19 th Annual AIAA/USU Conference on Small , Logan, Utah, USA, August 8-11, 2005 SSC05-VII-6 Session VII – Frank J. Redd Student Scholarship Competition

The Attitude Control System Concept for the Joint Australian Engineering Micro- (JAESat)

Aaron Dando * Advisor: Werner Enderle † Cooperative Research Center for Satellite Systems Queensland University of Technology GPO Box 2434, Brisbane, QLD, Australia, 4001

JAESat is a joint micro-satellite project between Queensland University of Technology (QUT), Australian Institute (ASRI) and other national and international partners including the Australian Cooperative Research Centre for Satellite Systems (CRCSS), Kayser-Threde GmbH, Concepts and Auspace who will contribute to this project. The JAESat micro-satellite project is an educational and GNSS technology demonstration mission. The main objectives of the JAESat mission are the design and development of a micro-satellite in order to educate and train students and also to generate a platform in space for technology demonstration and conduction of research on a low-cost basis. The main payload on-board JAESat will be a GPS receiver called SPARx (SPace Applications Receiver), developed by the Queensland University of Technology for attitude and orbit determination. In addition to the GPS based attitude sensor, a star sensor will be on-board JAESat for attitude determination. JAESat will be three-axis stabilized based on a zero-momentum approach using magnetic coil actuators. This paper will outline the Attitude Control System (ACS) concept for JAESat including: subsystem configuration and components, performance requirements, control mode definition, attitude dynamic modeling, control law development, and attitude determination concept. Performance of the JAESat ACS is predicted via simulations using a comprehensive ACS model developed in Matlab Simulink.

I. Introduction The JAESat micro-satellite project is an educational The JAESat mission will ultimately consist of two and Global Navigation Satellite Systems (GNSS) micro-satellites (master and slave) flying in formation technology demonstration mission, which will also (see Figure 1). The satellites will be coupled during the generate data for scientific use. The high level mission launch phase and then separated in space by a spring objectives 1,2 of JAESat are: mechanism. Following the separation the two satellites will drift away from each other with a low drift rate (~ • Design, develop, manufacture, test, launch and 0.01 m/s). A communication link between the two operate the educational/research micro-satellite satellites will be established in form of a RF Inter- JAESat Satellite Link (ISL). The master satellite will be a cube • Develop payloads with a technological and with side length 390 mm and approximate mass 25 kg. scientific relevance The slave satellite will be half the height of the master • Use JAESat as a sensor in space and GNSS satellite with dimensions 390 mm x 390 mm x 195 mm technology demonstrator mission and approximate mass 15 kg.

As can be seen from the high level mission objectives, Negotiations with prospective launch providers for a the education and training aspects play an important piggyback launch are ongoing and consequently the role in the JAESat mission. The GNSS mission final orbit of JAESat has not yet been established. objective is driven by the SPARx (Space Applications However, it is intended that JAESat will be placed into GPS Receiver), a development by QUT/CRCSS. a circular, near-polar sun-synchronous Low Earth Orbit Functions and performance of SPARx will be tested (LEO) with an orbit altitude between 600 km and 800 and validated in space within the JAESat mission. km. The operational life time of JAESat is expected to Another key element of the high level mission be approximately 12 months. Mission operations will objectives is the development and testing of a novel be conducted from a located at the integrated attitude sensor concept combining star Queensland University of Technology in Brisbane, sensor and GPS attitude sensor data. Australia. JAESat will be designed to possess a high ______degree of on-board autonomy and to conduct a variety * PhD Candidate, CRCSS at the Queensland University of of experiments based on the mode of interoperation Technology. Member IEEE. [email protected]. between the payloads on-board the two satellites. †Associate Professor, CRCSS at the Queensland University of Technology. Member IEEE. [email protected].

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Figure 1: Separation of master satellite (left) and slave satellite (right) in space

The main experiments 3,4 for JAESat include: are still coupled, the requirements for the master satellite specified in Table 1 may be used. The orbit- • Testing and evaluation of the QUT/CRCSS average power consumption requirement for the master SPARx, including attitude capability ACS in each operational mode is TBD. • Testing of a new integrated star sensor/GPS sensor concept for three-axis attitude Accuracy Jitter Settling Time determination MASTER 5 deg (per axis) TBD 6 orbits (per • Relative navigation between JAESat master and axis) slave satellites • SLAVE 10 deg (roll, pitch) TBD N/A Orbit determination concepts: 15 deg (yaw) o On-ground precise orbit determination based

on GPS code and carrier phase measurements Table 1: Minimum pointing requirements (3 σ ) with baseline ACS o On-board orbit determination based on GPS components receiver position solutions • Relative positioning between master and slave III. Attitude Control System Concept satellites The key driver for the Attitude Control System (ACS) • Establishment of stable RF inter-satellite link concept is the establishment of an ISL so that • Atmospheric research housekeeping and scientific data can be transferred between the two satellites. Communication with the SLAVE ground station (uplink satellite commands and (Gravity-Gradient downlink telemetry) will be via the master satellite Stabilised) only. The communications concept for the JAESat mission is illustrated in Figure 2. To meet the important ISL mission requirement an innovative ACS concept is proposed comprising a three-axis stabilized master satellite and gravity-gradient stabilized slave satellite in ISL Uplink / Downlink formation flight. This concept aims to reduce the cost and complexity of future formation flying missions. In MASTER addition, system modularity will be a key feature of the (Three-Axis Stabilised) low-cost master ACS design to accommodate modifications and improvements for different micro-

satellite missions conducted in LEO. Figure 2: Inter-satellite and ground station communications concept for the JAESat mission Master Satellite The master satellite will be three-axis stabilized using a II. ACS Requirements zero-momentum approach. The baseline ACS will A preliminary set of ACS performance requirements provide three-axis attitude determination and control has been derived for the Normal Mode of operations for large-angle tracking/slewing maneuvers and also for (see Table 1), driven primarily by the mission fine pointing. A block diagram of the master ACS is requirement to establish an ISL. The Detumbling Mode depicted in Figure 3. Component selection was based requirement is for the coupled satellite angular velocity on total system cost, requirements (functional and to be damped to less than 0.2 deg/sec (per axis). During performance), availability, and system compatibility. the Reorientation Mode in which the master and slave

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Sensor s Target Motion Actuators Time & Orbit Quaternion Propagator & Magnetic Field Model

GPS Attitude Sensor

Sensor Attitude Control Law Magnetic Selection Determination Coils Star Sensor

ACS Input ACS Output (Commands) (Telemetry)

Figure 3: ACS block diagram for the JAESat master satellite

The actuator complement consists of three magnetic unobstructed half sphere field-of-view. The ground link coil actuators with air cores (air coils) whose magnetic and inter-satellite link antennas will be located on the – moment vectors are aligned with the satellite body z and +x faces respectively. During the Normal Mode axes. Three additional coils may be implemented to the +z body axis will point in the zenith direction provide a certain level of redundancy in the ACS allowing for the best possible visibility of the GPS design. The air coils are to be designed and satellite constellation and reference stars so that a manufactured at QUT. Preliminary design parameters continuous attitude solution will be available. for the air coils using SWG-18 copper wire are summarized in Table 2. A KM-1303 star sensor 8 has been contributed by Kayser-Threde GmbH for use in the JAESat project MASS 0.57 kg (see Table 3). The major operational limitation of star 2 COIL AREA 0.15 m sensors is their sensitivity to large rotation rates (> 5 NO. OF TURNS 50

deg/sec) leading to star identification difficulties and RESISTIVITY OF COPPER WIRE 1.7e -8 m WIRE CROSS-SECTIONAL AREA 0.823e -6 m 2 consequently an erroneous attitude solution. However, COIL RESISTANCE 1.6 by using additional external information from the GPS SATURATION MOMENT 10 Am2 attitude sensor, the probability of correctly identifying MAX. COIL CURRENT 1.3 A star patterns will improve substantially. MAX. POWER CONSUMPTION 2.8 W AT SATURATION (COIL ONLY) DIMENSIONS Star sensor body: 112 × 115 × 45 mm

Table 2: Preliminary design parameters for a single air coil Height: 170 mm (with baffle) MASS Sensor unit: 0.58 kg 16 mm lens: 0.1 kg The baseline sensor complement consists of an Baffle: ca. 0.1 kg integrated attitude sensor comprising a star sensor and POWER Power consumption max.: 5 W multi-antenna GPS sensor. This concept has been Power consumption previously investigated 5-7 by the German Space typical at 12 VDC: 4.2 W Input voltage range: 12 - 15 VDC Operations Center (GSOC) for three-axis attitude Connector type: DSUB 9 pin determination. It aims to overcome many of the STAR SENSOR Field of view: 21° × 31° performance and functional limitations of conventional PERFORMANCE Sensitivity: MV = +6 to -2 attitude sensors whilst also providing critical navigation Update period: 250 ms data and precise timing. The boresight axis of the star Star acquisition time: 0.5 s (first acquisition) Accuracy: ± 0.02° (2 Sigma) sensor will be aligned with the +z body axis. The four GPS antennas will be located at the corners of the +z Table 3: Key characteristics and specifications of the Kayser-Threde face of the master (see Figure 10), thus providing an KM-1301 star sensor

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The GPS based three-axis attitude sensor will be mode the master will test and validate the novel attitude designed and manufactured at QUT. It will use the low- sensor concept for three-axis attitude determination. cost, lightweight SPARx technology which is based on the MITEL GP2021, GP2015, and GP2010 chip set and ORIGIN Master satellite mass center is a modification of the MITEL Orion9 GPS receiver X-AXIS Completes right-hand orthogonal set demonstrator. Each GPS SPARx unit is 95 mm x 50 Y-AXIS Direction of the specific angular momentum mm x 50 mm with 12 parallel channels and has a power vector or orbit plane normal ( rˆ × vˆ ) consumption of approximately 2W (including active Z-AXIS Direction of satellite position vector or antenna). zenith ( rˆ )

Table 5: Definition of reference frame 2 All ACS processing for the master will be performed by the on-board flight computer, a low-cost Slave Master commercial-off-the-shelf Intrinsyc ® Cerf™ Board. Specific features include a 192 MHz Intel™

Strongarm™ CPU, 16 MB flash memory, 32 MB ZR SDRAM, and three RS232 serial ports.

Slave Satellite YR The slave satellite will be gravity-gradient stabilised Star sensor & GPS antennas on –X face without using a deployable boom or libration damper. R XR Hence the satellite inertia matrix will need to be designed to ensure gravity-gradient stabilization within Earth the requirements specified in Table 1. Z R IV. ACS Operational Modes Detumbling Mode : This mode will use a simple B-dot Slave Master ISL control law to damp the angular rates of the coupled satellite (master and slave) relative to the earth’s XR magnetic field.

Star sensor & GPS antennas on +Z face Reorientation Mode : This mode will use a star sensor YR R based inertial attitude solution and a tracking maneuver control law to reorientate the coupled satellite so that it Earth tracks reference frame 1 (defined in Table 4), within the accuracy specified in Table 1. Once the tracking Figure 4: Reorientation and Normal ACS operational modes maneuver has been completed the slave satellite will be correctly orientated for gravity-gradient stabilisation V. Attitude Modeling and the two satellites will then be separated. This section will present (without derivation) the dynamic and kinematic equations of motion for the ORIGIN Master satellite mass center three-axis stabilised JAESat master satellite. Both X-AXIS − ˆ Toward the center of the earth or nadir ( r ) attitude regulation and attitude tracking will be Y-AXIS Direction of specific angular momentum considered. All vectors are expressed in the master vector or orbit plane normal ( rˆ × vˆ ) satellite body frame (origin at satellite mass center). Z-AXIS Completes right-hand orthogonal set

Table 4: Definition of reference frame 1 Kinematic Equations of Motion For attitude regulation the objective is to stabilise the Normal Mode : This is the primary mode for the satellite body frame with respect to an inertial (non- JAESat mission during which a number of experiments rotating) reference frame. For this case the kinematic will be conducted (see section I). Following the equations of motion are given by:

separation an ISL will be established between the two Ξ satellites. In this mode the slave will be gravity- 1 ω q& = (q) (1) gradient stabilised and the master will be three-axis 2 stabilised tracking reference frame 2 (defined in Table 5), within the accuracy specified in Table 1. The master [ ×]+ Ξ  q q I ×  will use the same control and attitude determination (q) = 13 4 3 3 (2)  − T  algorithms as the Reorientation Mode. Also during this  q13 

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ω

δ δ δ where is the angular velocity of the body frame Ξ 1 ω q& = ( q) (10) relative to the inertial frame, Inxn is an n x n identity 2 matrix, [ ×⋅ ] denotes the vector cross product operator, and q is a quaternion describing the orientation of the Dynamic Equations of Motion body frame relative to the inertial frame. The attitude The (rigid body) dynamic equations of motion for the quaternion is defined as: JAESat master satellite are given by:

& = − −1 [ ×] + −1 + −1 ω ω q13  J J ω J Tgg J T (11) q =   (3)  q 4 

where ω is the angular velocity of the body frame relative to the inertial frame, J is the satellite inertia   q1 Φ     matrix, T is the external control torque generated by the q = q = nˆ sin   (4a) magnetic coils, and T is the gravity-gradient 13  2   2  gg   disturbance torque. The control torque is defined by: q 3 

= [ ×] Φ T m B (12) =   q 4 cos   (4b)  2  where m is the magnetic moment generated by the magnetic coils, and B is the magnetic flux density of where nˆ is a unit vector in the direction of Euler axis the earth’s magnetic field at the satellite location (also

of rotation and Φ is the angle of rotation. Since the called geomagnetic field vector). The earth’s main quaternion is a non-minimal parameterisation of the magnetic field is modeled using a 10 th order IGRF attitude an additional parameter is required which is model with extrapolated J2005 coefficients. Each related through the constraint equation: magnetic coil produces a magnetic moment according to: T = T + 2 = q q q13 q13 q 4 1 (5) m = NAI (13)

For attitude tracking the objective is to stabilise the satellite body frame with respect to a rotating reference where N is the number of turns of wire, A is the area frame defined in terms of a desired quaternion q and formed by the coil, and I is the coil current. The d gravity-gradient torque is given by: angular velocity ω d. The error quaternion describing the orientation of the body frame relative to the rotating = ω2 [ ×] reference frame is defined as: Tgg 3 0 rˆ rJˆ (14)

δ = ⊗ −1 q q qd (6) where rˆ is a unit vector in the zenith direction, and ω 0 is the orbit angular velocity. For the case of attitude where the ⊗ operator denotes quaternion regulation Eq (11) can be used directly. For attitude multiplication 10 . This may also be expressed as: tracking the dynamic equations of motion are obtained

by differentiating Eq (9) and substituting the result into δ q = Ψ (q)q (7) Eq (11) which leads to the following expression: d

δ

& = − −1[ ×] − & + −1 + −1 ω ω ω ω

Ξ J J J T J T (15) − T  d gg

ψ (q) (q) = (8)  T   q  VI. Attitude Control Laws This section presents the control laws that will be The error angular velocity describing the angular implemented in the master ACS during each mode of velocity of the body frame relative to the rotating operation. For each control law (including derivations) reference frame is defined as: all vectors are expressed in the master satellite body

frame unless otherwise specified. Performance of the

δ

ω ω = − ω d (9) closed-loop system is simulated using a comprehensive ® ACS model developed in Matlab Simulink. For all Hence for attitude tracking the kinematic equations of simulations perfect attitude knowledge has been motion are given by: assumed, i.e. attitude determination using ideal sensors

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FEDSAT 1 27598U 02056B 05122.26089911 -.00000001 00000-0 17045-4 0 6439 2 27598 98.5672 196.2324 0009070 346.7664 13.3272 14.27886601124186

Table 6: Two-Line Element Set for FedSat

with zero noise, infinite bandwidth, and zero 1.4 misalignment. The satellite orbit state vector information is based on the SGP4 algorithm using 1.2

NORAD Two-Line Elements (TLE) for the 800km 1 sun-synchronous orbit of FedSat (see Table 6). 0.8

Detumbling Mode 0.6 Three-axis magnetic moment commands for the [Watts] Power magnetic coils are generated using a simple B-dot 0.4 11 control law : 0.2

0 = − & 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 m KB (16) Simulation Time [sec] where K is a positive definite diagonal gain matrix and Figure 5: Satellite angular rates and total magnetic coil power & consumption during Detumbling Mode B is the time derivative of the body frame components of the geomagnetic field vector. Reorientation Mode Significant research efforts have been made to develop Parameter Value Units 2 closed-loop control laws for large-angle attitude Inertia Matrix  8.1 0 0  kgm 12-18 =   J  0 0.2 0  tracking maneuvers. The control law to be  0 0 0.1  implemented in the JAESat master satellite was φ θ ψ = [ ] T Initial Attitude ( 0 , 0 , 0 ) 0.0 0.0 0.0 rad developed by Crassidis, Vadali, and Markley for the

(1-2-3 Euler Angles) NASA Microwave Anisotropy Probe (MAP) ω δ = [ ] T Initial Angular Rates 0 .0 052 .0 052 .0 052 rad/sec 17,18 2 mission. It is based on a variable-structure (sliding  e5.2 6 0 0  Gain Matrix   Am s/T K =  0 e5.2 6 0  mode) control approach with optimal switching  6   0 0 e5.2  surfaces and is an extension of previous research 19 Simulation Duration 18000 sec conducted by Vadali on large-angle slew maneuvers. The principle advantage of sliding mode control is its Table 7: Parameters for Detumbling Mode simulation robustness with respect to satellite modeling uncertainties and unexpected disturbance torques. Figure 5 shows that the coupled satellite angular velocities are damped to less than 0.2 deg/sec (with To obtain the optimal switching surfaces a control law

respect to reference frame 1) in approximately one ω of the form ω = (q) is sought which minimizes the orbit, which is compliant with the requirements following performance index: specified in section II. The values in Table 7 were selected by trading-off system settling time for

 t 

δ δ δ δ Π

magnetic coil power consumption. = 1 ()ρ T + T ω  ω  lim t→∞ ∫ q13 q13 dt (17) 2   t s  5 wx wy 4 ρ wz subject to Eq (1), where is a scalar gain and ts is the 3 time of arrival at the sliding manifold. The Hamiltonian 2 associated with minimising Eq (17) is defined as:

1

δ δ δ δ 0 λ

= 1 ρ T + 1 T + T & ω ω

Rates [deg/sec] Rates H q13 q13 q (18) -1 2 2

-2 λ

-3 where is the costate vector associated with q.

-4 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Simulation Time [sec]

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δ δ The necessary conditions for optimality according to 1 δ q& = ksgn []q (1− q 2 ) (23b) Pontryagin’s Minimum Principle are: 4 2 4 4

∂ & = H The trajectory in the state-space that slides on the q λ (19a) ∂ sliding manifold can be shown to be asymptotically

λ & ∂H stable using Lyapunov’s Direct Method. The following = − (19b) ∂q candidate Lyapunov function is proposed:

∂H δ

= δ 0 ω (19c) = 1 T ∂ V q13 q13 (24) 2

Using Eqs (1), (7), (8), (9), and (18) the solution of Eqs Substituting Eq (23a) into the derivative of Eq (24) (19a)-(19c) leads to the following two-point boundary- leads to the following expression: value problem:

δ δ

& 1 T δ ω Ξ = − & 1 V k q4 q13 q13 (25) q = (q) (20a) 2 2

ω

λ λ

Ξ Ξ Ξ & = − ρ T + 1 (qd ) (qd )q ( ) (20b) which is clearly negative definite provided k > 0.

2

ω ω λ = − 1 Ξ T + A control law is required so that the closed-loop system (q) d (20c) 2 can asymptotically track a desired quaternion qd and ω 19,20 angular velocity d. The equivalent control method It can be shown based on analysis that the following is used to develop a control law that induces ideal choice for the sliding manifold minimizes Eqs (17) and sliding based on external control torque inputs. It will (18): be subsequently proven using a Lyapunov stability

analysis that the same control law can also be used to

ω δ = + δ = s kq13 0 (21) force the state trajectory onto the sliding manifold. The dynamic and kinematic equations of motion for the = ± attitude tracking case, given by Eqs (10) and (15), can provided that k ρ where k is a scalar gain. It is be expressed as a system of n equations, linear in the m

critical to note that for the quaternion parameterisation controls: δ of the attitude δ q and – q represent the same orientation, although the former gives the shortest x& = f (x) + BT (26) distance to the sliding manifold whilst the later gives the longest distance. The sliding manifold defined by where Eq (21) must therefore be modified to ensure that the attitude tracking maneuver follows the shortest possible δ  q

= ω path to the sliding manifold (and also to the reference x δ  (27) trajectory):  

ω

δ

δ δ = + [ ] = δ

s ksgnq4 t( s ) q13 0 (22)  q& 

ω ω = ω f (x) − −1[]× + −1 − &  (28)  J J J Tgg d 

It can be shown based on analysis and simulation that δ δ sgn[ q4(t s)] may be replaced with sgn[ q4(t)] (denoted δ 0  herein by sgn[ q4]) which also produces a maneuver to = 4x3 B  −1  (29) the reference trajectory in the shortest possible  J  distance. Substituting Eq (22) into the derivative of Eq (7) leads to the following kinematic equations for During ideal sliding on the sliding manifold defined by “ideal sliding” on the sliding manifold: s = 0, s& can be set to zero:

δ δ & 1 δ & & q = − k q q s = Px = fP (x) + PB Teq = 0 (30) 13 2 4 13

ω

δ δ + 1 [][]× ()− δ q13 2 d ksgn q 4 q13 (23a) where Teq is the equivalent control torque and P is a m 2 x n Jacobian matrix defined by:

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∂ ∂ can be assessed using Lyapunov’s Direct Method. The  s1 s1   L  ∂x ∂x following candidate Lyapunov function is proposed: ∂s  1 7  P = =  M O M  (31) ∂x ∂ ∂ 1  s3 L s3  V = sTs (38) ∂ ∂  2 x1 x7   The derivative of the sliding vector in Eq (22) is Using the definition of the sliding vector in Eq (22), the obtained by assuming that sgn[ δ q4] is constant with Jacobian matrix defined by Eq (31) reduces to: respect to time:

δ

δ δ P = [ksgn [ q ]I × 0 × I × ] (32) & = ω & + [ ] & 4 3 3 3 1 3 3 s ksgn δ q4 q13 (39)

The equivalent control torque can be obtained by The dynamic equations of motion for the attitude rearranging Eq (30): tracking maneuver are obtained by substituting Eq (36) into Eq (15): 1-

Teq = []PB fP (x) (33)

δ δ ω & = − [ ] & − υ ksgn δ q 4 q13 G (40) which has a solution provided that [PB ] is non- singular. Substituting Eqs (28), (29), and (32) into Eq Substituting Eqs (39) and (40) into the derivative of Eq (33) provides an expression for the equivalent control (38) produces the following expression: torque:

& 1 T υ

ω ω ω δ

δ = − = [ ×] − + [& − [ ] & ] V s G (41) Teq J Tgg J d ksgn q4 q13 (34) 2 where δ q& is given by Eq (23a). Under the influence which is negative definite provided that G is a positive 13 definite matrix. Hence the control law given by Eq (36) of additional external disturbance torques, unmodeled may also be used to asymptotically force the state dynamics, parameter uncertainties and parameter trajectory towards the sliding manifold. variations, the equivalent control torque given by Eq (34) will not be sufficient to exactly maintain the ideal As stated above, Eq (36) has been developed sliding motion. Hence it is necessary to modify Teq in specifically for continuous external control torque order to account for these non-ideal effects so that the inputs. The JAESat master satellite will use three state trajectory will remain close to the sliding orthogonal magnetic coils to generate the control manifold. The modified control law is selected as: torque. The fundamental limitation of using magnetic υ coils is that only the component of Eq (36) = − T Teq JG (35) perpendicular to the geomagnetic field vector can be or generated. This constraint follows directly from the

definition of the magnetic torque given by Eq (12). The ω = [ ×] ω − implication for closed-loop stability is that Lyapunov

T J Tgg

υ

ω δ δ stability results given by Eqs (25) and (41) will not in + [& − [ ] & − ] J d ksgn q 4 q13 G (36) general be valid for the magnetically actuated system. The following expression is used to generate the three-

υ where G is a 3 x 3 positive definite diagonal matrix and axis magnetic coil commands: is a saturation function defined by: ⊥ B × T + > ε m = (42) 1 for si 2

 B υ s ε =  i for s ≤ i = 1,2,3 (37) i ε i  − < − ε ⊥  1 for si where T is the component of Eq (36) perpendicular

to the geomagnetic field vector. The control law ε where ε is a small positive scalar. The saturation parameters k, , and G must be carefully selected function is used to minimise chattering in the control empirically so that the closed-loop system is torque. The asymptotic stability of the closed-loop asymptotically stable and the minimum performance system using the control law given by Equation (36) requirements specified in Table 1 are achieved.

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Parameter Value Units Figure 6 depicts the simulation of the large-angle Inertia Matrix  8.1 0 0  kgm 2 =   tracking maneuver. The settling time for all axes is J  0 0.2 0   0 0 0.1  approximately the same and less than 30000 seconds. T  π  Initial Attitude π rad The steady-state pointing performance is depicted in (φ0 ,θ0,ψ0) =  0 

(1-2-3 Euler Angles)  2  Figure 7 and vastly exceeds the specified accuracy ω δ = [ ] T Initial Angular Rates 0 .0 0035 .0 0035 .0 0035 rad/sec requirements. The magnetic coil power consumption is = Scalar Gain k .0 001 rad/sec very low throughout the attitude tracking maneuver Deadband Constant ε = 01.0 - −5 which is a key advantage of magnetic actuation. Saturation Gain  e2 0 0  /sec  −  Matrix G =  0 e2 5 0   −5   0 0 e2  Normal Mode Simulation Duration 40000 sec The control law defined by Eqs (36) and (42) can also

Table 8: Parameters for Reorientation Mode simulation be used for three-axis stabilisation of the master satellite during the Normal Mode.

200 Roll Parameter Value Units Pitch   2 150 Inertia Matrix 86.0 0 0 kgm Yaw =   J  0 9.0 0    100  0 0 8.0 

T Initial Attitude (φ0,θ0,ψ0) = [ 35.0 35.0 35.0 ] rad 50

(1-2-3 Euler Angles) ω 0 δ = −5 [ ] T Initial Angular Rates 0 e1 * 75.1 75.1 75.1 rad/sec = -50 Scalar Gain k .0 001 rad/sec Euler Angles [deg] Angles Euler Deadband Constant ε = 01.0 - -100 − Saturation Gain  e4 5 0 0  /sec  −  Matrix G =  0 e4 5 0  -150  −5   0 0 e4 

-200 0 0.5 1 1.5 2 2.5 3 3.5 4 Simulation Duration 40000 sec Simulation Time [sec] 4 x 10 Table 9: Parameters for Normal Mode simulation 0.016

0.014 20 Roll 0.012 15 Pitch Yaw 10 0.01

5 0.008 0

Power [Watts] Power 0.006 -5

0.004 -10 Euler Angles [deg] Angles Euler

0.002 -15

-20 0 0 0.5 1 1.5 2 2.5 3 3.5 4 -25 Simulation Time [sec] 4 x 10 -30 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 6: Simulation Time [sec] 4 Satellite attitude and total magnetic coil power x 10 -5 consumption during Reorientation Mode x 10 3.5

3 0.04 Roll Pitch 2.5 0.03 Yaw

0.02 2

0.01 1.5 Power [Watts] Power 0 1 -0.01 Euler Angles [deg] Angles Euler 0.5 -0.02

0 -0.03 0 0.5 1 1.5 2 2.5 3 3.5 4 Simulation Time [sec] 4 x 10 -0.04 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 Simulation Time [sec] 5 Figure 8: Satellite attitude and total magnetic coil power x 10 consumption during Normal Mode Figure 7: Steady-state pointing during Reorientation Mode

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0.05 previous quaternion estimates. The Euler angle is then Roll 0.04 Pitch divided by the sampling period and low- filtered to Yaw 0.03 obtain the angular velocity about the Euler axis. Since

0.02 the star sensor delivers directly an attitude solution, this

0.01 section will concentrate on a brief description of the applied attitude determination concept based on GPS. 0

-0.01

Euler Angles [deg] Angles Euler GPS Based Attitude Determination -0.02 The fundamental physical principle of the GPS based -0.03 attitude determination process is the interferometric -0.04 principle, which is depicted in Figure 10. The GPS -0.05 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 Simulation Time [sec] 5 receiver measures single (SD) or double (DD) carrier x 10 phase differences. As discussed in section 3, the master Figure 9: Steady-state pointing during Normal Mode satellite will have four antennas arranged in a rectangular configuration on the +z satellite face (see Results of the Normal Mode simulation provided in Figure 10). It is intended that the master will use SD Figures 8 and 9 clearly show that the ACS is able to observations for the attitude determination process. The easily meet the performance requirements specified in basic equations for the SD attitude determination Table 1. Alternative control strategies, for example algorithm 5,6 using Euler angles (rotation sequence 3-1- proportional-derivative control or a Linear Quadratic 2) as attitude parameters will be presented below. The Regulator (LQR) approach, will also be considered in ideal observation equation (i.e. neglecting any error) is future research in an attempt to improve the steady- given by:

state pointing performance with respect to Figure 9. λ ∆Φ i = ∆ρ i − ∆ N i (43) Simulations were also performed for both the m m m

Reorientation and Normal modes to investigate the ρ

where ∆ Φ is the SD carrier phase, ∆ is the SD slant robustness of the control law with respect to initial λ conditions and the satellite inertia matrix. Uncertainties distance, is the wavelength of the GPS L1 signal, ∆ N of up to 15% in the moments of inertia and up to 25% is the difference of the initial number of carrier phase in the rates were acceptable although the settling time cycle ambiguities, m is the index for the baseline, and i was considerably degraded. is the index for the GPS satellite pair that has been used to generate the SD observation. The SD slant distance VII. Attitude Determination as a function of the line-of-sight unit vector and the corresponding baseline vector is given by: Three-axis attitude determination (attitude and rates) for the master satellite will be performed by the star ρ i = [ φ, θ, ψ) i ]• sensor and also by the attitude sensor based on GPS. ∆ m A( u m b m (44) However, the star sensor will be used predominately to provide attitude state information to the control where A(φ,θ,ψ) is the attitude matrix (reference frame algorithm. The GPS attitude sensor on the other hand to body frame), u is the line-of-sight unit vector will be used to test and validate different algorithms for expressed in the reference frame, and b is the baseline GPS-based attitude determination and carrier phase vector expressed in the body frame. Substituting Eq cycle ambiguity resolution. It is also intended to (44) into (43) and also considering an error term, leads conduct a series of tests where the benefits of this to the general observation equation for SD carrier phase combination of attitude sensors will be clearly measurements:

demonstrated. The star sensor directly outputs a λ quaternion estimate of the satellite three-axis attitude i = [ φ, θ, ψ) i ]• − i + ε ∆ Φ A( u b ∆ N ∆ (45) relative to an inertial reference frame (see Ref. 8 for m m m m further details). It also has the capability to provide star ∆ε data information so that new algorithms can be tested where represents errors such as line bias, receiver for advanced star identification and attitude noise and multipath effects. The carrier phase cycle ambiguity ∆N will be solved in an initialisation step for determination. Since rate gyros will not be included in 7 the ACS design, the body rates will be derived from the the attitude determination process with the STAR star sensor output using the on-board flight computer. algorithm based on spherical trigonometry. This One possible method is to calculate the Euler axis/angle method will also provide an initial guess for the state parameters for the angular motion between each vector x defined as: sampling time step by differencing the current and

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∆Φ∆Φ∆Φ Planar ∆∆∆ρρρ Wave front

λλλ ∆∆∆N u s γγγ b

Antenna A Antenna B λλλ (Master) (Slave)

Figure 10: Interferometric principle and GPS attitude sensor array

θ x = [φ ψ ]T (46) multipath strongly depends on the design of the satellite, in particular possible points and surfaces for

the reflection of signals. Tests will be conducted to gain A minimum of two visible GPS satellites are required an understanding about the multipath effects for the in order to be able to perform a deterministic attitude JAESat master satellite. Currently a Gaussian solution. If more than two GPS satellites are visible, the distributed multipath error of 5mm (1 σ, SD) has been number of measurements is larger than the number of assumed for simulation purposes. The one sigma unknown parameters, and therefore a Least Squares attitude solution accuracy can be calculated using the Solution (LSQ) will be used in a sequential way. The following expressions: attitude determination solution will be obtained through

the following expression: σ

σ = SD Yaw YawDOP (50a) −1 b ∆ = [ T ] T x M M M l (47) σ σ = SD RollDOP (50b) Roll b

where ∆ x = [ ∆ φ ∆ θ ∆ ψ]T is the solution vector, l is a σ vector containing the residuals between calculated and σ = SD Pitch PitchDOP (50c)

measured observation, and M is a design matrix b σ

defined by: σ = SD AttDOP (50d) Att b ∂ ∆ Φ M = (48) ∂ ∗ where the attitude Dilution of Precision (DOP) is given x x=x by:

where x* is the initial state vector. The LSQ attitude determination solution is given by: YawDOP 2 + PitchDOP 2 ATTDOP = (51) 2 ∗ + RollDOP x = x + ∆x (49)

The DOP values are obtained from the covariance The accuracy of GPS based attitude determination is T -1 dependant upon the geometry, the baseline length and matrix Cov = [M M] of the attitude solution. the measurement errors. It is commonly known that the most significant measurement error on the carrier phase results from multipath effects. The magnitude of the

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VIII. Conclusion In summary, an innovative low-cost ACS concept has [7] Arbinger, C., W. Enderle, L. Fraiture, and O. Wagner. (1999). been proposed for the JAESat mission. The proposed A new algorithm (STAR) for cycle ambiguity resolution control law based on a sliding mode approach provides within GPS based attitude determination. Proceedings of the optimal large-angle slewing/tracking maneuvers and th sub-degree steady-state pointing capabilities. The 4 ESA International Conference on Guidance, Navigation, attitude determination concept is primarily based on the and Control Systems , 345-347, Noordwijk, The Netherlands. star sensor, but a novel integrated three-axis attitude [8] Kayser-Threde GmbH. (1998). KM-1301 Star Sensor User sensor concept can be applied as well. The system Manual. performance of the JAESat master ACS has been [9] MITEL Semiconductors. (2001). GPS Orion 12 Channel GPS verified through simulations and exceeds the specified minimum performance requirements, although the Reference Design AN4808. effect of using practical attitude sensors needs to be [10] Shuster, M.D. (1993). A survey of attitude representations. The assessed. Further research and development will Journal of the Astronautical Sciences , 41(4): 439-517. include implementation of the ACS software in the on- [11] Stickler, A.C., and K.T. Alfriend (1976). Elementary magnetic board flight computer and development of the attitude control system. Journal of and , integrated attitude sensor. 13(5): 282-287. IX. Acknowledgements [12] Vadali, S.R., and J.L. Junkins. (1984). Optimal open-loop and The JAESat ACS R&D activities are being conducted stable feedback control of rigid spacecraft attitude maneuvers. at the Cooperative Research Centre for Satellite The Journal of the Astronautical Sciences , 32(2): 105-122. Systems, Queensland University of Technology. [13] Wen, J. T-Y., and K. Kreutz-Delgado. (1991). The attitude Special thanks to the CRCSS and all external project control problem. IEEE Transactions on Automatic Control , partners for their valuable contributions to the JAESat project. 36(10): 1148-1162. [14] Lo, S.C., and Y.P. Chen. (1995). Smooth sliding-mode control Finally, I would like to thank Werner Enderle of QUT for spacecraft attitude tracking maneuvers. Journal of for reviewing the paper and providing critical feedback. Guidance, Control, and Dynamics , 18(6): 1345-1349.

[15] Crassidis, J.L., and F.L. Markley. (1996). Sliding mode control X. References [1] Enderle, W. (2003). JAESat Project Documentation JAESat- using modified Rodrigues parameters. Journal of Guidance, GPS-PO-0001. QUT Project Documentation , Brisbane, Control and Dynamics, 19(6): 1381-1393. Australia. [16] Robinett, R.D., and G.G. Parker. (1997). Least squares sliding [2] Enderle, W. (2004). JAESat Mission Concept Description. mode control tracking of spacecraft large angle maneuvers. QUT Project Documentation , Brisbane, Australia. The Journal of the Astronautical Sciences , 45(4): 433-450. [3] Enderle, W., C. Boyd, and J.A. King. (2004). Joint Australian [17] Crassidis, J.L., S.R. Vadali, and F.L. Markley. (1999). Optimal Engineering (Micro) Satellite (JAESat) - A GNSS technology variable-structure control tracking of spacecraft maneuvers. demonstration mission. Proceedings of the Global Navigation Proceedings of the NASA Flight Mechanics/Estimation Theory Satellite Systems , Sydney, Australia. Symposium , 201-214, Greenbelt, Maryland, USA. [4] Enderle, W., A. Dando, and T. Bruggemann. (2005). The orbit [18] Crassidis, J.L., S.R. Vadali, and F.L. Markley. (2000). Optimal and attitude determination concept for the Joint Australian variable-structure control tracking of spacecraft maneuvers. Engineering (Micro) Satellite - JAESat. Proceedings of the Journal of Guidance, Control, and Dynamics , 23(3): 564-566. 28 th Annual AAS Guidance and Control Conference , [19] Vadali, S.R. (1986). Variable-structure control of spacecraft Breckenridge, Colorado, USA. large angle maneuvers. Journal of Guidance, Control, and [5] Arbinger, C., and W. Enderle. (2000). Spacecraft attitude Dynamics , 9(2): 235-239. determination using a combination of GPS attitude sensor and [20] Utkin, V.I. (1977). Variable structure systems with sliding star sensor measurements. Proceedings of the ION GPS modes. IEEE Transactions on Automatic Control , 22(2): 212- Conference , 2634-2642, Salt Lake City, Utah, USA. 222. [6] Arbinger, C., W. Enderle, and O. Wagner. (2001). GPS based [21] Hughes, P.C. (1986). Spacecraft Attitude Dynamics . Canada: star sensor aiding for attitude determination in high dynamics. John Wiley and Sons. Proceedings of the ION GPS Conference , 2952-2959, Salt [22] Wertz, J.R., ed. (2000). Spacecraft Attitude Determination and Lake City, Utah, USA. Control . Holland: Kluwer Academic Publishers.

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