratitude to my team leader Xavier LANGEVIN for the very pleasant and motivating working atmosphere he managed to set within the Earth Observation team. I want to thank all my colleagues, with a special mention for Anais ARDAN, Alexandre JULLIEN, Jean-Michel TRAVERT and Karl DEVILLIERES for the guidance and the expertise they shared with me throughout my internship. Thanks also to Nicolas CUILLERON and Jean SPERANDEI from the Advanced Studies Dept. for involving me into their ongoing studies.

Last but not least, I would like to thank my KTH supervisor Gunnar TIBERT for his help regarding several complex physical aspects of my study, as well as his understanding of the security and confidentiality issues I came across during this master thesis.

References

Reports, Theses, and Individual Papers

[1] AS250 Generic – RW Cluster Management Function Design, EADS Astrium unpublished

[2] AS250 Generic – Gyroless Attitude Estimation Function Design, EADS Astrium unpublished

[3] Etude Interne Filtrage Tacho-Stellaire – Hybridation tacho-stellaire pour restitution d’attitude gyroless, EADS Astrium unpublished

[4] Analyse de covariance – Quelques éléments de modélisation pour l’analyse de covariance, EADS Astrium unpublished

[5] AS250 – RW Cluster Management Function Tuning, EADS Astrium unpublished

[6] AS250 – Mission specific AOCS numerical reference document, EADS Astrium unpublished

Books

[7] D. Alazard, Introduction au filtre de Kalman, version 0.0, Supaero, January 2005

[8] T. Glad, L. Ljung, Control Theory: Multivariable and Nonlinear Methods, 2000, Taylor & Franc