SAEM . . . Système Automatique d'équilibrage de Masse. ⢠ABMS . ... rational are : Surrey in England, Swiss cube by the Polytechnique Institut in ...... project. I cant also forget Said with whom talking, I learned a lot of new ways to approach the.
Developement of the Control laws for the active Attitude and Orbit Control System (AOCS) of a Picosatellite (Développement de lois de contrôle pour un système de contrôle d'attitude actif pour pico-satellite)
2011/2012 by
OVIE EseOghene
Laboratoire : AMPERE-UMR 5005-LYON (France)
Encadre Par : Richard MOREAU et Damien EBERARD
1
Table des matières 1 Introduction 1.1
8
Architechture of a Picosatellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 CMG system for attitude control 2.1
9
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Prototype description and connection 3.1
8
9
11
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 CMG Simulation and Experiments
11
13
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.2
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.3
Identi cation of Gimbal Motor parameters . . . . . . . . . . . . . . . . . . . . . . . .
14
4.4
System gains using Transfer function
. . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.5
Controller selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.6
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.7
Analysis of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5 Automatic Mass Balancing System AMBS
24
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.2
Identi cation and Sizing of Linear Motor . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.3
Choice of parameters[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.4
Simulation experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.5
Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.6
Acceleration obtained
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.7
Calculation of maximum force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5.8
Chosen values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5.9
Motor selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5.10 Velocity Pro les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
5.11 Kinematics of the LM for the chosen pro le . . . . . . . . . . . . . . . . . . . . . . .
28
2
5.12 Force calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5.13 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6 Singular conditions & Singularity Avoidance
34
6.1
Software In the Loop (SIL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
6.2
Singularity control & steering laws . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
6.3
Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
7 Conclusion
39
7.1
Achievements and progress made . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
7.2
Future improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
7.3
Thanks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
7.4
Resume en francais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Appendices
42
A Wireless connection(Realtime Hardware side)
42
B Wireless connection(Prototype side)
42
C Direct connection
42
D Motor controller hardware
42
E Unit conversion
42
3
Table des gures 1
Cubesat speci cation [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2
Single gimbal [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3
CMG Pyramidal con guration [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4
CMG Pyramidal con guration [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
5
Global AOCS schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
6
Gimbal DC motor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
7
Second order response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
8
Simulation o ine experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
9
Simulation o ine experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
10
Gimbal DC motor model
21
11
Experiment 1 : simultaneous common reference input of motor 2 and 4
. . . . . . .
22
12
Experiment 1 : simultaneous common reference input of motor 2 and 4
. . . . . . .
22
13
Experiment 3 : simultaneous common reference input of motor 2 and 4 . . . . . . . .
23
14
Schematic of AMBS with test platform
. . . . . . . . . . . . . . . . . . . . . . . . .
25
15
AMBS parameters and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
16
Pro l de vitesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
17
Forces sur le moteur lineaire
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
18
Simulated kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
19
Simulated kinematic [] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
20
Global AOCS schematic : conf. gure5 . . . . . . . . . . . . . . . . . . . . . . . . . .
34
21
SIL model implemented in Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
22
Singular points generated by 3 axes torques . . . . . . . . . . . . . . . . . . . . . . .
37
23
Generated by 3 axes torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
24
Platform velocity response
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
25
Platform angular response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Liste des tableaux 1
Gimbal motor parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2
ChienHronesRenswick tuning formulas[2] . . . . . . . . . . . . . . . . . . . . . . .
18
3
Synthesized parameters from Identi cation . . . . . . . . . . . . . . . . . . . . . . . .
19
4
Simulation only vs Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5
Motor parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
6
Comparison of LM parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5
Acronymes • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
SGCMG . . . Single Gimbaled micro Control Moment Gyroscope IMS . . . Inertial Measurement System IMU . . . Inertial Measurement Unit RTH . . . Real Time Hardware AOCS . . . Attitude and Orbit Control System COM . . . Centre de Masse COR . . . Centre of Rotation COG . . . Centre of Gravity OBC . . . On Board Computer SCA . . . Système de Contrôle d'Attitude SAEM . . . Système Automatique d'équilibrage de Masse ABMS . . . Automatic Mass Balancing System
Parameter nomenclature & Symbols
LM . . . Linear Motor L. . . Inductance R. . . Resistance V. . . Voltage i,I. . . Current ω . . . Angular velocity Ke . . . back EMF constant Km . . . Torque constant J. . . Rotor Inertia Jch . . . load inertia B. . . Damping constant θ. . . Angular displacement G. . . Transfer function gain Gsys . . . global System gain τe . . . Electric time constant τm . . . Mechanical time constant mj . . . LM mass β . . . slope angle of LM with vertical Kf . . . Femax . . . Maximum force Im ax. . . Maximum current s. . . linear distance t. . . time D. . . Laplacian matrice SVD. . . Singular Value Decomposition SIL. . . System In-the Loop U,S,V. . . SVD composants of matrix D D‡ . . . SDA pseudo inverse of D SSDA . . . SDA pseudoinverse of matrix S Spseudo . . . generalised pseudoinverse of matrix S σ . . . gimbal angular position
6
Résumé
This report looks at the use of SGCMG for the AOCS of a cube satellite and explains some of the experimental procedures put in place to make this possible. The main work in this report centers around continuing the process of building up a laboratory prototype and also the accompanying models to be used for simulation, control & command experiments. The research work carried out is divided into theoretical and practical parts. Chapters 1 & 2 look at the pico-satellite, the CMG, its structure and torque ampli cation method, Chapter 3 of the report looks at the picosatellite prototype with which all the practical experiments are carried out describing the various composants in the system. Chapter 4 begins the practical work done on the system. It looks at the System identi cation of the actuator i.e. gimbals the recovering of systems parameters and the simulation and control experiments performed to validate the models used. Also discussed are the exigencies of using a CMG actuator system for spacecraft attitude control which relates to singularities. Chapter 5 explains the work done on the AMBS and the sizing of the linear motors to be used. Motor selection is explained and the criteria for the choice. The various simulation experiments are clearly shown. Chapter 6 touches the subject of singularity , explains its eect and importance in a CMG actuated system. The experiments done here centered around Software inthe Loop implementation of the singularity avoidance law. Visualisation of the Singular momentum envelope is also done in simulation. Chapter 7 concludes the report and details some of the achievements, challenges faced during implementation and also perspectives for the future.
7
1 Introduction The study of picosatellites was initiated by researchers in the US. Principally by scientists and researchers from Stanford and Calpoly. However today researchers from other countries particularly France, Germany, Switzerland, Japan, China and the UK are also contributing to the this eld of research.
1.1 Architechture of a Picosatellite A picosatellite can basically be de ned as a cube of dimensions measuring 10 × 10 × 10cm, see gure 4. This standard is conventionally referred to by the abbreviation 1U, that represents 1 unit of size. Other standard sizes can be found for cubesatellites with dimensions 2U (10 × 10 × 20cm) and 3U (10 × 10 × 30cm). For the picosatellite currently under study, the following dimensions are obtained : 1. Dimension . . . 10cm × 10cm × 10cm 2. Weight . . . 1kg 3. Energy consumption . . . 1watt
Figure 1 Cubesat speci cation [15] The research in AOCS of a cubesat is as was earlier stated is going on in many universities and laboratories all over the world, principally in the US, Europe, Japan, China . . . In Europe, notable cubesat projects especially amongst those that have been launched and are operational are : Surrey in England, Swiss cube by the Polytechnique Institut in Lausanne, UWE1-2-3 by the University of Würzburg in Germany, University of Montpellier, Instituto Superior Técnico of Lisbon, Aalborg University Denmark, programme ESF COSY.
8
2 CMG system for attitude control 2.1 Introduction CMGs are used to provide the fast and high torque[5][12][17] required for certain operational modes or maneuvers of a satellite. Some of these modes are pointing, scanning and tracking. These maneuvers are carried out around the 3 axes of the satellite while at all times trying to maintain a balance about the COM of the satellite. The possible attitude changes aect the roll, pitch and yaw axis directions and is carried out by applying torques to the x, y and z axes respectively (Tx , Ty & Tz ). The basic parts of a Single Gimbaled Gyroscope is shown in Figure 2 :
Figure 2 Single gimbal [7] CMG works by momentum transfer between the spinning wheel of the gyroscope and the body of the satellte or host vehicle. This type of noncontact momentum transfer whose change (momentum dierential) generates a torque that acts on the body of the satelite is described as the Coriolis eect. The control of the satellites attitude by the commanded torque input is made through an input command to the CMG gimbal motors. The input command tracks a given steering sequence in order to make the desired attitude change. The factors that are important for the study of the steering laws all relate very directly to the CMG characteristic parameters. These are the torque (Lr), angular momentum (H ) produced in the ywheel, gimbal angular displacement (σ ), gimbal rates (σ˙ ) and 9
the vehicle angular rate (velocity ω ) values. Such detection and control process with the help of attitude (angles) and rate (velocities) sensors, helps the system to calculate its current pose (position and orientation), and correct for desired pointing direction. To achieve this, the well known con guration (see Figure 3) of 4 CMGs arranged in pyramidal con guration[14].
Figure 3 CMG Pyramidal con guration [7] The choice of 4 is based on previous research[5] which has been done to verify that 6 CMG work ne for complete singularity avoidance in all 3 axes of orientation and command but also means that higher cost and greater mass is placed on a satellite which is supposed to be small, light weight, have low energy consumption and low cost. The pyramidal con guration with 4 similar (i.e. of equal mass and geometry) CMGs arranged around the four faces of the pyramid will produce a uniformly smooth and spherical momentum envelope where the angular momentum of each about the gimbal spin-axis is equal[10]. In this case even though we have one more CMG than is necessary for three axes control,the fourth adds the necessary dynamics required to maneuver around and away from a a singular condition. The process for achieving this is treated in a latter part of this report.
10
3 Prototype description and connection 3.1 Introduction
Figure 4 CMG Pyramidal con guration [14] The prototype for the picosatellite used during this experiment has been put in place with all the basic hardware and software components implemented. The following components on the prototype are critical to perform the experiments on the AOCS while online and in real time : 1. Set of 4 SGCMG 2. Controller electronics for the SGCMG and necessary power supply module 3. FPGA card for registering encoder count data 4. Wireless master and slave modules for communication between RTH and prototype 5. dSPACE RTH for data acquisition and signal processing Having all the aforementioned components on board the system, some adjustments still needed to be done in order to make the encoder data available to the RTH and also in order to implement redundancy in the two way communication between the satellite and prototype. The redundancy implemented ensured that communication can be done either wirelessly or by direct connection between thez satellite and the RTH. It must be stated however that for the nal experiments to be performed on the complete system i.e. satellite, AMBS, AOCS ; only the wireless communication 11
system will enable the experiments to be smoothly implemented. Another hardware adjustment which was made on the system involved the recovering of encoder data from motors and sending same to the RTH for visualisation and control needs. For this it was necessary to make hardwire connection between the motor sensor output and an FPGA counter board.
12
4 CMG Simulation and Experiments 4.1 Introduction To get a better insight into the work entailed in this internship, the ACS must be put in perspective especially as it concerns the control system to be put in place. The global model is shown in Figure5.
Figure 5 Global AOCS schematic 4.2 Simulation The experiments performed on the actuator system was done in two main categories as highlighted below : Simulation experiments(o ine) Command experiments(online real-time) The simulation experiments consisted of using the already well known models of DC servo motor to analyse the gimbal motor parameters especially as to the time constants, stability of the model,open and closed loop behaviour. This experiment could be carried out in matlab simulink and any of the following models used : 1. Transfer function model 2. State space model 3. Dierence equation model For my work, I have used the dierence equation model in running the simulation only experiments. This enabled me to take into accout all the variables which normally are found in the dynamics 13
of a DC motor but are not always stated in the manufacturers datasheet. These variables are the backlash value, frictional eects of moving gears (because the gimbal actuator motors were gear reduction motors and moving rotor always has static and viscous frictional eects occuring). For the Control real time experiments, the need for a real time hardware platform was required. For this dSPACE real time acquisition card and processing unit was used in conjunction with matlab/simulink. To accomplish the task of controlling the physical actuator system, a good knowledge of the motor design and operational structure is needed. The supplied motors which have already been installed in the desired pyramidal arrangement in previous work done in the Laboratoire Ampere, make use of optical encoders for capturing the motor revolutions. It is this data that is used in the calibrating the angular displacement needed in degrees or radians. Encoder data also gives the measured angular displacement used as feedback to the motor model.
Figure 6 Gimbal DC motor model 4.3 Identi cation of Gimbal Motor parameters The gimbal motors used are from Faulhaber[19] Series 1512, with drive electronics type SC1801 DC gear motors. The following equations are used to manually calculate for the CMG system parameters for the system identi cation exercise : Let θ be the angular displacement of the motor rotor and ω the angular velocity and i the input current to the motor coils. Then from the equations below for a standard DC motor Electrical side :
L
di = Ri − Kω − V dt
(1)
dω = ΣT dt
(2)
Mechanical side :
J
14
where ΣT is the sum of all the torques acting on the rotor of the Motor. They are summarised as
ΣT = Ki − Bω − Tload
(3)
Therefore ,
J
dω = Ki − Bω − Tload dt
(4)
and nally the angular displacement of the rotor is given by :
dθ =ω dt
(5)
After simplifying assumptions ( let x1=θ, x2=i and x3=ω ) we get the state space form of the system as :
˙ 0 0 x1 x2 ˙ = 0 − R L ˙ x3 0 KJm
1 x1 0 − KLe x2 + L1 Vin x3 0 − Jb
(6)
It should be noted here that frictional eects (dry and viscous) have not been taken into consideration as they would naturally introduce non-linearities to the system. Ignoring such system dynamics will be a possible source of errors later on and will be commented on.
4.4 System gains using Transfer function After taking laplace transforms and computing the transfer function for the DC motor , relating the input current to the output velocity , we have the following system transfer function which we refer to as the gain of the motor. 1 W (s) = J I(s) s+
(7)
b J
After taking the laplace transforms of the motor equations and recall that Tm = Kw and Te = Ki we have :
W (s) KM = I(s) Js + B I(s) = V (s)
1 (Ls + R)
15
(8)
(9)
Then the global transfer function of the motor becomes :
W (s) I(s) W (s) ∗ = = I(s) V (s) V (s) Gsys =
KM 1 ∗ (Js + B) (Ls + R)
W (s) KM = 2 V (s) s LJ + (LB + RJ) s + RB + Ke KM
(10)
(11)
The calculation of time constants is made so as to have a good estimate of the possible reponse time of the system when it is excited by a command signal. The calculated time constants follow the equations given below :
La Ra
(12)
Ra × J KE × KM
(13)
τe =
τm =
As seen is a second order transfer function with damping, resistance, inertia and inductance terms. Calculating the gain for controlller parameter identi cation we have the values obtained from the manufacturer and calculated values used in the DC gear motor with intergrated encoder shown in the table below : Parameter
Model values
Units
Nominal Voltage terminal resistance R Back-EMF constant kE Torque constant kM Rotor inductance L Rotor inertia J
12 130 0.749 7.15 2100 9 × 10−9
volt Ω mV/rpm mNm/A µH kgm2
τe τm Jch
1.62 × 10−5 0.22 × 10−6 154 × 10−9
sec sec Kgm2
Manufacturer datasheet
Calculated values
Table 1 Gimbal motor parameters
16
4.5 Controller selection The system identi cation proceeds to identify the parameters of importance to the system response such as : • Maximum error • Static error • Response time or time constant • System delay The method used in identifying the system parameters is the System Identi cation method [3] and the Broida Method as explained in Hamiti et al[2]. In this method, an impulse response is searched for by injecting into the plant (gimbal motor model) a step input and then observing its response. This is then used to specify the model that mapped one to the other. It is not a perfect model but good model information is obtained from such direct experimentation with the plant. This is a form of Hardware In-the Loop (HIL) experiment. From the rst tests performed, the system is seen to be very well a second order system which is overdamped :
Figure 7 Second order response
17
From the test, one consideration was to have as high as possible a gain that brings the system close to overshoot condition. This describes the job of a proportional gain within a controller. From the response curve obtained, Broida's method was used for the system identi cation especially as this method is well detailed for problems of this type with some hidden dynamics and static nonlinearities. A similar problem is solved by this method in a paper by Hamiti et al.[2], even though the problem to be solved was for a pneumatic system. The Broida method oers the advantage of determination of two operating points t1 and t2 without plotting the curves gets rid of the tangent method(Ziegler and Nichols, Borne) which is sometimes di cult to plot on inaccurate curves, as in this case due to non-linearities(friction) and hidden dynamics By using Broida's method we apply the following expressions :
t1 = 0.28 × δy
(14)
t2 = 0.4 × δy
(15)
These times will be used with Broidas' formula shown below (which works well within a chosen linear region of system response curve)
T = 2.8 × t1 − 1.8 × t2
(16)
τ = 5.5 × (t2 − t1)
(17)
And making use of the formulas below in Table 2 : Controllers Type PI
Overshoot = 0 with minimal tresponse Parameters Kp Ti
regulation 0.6 Tτ 4τ
Table 2 ChienHronesRenswick tuning formulas[2] The following values were obtained for the parameter identi cation 3
18
Tracking 0.35 Tτ 1.2T
From
Parameter Using Broida t1 t2 T τ Kp regulation Ti regulation Kp tracking Ti tracking
Values
Units
8.4 12.0 1.92 19.8 0.0582 79.2 0.034 2.304
sec sec sec sec no unit no unit no unit no unit
τps τs delaymax Errormax
0.15 0.55 14 1.46
sec sec msec degree
Direct System Identi cation
Table 3 Synthesized parameters from Identi cation 4.6 Results From the values shown in table3, using the Broida method to compare with the System Identi cation method shows a very signi cant dierence. And using the gains for Kp and Ti obtained from the Broidas method did not work when tuning the PID controller for the gimbal motor in actual experiment. I therefore had to settle for the values obtained from direct experiments with the prototype, while discovering through this process that there were some behaviors of the gimbal motors which are not evident from the data sheet. From the gimbal motor response in open loop, a good choice of possible values for controller gains was made. The controller parameters were chosen to give the controlled system the following properties which improved the functionality of the AOCS : 1. Speed with respect to the electrical and mechanical time constant 2. good tracking ability 3. smoothness and repeatability 4. Stability 5. Precision In the rst instance, using a PI controller was su cient for controlling the gimbal motors in position. The position control is implemented by using the encoder count data from the optical encoders integrated in each gimbal motor. After certain calculations relating sensor (encoder) count data to number of revolutions, a control model was implemented that precisely controlled the gimbal motors in both forward and reverse directions for any angular position command given. For each of the experimental categories earlier stated, the response to a series of step input signals obtained from the closed loop experiments are shown below : 19
Figure 8 Simulation o ine experiments
(a) Reference and command
(b) Reference, measure and error
Figure 9 Simulation o ine experiments 20
The experiments carried out on the actual hardware constituted the most part of the work due to the need to have a working prototype. The level of development of the prototype has reached an advanced stage with work originally started by Velasquez (a Post doc ) and continued by Perion[7]. For the type of experiments needed to test the AOCS system, the prototype must be free to rotate around the principal Z-axes or yaw without having any hinderances. The cause of hindrance being connected cabling and eect of gravity torque. To avoid the negative eects just spoken of, the experiments are previewed to be carried out in a wireless environment. Such wireless communication was ideal as it also describes in better detail the way satellites in orbit use uplink and telemetry data signals for communicating between satellite and ground station. Work on the wireless system has commenced with previous work done by this author[14]. But recent developments has made working with the wireleess system very unpredictable as the signals are intermittent . Being able to resolve this problem and having constant wireless communication established will make the AOCS experiments more realistic. In order not to loose all the time rectifying the wireless system, the experiments carried out were done with cabling between the prototype and RTH. In this con guration, non of the wireless problems were observed and the system worked perfectly. This allowed the work to continue and complete experiments were done with results as would be shown below :
Figure 10 Gimbal DC motor model
21
The results obtained from the closed loop control model while running the control online is given below with xaxis always time and yaxis always angular displacement :
Figure 11 Experiment 1 : simultaneous common reference input of motor 2 and 4
Figure 12 Experiment 1 : simultaneous common reference input of motor 2 and 4 a third experiment using dierent angular displacement commands is shown below just to test the ability of the control to yield the desired response for any given command
22
(a) Motor 2
(b) Motor 4
Figure 13 Experiment 3 : simultaneous common reference input of motor 2 and 4 4.7 Analysis of results The analysis of the results mainly follows from the experiments performed in simulation-only experiments and Control experiments. The metrics used to analyse the results is by comparing the various parameters ( Maximum error, Static error, Response time or time constant, System delay) listed earlier and quantifying the error. Parameters Max error Static error Response time Delay
Simulation only 1.46 deg 0.6875deg 0.163sec 0.014
Real time Control 0.82 deg 0.653 deg 0.13sec 0.01sec
Table 4 Simulation only vs Control
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dierence 0.64 0.0345 0.033 4e−3 sec
5 Automatic Mass Balancing System AMBS 5.1 Introduction The AMBS implemented is meant to compensate for gravity torque eects which cause perturbations to the satellite experimental platform during the AOCS experiments. The immediate eect of this perturbation is the tilting of the platform and subsequent deviation between the Center of Rotation (center of mass) and the Center of Gravity of the platform. To compensate for these torque eects, the mass balancing system was conceived and preliminary designs for the system have been done during this internship to have a working prototype. The set of tasks leading to the nal implementation of the 3 axes AMBS are the following : 1. System identi cation for motor sizing 2. Choosing or selection (from COTS components) of linear motors based on identi ed parameters in the previous step 3. Construction of the working prototype and tests
5.2 Identi cation and Sizing of Linear Motor Identi cation of motor parameters was done using the simulation model previously implemented in the Masters thesis of Perion[7]. Using this model,the parameters chosen for simulated experiments were made to be as close to the experimental values obtained from Perion's work. The values were chosen based on the knowledge of the existing prototype dimensions, the air bearing test bench to be used and the geometric con guration of the proposed AMBS.
5.3 Choice of parameters[7] The choice of parameters is made from studies of an existing mode. The model used for the AMBS is well discussed in Perions'[7] work and forms part of the previous work done at the Laboratory Ampere. A schematic representation of the AMBS is shown below 14 and primarily consists of 3 moving masses linked with the linear motor, arranged in a umbrella con guration. Each motor having 2 degrees of freedom along its railings.
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Figure 14 Schematic of AMBS with test platform The parameters used for dimensioning are : The force acting on the linear motor The maximum linear velocity and acceleration attainable from simulation use of the speedtime or "Velocity pro le plot" to nd accelaration from the acceleration found,we calculate the force fournished by the motors The objective of this study is to dimension the motors in order to satisfy the values obtained from simulation. The experiments to determine these dimensions are discussed next. The selection of linear motors will be made from the results obtained from simulation.
5.4 Simulation experiments
Figure 15 AMBS parameters and dimensions
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The table5 below shows 4 experiments done to get out the parameters used in Linear motor dimensioning Mass parameters mj : 39,Msat : 922 mj : 39,Msat : 922 mj : 20,Msat : 1000 mj : 20,Msat : 1000
β 55 60 55 60
Radius and height 120,40 100,40 120,40 100,40
rsat [-0.4 0.5 2] [1 -1 2] [-0.4 0.5 2] [1 -1 2]
Table 5 Motor parameters Where as shown in the Figure15 :
∗ ∗ ∗ ∗
β is the angle between the umbrella arm and the vertical Radius is the distance between moving mass and center or rotation rsat position vector of the mass h height of mass from horizontal
5.5 Results and analysis The interesting results are those that show the maximum velocity attainable by the linear motors. 1. With the parameters Mmj = 20g , β = 55degr , β = 60degr the maximum speeds are : 19 mm/s et 45.mm/s respectively . 2. With the parameters Mmj = 39g , β = 55degr , β = 60degr the maximum speeds are : 9.2 mm/s et 20.1mm/s respectively From the results,we can conclude naturally that for :the mass Mmj = 20g , the velocity is greater that the velocity for the mass of Mmj = 39g for the same angle of β .
5.6 Acceleration obtained The attainable acceleration used in simultion is bound by the maximum velocity : 1. With the parameters Mmj = 20g , β = 55degr , β = 60degr the maximum acceleration are : 2200mm/s2 et 5500mm/s2 respectively. 2. With the parameters Mmj = 39g , β = 55degr , β = 60degr the maximum acceleration are : 980mm/s2 et 2590mm/s2 respectively
Note : I discover that the acceleration can be obtained also from choosing a velocity pro le which can be triangular, trapezoidal or a shape that satis es the commutation/switching method to be used for actuating the motors. 26
5.7 Calculation of maximum force After the calculation of the maximum velocity and acceleration, the calculation of the maximum force fournishable by the linear motors can be easily computed :
F = mass ∗ acceleration
(18)
5.8 Chosen values The following values are obtained from the simulation. The force is parameterised in Newtons ( Kg∗m ). To make a better and safer choice of the motors, some "Oversizing" is necessary, a factor s2 of 0.20*FC has therefore been added to the calculated force. This is to avoid any problems that arise from changing the mass in the future. Simulator parameters mj : 39,Msat : 922,55,120 mj : 39,Msat : 922,60,100 mj : 20,Msat : 1000,55,120 mj : 20,Msat : 1000,60,100
velocity 9.2mm/s 20.1mm/s 19mm/s 45mm/s
acceleration 980mm/s2 2590mm/s2 2200mm/s2 5500mm/s2
force generated (FG) 0.038 N 0.101 N 0.044 N 0.11 N
FG + 0.20*FG 0.0456 N 0.1212 N 0.0528 N 0.132 N
Table 6 Comparison of LM parameters. Using the values in the last column of Table 6 we can start searching for a lineaire motor of the desired speci cation.
5.9 Motor selection After a careful study of dierent linear motors, the choice for Faulhaber linear actuator was made. Another little but no-less important fact is that the hardware motor controllers already being used on the prototype are all from the same manufacturer (Faulhaber) , thus giving some level of redundancy to the system. Other more speci c criteria used in choosing the linear motors is made from calculation based on the manufacturers data sheet and using standard known formulas : 1. Continuous force at noload :
F emax = Kf × Iemax
(19)
2. force maximum : attainable maximum force linked with maximum current
F pmax = Kf × Ipmax
27
(20)
3. Continous current : necessary current to sustain the continuous force s T125 − T22 × 0.8164 Iemax = R × (1 + α22 (T125 − T22 )) × (Rth1 + 0.45 × Rth2 ) 4. Induced EMF constant :
Ke =
2 × Kf √ 6
5. acceleration at noload :
V emax =
(22)
F emax Mm
(23)
p aemax × Smax
(24)
aemax = 6. velocity at noload :
(21)
5.10 Velocity Pro les Before any determination of the force fournishable by the LM we must rstly choose a velocity pro le which the LM will follow when loaded. For this, three types of velocity pro les are looked at : Triangular pro le Trapezoidal pro le
(a) Triangular pro le
(b) Trapezoidal pro le
Figure 16 Pro l de vitesse hybrid(combination of triangular or trapezoidal) . . .
5.11 Kinematics of the LM for the chosen pro le For the triangular pro le, the calculation of the kinematic parameters use the following equations : 1. Displacement
s= 2. Velocity
1 1 × v × tou × a × t2 2 2
(25)
s a×t v = 2 × ou t 2
(26)
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3. acceleration
a=4×
29
s v2 ou t2 s
(27)
For the trapézoïdal pro le, the calculation of the kinematic parameters use the following equations : 1. Displacement
s= 2. Velocity
2 1 × v × tou × a × t2 3 4, 5
(28)
s a×t v = 1, 5 × ou t 3
(29)
3. acceleration
a = 4.5 ×
v2 s ou2 × t2 s
(30)
5.12 Force calculation The forces acting on and also produced by the LM is shown in the gure 17 below
Figure 17 Forces sur le moteur lineaire Forces considered are obtained from the formulas below :
ΣF = m × a
(31)
Fe − Fext − Ff − Fx = m × a
(32)
Ff = m × g × µ × cos(β)
(33)
30
Fx = m × g × sin(β)
Figure 18 Simulated kinematic from these values, a calculation of force generated (during a period of 10 seconds) is made : 1. For the trapezoidal case (20g mass, 0.020m/s2 acceleration, angle β of 55 degré) Fe =0.18648 N 2. For the triangular (20g mass, 0.016m/s2 acceleration, β of 55 degré) Fe =0.1828 N 3. For the trapezoidal case (39g mass, 0.020m/s2 acceleration, β of 55 degré) Fe =0.18326 N Fe =0.184 N 4. triangular pro le (39g mass, 0.016m/s2 acceleration, β of 55 degré) Fe =0.183 N from these values, a calculation of force generated (during a period of 500 milli sec) is made : 1. For the trapezoidal case (20g mass, 8.1m/s2 acceleration, β of 55 degré) Fe =0.344 N 2. For the trapezoidal case (20g masse, 8.1m/s2 acceleration, β of 60 degré) Fe =0.3513 N 3. For the triangular pro le (20g mass, 6.4m/s2 acceleration, β of55 degré) Fe =0.31048 N 31
(34)
Figure 19 Simulated kinematic [] 4. For the triangular pro le (20g mass, 6.4m/s2 acceleration, β of 60 degré) Fe =0.3173 N 5. For the trapezoidal case (39g mass, 8.1m/s2 acceleration, β of 55 degré) Fe =0.4984 N 6. For the trapezoidal case (39g mass, 8.1m/s2 acceleration, β of 60 degré) Fe =0.5052 N 7. For the triangular pro le (39g mass, 6.4m/s2 acceleration, β of 55 degré) Fe =0.432 N 8. For the triangular pro le (39g mass, 6.4m/s2 acceleration, β of 60 degré) Fe =0.4389 N
5.13 Discussion The experiments on the AMBS have not yet been performed because of the problems posed by the communication system as highlighted earlier. This will be done in future work in order to carry out some experiments to validate the ACS. For the motor selection process, reference was made to several datasheets from linear motor manufacturers, and then a choice was made guided by : 1. the displacement time of the masses (mass, velocity, acceleration) 32
2. the velocity pro le desired 3. stroke lenght of the linear motor During simulation, the time plays a strong role in the determination of the maximum force attainable and the best choice for choosing the time is made from simulation experiments. Because of the problems encountered with the wireless communication system, the actual implementation of a mass balancing prototype is still pending. Not withstanding, work continued on the control aspect of the AOCS actuators.This has been well detailed in the previous and next chapters of this report. Also all the experiments carried out on the actuator system made use of only 2 out of 4 CMGs. For normal 3 axes control as stated previously, we need at least 3 CMGs. But from studies done by Kurokawa[5], Ford and Hall[18], Joon and Tsiostras[17], it is established that 4 CMGs will perfrom the task of Attitude control if a robust enough control law that avoids singular states is implemented. Because of this fact that singular states can be encountered and our use of 4 MCMGs we had to do some studies of the steering (velocity) laws which will be used to produce the needed body torques to the satellite and avoid the uncontrollable singular states mentioned. This is the subject of the next chapter.
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6 Singular conditions & Singularity Avoidance Haven described the challenges faced with implementing the AMBS, the project continued with the study of the Singular states found to occur when certain combination of gimbal angle commands are given. This is especially important as for this internship, the model put in place for the dynamics of the CMG gimbal motor uses angular reference values as command. To understand the global nature of the control law to be put in place, the next section gives a schematic overview of the various subsystems and their interrelationship. Especially as it relates to the various subsystem modules and the inputoutput signals involved with each subsystem.
6.1 Software In the Loop (SIL)
Figure 20 Global AOCS schematic : conf. gure5 The experiments made here follows the AOCS schematic diagram earlier shown in Figure5 and repeated here for clarity. The aim is to synthesize the Outer (A) and inner (C) loops which deal with
◦ Global Vehicle control law (outer loop) [7][17] ◦ CMG steering law for singularity avoidance [17][18] (loop C in gure 20) The control laws implemented here are also dierent for the two loops just described ; ◦ Global Vehicle control law (outer loop) uses a stabilizing control law requiring the synthesis of a robust dynamic state feedback controller ◦ CMG steering law for SDA uses a lyapunov based SDA control
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Below is a standard model of the SIL system implemented in matlab and a description of the model subsystems :
Figure 21 SIL model implemented in Simulink From the gure 21 : ◦ The blue block is the stabilizing state feedback controller. ◦ The yellow block is the CMG steering logic and actuator system.
◦ The magenta block is the spacecraft system.
6.2 Singularity control & steering laws From literature [5][17][18] we nd the existence of several Singularity control and steering laws used in spacecraft attitude control. The distinguishing factor between all these laws remains the adaptability to dierent attitude control systems. By dierent attitude control systems is meant, reaction wheel actuated systems, CMG actuated systems, the con guration of these actuators, wheather single gimbal, double gimbal, rooftop arrangement, pyramidal formation e.t.c. For the most part, the study of pyramidal CMG con guartion and the singular states it generates for some gimbal angle combinations has not been widely researched inspite of the bene ts which such con guartion gives in terms of fewer numbers, reduced mass, low energy, redundancy, compactness. In [17] a comparison of various CMG steering laws is made. Amongst the laws which are known to be used, three which are in common use are compared. These are :
∗ Singularity Robust(SR) steering logic ∗ Singular Direction Avoidance(SDA) ∗ O Diagonal Singularity Robust(oDSR) steering logic In all of these methods, for singularity avoidance, a generalised inverse is used. Where this generali35
sed inverse is obtained from the dynamics of the actuator system(its usually a matrix) relating the gimbal velocities and produced torques. It should be recalled that the produced gimbal torques are in turn transfered(by the process of Coriolis) to the body of the satellite(forming the satellite body torques) which is used to regulate the attitude of the satellite. For this work, after studying the three steering laws, a choice was made to implement the SDA steering law because of the following reasons :
◦ less computational eort ◦ produces nite gimbal rates ◦ smaller torque error in comparison to SR steering law The form of the controller described by the SDA steering law is is obtained either from the generalised inverse or the SVD of the matrix which relates the gimbal input commands (angle or velocity) to the generated 3-axes satellite torque. The method for controller synthesis used in avoiding the singularities is described below :
D = U SV T
(35)
DT = V SU T
(36)
and for the pseudoinverse of D we make the computation
Spseudo = S T SS T + αI
−1
(37)
where I is an identity matrix IRn×n and we have the pseudoinverse of D as D‡
D = V Spseudo U T
(38)
and for the SDAinverse of D we make the following computation in designing a control law to avoid the singularity : ‡ SSDA = diag(
1 1 S33 , , 2 ) S11 S22 S33 + α
(39)
and nally we have the SDA pseudoinverse of D as D‡ ‡ D‡ = V SSDA UT
the proposed controller is of the form : 36
(40)
σ˙ = D‡ × Lr
(41)
6.3 Simulation and results The results for the SIL simulation experiments are shown below : Also shown is the schematic of a Singular momentum envelope and singular points generated by the 3 axes torques plotted against the changing gimbal angles commands. What this plot shows is the 3D geometry Torque points.
Figure 22 Singular points generated by 3 axes torques And the experiments for the SDA have the following results : A brief comment on these results must be made here because the implementation of the SDA control law has been made from literature[18][17] with a lot of uncertainty regarding some of the parameters used. The results shown here have been made from correctly implementaing the global AOCS model described in gure5.
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Figure 23 Generated by 3 axes torques
Figure 24 Platform velocity response Although the results are not intuitive, work is continuing on this part to correctly reproduce the simulation results as seen in literature[5][18][17]. This will require ne tuning the parameter values used for the simulation.
38
Figure 25 Platform angular response
7 Conclusion This report has looked at several aspects in the practical implementation of of the AOCS test bench which is to be installed at the Laboratoire Ampere. It has covered a wide range of practical subjects and used tools needed which with a control systems engineer should be conversant with. It is particularly interesting because the Masters thesis for which I am registered is a research masters and I was priviledged to have worked in a purely research based subject, in a laboratory equipped with the tools to get the job done. In the next section, I proceed to highlight some of the and also state the speci c challenges encountered.
achievements gained
during this project
7.1 Achievements and progress made Duringf this internship work, the goals originally set have changed somewhat due to the fact that for such practical implementation of systems, unforseen challenges can not be ruled out. Nevertheless, A number of very distinct advances have been made in studyiong the system and adding to the already existing work. The goals achieved so far are listed below :
∗ Identi cation of the actuator system using standard System identi cation method ∗ Detailed simulation models in matlab/simulink to obtain plots and reference response ∗ Detailed control/command models in simulink/dSPACE for experiments on prototype with results plotted ∗ Sizing of linear motors for use in the AMBS ∗ Theoretical studies of the nature of singularity eects on the desired command ∗ simulation and visualisation of singular envelope in simulink. 39
The issues or
challenges which were encountered are :
∗ Wireless communication system which operates intermitently, therefore preventing robust study of the control laws implemented using the air bearing system. ∗ The implementation of the complete simulation model for SDA steering law which has been completed in simulink needs a lot of ne tuning on the parameters before being validated.
7.2 Future improvement It is previewed that even though the internship is ended, work is going to continue on the wireless system of communication and the SDA steering law. Also a manual of various connections made during this internship will be put in place to help future participants in the project have a good reference to work with.
7.3 Thanks My thanks go to everyone who has been instrumental in making this project come this far. I am skeptical to call it a success as there is still a lot of work to be done and also because it is better the credit comes from my supervisors. But I can be sure when I say I have learnt a lot in terms of practical engineering than at any other time in my life and for this I will be eternally grateful to Carlos Velasquez, PierreAlexandre, who are two previous students who did a lot of work on this project. I cant also forget Said with whom talking, I learned a lot of new ways to approach the problems I faced. I am particularly grateful to the Professors with whom I had the opportunity to speak with and gain from their very unique insight into the problems I faced. I want to thank, Jean-Pierre Simon for his uncommon enthusiasm, Xavier Brun for pointing out the "later obvious" to an ignorant student(myself), Eric Bideaux for his comments, Arnaud Leleve for his advice and nally the two individuals who have been my constant and ever ready resource for knowledge and guidance in the persons of my Supervisors(encadrants) Damien Eberard and Richard Moreau. Thank you for everything, especially making me better at asking the right questions.
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7.4 Resume en francais
41
A Wireless connection(Realtime Hardware side) The sub D(3B) cable handles the wireless connection of the RTH side. From the wireless module(WM) the pins 18 and 19 are used for Transmission and reception respectively.These two pins interface to the dSPACE board throuigh the pins 13 and 14 of the max232 transistor IC(supplies the logic for driving the wireless module)
B Wireless connection(Prototype side) On the satellite side,the encoder counts which are retrieved by the FPGA RS232 module are sent to the wireless module through the max232 IC for this slave WM.The pin connections are as follows : The RS232 is a female connector having pins 2 and 3 for transmission and reception respectively.These two pins are connected to the pins 14(receive) and 13(transmit) respectively on the max232 IC.
C Direct connection During experiments, when making a direct connection between the prototype and the RTH. The following connection scheme can be used but not mandatorily so. The sub-D connector 1A of dSPACE is used.The pins 25 and 27 of the connector 1A is connected to the unsoll (speed control) and dir (direction control) inputs of the motor controller (mot-cntrl-n) on the prorotype.The pins 42 and 10 of the connector 1A were connected to the speed control and direction control inputs of the second motor controller.
D Motor controller hardware The gimbal motor 2 appears not to be functioning properly as the velocity response is null giving constant velocity all the time for whatever voltage input to the motor windings.Thgis fault is not a function of the motor but its controller. The gimbal motor 4 responds correctly to the voltage input and gives a correct velocity read out.Its response is consistent with proper operation both in forward and reverse motion.
E Unit conversion 10mm=1cm=0.01m
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Références [1] Takeya SHIMA , Mitsunori SAITO , Kazuhiko FUKUSHIMA , Katsuhiko YAMADA, Automatic Balancing for a Three Axis Spacecraft Simulator. Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan, Department of Aerospace Engineering, Nagoya University, Nagoya, Japan, 2009. [2] K. Hamiti, A. Voda-Besanqon and H. Roux-Buisson, POSITION CONTROL OF A PNEUMATIC ACTUATOR UNDER THE INFLUENCE OF STICTION. UJFG-LIME, B.P. 53, 38041 Grenoble Cedex, France, Laboratoire d'Automatique de Grenoble, ENSIEG, B.P. 46, 38402 Saint Martin d'Heres, France, 1996. [3] Kevin M. Passino and Nicanor Quijano, Modeling and System Identi cation for a DC Servo. Dept. Electrical Engineering, The Ohio State University, 2004. [4] J. Prado., G. Bisiacchi., L. Reyes., E. Vicente., F. Contreras., M. Mesinas., and A. Juárez. THREE-AXIS AIR BEARING BASED PLATFORM FOR SMALL SATELLITE ATTITUDE DETERMINATION AND CONTROL SIMULATION. Instituto de Ingeniería, UNAM, Cd. Universitaria Coyoacán, 04510, México D.F. México, 2005. [5] Haruhisa Kurokawa, A Geometric Study of SGCMG : Singularity problems and Steering laws. Report of Mechanical Engineering laboratory, No. 175, p. 108, 1998. [6] Jae-Jun Kim and Brij N. Agrawal. System Identi cation and Automatic Mass Balancing of Ground-Based Three-Axis Spacecraft Simulator. Department of Mechanical and Astronautical Engineering, Naval Post Graduate School Monterey CA 93943, 2006. [7] Pierre-Alexandre Perion, Instrumentation et commande d'une pico-satellite. Rapport Master, Ampere UMR 5005 LYON (FRANCE), 2010. [8] V.J. Lappas,W.H. Steyn, C.I. Underwood, Practical Results on the Development of a Control Moment Gyro based Attitude Control System for Agile Small Satellites. Surrey space center, University of Surrey, Guilford, Surrey GU2 5XH, UK University of Stellenbosch, Stellenbosch , 7602, South Africa [9] B.T. Costic,D.M. Dawson, M.S. De Quieroz and V. Kapila, A Quaternion-Based Adaptive Attitude Tracking Controller Without Velocity Measurements. 43
Department of Electrical and Computer engineering, Clemson university, Department of Mechanical Engineering, Lousiana state university,Baton Rouge ; Department of Mechanical Engineering, Polytechnic New york 2007 [10] Jingrui Zhang, Output torque estimation of MCMG's for agile satellites. The Chinese Society of Theoretical and Applied Mechanics, 2009. [11] W. MacKunis,K. Dupree,N. Fitz-Coy and W.E. Dixon, Adaptive satellite attitude control in presence of Inertia and gimbal uncertainties. The Journal of the Astronautical Sciences, Vol. 56, No. 1, January March 2008, pp. 121134, 2008. [12] Jacques Busseuil,Michel Libre and Xavier Rozer, High Precision Mini CMGs and their Spacecraft Applications. Aerospatiale France AS 98-006. 1998 [13] Nicolas PRALY, Contrôle d'attitude trois axes d'un Pico-satellite. Rapport Master, Ampere UMR 5005 LYON (FRANCE), 2009. [14] Ovie EseOghene, Development of a Benchmark for an AOCS for CubeSat. Report Master 1,International Masters in Embedded Systems and medical Image engineering (IMESI) INSA de Lyon June, 2011. [15] David M. Meissner, A Three Degrees of Freedom Testbed for Nanosatellite and Cubesat Attitude Dynamics, Determination and Control. December 2009. [16] Pedro Tavares, Paulo Tabuada, Pedro Lima, PROJECT CONSAT : CONTROL OF SMALL SATELLITES. Instituto de Sistemas e Robótica, pólo do Instituto Superior Técnico June, 1998. [17] Dongwon Jung and Panagiotis Tsiotras, An Experimental Comparison of CMG Steering Control Laws, Georgia Institute of Technology, Atlanta, GA 30332-0150 AIAA circa 2004. [18] Kevin A. Ford and Christopher D. Hall, Singular Direction Avoidance Steering for Control-Moment Gyros, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, Dayton, 45433 Journal of Guidance, control and dynamics 2000. [19] Dr. Fritz Faulhaber, Technical data Sheet for SC1801, GmbH & Co. KGDaimlerstr. 23 / 25 . 71101 Schonaich Technical information data, Aug. 11 2009. 44
[20] Lamport, L., LaTeX : A Documentation Preparation System User's Guide and Reference Manual, Addison-Wesley Pub Co., 2nd edition, August 1994.
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