Attitude Control Investigation Using Spacecraft Hardware-In-The-Loop ...
ATTITUDE CONTROL INVESTIGATION USING SPACECRAFT HARDWARE-IN-THE-LOOP SIMULATOR Scott A. Kowalchuk∗ Advisor: Dr. Christopher D. Hall† Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University
Abstract The Distributed Spacecraft Attitude Control System Simulator (DSACSS) testbed at Virginia Polytechnic Institute and State University facilitates investigation of various control strategies for single and multiple spacecraft. DSACSS is comprised of two independent hardware-in-the-loop simulators and one software spacecraft simulator. The two hardware-in-theloop spacecraft simulators have similar subsystems as flight-ready spacecraft, mounted on independent spherical air-bearing platforms. The nonlinear attitude control investigation uses one hardware-in-theloop simulator with three reaction wheels as the actuator for attitude control. A Lyapunov based angular rate controller and Modified Rodrigues Parmater (MRP) attitude controller are evaluated using the hardware simulator. The angular rate controller resulted in successfully driving the angular rates of the simulator to zero. The MRP attitude controller resulted in successfully driving the angular rates and MRP attitude vector of the simulator to zero. The angular rate and MRP controllers demonstrate the capability of investigating nonlinear attitude control strategies in real-time with hardware similar to flightready spacecraft.
Introduction There are many strategies to control the attitude and orbital position of a single spacecraft. The control
effort required depends on the specific mission objectives. In the past few years research has been conducted with regards to controlling the relative position between spacecraft using classical orbital elements (COE).1,2,3,4 Recently, spacecraft formation flying research is directed toward the coupled relative attitude and orbit control of spacecraft within the formation. Much of the relative attitude and orbit control research includes development of estimation techniques.5,6,7,8 Space demonstrator missions are typically high risk, high visibility, and high cost, which ultimately hinders the development of relative attitude and orbit control research. The trend for spacecraft related research has been to simulate space systems in a laboratory environment to reduce risk and cost. Simulations provide valuable knowledge for validating concepts and missions. Many of these simulations are software-based architectures. At Virginia Tech we have developed a hardware-in-the-loop Distributed Spacecraft Attitude Control System Simulator (DSACSS) to provide a more realistic demonstration of expected performance of distributed space systems as compared with software-based architectures. The spacecraft attitude control investigation presented in this paper evaluates two Lyapunov based attitude control laws using a hardware-in-the-loop spacecraft attitude control simulator that is part of the DSACSS facility. The following sections provide an overview of DSACSS, the development of the control laws, the simulation results, and future work.
∗
Graduate Research Assistant, [email protected], Virginia Polytechnic Institute and State University Aerospace Engineering Department, 215 Randolph Hall, Blacksburg VA, 24061. † Professor, [email protected], Virginia Polytechnic Institute and State University Aerospace Engineering Department, 215 Randolph Hall, Blacksburg VA, 24061.
DSACSS The DSACSS facility housed in the Space Systems Simulation Laboratory (SSSL) at Virginia Polytech-
Figure 1: Hardware-in-the-Loop Distributed Spacecraft Attitude Control System Simulator (DSACSS), Left: Whorl−I (Tabletop Simulator Configuration), Middle: Whorl−III (Software Simulator), Right: Whorl−II (Dumbbell Simulator Configuration)
nic Institute and State University provides a high fidelity solution for Earth-based hardware-in-the-loop simulations of space vehicles. DSACSS allows individual or component level development and testing of hardware and software interfaces. The DSACSS facility is comprised of two hardware/software simulators and one simulator based on software. The simulators are shown in Figure 1. The two hardware-in-the-loop spacecraft simulators are individually mounted on spherical air bearing platforms, which provide a nearly torque-free environment for simulating attitude control. The software simulator is a PC/104 stack, which is identical to the flight computers mounted on the hardware spacecraft simulators. The software simulator is also used as a backup flight computer for the hardware simulators. Each hardware spacecraft simulator has a unique configuration as can be seen in Figure 1. For clarification the spacecraft “tabletop” style configuration is referred to as Whorl−I, the “dumbbell” style configuration is referred to as Whorl−II, and the software simulator is referred to as Whorl−III. The yaw axis of Whorl−I has a rotational freedom of 360◦ while the roll and pitch axes have a bounded rotation of ± 5◦ . The yaw and roll axes of Whorl−II have rotational freedom of 360◦ while the pitch axis has a bounded rotation of ± 30◦ . Each simulator has a maximum payload of 300 lbs.9 Whorl−I is selected as the hardware simulator used to evaluate the attitude feedback controllers. For this reason only Whorl−I is discussed further. For more information on Whorl−II and Whorl−III refer to Ref. 9 and 10. Figure 2 is a block diagram of the subsystems Kowalchuk
for Whorl−I. Whorl−I has command and data handling; communications; attitude determination and control; power; payload; and guidance and navigation subsystems. Whorl−I operates on a 24 volt battery system that provides 24 amps of current. Whorl−I is controlled by a flight computer, which is a PC/104 stack comprised of a 733 MHz processor, 512 MB of RAM, and a 2 GB flash drive. The flight computer is using Scientific Linux 4 as the operating system. Analog and digital devices interface with the PC/104 through a DAQ card. The flight computer can communicate with other laboratory resources using a wireless bridge to access the private network. Whorl−I is equipped with a BEI MotionPak II unit that has three-axis accelerometers and rate gyros for attitude determination. A Honeywell HMR2300 three axis strap down magnetometer is also used for attitude determination. The attitude of Whorl−I is controlled by using momentum wheels, nitrogen gas (N2 ) attitude thrusters, or a control moment gyro (CMG). Linear actuators are used to adjust the center-of-mass of the simulator.9 Whorl−I has three reaction wheels aligned along the x, y, and z body axes. For this study the attitude is controlled by using the reaction wheels. The reaction wheel motor is a SM3430 SmartMotor produced by the Animatics Corporation and is supplied with 24 volts of power as indicated in Figure 2. The motor is controlled by the flight computer through a serial interface. The motor has a built in logic computer with a 4000 count encoder. Figure 3 is an illustration of the reaction wheel on Whorl−I with the transverse and spin axes labeled. 2
Figure 2: Whorl−I System Block Diagram
Lyapunov Based Attitude Control Laws
the angular rate and attitude of Whorl−I. The objective of the angular rate controller is to drive the The equations of motion of a rigid spacecraft with angular rates to zero. The objective of the modified N reaction wheels are provided in Eq. (1) from Rodrigues parameter (MRP) controller is to drive the Ref. (11). Where (·)× is the skew operator, [Ws ] is a angular rates and MRP vector to zero . matrix defining the spin axis orientation of the reaction wheels, L is an external torque. Eq. (2) defines Angular Rate Controller the body fixed inertia matrix ([I]) and Eq. (3) defines The objective of the angular rate controller is to drive hs . The transverse axes of the reaction wheel are inthe angular rates of Whorl−I to zero. The angudicated by w ˆt and w ˆg . The reaction wheel spin axis lar velocity error is expressed in the body frame in inertia is represented as Js and the wheel speed is Eq. (4), where ωr is the angular velocity in the referrepresented as Ω. ence frame and RBR is the rotation matrix from the reference frame to the body frame. × [I]ω˙ = −ω ([I]ω + [Ws ]hs ) + [Ws ]u + L (1) B δω = B ω − RBR R ωr (4)
[I] = [Is ] +
N X
Jti w ˆti w ˆtTi
+
Jgi w ˆ gi w ˆgTi
i=1
hsi = Jsi (ωsi + Ωi )
The reference angular velocity is zero for the desired situation where the angular rates go to zero. (2) Therefore Eq. (4) simplifies to B δω = B ω. The candidate Lyapunov function is provided in (3) Eq. (5).11
Two Lyapunov based control laws are developed for using reaction wheels as the actuator to control Kowalchuk
1 V (ω) = ω T [I]ω 2
(5) 3
determined from Eq. (9). The relationships between the other attitude representations and the MRP vector can be found in Ref 11. Φ e ˆ (9) 4 The MRP vector is a convenient attitude representation for attitude feedback control. By using the mapping relationship in Eq. (10) the MRP vector can always represent a principal rotation angle less than or equal to 180◦ .11 Therefore the MRP mapping ensures the shortest distance to a principal rotation angle of zero degrees. The MRP mapping is exploited to develop the Lyapunov based MRP controller. σ = tan
−σ1,2,3 (10) σT σ The angular velocity error is expressed in Figure 3: Illustration of Whorl−I Reaction Wheel Eq. (4). The MRP vector error is expressed in Eq. (11).11 Taking the derivative of Eq. (5) results in Eq. (6). 0 |2 )σ−(1−|σ|2 )σ 0 Note that ω T ω × [I]ω = 0. δσ = (1−|σ 1+|σ 0 |2 |σ|2 +2σ 0 ·σ (11) 0 + 1+|σ0 |2σ×σ 2 |σ|2 +2σ 0 ·σ V˙ (ω) = −ω T ω × [I]ω+ (6) The reference MRP vector is zero for the desired ω T (−ω × [Ws ]hs + [Ws ]u + L) situation where the MRP vector goes to zero. ThereSetting Eq. (6) equal to to Eq. (7) and requiring fore Eq. (11) simplifies to δσ = σ. The candidate Lyapunov function is shown in the gain ([P ]) to be positive definite guarantees stability by forcing the Lyapunov derivative to be nega- Eq. (12). Where K is a positive scalar gain quantity.11 tive semidefinite. s σ1,2,3 =
V˙ (ω) = −ω T [P ]ω
(7)
Performing the substitution and rearranging the expression results in the angular rate control law shown in Eq. (8). [Ws ]u = −[P ]ω + ω × [Ws ]hs − L
1 V (ω, σ) = ω T [I]ω + 2K log 1 + σ T σ (12) {z } |2 {z } | V (ω)
V (σ)
Eq. (12) can be broken into V (ω) and V (σ). Taking the derivative of Eq. (12) results in the fol(8) lowing.
V˙ (ω, σ) = V˙ (ω) + V˙ (σ) (13) The first term V˙ (ω) is presented in the anguMRP Controller lar rate development (Eq. (6)). The second controller ˙ term V (σ) is solved by substituting the MRP rate The objective of the MRP controller is to drive the vector (Eq. (14)) into the equation and re-grouping angular rates and MRP vector of Whorl−I to zero. 11 T × The MRP vector can be defined in terms of terms. Note that σ σ = 0. quaternions, classical Rodrigues parameters, and principal rotation elements. Eq. (9) expresses the σ˙ = (1 − σ 2 )I + 2σ × + 2σσ T ω/4 (14) MRP vector (σ) in terms of the principal rotation elements (Φ, e ˆ). The MRP vector has a singularThe substitution and re-grouping results in the ity at a principal rotation angle of ±360◦ as can be elegant form in Eq. (15). The control vector (u) may be solved by using a psuedoinverse or minimum norm inverse.
Kowalchuk
4
The mass for Whorl−I is estimated to be (15) 88.46 kg from the CAD drawings of Whorl−I. The results of running the controllers on The substitution of Eq. (6) and Eq. (15) into Whorl−I along with a numerical simulation comparEq. (13) results in Eq. (16). ing the results are presented in the following subsections. The numerical simulations were conducted using Matlab. The numerical simulation results in each V˙ (ω, σ) = ω T [I]ω˙ + ω T Kσ T × T ˙ (16) figure are indicated by (·)(·)Sim . V (ω, σ) = −ω ω [I]ω + ω (Kσ −ω × [Ws ]hs + [Ws ]u + L) V˙ (σ) = ω T Kσ
Angular Rate Controller
Setting Eq. (16) equal to to Eq. (7) and requiring the gain ([P ]) to be positive definite guarantees stability by forcing the Lyapunov derivative to be negative semidefinite. Performing the substitution and rearranging the expression results in the MRP control law shown in Eq. (17).
Assuming no external torque and substitutingthe re ˆ s] action wheel spin axis orientation matrix [W into the control law (Eq. (8)) simplifies to the following. u = −[P ]ω + ω × hs
[Ws ]u = −[P ]ω − Kσ + ω × [Ws ]hs − L
(17)
The control vector (u) may be solved by using a psuedoinverse or minimum norm inverse as previously mentioned.
Simulation Results Whorl−I has three reaction wheels aligned along each body axis. The estimated reaction wheel spin axis orientation matrix is provided as follows. Note that (ˆ·) refers to estimate. 1 0 0 ˆ s] = 0 1 0 [W 0 0 1 The inertia matrix estimate is obtained from CAD drawings of Whorl−I and provided as follows. 8.8325 0.6854 −0.1579 ˆ = 0.6854 7.6989 −0.1035 kg-m2 [I] −0.1579 −0.1035 12.9964 The control laws specified in Eq. (8) and (17) do not require knowledge of the inertia matrix. The estimated inertia matrix is used for the numerical simulation comparing the performance of each controller with the results from controlling Whorl−I. The spin axis inertia for each reaction wheel is also determined from CAD drawings and is provided as follows. Jˆsx,y,z = 0.075 kg-m2 Kowalchuk
(18)
The angular rate controller parameters are provided in Table 1. Table 1: Angular Rate Controller Parameters Parameter Value Control Rate 2 Hz Observe Rate 2 Hz Wheel Speed Query 10 Hz ω(t0 ) (0.002, 0.010, 0.194) rad/s [P ] diag( 3.64, 3.40, 4.41 ) kg-m2 /s Ωx 0 rad/s Ωy 0 rad/s Ωz 0 rad/s The angular rate controller results are shown in Figures 4 through 6. Whorl−I was given an initial angular rate about the z body axis and then the angular rate controller was turned on. The 16 second delay in the start of the controller at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels. Figure 4 provides the angular rate time history. Whorl−I has approximately zero angular rate in the z axis after 30 seconds into the simulation. The numerical simulation has approximately zero angular rate after 25 seconds into the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by the numerical simulation. After 30 seconds the noise from the angular rate measurements can be clearly seen. Figure 5 shows the required control torque specified by the angular rate controller. The control 5
Figure 4: Whorl−I and Numerical Simulation Angular Rates for Angular Rate Controller
Figure 5: Whorl−I and Numerical Simulation Control Torque for Angular Rate Controller
Figure 6: Whorl−I and Numerical Simulation Reaction Wheel Rates for Angular Rate Controller
Kowalchuk
6
torques specified by the angular rate controller for Whorl−I have the same increasing control torque trend to zero as the numerical simulation. Figure 6 shows the reaction wheel speed time history for the controller running on Whorl−I and the numerical simulation. The reaction wheel aligned along the y axis is in good agreement between the numerical simulation and Whorl−I results. The reaction wheel aligned along the x axis has an increasing wheel speed. After further inspection of the angular rate measurements, there is a small angular rate bias when the simulator is at rest. The angular rate gyros were re-calibrated to correct for the bias. The reaction wheel aligned along the z axis for Whorl−I has a similar trend as the numerical simulation. The numerical simulation ended with an angular velocity of approximately 35 rad/s as compared with 40 rad/s for the reaction wheel along the z axis. The discrepancy in wheel speeds between Whorl−I and the numerical simulation may be contributed to the simplified reaction wheel motor torque model used in the numerical simulation and shown in Eq. (19).11 Ref. 12 provides a more developed dynamic model of a reaction wheel motor. The model takes into account the electrical resistance of the motor armature, voltage limits, current limits, viscosity damping, and static friction of the rotor. ˆsTi ω ˙ usi = Jsi Ω˙ i + w
(19)
MRP Controller Assuming no external torque and substituting the ˆ s] reaction wheel spin axis orientation matrix [W into the control law (Eq. (17)) simplifies to the following. u = −[P ]ω − Kσ + ω × hs
(20)
The MRP controller parameters are provided in Table 2. The angular rate controller results are shown in Figures 7 through 10. Whorl−I was given an initial angular rate about the z body axis and then the MRP controller was turned on. The 16 second delay in the start of the controller at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels. Figure 7 provides the angular rate time history. Whorl−I has approximately zero angular rate in the Kowalchuk
Table 2: MRP Controller Parameters Parameter Value Control Rate 2 Hz Observe Rate 2 Hz Wheel Speed Query 10 Hz σ(t0 ) (0.002, 0.009, 0.069) ω(t0 ) (-0.015, -0.021, 0.199) rad/s K 1.5 kg-m2 /s2 [P ] diag( 3.64, 3.40, 4.41 ) kg-m2 /s Ωx 0 rad/s Ωy 0 rad/s Ωz 0 rad/s
z axis after 50 seconds into the simulation. The numerical simulation has approximately zero angular rate after 48 seconds into the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by the numerical simulation. After 50 seconds the noise from the angular rate measurements can be clearly seen. Figure 8 shows the MRP vector time history for Whorl−I and the numerical simulation. Whorl−I has approximately a zero MRP vector at 50 seconds into the simulation. The numerical simulation has approximately a zero vector at 48 seconds. The MRP vector for Whorl-I agrees very closely with the numerical simulation results. In Figure 8 the attitude observer returned two erroneous attitudes for Whorl−I as can be seen for σ3 at 21 and 42 seconds. The two erroneous attitudes are also seen in the control history. Figure 9 shows the required control torque specified by the angular rate controller. The control torques specified by the angular rate controller for Whorl−I have the same increasing control torque trend to zero as the numerical simulation. Figure 6 shows the reaction wheel speed time history for the controller running on Whorl−I and the numerical simulation. The reaction wheel aligned along the z axis for Whorl−I has a similar trend as the numerical simulation. The numerical simulation ended with an angular velocity of approximately 36 rad/s as compared with 40 rad/s for the z reaction wheel. As previously discussed in the angular rate controller results, the discrepancy in wheel speeds between Whorl−I and the numerical simulation may be contributed to the simplified reaction wheel mo7
Figure 7: Whorl−I and Numerical Simulation Angular Rates for MRP Controller
Figure 8: Whorl−I and Numerical Simulation MRP for MRP Controller
Figure 9: Whorl−I and Numerical Simulation Control Torque Results for MRP Controller
Kowalchuk
8
Figure 10: Whorl−I and Numerical Simulation Reaction Wheel Rates for MRP Controller tor torque model used in the numerical simulation A Global Positioning System (GPS) closed-loop and shown in Eq. (19). hardware-in-the-loop simulator has been developed and integrated into DSACSS to investigate orbital control strategies.13,14 Whorl−I combined with the Conclusions GPS simulator will provide hardware-in-the-loop simulations that will facilitate development and inWhorl−I was given an initial angular rate, primar- vestigation of coupled attitude and orbital control ily about the body z axis, to investigate the per- strategies . formance of the angular rate and MRP controllers. The angular rate controller resulted in successfully driving the angular rates of Whorl−I to zero. The References control and angular rate time history matched the trends observed from a numerical simulation of the [1] B. J. Naasz, R. D. Burns, D. Gaylor, and angular rate controller with the same initial condiJ. Higinbotham, “Hardware-in-the-loop Testtions as Whorl−I. The MRP controller successfully ing of Continuous Conrol Algorithms for drove the MRP attitude vector of Whorl−I to zero. a Precision Formation Flying Demonstration The angular rates, MRP vector, and control time hisMission,” 18th International Symposium on tory closely matched the numerical simulation of the Space Flight Dynamics, Haus der Bayerischen MRP controller. Wirtschaft, Munich, Germany, October 11–15, The DSACSS framework is a flexible testbed 2004. for investigation of a variety of spacecraft control techniques, such as Lyapunov based attitude control laws. The advantage of DSACSS over software sim- [2] B. J. Naasz, C. D. Karlgaard, and C. D. Hall, “Application of Several Control Techniques for ulator architectures is that hardware-in-the-loop simthe Ionospheric Observation Nanosatellite Forulations for testing and validation of concepts promation,” 2002 AAS/AIAA Space Flight Mevide a more realistic demonstration of expected perchanics Conference, San Antonio, Texas, Janformance of real systems. uary 27–30, 2002.
Future Work Future plans are to develop and implement nonlinear attitude tracking controllers for Whorl−I and compare the performance to numerical simulations. Kowalchuk
[3] H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, “Spacecraft Formation Flying Control Using Mean Orbit Elements,” Journal of the Astronautical Sciences, Vol. 48, No. 1, 2000, pp. 69–89. 9
[4] S. Leung, E. Gill, O. Montenbruck, and [12] M. J. Sidi, Spacecraft Dynamics and Control. Cambridge University Press, 1997. S. Montenegro, “A Navigation Processor for Flexible Real-Time Formation Flying Applications,” International Symposium Formation [13] S. A. Kowalchuk and C. D. Hall, “Hardwarein-the-Loop Simulation of Classical Element Flying Missions and Technologies, Centre NaFeedback Controller,” Goddard Flight Metional d’Etudes Spatiales Toulouse Space Cenchanics Symposium, Greenbelt, Maryland, Octre - France, October 29–31, 2002. tober 18–20, 2005. [5] S. Kim, J. L. Crassidis, Y. Cheng, A. M. [14] S. A. Kowalchuk and C. D. Hall, “GPS Fosbury, and J. L. Junkins, “Kalman FilterHardware-in-the-loop Spacecraft Formation ing for Relative Spacecraft Attitude and PosiFlying Simulation,” AAS/AIAA Space Flight tion Estimation,” AIAA Guidance, Navigation, Mechanics Meeting, Sedona, Arizona, Jan. 28– and Control Conference and Exhibit, San FranFeb. 1, 2007. cisco, California, August 15–18, 2005. [6] M. C. VanDyke and C. D. Hall, “Decentralized Coordinated Attitude Control of a Formation of Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1101– 1109. [7] G. Q. Xing and S. A. Parvez, “Relative Attitude Kinematics and Dynamics Equations and Its Applications to Spacecraft Attitude State Capture and Tracking in Large Angle Slewing Maneuvers,” 1999 Space Control Conference, MIT Lincoln Laboratory, 1999. April 13-15. [8] F. L. Markley, J. L. Crassidis, and Y. Cheng, “Nonlinear Attitude Filtering Methods,” AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, August 15–18, 2005. [9] S. A. Kowalchuk and C. D. Hall, “Distributed Spacecraft Attitude Control System Simulator: Feedback Control Capabilities and Visualization Techniques,” 7th International Quantitative Feedback Theory and Robust Frequency Domain Symposium, University of Kansas, Lawrence, Kansas, August 5–7, 2005. [10] J. L. Schwartz and C. D. Hall, “The Distributed Spacecraft Attitude Control System Simulator: Development, Progress, Plans,” 2003 Flight Mechanics Symposium, NASA Goddard Space Flight Center, Greenbelt, Maryland, October 28–30, 2003. [11] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems. AIAA Education Series, 2003. Kowalchuk
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Abstract The Distributed Spacecraft Attitude Control System Simulator (DSACSS) testbed at Virginia Polytechnic Institute and State University facilitates investigation of various control strategies for single and multiple spacecraft. DSACSS is comprised of two independent hardware-in-the-loop simulators and one software spacecraft simulator. The two hardware-in-theloop spacecraft simulators have similar subsystems as flight-ready spacecraft, mounted on independent spherical air-bearing platforms. The nonlinear attitude control investigation uses one hardware-in-theloop simulator with three reaction wheels as the actuator for attitude control. A Lyapunov based angular rate controller and Modified Rodrigues Parmater (MRP) attitude controller are evaluated using the hardware simulator. The angular rate controller resulted in successfully driving the angular rates of the simulator to zero. The MRP attitude controller resulted in successfully driving the angular rates and MRP attitude vector of the simulator to zero. The angular rate and MRP controllers demonstrate the capability of investigating nonlinear attitude control strategies in real-time with hardware similar to flightready spacecraft.
Introduction There are many strategies to control the attitude and orbital position of a single spacecraft. The control
effort required depends on the specific mission objectives. In the past few years research has been conducted with regards to controlling the relative position between spacecraft using classical orbital elements (COE).1,2,3,4 Recently, spacecraft formation flying research is directed toward the coupled relative attitude and orbit control of spacecraft within the formation. Much of the relative attitude and orbit control research includes development of estimation techniques.5,6,7,8 Space demonstrator missions are typically high risk, high visibility, and high cost, which ultimately hinders the development of relative attitude and orbit control research. The trend for spacecraft related research has been to simulate space systems in a laboratory environment to reduce risk and cost. Simulations provide valuable knowledge for validating concepts and missions. Many of these simulations are software-based architectures. At Virginia Tech we have developed a hardware-in-the-loop Distributed Spacecraft Attitude Control System Simulator (DSACSS) to provide a more realistic demonstration of expected performance of distributed space systems as compared with software-based architectures. The spacecraft attitude control investigation presented in this paper evaluates two Lyapunov based attitude control laws using a hardware-in-the-loop spacecraft attitude control simulator that is part of the DSACSS facility. The following sections provide an overview of DSACSS, the development of the control laws, the simulation results, and future work.
∗
Graduate Research Assistant, [email protected], Virginia Polytechnic Institute and State University Aerospace Engineering Department, 215 Randolph Hall, Blacksburg VA, 24061. † Professor, [email protected], Virginia Polytechnic Institute and State University Aerospace Engineering Department, 215 Randolph Hall, Blacksburg VA, 24061.
DSACSS The DSACSS facility housed in the Space Systems Simulation Laboratory (SSSL) at Virginia Polytech-
Figure 1: Hardware-in-the-Loop Distributed Spacecraft Attitude Control System Simulator (DSACSS), Left: Whorl−I (Tabletop Simulator Configuration), Middle: Whorl−III (Software Simulator), Right: Whorl−II (Dumbbell Simulator Configuration)
nic Institute and State University provides a high fidelity solution for Earth-based hardware-in-the-loop simulations of space vehicles. DSACSS allows individual or component level development and testing of hardware and software interfaces. The DSACSS facility is comprised of two hardware/software simulators and one simulator based on software. The simulators are shown in Figure 1. The two hardware-in-the-loop spacecraft simulators are individually mounted on spherical air bearing platforms, which provide a nearly torque-free environment for simulating attitude control. The software simulator is a PC/104 stack, which is identical to the flight computers mounted on the hardware spacecraft simulators. The software simulator is also used as a backup flight computer for the hardware simulators. Each hardware spacecraft simulator has a unique configuration as can be seen in Figure 1. For clarification the spacecraft “tabletop” style configuration is referred to as Whorl−I, the “dumbbell” style configuration is referred to as Whorl−II, and the software simulator is referred to as Whorl−III. The yaw axis of Whorl−I has a rotational freedom of 360◦ while the roll and pitch axes have a bounded rotation of ± 5◦ . The yaw and roll axes of Whorl−II have rotational freedom of 360◦ while the pitch axis has a bounded rotation of ± 30◦ . Each simulator has a maximum payload of 300 lbs.9 Whorl−I is selected as the hardware simulator used to evaluate the attitude feedback controllers. For this reason only Whorl−I is discussed further. For more information on Whorl−II and Whorl−III refer to Ref. 9 and 10. Figure 2 is a block diagram of the subsystems Kowalchuk
for Whorl−I. Whorl−I has command and data handling; communications; attitude determination and control; power; payload; and guidance and navigation subsystems. Whorl−I operates on a 24 volt battery system that provides 24 amps of current. Whorl−I is controlled by a flight computer, which is a PC/104 stack comprised of a 733 MHz processor, 512 MB of RAM, and a 2 GB flash drive. The flight computer is using Scientific Linux 4 as the operating system. Analog and digital devices interface with the PC/104 through a DAQ card. The flight computer can communicate with other laboratory resources using a wireless bridge to access the private network. Whorl−I is equipped with a BEI MotionPak II unit that has three-axis accelerometers and rate gyros for attitude determination. A Honeywell HMR2300 three axis strap down magnetometer is also used for attitude determination. The attitude of Whorl−I is controlled by using momentum wheels, nitrogen gas (N2 ) attitude thrusters, or a control moment gyro (CMG). Linear actuators are used to adjust the center-of-mass of the simulator.9 Whorl−I has three reaction wheels aligned along the x, y, and z body axes. For this study the attitude is controlled by using the reaction wheels. The reaction wheel motor is a SM3430 SmartMotor produced by the Animatics Corporation and is supplied with 24 volts of power as indicated in Figure 2. The motor is controlled by the flight computer through a serial interface. The motor has a built in logic computer with a 4000 count encoder. Figure 3 is an illustration of the reaction wheel on Whorl−I with the transverse and spin axes labeled. 2
Figure 2: Whorl−I System Block Diagram
Lyapunov Based Attitude Control Laws
the angular rate and attitude of Whorl−I. The objective of the angular rate controller is to drive the The equations of motion of a rigid spacecraft with angular rates to zero. The objective of the modified N reaction wheels are provided in Eq. (1) from Rodrigues parameter (MRP) controller is to drive the Ref. (11). Where (·)× is the skew operator, [Ws ] is a angular rates and MRP vector to zero . matrix defining the spin axis orientation of the reaction wheels, L is an external torque. Eq. (2) defines Angular Rate Controller the body fixed inertia matrix ([I]) and Eq. (3) defines The objective of the angular rate controller is to drive hs . The transverse axes of the reaction wheel are inthe angular rates of Whorl−I to zero. The angudicated by w ˆt and w ˆg . The reaction wheel spin axis lar velocity error is expressed in the body frame in inertia is represented as Js and the wheel speed is Eq. (4), where ωr is the angular velocity in the referrepresented as Ω. ence frame and RBR is the rotation matrix from the reference frame to the body frame. × [I]ω˙ = −ω ([I]ω + [Ws ]hs ) + [Ws ]u + L (1) B δω = B ω − RBR R ωr (4)
[I] = [Is ] +
N X
Jti w ˆti w ˆtTi
+
Jgi w ˆ gi w ˆgTi
i=1
hsi = Jsi (ωsi + Ωi )
The reference angular velocity is zero for the desired situation where the angular rates go to zero. (2) Therefore Eq. (4) simplifies to B δω = B ω. The candidate Lyapunov function is provided in (3) Eq. (5).11
Two Lyapunov based control laws are developed for using reaction wheels as the actuator to control Kowalchuk
1 V (ω) = ω T [I]ω 2
(5) 3
determined from Eq. (9). The relationships between the other attitude representations and the MRP vector can be found in Ref 11. Φ e ˆ (9) 4 The MRP vector is a convenient attitude representation for attitude feedback control. By using the mapping relationship in Eq. (10) the MRP vector can always represent a principal rotation angle less than or equal to 180◦ .11 Therefore the MRP mapping ensures the shortest distance to a principal rotation angle of zero degrees. The MRP mapping is exploited to develop the Lyapunov based MRP controller. σ = tan
−σ1,2,3 (10) σT σ The angular velocity error is expressed in Figure 3: Illustration of Whorl−I Reaction Wheel Eq. (4). The MRP vector error is expressed in Eq. (11).11 Taking the derivative of Eq. (5) results in Eq. (6). 0 |2 )σ−(1−|σ|2 )σ 0 Note that ω T ω × [I]ω = 0. δσ = (1−|σ 1+|σ 0 |2 |σ|2 +2σ 0 ·σ (11) 0 + 1+|σ0 |2σ×σ 2 |σ|2 +2σ 0 ·σ V˙ (ω) = −ω T ω × [I]ω+ (6) The reference MRP vector is zero for the desired ω T (−ω × [Ws ]hs + [Ws ]u + L) situation where the MRP vector goes to zero. ThereSetting Eq. (6) equal to to Eq. (7) and requiring fore Eq. (11) simplifies to δσ = σ. The candidate Lyapunov function is shown in the gain ([P ]) to be positive definite guarantees stability by forcing the Lyapunov derivative to be nega- Eq. (12). Where K is a positive scalar gain quantity.11 tive semidefinite. s σ1,2,3 =
V˙ (ω) = −ω T [P ]ω
(7)
Performing the substitution and rearranging the expression results in the angular rate control law shown in Eq. (8). [Ws ]u = −[P ]ω + ω × [Ws ]hs − L
1 V (ω, σ) = ω T [I]ω + 2K log 1 + σ T σ (12) {z } |2 {z } | V (ω)
V (σ)
Eq. (12) can be broken into V (ω) and V (σ). Taking the derivative of Eq. (12) results in the fol(8) lowing.
V˙ (ω, σ) = V˙ (ω) + V˙ (σ) (13) The first term V˙ (ω) is presented in the anguMRP Controller lar rate development (Eq. (6)). The second controller ˙ term V (σ) is solved by substituting the MRP rate The objective of the MRP controller is to drive the vector (Eq. (14)) into the equation and re-grouping angular rates and MRP vector of Whorl−I to zero. 11 T × The MRP vector can be defined in terms of terms. Note that σ σ = 0. quaternions, classical Rodrigues parameters, and principal rotation elements. Eq. (9) expresses the σ˙ = (1 − σ 2 )I + 2σ × + 2σσ T ω/4 (14) MRP vector (σ) in terms of the principal rotation elements (Φ, e ˆ). The MRP vector has a singularThe substitution and re-grouping results in the ity at a principal rotation angle of ±360◦ as can be elegant form in Eq. (15). The control vector (u) may be solved by using a psuedoinverse or minimum norm inverse.
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The mass for Whorl−I is estimated to be (15) 88.46 kg from the CAD drawings of Whorl−I. The results of running the controllers on The substitution of Eq. (6) and Eq. (15) into Whorl−I along with a numerical simulation comparEq. (13) results in Eq. (16). ing the results are presented in the following subsections. The numerical simulations were conducted using Matlab. The numerical simulation results in each V˙ (ω, σ) = ω T [I]ω˙ + ω T Kσ T × T ˙ (16) figure are indicated by (·)(·)Sim . V (ω, σ) = −ω ω [I]ω + ω (Kσ −ω × [Ws ]hs + [Ws ]u + L) V˙ (σ) = ω T Kσ
Angular Rate Controller
Setting Eq. (16) equal to to Eq. (7) and requiring the gain ([P ]) to be positive definite guarantees stability by forcing the Lyapunov derivative to be negative semidefinite. Performing the substitution and rearranging the expression results in the MRP control law shown in Eq. (17).
Assuming no external torque and substitutingthe re ˆ s] action wheel spin axis orientation matrix [W into the control law (Eq. (8)) simplifies to the following. u = −[P ]ω + ω × hs
[Ws ]u = −[P ]ω − Kσ + ω × [Ws ]hs − L
(17)
The control vector (u) may be solved by using a psuedoinverse or minimum norm inverse as previously mentioned.
Simulation Results Whorl−I has three reaction wheels aligned along each body axis. The estimated reaction wheel spin axis orientation matrix is provided as follows. Note that (ˆ·) refers to estimate. 1 0 0 ˆ s] = 0 1 0 [W 0 0 1 The inertia matrix estimate is obtained from CAD drawings of Whorl−I and provided as follows. 8.8325 0.6854 −0.1579 ˆ = 0.6854 7.6989 −0.1035 kg-m2 [I] −0.1579 −0.1035 12.9964 The control laws specified in Eq. (8) and (17) do not require knowledge of the inertia matrix. The estimated inertia matrix is used for the numerical simulation comparing the performance of each controller with the results from controlling Whorl−I. The spin axis inertia for each reaction wheel is also determined from CAD drawings and is provided as follows. Jˆsx,y,z = 0.075 kg-m2 Kowalchuk
(18)
The angular rate controller parameters are provided in Table 1. Table 1: Angular Rate Controller Parameters Parameter Value Control Rate 2 Hz Observe Rate 2 Hz Wheel Speed Query 10 Hz ω(t0 ) (0.002, 0.010, 0.194) rad/s [P ] diag( 3.64, 3.40, 4.41 ) kg-m2 /s Ωx 0 rad/s Ωy 0 rad/s Ωz 0 rad/s The angular rate controller results are shown in Figures 4 through 6. Whorl−I was given an initial angular rate about the z body axis and then the angular rate controller was turned on. The 16 second delay in the start of the controller at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels. Figure 4 provides the angular rate time history. Whorl−I has approximately zero angular rate in the z axis after 30 seconds into the simulation. The numerical simulation has approximately zero angular rate after 25 seconds into the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by the numerical simulation. After 30 seconds the noise from the angular rate measurements can be clearly seen. Figure 5 shows the required control torque specified by the angular rate controller. The control 5
Figure 4: Whorl−I and Numerical Simulation Angular Rates for Angular Rate Controller
Figure 5: Whorl−I and Numerical Simulation Control Torque for Angular Rate Controller
Figure 6: Whorl−I and Numerical Simulation Reaction Wheel Rates for Angular Rate Controller
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torques specified by the angular rate controller for Whorl−I have the same increasing control torque trend to zero as the numerical simulation. Figure 6 shows the reaction wheel speed time history for the controller running on Whorl−I and the numerical simulation. The reaction wheel aligned along the y axis is in good agreement between the numerical simulation and Whorl−I results. The reaction wheel aligned along the x axis has an increasing wheel speed. After further inspection of the angular rate measurements, there is a small angular rate bias when the simulator is at rest. The angular rate gyros were re-calibrated to correct for the bias. The reaction wheel aligned along the z axis for Whorl−I has a similar trend as the numerical simulation. The numerical simulation ended with an angular velocity of approximately 35 rad/s as compared with 40 rad/s for the reaction wheel along the z axis. The discrepancy in wheel speeds between Whorl−I and the numerical simulation may be contributed to the simplified reaction wheel motor torque model used in the numerical simulation and shown in Eq. (19).11 Ref. 12 provides a more developed dynamic model of a reaction wheel motor. The model takes into account the electrical resistance of the motor armature, voltage limits, current limits, viscosity damping, and static friction of the rotor. ˆsTi ω ˙ usi = Jsi Ω˙ i + w
(19)
MRP Controller Assuming no external torque and substituting the ˆ s] reaction wheel spin axis orientation matrix [W into the control law (Eq. (17)) simplifies to the following. u = −[P ]ω − Kσ + ω × hs
(20)
The MRP controller parameters are provided in Table 2. The angular rate controller results are shown in Figures 7 through 10. Whorl−I was given an initial angular rate about the z body axis and then the MRP controller was turned on. The 16 second delay in the start of the controller at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels. Figure 7 provides the angular rate time history. Whorl−I has approximately zero angular rate in the Kowalchuk
Table 2: MRP Controller Parameters Parameter Value Control Rate 2 Hz Observe Rate 2 Hz Wheel Speed Query 10 Hz σ(t0 ) (0.002, 0.009, 0.069) ω(t0 ) (-0.015, -0.021, 0.199) rad/s K 1.5 kg-m2 /s2 [P ] diag( 3.64, 3.40, 4.41 ) kg-m2 /s Ωx 0 rad/s Ωy 0 rad/s Ωz 0 rad/s
z axis after 50 seconds into the simulation. The numerical simulation has approximately zero angular rate after 48 seconds into the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by the numerical simulation. After 50 seconds the noise from the angular rate measurements can be clearly seen. Figure 8 shows the MRP vector time history for Whorl−I and the numerical simulation. Whorl−I has approximately a zero MRP vector at 50 seconds into the simulation. The numerical simulation has approximately a zero vector at 48 seconds. The MRP vector for Whorl-I agrees very closely with the numerical simulation results. In Figure 8 the attitude observer returned two erroneous attitudes for Whorl−I as can be seen for σ3 at 21 and 42 seconds. The two erroneous attitudes are also seen in the control history. Figure 9 shows the required control torque specified by the angular rate controller. The control torques specified by the angular rate controller for Whorl−I have the same increasing control torque trend to zero as the numerical simulation. Figure 6 shows the reaction wheel speed time history for the controller running on Whorl−I and the numerical simulation. The reaction wheel aligned along the z axis for Whorl−I has a similar trend as the numerical simulation. The numerical simulation ended with an angular velocity of approximately 36 rad/s as compared with 40 rad/s for the z reaction wheel. As previously discussed in the angular rate controller results, the discrepancy in wheel speeds between Whorl−I and the numerical simulation may be contributed to the simplified reaction wheel mo7
Figure 7: Whorl−I and Numerical Simulation Angular Rates for MRP Controller
Figure 8: Whorl−I and Numerical Simulation MRP for MRP Controller
Figure 9: Whorl−I and Numerical Simulation Control Torque Results for MRP Controller
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Figure 10: Whorl−I and Numerical Simulation Reaction Wheel Rates for MRP Controller tor torque model used in the numerical simulation A Global Positioning System (GPS) closed-loop and shown in Eq. (19). hardware-in-the-loop simulator has been developed and integrated into DSACSS to investigate orbital control strategies.13,14 Whorl−I combined with the Conclusions GPS simulator will provide hardware-in-the-loop simulations that will facilitate development and inWhorl−I was given an initial angular rate, primar- vestigation of coupled attitude and orbital control ily about the body z axis, to investigate the per- strategies . formance of the angular rate and MRP controllers. The angular rate controller resulted in successfully driving the angular rates of Whorl−I to zero. The References control and angular rate time history matched the trends observed from a numerical simulation of the [1] B. J. Naasz, R. D. Burns, D. Gaylor, and angular rate controller with the same initial condiJ. Higinbotham, “Hardware-in-the-loop Testtions as Whorl−I. The MRP controller successfully ing of Continuous Conrol Algorithms for drove the MRP attitude vector of Whorl−I to zero. a Precision Formation Flying Demonstration The angular rates, MRP vector, and control time hisMission,” 18th International Symposium on tory closely matched the numerical simulation of the Space Flight Dynamics, Haus der Bayerischen MRP controller. Wirtschaft, Munich, Germany, October 11–15, The DSACSS framework is a flexible testbed 2004. for investigation of a variety of spacecraft control techniques, such as Lyapunov based attitude control laws. The advantage of DSACSS over software sim- [2] B. J. Naasz, C. D. Karlgaard, and C. D. Hall, “Application of Several Control Techniques for ulator architectures is that hardware-in-the-loop simthe Ionospheric Observation Nanosatellite Forulations for testing and validation of concepts promation,” 2002 AAS/AIAA Space Flight Mevide a more realistic demonstration of expected perchanics Conference, San Antonio, Texas, Janformance of real systems. uary 27–30, 2002.
Future Work Future plans are to develop and implement nonlinear attitude tracking controllers for Whorl−I and compare the performance to numerical simulations. Kowalchuk
[3] H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, “Spacecraft Formation Flying Control Using Mean Orbit Elements,” Journal of the Astronautical Sciences, Vol. 48, No. 1, 2000, pp. 69–89. 9
[4] S. Leung, E. Gill, O. Montenbruck, and [12] M. J. Sidi, Spacecraft Dynamics and Control. Cambridge University Press, 1997. S. Montenegro, “A Navigation Processor for Flexible Real-Time Formation Flying Applications,” International Symposium Formation [13] S. A. Kowalchuk and C. D. Hall, “Hardwarein-the-Loop Simulation of Classical Element Flying Missions and Technologies, Centre NaFeedback Controller,” Goddard Flight Metional d’Etudes Spatiales Toulouse Space Cenchanics Symposium, Greenbelt, Maryland, Octre - France, October 29–31, 2002. tober 18–20, 2005. [5] S. Kim, J. L. Crassidis, Y. Cheng, A. M. [14] S. A. Kowalchuk and C. D. Hall, “GPS Fosbury, and J. L. Junkins, “Kalman FilterHardware-in-the-loop Spacecraft Formation ing for Relative Spacecraft Attitude and PosiFlying Simulation,” AAS/AIAA Space Flight tion Estimation,” AIAA Guidance, Navigation, Mechanics Meeting, Sedona, Arizona, Jan. 28– and Control Conference and Exhibit, San FranFeb. 1, 2007. cisco, California, August 15–18, 2005. [6] M. C. VanDyke and C. D. Hall, “Decentralized Coordinated Attitude Control of a Formation of Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1101– 1109. [7] G. Q. Xing and S. A. Parvez, “Relative Attitude Kinematics and Dynamics Equations and Its Applications to Spacecraft Attitude State Capture and Tracking in Large Angle Slewing Maneuvers,” 1999 Space Control Conference, MIT Lincoln Laboratory, 1999. April 13-15. [8] F. L. Markley, J. L. Crassidis, and Y. Cheng, “Nonlinear Attitude Filtering Methods,” AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, August 15–18, 2005. [9] S. A. Kowalchuk and C. D. Hall, “Distributed Spacecraft Attitude Control System Simulator: Feedback Control Capabilities and Visualization Techniques,” 7th International Quantitative Feedback Theory and Robust Frequency Domain Symposium, University of Kansas, Lawrence, Kansas, August 5–7, 2005. [10] J. L. Schwartz and C. D. Hall, “The Distributed Spacecraft Attitude Control System Simulator: Development, Progress, Plans,” 2003 Flight Mechanics Symposium, NASA Goddard Space Flight Center, Greenbelt, Maryland, October 28–30, 2003. [11] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems. AIAA Education Series, 2003. Kowalchuk
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