- Email: [email protected]
System Identification of a Spacecraft Antenna Pointing Mechanism
19th IFAC Symposium on Automatic Control in Aerospace September 2-6, 2013. Würzburg, Germany
System Identification of a Spacecraft Antenna Pointing Mechanism Andrei Kornienko*, Robert Wuseni* Bernhard Specht* *Astrium GmbH, D-88039 Friedrichshafen, Germany (e-mail: andrei.kornienko, robert.wuseni, [email protected]). Abstract: This paper describes results of identification of a spacecraft antenna pointing mechanism. Accurate modelling of the physical effects that occur during the mechanism operation is mandatory for the realistic assessment of the microvibration budget as well for the control design of the mechanism electronics. System identification is seen as a very attractive methodology in characterizing the entire mechanism, including stepper motor electronics, electromagnetic effects that occur during stepping and microstepping operation and accurate modelling and parameterization of the mechanical drive train. The results achieved throughout model identification show a very good agreement between the model and experimental data in wide operational ranges in time and in frequency domains. Keywords: System Identification, Stepper Motor, Modelling, Experiment Design, Optimization 1
results chapter. Finally, some conclusions of the work and future perspectives will be given.
INTRODUCTION
Continuously increasing demands on the performance accuracy of the modern Earth observation and science satellite missions require elaborated analysis approaches. It is well known that typical satellite on-board actuators, such as reaction wheels or drive mechanisms, can significantly degrade performance of the onboard optical or scientific payload. To be able to characterize the performance that can be achieved by the spacecraft in its in-orbit operation it is mandatory to acquire sufficient knowledge of the spacecraft dynamics, as well as its internal components. Reaction wheels, antennas, solar array drive mechanism, compressors and other actuators, are potential sources of the microvibration disturbances that may significantly affect inorbit performance of the satellite (Galeazzi, et al. (1996)). Thus, knowing perturbation levels of the actuators is quite fundamental for the spacecraft design. In order to assess the microvibration budget, elaborated simulation models of the on-board actuators are developed (Vitelli, et al. (2012)). The simulation models typically contain uncertain parameters that can be derived either from analytical considerations, using supplier specifications or determined experimentally. One of the possibilities to determine the model parameters from experimental data is based on the system identification approach (Ljung (1999)). This paper is focused on presenting results of identification of the satellite antenna pointing mechanism.
2
System identification is a process to determine mathematical models of dynamic systems based on experimental data. Its fundamental concept can be best described by the diagram shown in Fig. 1. Input
Physical Plant Sensors and Recorder
Meas. Input
Tunable Plant Model
Output
Meas. Output
Adaptation Algorithm Model Output
Model Update
Fig. 1: A schematic diagram of the system identification process. The tuneable plant model is a mathematical model reflecting the real physical plant or process. It is subject to the measured inputs from the real process. The identification error, i.e. the difference between measured outputs (subject to errors/noise) and the model output, is applied to improve the mathematical model. The goal is to achieve an agreement between the plant and the model to a certain accuracy (Ljung (1999)).
Following a brief description of the system identification process, details of mathematical model that represents the antenna pointing mechanism will be given. Experimental design aspects including measurement hardware setup and conduction of experiments will be shortly outlined. A comparisson between the model output and the experimental data in the time and frequency domain as well as specified versus estimated parameter values will be presented in the
978-3-902823-46-5/2013 © IFAC
SYSTEM IDENTIFICATION AS EXPERIMENTALLY BASED MODELLING
Identification of control-relevant design models from detailed simulation models allows efficient approaches for the design, tuning, and validation of control algorithms and flight software of satellites. The improvement of technical expertise 172
10.3182/20130902-5-DE-2040.00057
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
Test mass states
Stepping rate
Voltage Stepping Command (VST)
Phase voltage
Two Phase E-Motor (E2P)
Motor torque
Rotor and Gear (RAG)
Applied torque Residual torque
Rotor states
Test Mass
Measured torque
Fig. 2: Top-level architecture of the simulation model αshift represents the geometric relationship between the two motor phases. In the standard case a value of αshift = 90° for one phase is appropriate.
in the area of system and parameter identification is a key element for the mastery and realization of future highaccuracy satellite missions, and thus allows both precision and efficiency increases while lowering development times and costs. 3
The total generated torque on the rotor is a sum of the individual coil torques and the detent torque. The individual torques are calculated as:
ANTENNA MECHANISM MODELLING AND PARAMETERISATION
p TA/ B = iA/ B ⋅ km ⋅ sin ⋅ α + α shift 2
3.1 Mathematical Model
with km denoting the motor torque constant.
The mathematical representation (Athani (1997)) of the antenna drive mechanism is based on four major functional blocks (see Fig. 2) and realised using MATLAB/Simulink™ simulation environment. The details of individual functional blocks are described below.
The detent torque is a result of the motors permanent magnets. Its shape follows a sinusoidal profile with its maximum between two poles according to the equation (4). In order to take imperfections of the rotor and stator into account, the detent torque is calculated as a sum of multiple torque harmonics with different periods over a full revolution. In this analysis a total of two harmonics have been considered.
Voltage Stepping Command (VST): The VST Block serves as a signal generator for a two phase bi-polar stepper motor. With respect to the selected mode of operation, which can be either full stepping or microstepping with multiple sub-steps, this block generates the required voltage profiles for the commutation of the motor. The stepping rate is applied at the input. The block operates as an ideal voltage source with a preselected peak voltage. No current regulation is implemented.
360° TD = ∑k K Dk ⋅ sin α ⋅ α det k
The overall motor torque is then computed by combining equations (3) and (4):
The E2P Block is the heart of the stepper motor. It accommodates the electrical section of the stepper motor. The currents within the two coils are integrated with the differential equation
𝑇𝑚𝑜𝑡 = 𝑇𝐴 + 𝑇𝐵 − 𝑇𝐷
All mechanical components of the stepper motor and the attached gear unit are represented by the Rotor And Gear (RAG) block. The input to this block is the torque, generated by the electromagnetic forces.
(1)
The net residual torque Tres which forces the rotor to move is calculated by subtracting the appropriate friction and reaction torque losses:
The back-EMF voltage uemf of each phase is proportional to the rotor rate times the back-EMF constant Kv. It depends further on the angular position α of the rotor with its p magnetic poles according to the following formula: 𝑝 2
(5)
Rotor and Gear (RAG):
With L denoting the coil Inductance, i the phase current, R coil and input resistance. The right hand side of equation (1) is the difference between the applied voltage uapp and the back-Electro Magnetic Field (EMF) voltage uemf.
𝑢𝑒𝑚𝑓 = 𝛼 ̇ ∙ 𝐾𝑉 ∙ 𝑠𝑖𝑛 � ∙ 𝛼 + 𝛼𝑠ℎ𝑖𝑓𝑡 �
(4)
where KDk is the torque amplitude of harmonic k, and αdetk is the torque period of harmonic k.
Two Phase E-Motor (E2P):
di L ⋅ + R ⋅= i uapp − uemf dt
(3)
𝐽𝑟𝑜𝑡 𝛼̈ = 𝑇𝑟𝑒𝑠 = 𝑇𝑚𝑜𝑡 − 𝑇𝑓𝑟 − 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟
(6)
with Trot_gear denoting the gear reaction torque from the gear output shaft (αgear)to the rotor: 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟 =
(2)
𝐾𝑠 𝐺²
�𝛼 − 𝐺𝛼𝑔𝑒𝑎𝑟 � +
𝐷
𝐺²
�𝛼̇ − 𝐺𝛼̇ 𝑔𝑒𝑎𝑟 �
(7)
where Ks and D denoting the gear stiffness and damping at output respectively, G is the transmission ratio of the gear. 173
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
The friction torque Tfr is modelled by incorporating the static Tfrs (stiction) and dynamic Tfrd (coloumb) friction phenomena together with the viscous friction component: 𝑇𝑓𝑟𝑠 , 𝑇𝑓𝑟 = � 𝑇𝑓𝑟𝑑 + 𝑇𝑣𝑖𝑠𝑐 ∙ 𝛼̇ ,
|𝛼| = 0 |𝛼| > 0
(8)
The test mass attached to the gear output is modelled using dynamics of a rigid body: (10)
The modelling, described in Section 3.1 condenses many physically meaningful parameters. In general, all model parameters can be categorised in two different sets. The first set is composed by parameters related to electro-magnetic phenomena of the motor, whereas the second set of parameters characterise the mechanical drive-train parameters.
EXPERIMENT DESIGN
Along with limitations posed by the measurement hardware, the choice of the sampling rate of the measurements is usually predominated by the subsequent microvibration analysis. The simulation model is coupled with structural transfer functions to be able to assess the microvibration levels at the payload locations. To be able to identify the model at wide frequency ranges, the torque measurements have been sampled and recorded at 10 kHz.
The electrical motor and mechanical drive-train parameters with a priori values based on supplier’s catalogue data are listed in Table 1 and Table 2 respectively. Table 1. Electrical Motor Data
Number of poles [-] Coil resistance [Ohm] Coil inductance [H] Torque constant [Nm/A] Detent torque angle [deg] Detent torque amplitude [Nm]
Torque period of the detent harmonics αdet
The test is conducted by commanding the motor in an operation mode, leading to a rotation of the test mass. The torque transducer measures the induced reaction torque at the basis of the test adapter. The motor is driven with a voltage controlled input signal in full-stepping and micro-stepping mode. The antenna pointing mechanism allows operation at different stepping rates and the rate change according to some predefined profile.
3.2 Model Estimation Parameters
p R L km αdet Kd
•
The test set-up consists of a Kistler torque transducer with a cylindrical test adapter and the mounted test article as shown in Fig. 3. The Motor Gear Unit (MGU) consists of the Phytron VSS 43.200 stepper motor (Phytron-Elektronik GmbH) together with the HFUC 11 harmonic drive (Harmonic Drive AG).
Test Mass:
Description
Inertia of test mass Jm
4.1 Experimental Setup
The final measurement torque is computed using the negative sum of the residual and the applied torques.
Parameter
•
4 (9)
𝐽𝑚 𝛼̈ 𝑔𝑒𝑎𝑟 = 𝑇𝑎𝑝𝑝𝑙𝑖𝑒𝑑
Gear transmission ratio G
Only the remaining parameters are subject to estimation.
The applied torque to rotate the test mass is calculated by: 𝑇𝑎𝑝𝑝𝑙𝑖𝑒𝑑 = 𝐺 ∙ 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟
•
Nominal Value 100 9.5 0.0229 0.277 [1.8, 3.6] [0.01, 0.0]
Table 2. Mechanical Drive-Train Data Nominal Value G Transmission ratio [-] 50 0.03/ Tfrs/Tfrd Static/Dynamic friction [Nm] 0.0265 Tvisc Viscous friction [Nms/rad] 9.74E-4 Ks Gear stiffness at output [Nm/rad] 2200 D Gear damping at output [Nms/rad] 0.48 Jrot Rotor inertia [kgm²] 8.9E-6 Jm Test mass inertia [kgm²] 0.010534 This list contains parameters which configure the generic model as a representation of specific physical drive unit and therefore are well known or can be measured easily. These include: Parameter
Description
•
Number of motor poles p
•
Rotor Inertia Jrot
Fig. 3: Experimental setup for antenna pointing mechanism testing with Kistler torque transducer with torque sensor 4.2 Input Design To maximize the visibility of the drive unit’s electromechanical properties in the torque output signal, the commanded stepping rate has to be set to frequencies that excite the eigenfrequencies of the drive chain. A sweep of the stepping rates within the operational constraints of the 174
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
mechanism has been used to detect these frequencies of interest (see Fig. 4). For the selected drive unit high amplitudes can be seen at 23 Hz, 46 Hz and 92 Hz.
assumed to be correct, the measured torque showed a slower ramping-up profile. This was an indirect indication that the phase inductance has a larger value than specified. For this reason the phase resistance R and inductance L have been also estimated in this measurement record and kept at estimated values for the subsequent measurements to avoid high correlation with other model parameters. 1
torque [Nm]
0.5 0 -0.5
Fig. 4: Torque amplitudes of the full step sweep input The measurement data is gathered with a constant stepping rate at these selected frequencies for both, full stepping and microstepping commanding. 5
Measurement Model
-1
0
0.2
0.4
time [s]
0.6
0.8
1
Fig. 5: Torque profiles at constant velocity 46 steps per second with full speed operation
RESULTS AND ANALYSIS
0.7
Due to inherently nonlinear input/output relationships of the model dynamics the output-error identification setup has been applied. Under assumption of Gaussian distribution of the measured torque noise, the maximization of the likelihood function can be used as estimation criteria. However, since only the induced reaction torque measurements from the experimental data was available, it has allowed an equivalent form of a simple optimization criteria based on the sum of the squared error between measured and modelled outputs. For minimization of the cost criteria a constrained optimization algorithm from the Matlab optimization toolbox (Coleman, et. al. (2013)) has been utilized.
Meas Model
0.6
Torque [Nm]
0.5 0.4 0.3 0.2 0.1 0
0
100
200 300 Frequency [Hz]
400
500
Fig. 6: Amplitude spectrum at constant velocity 46 steps per second with full speed operation
The overall number of estimation parameters is relatively large and, potentially, it can cause numerical optimization problems due to possible linear dependency between some of the model parameters. It has been therefore decided to perform parameter identification in several steps. For this reason several measurement records with different rotor speeds, operational modes and interval lengths have been used for estimating the model parameters.
Results of the estimated model performance versus measurement data are shown in Fig. 5 and in Fig. 6. As can be seen, a very good agreement with the measured torque is achieved in time domain. In the frequency domain the model correctly describes the amplitude of the stepping perturbation coupled with torsional mode at 46Hz, whereas Slight performance degradation can be observed for higher stepping harmonics at frequencies 92Hz and 138Hz.
5.1 Full Stepping Mode
It should be also stressed that a good match between the measured and modelled outputs is also achieved at the first 0.3 seconds of the measurement interval, where the rotor is accelerating.
While operating in full stepping mode the motor generates high level torques as a sequence of sharp peaks at the beginning of each motor step. This can influence the estimation quality of some of the model parameters in a positive way.
5.2 Micro-Stepping Mode The ability to characterise the mechanism disturbances in the microstepping commanding model is the most important for the micro-vibration budget calculation, and hence, for the overall mission performance assessment.
The equivalent stiffness of the mechanism (i.e. the motor and the gearbox) does not depend on the rotational speed and can be determined from a short measurement record with constant speed operation. Furthermore, the motor torque can be generated at zero and very low rotor rates, where many other effects such as viscous friction, have low influence. It is therefore advantageous to consider the beginning of the measurement interval for determining the motor torque constant.
In the micro-stepping commanding mode the amount of current of each electrical phase can be different, which forces the motor to make a step between its natural full step positions. The magnitude of the electromagnetic stiffness is slightly reduced when comparing with the full stepping mode. On the other hand, the detent torque effect becomes more apparent and hence, is better observable in the
Although a priori values of the motor windings parameters, i.e the phase resistance R and the phase inductance L were 175
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
measurement data. The estimated model performance when commanding a constant speed of 46 steps per second in microstepping mode is shown in Fig. 7.
approximates the structural oscillation of the housing, as shown in Fig. 9. Residual torque
0.08
s 2 + 2ζω0 s + ω02
Meas Model
0.06
torque [Nm]
0.04
Applied torque
0.02
Fig. 9: Incorporation of flexibility of the mechanism housing into simulation model
0 -0.02
Along with the other model parameters, the natural frequency ω0, damping ζ and the resonance amplification factor Kres has been estimated. Inclusion of these three additional parameters in the parameter estimation has not, however, affected the convergence of numerical optimisation. The performance improvement of the extended model is shown in Fig. 10. As can be observed the resonance oscillations are now adequately described by the extended model.
-0.04 -0.06
Measured torque
K res
0
0.2
0.4
time [s]
0.6
0.8
1
Fig. 7: Estimated model fit to measurement data at microstepping commanding with 46 steps per second As stated above, the electronic part and the hardware setup of the antenna pointing mechanism allows operation at different speeds and a speed change during one measurement interval. For this reason a velocity sweep commanding between 0 and 120 steps per second in the micro-stepping commanding has been applied. This allows an excitation of the physical plant at different frequencies, and, hence verification of the adequacy of the chosen mathematical model at wide operational speeds.
0.06
Meas Model Model ext.
torque [Nm]
0.04
The time histories shown in Fig. 8 indicate a good agreement between the measurements and the model over the whole measurement interval with duration of 60 seconds. Only at rotor speeds around 46 steps per second (22 seconds in measurement interval), the mechanism undergoes additional high torque amplitudes of about 0.1 Nm that could not be well described by the model. This can be attributed to the fact that different detent harmonics may have different phase shifts.
0.02 0 -0.02 -0.04 -0.06 0.1
0.15
0.2 time [s]
0.25
0.3
Fig. 10: Time responses of original and extended model versus measurement data at microstepping commanding (46 steps per second) 0.03
Meas Model
Torque [Nm]
0.025
0.6 Meas Model
0.4
torque [Nm]
0.2 0
0.015 0.01 0.005
-0.2
0
-0.4 -0.6 -0.8
0.02
0
10
20
30 time [s]
40
50
0
50
100
150
200
250
300
Frequency [Hz]
350
400
450
500
Fig. 11: Amplitude spectrum of extended model versus measurement data at microstepping commanding (46 steps per second)
60
Fig. 8: Measured vs. modelled torque profiles for a sweep measurement 0 to 120 steps per second in micro-stepping commanding
5.4 Linear Interdependence between Estimated Parameters The covariance of the estimated parameters, computed from the Fisher information matrix (Ljung 1999), can deliver important information about the parametric interdependency (correlation) of the model parameters in the statistical sense (Jategaonkar 2007). The correlation between two estimated parameters was computed in the numerical range between 0 (no correlation) and 1 (linear dependency). A consolidated treatment of all correlation results has been considered while performing the numerical optimization. If a nearly linear dependency between two or more model parameters was
5.3 Incorporation of flexible effects of the mechanism housing At the rotor speeds around 46 steps per second some high frequency oscillations have been observed in the measurement data. It was assumed that this effect is caused by high structural stiffness of the actuator housing. In order to incorporate this effect in the simulation model, it was necessary to pass the mechanism residual torque computed in equation (6) through a linear second order system that 176
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
detected after optimization run, the less important of them were kept at the constant values.
lowering of the resonance frequency to 46 Hz as confirmed by the spectral plot shown in Fig. 11. The viscous friction torque could be best identified when the decaying behaviour of the torque measurements were studied during the sweep profile experiment (see Fig. 8).
5.5 Analysis of Estimated Model Parameters The nominal model parameter values versus their experimentally determined counterparts are given in Table 3 and Table 4.
6
The systematic approach involving system identification methodology was applied for characterization of a spacecraft antenna pointing mechanism. Based on achieved results, the adopted physical modelling and the estimated parameters allow an adequate characterization of the physical plant for the subsequent assessment of the microvibration budget.
Table 3. Nominal vs. Estimated Electrical Motor Parameters Parameter R L km Kd
Description Coil resistance [Ohm] Coil inductance [H] Torque constant [Nm/A] Detent torque amplitude [Nm]
Nominal Value 9.5 0.0229 0.277 [0.01, 0.0]
Estimated Value 9.9 0.045 0.384 [0.0047, 0.0231]
Some of experimentally determined model parameters were, however, significantly different from the specified ones. These findings highlight the importance of experimental tests and of system identification in particular, while designing the satellite mechanisms and verification of the spacecraft performance.
Table 4. Nominal vs. Estimated Mechanical Drive-Train Parameters Parameter Tfrs/Tfrd Tvisc Ks D Kres ω0 ζ
Description Static/Dynamic friction [Nm] Viscous friction [Nms/rad] Gear stiffness [Nm/rad] Gear damping [Nms/rad] Resonance factor [-] Resonance frequency [rad/s] Resonance damping [-]
Nominal Value 0.03/ 0.0265 9.74E-4 2200 0.48 N/A
Estimated Value 0.008/ 0.0013 0.0061 913.47 0.39 4.6
N/A
5578
N/A
0.095
CONCLUSIONS AND PERSPECTIVES
The presented work is limited to the system identification aspects of the antenna pointing mechanism. To obtain an estimation of the expectable microvibration budget as a result of mechanism operation, a coupled analysis is performed with a global structural and the identified mechanism. This systematic approach will be applied for a number of future earth observation and science missions. ACKNOWLEDGMENTS Part of this work related to system identification has been financially supported by the Space Agency of the German Aerospace Center (DLR, Deutsches Zentrum für Luft- und Raumfahrt e.V.) with means of the German Ministry of Economy and Technology under support number 50 RR 1201
As noticed earlier, the phase coil inductance has been estimated to be by factor two greater than specified. In addition, also the motor torque constant has an increase of about 38%. One interesting result is that the motor detent torque is dominated by the second harmonic torque component, which was somewhat different than expected. This can be also confirmed by looking at the measurements of the sweep profile on Fig. 8. The maximum perturbation torque amplitude occurs at measurement time around 46 seconds, which corresponds to the rotation speed of the motor of about 92 steps per second. At this rate the second (3.6 deg) detent harmonic component reaches the perturbation frequency of 46 Hz. This frequency coincides also with eigenfrequency of the mechanical drive-train.
REFERENCES Athani. A. A. (1997). Stepper Motors: Fundamentals, Applications and Design. New Age International Publishers, New Delhi. Galeazzi, C, et. al. (1996). Experimental Activities on ARTEMIS for the Microvibration Verification, Proceedings of Space Structures, Materials and Mechanical Testing Conference. Jategaonkar, R. (2006). Flight Vehicle System Identification: A Time Domain Methodology, Progress in Aeronautics and Astronautics. Ljung, L. (1999). System Identification: Theory for the User, Prentice Hall PTR, New Jersey. Coleman, T. F., Zhang, Y. (2013). Matlab Optimization Toolbox: User’s Guide, The Mathworks. Harmonic Drive AG. (2010). General Catalogue, HFUC-2A, Harmonic Drive AG, Limberg/Lahn. Phytron-Elektronik GmbH (1997). VSS/VSH Extreme Enviroment Stepper Motors, Phytron-Elektronik GmbH, Gröbenzell. Vitelli, M., Specht, B., Boquet, F. (2012). A Process to Verify the Microvibrtion and Pointing Stability Requirements for the BepiColombo Mission, Workshop on Instrumentation for Interplanetary Missions, Greenbelt, Maryland
When looking at the mechanical drive-train parameters (Table 4) the following can be observed. The parameters that correspond to the static and dynamic damping of the mechanism were underestimated. This, however, was caused by a low sensitivity of these parameters on the modelled torque output. The most significant difference between the specified and the estimated parameters occurs in characterisation of the gear stiffness and the viscous friction components. The estimated stiffness is factor 2.4 less than was initially specified. As a result of the reduction of the torsional stiffness is the 177
System Identification of a Spacecraft Antenna Pointing Mechanism Andrei Kornienko*, Robert Wuseni* Bernhard Specht* *Astrium GmbH, D-88039 Friedrichshafen, Germany (e-mail: andrei.kornienko, robert.wuseni, [email protected]). Abstract: This paper describes results of identification of a spacecraft antenna pointing mechanism. Accurate modelling of the physical effects that occur during the mechanism operation is mandatory for the realistic assessment of the microvibration budget as well for the control design of the mechanism electronics. System identification is seen as a very attractive methodology in characterizing the entire mechanism, including stepper motor electronics, electromagnetic effects that occur during stepping and microstepping operation and accurate modelling and parameterization of the mechanical drive train. The results achieved throughout model identification show a very good agreement between the model and experimental data in wide operational ranges in time and in frequency domains. Keywords: System Identification, Stepper Motor, Modelling, Experiment Design, Optimization 1
results chapter. Finally, some conclusions of the work and future perspectives will be given.
INTRODUCTION
Continuously increasing demands on the performance accuracy of the modern Earth observation and science satellite missions require elaborated analysis approaches. It is well known that typical satellite on-board actuators, such as reaction wheels or drive mechanisms, can significantly degrade performance of the onboard optical or scientific payload. To be able to characterize the performance that can be achieved by the spacecraft in its in-orbit operation it is mandatory to acquire sufficient knowledge of the spacecraft dynamics, as well as its internal components. Reaction wheels, antennas, solar array drive mechanism, compressors and other actuators, are potential sources of the microvibration disturbances that may significantly affect inorbit performance of the satellite (Galeazzi, et al. (1996)). Thus, knowing perturbation levels of the actuators is quite fundamental for the spacecraft design. In order to assess the microvibration budget, elaborated simulation models of the on-board actuators are developed (Vitelli, et al. (2012)). The simulation models typically contain uncertain parameters that can be derived either from analytical considerations, using supplier specifications or determined experimentally. One of the possibilities to determine the model parameters from experimental data is based on the system identification approach (Ljung (1999)). This paper is focused on presenting results of identification of the satellite antenna pointing mechanism.
2
System identification is a process to determine mathematical models of dynamic systems based on experimental data. Its fundamental concept can be best described by the diagram shown in Fig. 1. Input
Physical Plant Sensors and Recorder
Meas. Input
Tunable Plant Model
Output
Meas. Output
Adaptation Algorithm Model Output
Model Update
Fig. 1: A schematic diagram of the system identification process. The tuneable plant model is a mathematical model reflecting the real physical plant or process. It is subject to the measured inputs from the real process. The identification error, i.e. the difference between measured outputs (subject to errors/noise) and the model output, is applied to improve the mathematical model. The goal is to achieve an agreement between the plant and the model to a certain accuracy (Ljung (1999)).
Following a brief description of the system identification process, details of mathematical model that represents the antenna pointing mechanism will be given. Experimental design aspects including measurement hardware setup and conduction of experiments will be shortly outlined. A comparisson between the model output and the experimental data in the time and frequency domain as well as specified versus estimated parameter values will be presented in the
978-3-902823-46-5/2013 © IFAC
SYSTEM IDENTIFICATION AS EXPERIMENTALLY BASED MODELLING
Identification of control-relevant design models from detailed simulation models allows efficient approaches for the design, tuning, and validation of control algorithms and flight software of satellites. The improvement of technical expertise 172
10.3182/20130902-5-DE-2040.00057
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
Test mass states
Stepping rate
Voltage Stepping Command (VST)
Phase voltage
Two Phase E-Motor (E2P)
Motor torque
Rotor and Gear (RAG)
Applied torque Residual torque
Rotor states
Test Mass
Measured torque
Fig. 2: Top-level architecture of the simulation model αshift represents the geometric relationship between the two motor phases. In the standard case a value of αshift = 90° for one phase is appropriate.
in the area of system and parameter identification is a key element for the mastery and realization of future highaccuracy satellite missions, and thus allows both precision and efficiency increases while lowering development times and costs. 3
The total generated torque on the rotor is a sum of the individual coil torques and the detent torque. The individual torques are calculated as:
ANTENNA MECHANISM MODELLING AND PARAMETERISATION
p TA/ B = iA/ B ⋅ km ⋅ sin ⋅ α + α shift 2
3.1 Mathematical Model
with km denoting the motor torque constant.
The mathematical representation (Athani (1997)) of the antenna drive mechanism is based on four major functional blocks (see Fig. 2) and realised using MATLAB/Simulink™ simulation environment. The details of individual functional blocks are described below.
The detent torque is a result of the motors permanent magnets. Its shape follows a sinusoidal profile with its maximum between two poles according to the equation (4). In order to take imperfections of the rotor and stator into account, the detent torque is calculated as a sum of multiple torque harmonics with different periods over a full revolution. In this analysis a total of two harmonics have been considered.
Voltage Stepping Command (VST): The VST Block serves as a signal generator for a two phase bi-polar stepper motor. With respect to the selected mode of operation, which can be either full stepping or microstepping with multiple sub-steps, this block generates the required voltage profiles for the commutation of the motor. The stepping rate is applied at the input. The block operates as an ideal voltage source with a preselected peak voltage. No current regulation is implemented.
360° TD = ∑k K Dk ⋅ sin α ⋅ α det k
The overall motor torque is then computed by combining equations (3) and (4):
The E2P Block is the heart of the stepper motor. It accommodates the electrical section of the stepper motor. The currents within the two coils are integrated with the differential equation
𝑇𝑚𝑜𝑡 = 𝑇𝐴 + 𝑇𝐵 − 𝑇𝐷
All mechanical components of the stepper motor and the attached gear unit are represented by the Rotor And Gear (RAG) block. The input to this block is the torque, generated by the electromagnetic forces.
(1)
The net residual torque Tres which forces the rotor to move is calculated by subtracting the appropriate friction and reaction torque losses:
The back-EMF voltage uemf of each phase is proportional to the rotor rate times the back-EMF constant Kv. It depends further on the angular position α of the rotor with its p magnetic poles according to the following formula: 𝑝 2
(5)
Rotor and Gear (RAG):
With L denoting the coil Inductance, i the phase current, R coil and input resistance. The right hand side of equation (1) is the difference between the applied voltage uapp and the back-Electro Magnetic Field (EMF) voltage uemf.
𝑢𝑒𝑚𝑓 = 𝛼 ̇ ∙ 𝐾𝑉 ∙ 𝑠𝑖𝑛 � ∙ 𝛼 + 𝛼𝑠ℎ𝑖𝑓𝑡 �
(4)
where KDk is the torque amplitude of harmonic k, and αdetk is the torque period of harmonic k.
Two Phase E-Motor (E2P):
di L ⋅ + R ⋅= i uapp − uemf dt
(3)
𝐽𝑟𝑜𝑡 𝛼̈ = 𝑇𝑟𝑒𝑠 = 𝑇𝑚𝑜𝑡 − 𝑇𝑓𝑟 − 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟
(6)
with Trot_gear denoting the gear reaction torque from the gear output shaft (αgear)to the rotor: 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟 =
(2)
𝐾𝑠 𝐺²
�𝛼 − 𝐺𝛼𝑔𝑒𝑎𝑟 � +
𝐷
𝐺²
�𝛼̇ − 𝐺𝛼̇ 𝑔𝑒𝑎𝑟 �
(7)
where Ks and D denoting the gear stiffness and damping at output respectively, G is the transmission ratio of the gear. 173
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
The friction torque Tfr is modelled by incorporating the static Tfrs (stiction) and dynamic Tfrd (coloumb) friction phenomena together with the viscous friction component: 𝑇𝑓𝑟𝑠 , 𝑇𝑓𝑟 = � 𝑇𝑓𝑟𝑑 + 𝑇𝑣𝑖𝑠𝑐 ∙ 𝛼̇ ,
|𝛼| = 0 |𝛼| > 0
(8)
The test mass attached to the gear output is modelled using dynamics of a rigid body: (10)
The modelling, described in Section 3.1 condenses many physically meaningful parameters. In general, all model parameters can be categorised in two different sets. The first set is composed by parameters related to electro-magnetic phenomena of the motor, whereas the second set of parameters characterise the mechanical drive-train parameters.
EXPERIMENT DESIGN
Along with limitations posed by the measurement hardware, the choice of the sampling rate of the measurements is usually predominated by the subsequent microvibration analysis. The simulation model is coupled with structural transfer functions to be able to assess the microvibration levels at the payload locations. To be able to identify the model at wide frequency ranges, the torque measurements have been sampled and recorded at 10 kHz.
The electrical motor and mechanical drive-train parameters with a priori values based on supplier’s catalogue data are listed in Table 1 and Table 2 respectively. Table 1. Electrical Motor Data
Number of poles [-] Coil resistance [Ohm] Coil inductance [H] Torque constant [Nm/A] Detent torque angle [deg] Detent torque amplitude [Nm]
Torque period of the detent harmonics αdet
The test is conducted by commanding the motor in an operation mode, leading to a rotation of the test mass. The torque transducer measures the induced reaction torque at the basis of the test adapter. The motor is driven with a voltage controlled input signal in full-stepping and micro-stepping mode. The antenna pointing mechanism allows operation at different stepping rates and the rate change according to some predefined profile.
3.2 Model Estimation Parameters
p R L km αdet Kd
•
The test set-up consists of a Kistler torque transducer with a cylindrical test adapter and the mounted test article as shown in Fig. 3. The Motor Gear Unit (MGU) consists of the Phytron VSS 43.200 stepper motor (Phytron-Elektronik GmbH) together with the HFUC 11 harmonic drive (Harmonic Drive AG).
Test Mass:
Description
Inertia of test mass Jm
4.1 Experimental Setup
The final measurement torque is computed using the negative sum of the residual and the applied torques.
Parameter
•
4 (9)
𝐽𝑚 𝛼̈ 𝑔𝑒𝑎𝑟 = 𝑇𝑎𝑝𝑝𝑙𝑖𝑒𝑑
Gear transmission ratio G
Only the remaining parameters are subject to estimation.
The applied torque to rotate the test mass is calculated by: 𝑇𝑎𝑝𝑝𝑙𝑖𝑒𝑑 = 𝐺 ∙ 𝑇𝑟𝑜𝑡_𝑔𝑒𝑎𝑟
•
Nominal Value 100 9.5 0.0229 0.277 [1.8, 3.6] [0.01, 0.0]
Table 2. Mechanical Drive-Train Data Nominal Value G Transmission ratio [-] 50 0.03/ Tfrs/Tfrd Static/Dynamic friction [Nm] 0.0265 Tvisc Viscous friction [Nms/rad] 9.74E-4 Ks Gear stiffness at output [Nm/rad] 2200 D Gear damping at output [Nms/rad] 0.48 Jrot Rotor inertia [kgm²] 8.9E-6 Jm Test mass inertia [kgm²] 0.010534 This list contains parameters which configure the generic model as a representation of specific physical drive unit and therefore are well known or can be measured easily. These include: Parameter
Description
•
Number of motor poles p
•
Rotor Inertia Jrot
Fig. 3: Experimental setup for antenna pointing mechanism testing with Kistler torque transducer with torque sensor 4.2 Input Design To maximize the visibility of the drive unit’s electromechanical properties in the torque output signal, the commanded stepping rate has to be set to frequencies that excite the eigenfrequencies of the drive chain. A sweep of the stepping rates within the operational constraints of the 174
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
mechanism has been used to detect these frequencies of interest (see Fig. 4). For the selected drive unit high amplitudes can be seen at 23 Hz, 46 Hz and 92 Hz.
assumed to be correct, the measured torque showed a slower ramping-up profile. This was an indirect indication that the phase inductance has a larger value than specified. For this reason the phase resistance R and inductance L have been also estimated in this measurement record and kept at estimated values for the subsequent measurements to avoid high correlation with other model parameters. 1
torque [Nm]
0.5 0 -0.5
Fig. 4: Torque amplitudes of the full step sweep input The measurement data is gathered with a constant stepping rate at these selected frequencies for both, full stepping and microstepping commanding. 5
Measurement Model
-1
0
0.2
0.4
time [s]
0.6
0.8
1
Fig. 5: Torque profiles at constant velocity 46 steps per second with full speed operation
RESULTS AND ANALYSIS
0.7
Due to inherently nonlinear input/output relationships of the model dynamics the output-error identification setup has been applied. Under assumption of Gaussian distribution of the measured torque noise, the maximization of the likelihood function can be used as estimation criteria. However, since only the induced reaction torque measurements from the experimental data was available, it has allowed an equivalent form of a simple optimization criteria based on the sum of the squared error between measured and modelled outputs. For minimization of the cost criteria a constrained optimization algorithm from the Matlab optimization toolbox (Coleman, et. al. (2013)) has been utilized.
Meas Model
0.6
Torque [Nm]
0.5 0.4 0.3 0.2 0.1 0
0
100
200 300 Frequency [Hz]
400
500
Fig. 6: Amplitude spectrum at constant velocity 46 steps per second with full speed operation
The overall number of estimation parameters is relatively large and, potentially, it can cause numerical optimization problems due to possible linear dependency between some of the model parameters. It has been therefore decided to perform parameter identification in several steps. For this reason several measurement records with different rotor speeds, operational modes and interval lengths have been used for estimating the model parameters.
Results of the estimated model performance versus measurement data are shown in Fig. 5 and in Fig. 6. As can be seen, a very good agreement with the measured torque is achieved in time domain. In the frequency domain the model correctly describes the amplitude of the stepping perturbation coupled with torsional mode at 46Hz, whereas Slight performance degradation can be observed for higher stepping harmonics at frequencies 92Hz and 138Hz.
5.1 Full Stepping Mode
It should be also stressed that a good match between the measured and modelled outputs is also achieved at the first 0.3 seconds of the measurement interval, where the rotor is accelerating.
While operating in full stepping mode the motor generates high level torques as a sequence of sharp peaks at the beginning of each motor step. This can influence the estimation quality of some of the model parameters in a positive way.
5.2 Micro-Stepping Mode The ability to characterise the mechanism disturbances in the microstepping commanding model is the most important for the micro-vibration budget calculation, and hence, for the overall mission performance assessment.
The equivalent stiffness of the mechanism (i.e. the motor and the gearbox) does not depend on the rotational speed and can be determined from a short measurement record with constant speed operation. Furthermore, the motor torque can be generated at zero and very low rotor rates, where many other effects such as viscous friction, have low influence. It is therefore advantageous to consider the beginning of the measurement interval for determining the motor torque constant.
In the micro-stepping commanding mode the amount of current of each electrical phase can be different, which forces the motor to make a step between its natural full step positions. The magnitude of the electromagnetic stiffness is slightly reduced when comparing with the full stepping mode. On the other hand, the detent torque effect becomes more apparent and hence, is better observable in the
Although a priori values of the motor windings parameters, i.e the phase resistance R and the phase inductance L were 175
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
measurement data. The estimated model performance when commanding a constant speed of 46 steps per second in microstepping mode is shown in Fig. 7.
approximates the structural oscillation of the housing, as shown in Fig. 9. Residual torque
0.08
s 2 + 2ζω0 s + ω02
Meas Model
0.06
torque [Nm]
0.04
Applied torque
0.02
Fig. 9: Incorporation of flexibility of the mechanism housing into simulation model
0 -0.02
Along with the other model parameters, the natural frequency ω0, damping ζ and the resonance amplification factor Kres has been estimated. Inclusion of these three additional parameters in the parameter estimation has not, however, affected the convergence of numerical optimisation. The performance improvement of the extended model is shown in Fig. 10. As can be observed the resonance oscillations are now adequately described by the extended model.
-0.04 -0.06
Measured torque
K res
0
0.2
0.4
time [s]
0.6
0.8
1
Fig. 7: Estimated model fit to measurement data at microstepping commanding with 46 steps per second As stated above, the electronic part and the hardware setup of the antenna pointing mechanism allows operation at different speeds and a speed change during one measurement interval. For this reason a velocity sweep commanding between 0 and 120 steps per second in the micro-stepping commanding has been applied. This allows an excitation of the physical plant at different frequencies, and, hence verification of the adequacy of the chosen mathematical model at wide operational speeds.
0.06
Meas Model Model ext.
torque [Nm]
0.04
The time histories shown in Fig. 8 indicate a good agreement between the measurements and the model over the whole measurement interval with duration of 60 seconds. Only at rotor speeds around 46 steps per second (22 seconds in measurement interval), the mechanism undergoes additional high torque amplitudes of about 0.1 Nm that could not be well described by the model. This can be attributed to the fact that different detent harmonics may have different phase shifts.
0.02 0 -0.02 -0.04 -0.06 0.1
0.15
0.2 time [s]
0.25
0.3
Fig. 10: Time responses of original and extended model versus measurement data at microstepping commanding (46 steps per second) 0.03
Meas Model
Torque [Nm]
0.025
0.6 Meas Model
0.4
torque [Nm]
0.2 0
0.015 0.01 0.005
-0.2
0
-0.4 -0.6 -0.8
0.02
0
10
20
30 time [s]
40
50
0
50
100
150
200
250
300
Frequency [Hz]
350
400
450
500
Fig. 11: Amplitude spectrum of extended model versus measurement data at microstepping commanding (46 steps per second)
60
Fig. 8: Measured vs. modelled torque profiles for a sweep measurement 0 to 120 steps per second in micro-stepping commanding
5.4 Linear Interdependence between Estimated Parameters The covariance of the estimated parameters, computed from the Fisher information matrix (Ljung 1999), can deliver important information about the parametric interdependency (correlation) of the model parameters in the statistical sense (Jategaonkar 2007). The correlation between two estimated parameters was computed in the numerical range between 0 (no correlation) and 1 (linear dependency). A consolidated treatment of all correlation results has been considered while performing the numerical optimization. If a nearly linear dependency between two or more model parameters was
5.3 Incorporation of flexible effects of the mechanism housing At the rotor speeds around 46 steps per second some high frequency oscillations have been observed in the measurement data. It was assumed that this effect is caused by high structural stiffness of the actuator housing. In order to incorporate this effect in the simulation model, it was necessary to pass the mechanism residual torque computed in equation (6) through a linear second order system that 176
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
detected after optimization run, the less important of them were kept at the constant values.
lowering of the resonance frequency to 46 Hz as confirmed by the spectral plot shown in Fig. 11. The viscous friction torque could be best identified when the decaying behaviour of the torque measurements were studied during the sweep profile experiment (see Fig. 8).
5.5 Analysis of Estimated Model Parameters The nominal model parameter values versus their experimentally determined counterparts are given in Table 3 and Table 4.
6
The systematic approach involving system identification methodology was applied for characterization of a spacecraft antenna pointing mechanism. Based on achieved results, the adopted physical modelling and the estimated parameters allow an adequate characterization of the physical plant for the subsequent assessment of the microvibration budget.
Table 3. Nominal vs. Estimated Electrical Motor Parameters Parameter R L km Kd
Description Coil resistance [Ohm] Coil inductance [H] Torque constant [Nm/A] Detent torque amplitude [Nm]
Nominal Value 9.5 0.0229 0.277 [0.01, 0.0]
Estimated Value 9.9 0.045 0.384 [0.0047, 0.0231]
Some of experimentally determined model parameters were, however, significantly different from the specified ones. These findings highlight the importance of experimental tests and of system identification in particular, while designing the satellite mechanisms and verification of the spacecraft performance.
Table 4. Nominal vs. Estimated Mechanical Drive-Train Parameters Parameter Tfrs/Tfrd Tvisc Ks D Kres ω0 ζ
Description Static/Dynamic friction [Nm] Viscous friction [Nms/rad] Gear stiffness [Nm/rad] Gear damping [Nms/rad] Resonance factor [-] Resonance frequency [rad/s] Resonance damping [-]
Nominal Value 0.03/ 0.0265 9.74E-4 2200 0.48 N/A
Estimated Value 0.008/ 0.0013 0.0061 913.47 0.39 4.6
N/A
5578
N/A
0.095
CONCLUSIONS AND PERSPECTIVES
The presented work is limited to the system identification aspects of the antenna pointing mechanism. To obtain an estimation of the expectable microvibration budget as a result of mechanism operation, a coupled analysis is performed with a global structural and the identified mechanism. This systematic approach will be applied for a number of future earth observation and science missions. ACKNOWLEDGMENTS Part of this work related to system identification has been financially supported by the Space Agency of the German Aerospace Center (DLR, Deutsches Zentrum für Luft- und Raumfahrt e.V.) with means of the German Ministry of Economy and Technology under support number 50 RR 1201
As noticed earlier, the phase coil inductance has been estimated to be by factor two greater than specified. In addition, also the motor torque constant has an increase of about 38%. One interesting result is that the motor detent torque is dominated by the second harmonic torque component, which was somewhat different than expected. This can be also confirmed by looking at the measurements of the sweep profile on Fig. 8. The maximum perturbation torque amplitude occurs at measurement time around 46 seconds, which corresponds to the rotation speed of the motor of about 92 steps per second. At this rate the second (3.6 deg) detent harmonic component reaches the perturbation frequency of 46 Hz. This frequency coincides also with eigenfrequency of the mechanical drive-train.
REFERENCES Athani. A. A. (1997). Stepper Motors: Fundamentals, Applications and Design. New Age International Publishers, New Delhi. Galeazzi, C, et. al. (1996). Experimental Activities on ARTEMIS for the Microvibration Verification, Proceedings of Space Structures, Materials and Mechanical Testing Conference. Jategaonkar, R. (2006). Flight Vehicle System Identification: A Time Domain Methodology, Progress in Aeronautics and Astronautics. Ljung, L. (1999). System Identification: Theory for the User, Prentice Hall PTR, New Jersey. Coleman, T. F., Zhang, Y. (2013). Matlab Optimization Toolbox: User’s Guide, The Mathworks. Harmonic Drive AG. (2010). General Catalogue, HFUC-2A, Harmonic Drive AG, Limberg/Lahn. Phytron-Elektronik GmbH (1997). VSS/VSH Extreme Enviroment Stepper Motors, Phytron-Elektronik GmbH, Gröbenzell. Vitelli, M., Specht, B., Boquet, F. (2012). A Process to Verify the Microvibrtion and Pointing Stability Requirements for the BepiColombo Mission, Workshop on Instrumentation for Interplanetary Missions, Greenbelt, Maryland
When looking at the mechanical drive-train parameters (Table 4) the following can be observed. The parameters that correspond to the static and dynamic damping of the mechanism were underestimated. This, however, was caused by a low sensitivity of these parameters on the modelled torque output. The most significant difference between the specified and the estimated parameters occurs in characterisation of the gear stiffness and the viscous friction components. The estimated stiffness is factor 2.4 less than was initially specified. As a result of the reduction of the torsional stiffness is the 177