SPHERES Development and Demonstrations of Close Proximity ...
SPHERES Development and Demonstrations of Close Proximity Formation Flight Maneuvers Brent E. Tweddle, Massachusetts Institute of Technology Alvar Saenz-Otero, Massachusetts Institute of Technology Space Systems Laboratory, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, 02139
spacecraft (e.g., all identical satellites are part of a sparse aperture, reconfigurable space based radar). The nature of these designs does not require precision formation flight, but rather requires coarse control to maintain a loose formation where the critical elements are to stay within range of each other and maintain a safe enough distance to prevent collisions and obstructions.
BIOGRAPHY Brent Tweddle is currently a Candidate for the Master of Science (S.M.) degree at the Massachusetts Institute of Technology (MIT) in the Department of Aeronautics and Astronautics and is a Research Assistant in the Space Systems Laboratory. Brent received a Bachelor’s of Applied Science (B.A.Sc. 2007) in Computer Engineering and Mechatronics from the University of Waterloo in Ontario, Canada.
This paper presents a method to remove the global navigation sensors for all but one satellite in the fractionated spacecraft system in order to reduce system cost and complexity. In this problem, all satellites must follow a globally defined trajectory in order to avoid collision with another object. Several tradeoffs and limitations associated with this approach and are discussed in this paper.
Dr. Alvar Saenz-Otero is a Research Scientist at the MIT Space Systems Laboratory and the SPHERES Lead Scientist. He obtained his doctoral degree from the MIT Department of Aeronautics and Astronautics in June 2005. He specializes in the use of the International Space Station for space technology maturation and the advancement of embedded system in aerospace.
INTRODUCTION The Synchronized Position Hold Engage Reorient Experimental Satellites (SPHERES) is an experimental hardware platform designed for the research and development of guidance, navigation and control algorithms. SPHERES currently operates aboard the International Space Station (ISS) [1]. The research problem presented in this paper is to develop and demonstrate a method to control the trajectory of an entire formation of satellites in the global frame using only relative navigation sensors [2] in order to minimize hardware requirements across the entire formation of spacecraft.
ABSTRACT The Synchronized Position Hold Engage Reorient Experimental Satellites (SPHERES) program, developed by the MIT Space Systems Laboratory, began operations aboard the International Space Station (ISS) in May 2006. SPHERES was designed as a research facility to demonstrate metrology, control, and autonomy algorithms for distributed satellite systems. By operating in the risktolerant environment of the ISS, SPHERES allows researchers to push the limits of their algorithms. The motivation for SPHERES formation flight tests arises from the desire to develop fractionated spacecraft, which use loosely coupled formation flight to maintain multiple satellites in close proximity to achieve a common goal. Fractionated spacecraft allow better adaptability, survivability and payload isolation than a large single-unit spacecraft. The system may consist of heterogeneous spacecraft (e.g., only one high-power communications satellite for ground telemetry, while all other use lowpower inter-satellite communications) or homogenous
ION NTM 2008, 28-30 January 2008, San Diego, CA
Typical challenges in spacecraft relative navigation involve controlling the separation between satellites such as in a space-based interferometer or autonomous docking system [2,3]. These types of missions are only concerned with the relative positions of the satellites in the formation while the global position of the satellites is not important. A number of missions exist where the global position of all of the satellites in the formation is more important than their relative positions. Future inspection missions might
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position and attitude. Twelve carbon dioxide thrusters are each capable of producing 125 mN of thrust and allow the satellites to maneuver with six degrees of freedom. The control algorithms are typically run at a rate of 1 Hz, but are capable of executing at 20 Hz.
require a formation of satellites to inspect another spacecraft or small asteroid. In this type of inspection mission it is more important to control the global position of each satellite in order to ensure there are no collisions than to control the relative position between each satellite. However, cost and complexity savings could be made if the majority of satellites utilized relative sensors. This paper discusses how to control the trajectory of an entire formation of satellites in the global frame using only relative navigation sensors. In this paper, we begin by presenting the motivation for the development of SPHERES, an overview of the hardware system and a brief outline of its pseudo-GPS global metrology system. This paper presents a navigation and control architecture to control the global position of a fractionated satellite system using only relative position sensors. A ‘leader-follower’ architecture is proposed that minimizes hardware complexity. Simulation and experimental results are presented from the air table testbed at the MIT Space Systems Laboratory. An investigation of the limitations of this approach is presented and methods to mitigate these limitations are described.
Figure 1: SPHERES Hardware The SPHERES hardware is capable of operating in multiple test facilities. The MIT Space Systems Laboratory has an air table, shown in Figure 2, which provides three degrees of freedom. The KC-135, parabolic flight aircraft provides short duration tests in a micro-gravity environment. The International Space Station (ISS) allows for long duration tests in a six degree-of-freedom micro-gravity environment. Figure 3 shows an astronaut operating the SPHERES platform in US Node 1.
SPHERES INTRODUCTION The primary purpose of the Synchronized Position Hold Engage Reorient Experimental Satellites is to mature and develop guidance, navigation and control (GN&C) algorithms. Many distributed satellite systems require new algorithms for maintaining their relative formation in order to complete their mission. SPHERES has been developed as a generic platform that can assist in the maturation of the algorithms required for these types of missions. In order to achieve this objective, SPHERES was designed to support the incremental maturation of guidance, navigation and control algorithms in a risktolerant and representative environment. One driving factor in the design of SPHERES is the capability to perform iterative research so that hypotheses can be developed, tested and modified in a simple and expedient manner. Additionally, the capability to support multiple researchers through a Guest Scientist Program has allowed scientists outside of MIT to run experiments on SPHERES. An expansion port that allows sciencespecific payloads to interface with the SPHERES onboard computer complements this capability. This type of system can be used to test a variety of technologies such as active docking, tethered flight or optical navigation [3].
Figure 2: Air Table at MIT
SPHERES HARDWARE A SPHERES micro-satellite system is shown in Figure 1. Each satellite weighs four kilograms, is 20 centimeters in diameter and uses cold gas propulsion to control its
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algorithms utilizes two SPHERES satellites. One satellite is designated as the leader satellite and utilizes global metrology. This simulates a typical modern satellite that would use GPS or radio tracking for positional navigation. A second SPHERES satellite is designated as the follower satellite and uses relative measurements for positional navigation. This simulates a satellite that does not use GPS or radio tracking and only knows its position relative to the leader satellite. In order to minimize hardware requirements, no control data can be transmitted between the leader and follower satellites. A few simplifications have been made to this problem. Rather than using actual relative navigation sensors on the follower satellite, the relative position is calculated by subtracting the global position of the follower satellite from the global position of the leader satellite. It is assumed that attitude navigation and control is performed in the global frame. Figure 4 shows the relative navigation geometry that is used in this approach. The leader and follower SPHERES are represented by octagons labeled with the letter ‘L’ and ‘F’ respectively. The orange curve depicts the desired global trajectory, which is time varying and is open loop trajectory (i.e. it does not depend on the current position of the satellites). The desired leader and follower global positions at a given instant in time are indicated with orange vectors. It is useful for later discussions to consider the angle between the leader and follower desired global positions as a phase difference. A typical proportion, integral derivative (PID) feedback loop is closed on the leader global error, which is shown in green in Figure 4.
Figure 3: SPHERES onboard ISS The SPHERES global metrology (i.e. position and attitude determination) system provides global navigation information that simulates a typical GPS/INS system; it provides full 6DOF state information with respect to an inertial frame [4]. and is designed to operate in a shirtsleeve environment. The system employs ultrasonic transmitters placed at known and fixed locations. Ultrasonic receivers are mounted on the SPHERES satellites and measure the time-of-flight of the signals in order to calculate position and attitude. The clocks on all components of the system are synchronized using infrared signals that are transmitted by one of the SPHERES satellite. Gyroscopes are incorporated in the global metrology system to enhance the state propagation algorithm. An extended Kalman filter is implemented to integrate the ultrasonic system and gyroscope measurements to provide a navigation solution at a maximum rate of 5 Hz with a position accuracy of +/- 5 mm and an attitude accuracy of +/- 1 degree.
The follower’s actual relative position, shown in purple, is the measurement that would be made by a relative navigation sensor. This can also be considered the leader satellite’s position with respect to the body (or local) frame of the leader satellite. The follower desired relative position is what that measurement should be based on the desired trajectory. The green vector labeled as the follower relative error is calculated by subtracting the follower desired relative position from the follower actual relative position. This assumes that the leader satellite perfectly tracks its desired trajectory and that the leader global error is zero. If the leader is slightly disturbed and has a non-zero error, the follower satellite will track this disturbance, resulting in an undesirable offset. Any errors in the position of the leader satellite will be propagated to the follower satellite.
RELATIVE NAVIGATION RESEARCH PROBLEM The principal research objective of the experiments discussed in this paper is to develop and demonstrate a method to control the trajectory an entire formation of satellites in the global frame using only relative sensors. This should be done while minimizing the hardware requirements across the entire formation of satellites. Typically, global sensors would be used to control global trajectories, however savings in cost and complexity can be made if relative sensors are used instead of global sensors. This approach is especially advantageous when considering large formations of satellites. The tradeoff of this approach is that this method introduces complexities in the GN&C system. RELATIVE NAVIGATION APPROACH The experimental approach used to develop and demonstrate these relative navigation and control
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blue global vectors respectively. The leader satellite has a constant offset that causes the actual trajectory to be outside the desired trajectory. This is due to a lack of feed-forward of the centripetal acceleration [5].
Figure 4: Relative Navigation Geometry SIMULATION RESULTS These navigation and control algorithms were implemented in a Matlab simulation of the SPHERES satellites that was developed by the MIT Space Systems Laboratory. Both satellites were commanded to follow a circular trajectory while maintaining a fixed attitude in the global frame.
Figure 6: Simulated Leader Global Trajectory Figure 7 shows the follower satellite’s global desired and actual trajectory, which began 180 degrees out of phase with the leader satellite. In Figure 4, these are the actual follower global position (blue) and the desired follower position (orange) respectively. In this case, the actual trajectory is inside the desired trajectory. This is due to the fact that the leader satellites global error vector was non-zero and was added to the follower satellite’s relative error. The vector addition of these errors results in a phase cancellation that, based on the phase difference, can either cause constructive or destructive interference. In the case shown in Figure 7 the actual trajectory results in a destructive interference that reduces the radius of the desired global circular trajectory.
Figure 5 shows the desired and actual follower relative position in the X and Y axis. These vectors are shown in purple in Figure 4. The plot shows that the desired relative trajectory is closely tracked and that the only minor errors are a result of additive white Gaussian noise and startup errors.
In addition to the phase cancellation, the actual global trajectory of the follower satellite contains more noise than the leader satellite. This is due to the fact that the noise of the leader satellites actual position is added to the noise of the follower satellite’s actual position. It is important to notice that the addition of an acceptable level of trajectory error in the follower relative position (shown in Figure 5) and an acceptable level of trajectory error in the leader global position (shown in Figure 6) result in an acceptable level of trajectory error in the follower global position (shown in Figure 7). In the experimental results shown in the next section, this will not be the case.
Figure 5: Simulated Follower Relative Position Figure 6 shows the leader satellite’s global desired and actual trajectory. In Figure 4, these are the orange and
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Figure 7: Simulated Follower Global Position Figure 8: Experimental Follower Relative Position EXPERIMENTAL RESULTS The plot in Figure 9 shows the global trajectory that was followed by the leader satellite. Although the trajectory has significant tracking errors it is interesting to observe how these errors were translated to the follower satellite.
These navigation and control algorithms were implemented on the SPHERES hardware and tested on the MIT Space Systems Laboratory’s air table. The trajectory the satellites were programmed to follow was a circle followed by a figure eight over the course of seven minutes. This required the SPHERES to travel with a constant speed of approximately two cm/s, which is a significantly fast rate for the satellites, and increases the magnitude of dynamic frictional forces. The MIT SSL air table provides a low friction testing environment where SPHERES satellites are capable of generating accurate trajectory tracking results. Previous tests [5] have shown between two SPHERES have demonstrated precision control. However high precision tests typically require a large effort to level the air table, minimize the effects of the curvature and maintain the flat surface smooth. The tests presented in this paper did not account for such details. As such the global position of the satellites was not expected to be precise; however, one should recall that fractionated spacecraft do not require global control. Figure 8 shows the actual and desired relative trajectory of the follower satellite with respect to the leader body frame. The desired trajectory was closely tracked with small errors that were primarily due to disturbances from the air table.
Figure 9: Experimental Leader Global Position The global trajectory of the follower satellite is shown in Figure 10. It is clear that the follower satellite has significant difficulties in following the global trajectory. These results show that the disturbances and noise are
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compounded in the follower satellite to the point where the actual trajectory is even close to the desired trajectory. Additionally, the effect of a phase difference of 120 degrees completely cancels the circle in the X-Axis. This destructive interference produces an almost perfectly straight horizontal line near the bottom of the plot.
In the simulation, all of the results were shown to have an acceptable level of trajectory tracking error. However, the experimental results show that the addition of an acceptable level of trajectory error in the follower relative position (shown in Figure 8) and an acceptable level of trajectory error in the leader global position (shown in Figure 9) result in an unacceptable level of trajectory error in the follower global position (shown in Figure 10). This is different from the simulation results, and although is likely due to the effects of the disturbances, this discrepancy warrants further investigation.
It is interesting to observe that the relative trajectory shown in Figure 8 is tracked much more closely than the global position of the leader satellite shown in Figure 9. It seems intuitive that the errors should be equal in both cases since they are performed on the same air table. Small scratches in the air table cause many of the tracking errors observed. These will typically slow the satellite down to a stop, where static friction becomes the primary force that opposes motion. When this occurs the direction of the leader’s error vector changes very slowly in direction, and the satellite continues to be stuck. However, the follower satellite’s error vector changes direction much more quickly and causes the satellite to fire its thrusters in a different direction. Since the static friction force caused by the small scratches are typically very directional, the follower satellite can typically free itself in a much shorter time frame than the leader satellite. While the air table is not applicable to space, the results do show that the way potential errors add from a leader to a follower may result in unacceptable results. Even if the leader trajectory is acceptable, and the following error is acceptable, it can result in an unacceptable global follower trajectory. At the same time, if one is only concerned with relative control between the satellites, the error can be acceptable.
CONCLUSION This paper introduces an approach for controlling global trajectories using a relative navigation system. A leaderfollower approach applicable to space missions that require a fractionated formation of spacecraft is presented. Simulation and hardware tests demonstrated the effects of error propagation from the leader satellite to the follower satellite. From the investigation of phase cancellation, noise addition and disturbance propagation it can be concluded that any satellite systems that utilize this leader-follower architecture should be designed to minimize the Leader’s trajectory errors. This can be done in a number of ways. For instance, the bandwidth of the desired trajectory can be reduced, the control authority can be increased, disturbances can be minimized or highly tuned control algorithms can be employed. FUTURE WORK Future work will include a number of different areas. Future research will investigate error performance experimentally by characterizing the addition of noise over a series of follower satellites and demonstrating accurate global tracking in the microgravity environment on the ISS. Other possibilities include transmitting the global error of the leader satellite, which allows determination of the “actual” relative distance (expand this idea, its important). Further, consideration will be given to use the relative sensors hardware available on the SPHERES satellites (rather than differencing the global positions). An additional area of future research is the investigation of global attitude navigation and control using only relative sensors. ACKNOWLEDGMENTS The authors wish to acknowledge the complete SPHERES team at MIT and Aurora Flight Sciences, as well as the DoD and its Space Test Programs office for their funding, launch opportunities, and continued operational support.
Figure 10: Experimental Follower Global Position
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REFERENCES [1] J. R. O´Donnell, Jr., M. Concha, et al, Space Technology 5 Launch and Operations, Paper AAS 07091, 2007 AAS Guidance, Navigation and Controls Conference, Breckenridge, CO (February 2007). [2] W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, (2003) [3] S. Nolet, A. Saenz-Otero, D. W. Miller, A. Fejzic, SPHERES Operations aboard the ISS: Maturation of GN&C Algorithms in Microgravity, Paper AAS 07-042, 2007 AAS Guidance, Navigation and Controls Conference, Breckenridge, CO (February 2007) [4] S. Nolet, The SPHERES Navigation System: from Early Development to On-Orbit Testing, AIAA Guidance, Navigation & Control, Hilton Head, SC (August 2007) [5] C. Mandy, A. Saenz-Otero, D. W. Miller, Satellite Formation Flight and Realignment Maneuver Demonstration aboard the International Space Station, SPIE Optics & Photonics, San Diego, CA (August 2007) [6] S. Nolet, D. W. Miller, Autonomous Docking Experiments Using the SPHERES testbed inside the ISS, SPIE Sensors and Systems for Space Applications, edited by R. T. Howard, R. D. Richards, SPIE Vol 6555 65550P1 (August 2007)
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spacecraft (e.g., all identical satellites are part of a sparse aperture, reconfigurable space based radar). The nature of these designs does not require precision formation flight, but rather requires coarse control to maintain a loose formation where the critical elements are to stay within range of each other and maintain a safe enough distance to prevent collisions and obstructions.
BIOGRAPHY Brent Tweddle is currently a Candidate for the Master of Science (S.M.) degree at the Massachusetts Institute of Technology (MIT) in the Department of Aeronautics and Astronautics and is a Research Assistant in the Space Systems Laboratory. Brent received a Bachelor’s of Applied Science (B.A.Sc. 2007) in Computer Engineering and Mechatronics from the University of Waterloo in Ontario, Canada.
This paper presents a method to remove the global navigation sensors for all but one satellite in the fractionated spacecraft system in order to reduce system cost and complexity. In this problem, all satellites must follow a globally defined trajectory in order to avoid collision with another object. Several tradeoffs and limitations associated with this approach and are discussed in this paper.
Dr. Alvar Saenz-Otero is a Research Scientist at the MIT Space Systems Laboratory and the SPHERES Lead Scientist. He obtained his doctoral degree from the MIT Department of Aeronautics and Astronautics in June 2005. He specializes in the use of the International Space Station for space technology maturation and the advancement of embedded system in aerospace.
INTRODUCTION The Synchronized Position Hold Engage Reorient Experimental Satellites (SPHERES) is an experimental hardware platform designed for the research and development of guidance, navigation and control algorithms. SPHERES currently operates aboard the International Space Station (ISS) [1]. The research problem presented in this paper is to develop and demonstrate a method to control the trajectory of an entire formation of satellites in the global frame using only relative navigation sensors [2] in order to minimize hardware requirements across the entire formation of spacecraft.
ABSTRACT The Synchronized Position Hold Engage Reorient Experimental Satellites (SPHERES) program, developed by the MIT Space Systems Laboratory, began operations aboard the International Space Station (ISS) in May 2006. SPHERES was designed as a research facility to demonstrate metrology, control, and autonomy algorithms for distributed satellite systems. By operating in the risktolerant environment of the ISS, SPHERES allows researchers to push the limits of their algorithms. The motivation for SPHERES formation flight tests arises from the desire to develop fractionated spacecraft, which use loosely coupled formation flight to maintain multiple satellites in close proximity to achieve a common goal. Fractionated spacecraft allow better adaptability, survivability and payload isolation than a large single-unit spacecraft. The system may consist of heterogeneous spacecraft (e.g., only one high-power communications satellite for ground telemetry, while all other use lowpower inter-satellite communications) or homogenous
ION NTM 2008, 28-30 January 2008, San Diego, CA
Typical challenges in spacecraft relative navigation involve controlling the separation between satellites such as in a space-based interferometer or autonomous docking system [2,3]. These types of missions are only concerned with the relative positions of the satellites in the formation while the global position of the satellites is not important. A number of missions exist where the global position of all of the satellites in the formation is more important than their relative positions. Future inspection missions might
217
position and attitude. Twelve carbon dioxide thrusters are each capable of producing 125 mN of thrust and allow the satellites to maneuver with six degrees of freedom. The control algorithms are typically run at a rate of 1 Hz, but are capable of executing at 20 Hz.
require a formation of satellites to inspect another spacecraft or small asteroid. In this type of inspection mission it is more important to control the global position of each satellite in order to ensure there are no collisions than to control the relative position between each satellite. However, cost and complexity savings could be made if the majority of satellites utilized relative sensors. This paper discusses how to control the trajectory of an entire formation of satellites in the global frame using only relative navigation sensors. In this paper, we begin by presenting the motivation for the development of SPHERES, an overview of the hardware system and a brief outline of its pseudo-GPS global metrology system. This paper presents a navigation and control architecture to control the global position of a fractionated satellite system using only relative position sensors. A ‘leader-follower’ architecture is proposed that minimizes hardware complexity. Simulation and experimental results are presented from the air table testbed at the MIT Space Systems Laboratory. An investigation of the limitations of this approach is presented and methods to mitigate these limitations are described.
Figure 1: SPHERES Hardware The SPHERES hardware is capable of operating in multiple test facilities. The MIT Space Systems Laboratory has an air table, shown in Figure 2, which provides three degrees of freedom. The KC-135, parabolic flight aircraft provides short duration tests in a micro-gravity environment. The International Space Station (ISS) allows for long duration tests in a six degree-of-freedom micro-gravity environment. Figure 3 shows an astronaut operating the SPHERES platform in US Node 1.
SPHERES INTRODUCTION The primary purpose of the Synchronized Position Hold Engage Reorient Experimental Satellites is to mature and develop guidance, navigation and control (GN&C) algorithms. Many distributed satellite systems require new algorithms for maintaining their relative formation in order to complete their mission. SPHERES has been developed as a generic platform that can assist in the maturation of the algorithms required for these types of missions. In order to achieve this objective, SPHERES was designed to support the incremental maturation of guidance, navigation and control algorithms in a risktolerant and representative environment. One driving factor in the design of SPHERES is the capability to perform iterative research so that hypotheses can be developed, tested and modified in a simple and expedient manner. Additionally, the capability to support multiple researchers through a Guest Scientist Program has allowed scientists outside of MIT to run experiments on SPHERES. An expansion port that allows sciencespecific payloads to interface with the SPHERES onboard computer complements this capability. This type of system can be used to test a variety of technologies such as active docking, tethered flight or optical navigation [3].
Figure 2: Air Table at MIT
SPHERES HARDWARE A SPHERES micro-satellite system is shown in Figure 1. Each satellite weighs four kilograms, is 20 centimeters in diameter and uses cold gas propulsion to control its
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algorithms utilizes two SPHERES satellites. One satellite is designated as the leader satellite and utilizes global metrology. This simulates a typical modern satellite that would use GPS or radio tracking for positional navigation. A second SPHERES satellite is designated as the follower satellite and uses relative measurements for positional navigation. This simulates a satellite that does not use GPS or radio tracking and only knows its position relative to the leader satellite. In order to minimize hardware requirements, no control data can be transmitted between the leader and follower satellites. A few simplifications have been made to this problem. Rather than using actual relative navigation sensors on the follower satellite, the relative position is calculated by subtracting the global position of the follower satellite from the global position of the leader satellite. It is assumed that attitude navigation and control is performed in the global frame. Figure 4 shows the relative navigation geometry that is used in this approach. The leader and follower SPHERES are represented by octagons labeled with the letter ‘L’ and ‘F’ respectively. The orange curve depicts the desired global trajectory, which is time varying and is open loop trajectory (i.e. it does not depend on the current position of the satellites). The desired leader and follower global positions at a given instant in time are indicated with orange vectors. It is useful for later discussions to consider the angle between the leader and follower desired global positions as a phase difference. A typical proportion, integral derivative (PID) feedback loop is closed on the leader global error, which is shown in green in Figure 4.
Figure 3: SPHERES onboard ISS The SPHERES global metrology (i.e. position and attitude determination) system provides global navigation information that simulates a typical GPS/INS system; it provides full 6DOF state information with respect to an inertial frame [4]. and is designed to operate in a shirtsleeve environment. The system employs ultrasonic transmitters placed at known and fixed locations. Ultrasonic receivers are mounted on the SPHERES satellites and measure the time-of-flight of the signals in order to calculate position and attitude. The clocks on all components of the system are synchronized using infrared signals that are transmitted by one of the SPHERES satellite. Gyroscopes are incorporated in the global metrology system to enhance the state propagation algorithm. An extended Kalman filter is implemented to integrate the ultrasonic system and gyroscope measurements to provide a navigation solution at a maximum rate of 5 Hz with a position accuracy of +/- 5 mm and an attitude accuracy of +/- 1 degree.
The follower’s actual relative position, shown in purple, is the measurement that would be made by a relative navigation sensor. This can also be considered the leader satellite’s position with respect to the body (or local) frame of the leader satellite. The follower desired relative position is what that measurement should be based on the desired trajectory. The green vector labeled as the follower relative error is calculated by subtracting the follower desired relative position from the follower actual relative position. This assumes that the leader satellite perfectly tracks its desired trajectory and that the leader global error is zero. If the leader is slightly disturbed and has a non-zero error, the follower satellite will track this disturbance, resulting in an undesirable offset. Any errors in the position of the leader satellite will be propagated to the follower satellite.
RELATIVE NAVIGATION RESEARCH PROBLEM The principal research objective of the experiments discussed in this paper is to develop and demonstrate a method to control the trajectory an entire formation of satellites in the global frame using only relative sensors. This should be done while minimizing the hardware requirements across the entire formation of satellites. Typically, global sensors would be used to control global trajectories, however savings in cost and complexity can be made if relative sensors are used instead of global sensors. This approach is especially advantageous when considering large formations of satellites. The tradeoff of this approach is that this method introduces complexities in the GN&C system. RELATIVE NAVIGATION APPROACH The experimental approach used to develop and demonstrate these relative navigation and control
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blue global vectors respectively. The leader satellite has a constant offset that causes the actual trajectory to be outside the desired trajectory. This is due to a lack of feed-forward of the centripetal acceleration [5].
Figure 4: Relative Navigation Geometry SIMULATION RESULTS These navigation and control algorithms were implemented in a Matlab simulation of the SPHERES satellites that was developed by the MIT Space Systems Laboratory. Both satellites were commanded to follow a circular trajectory while maintaining a fixed attitude in the global frame.
Figure 6: Simulated Leader Global Trajectory Figure 7 shows the follower satellite’s global desired and actual trajectory, which began 180 degrees out of phase with the leader satellite. In Figure 4, these are the actual follower global position (blue) and the desired follower position (orange) respectively. In this case, the actual trajectory is inside the desired trajectory. This is due to the fact that the leader satellites global error vector was non-zero and was added to the follower satellite’s relative error. The vector addition of these errors results in a phase cancellation that, based on the phase difference, can either cause constructive or destructive interference. In the case shown in Figure 7 the actual trajectory results in a destructive interference that reduces the radius of the desired global circular trajectory.
Figure 5 shows the desired and actual follower relative position in the X and Y axis. These vectors are shown in purple in Figure 4. The plot shows that the desired relative trajectory is closely tracked and that the only minor errors are a result of additive white Gaussian noise and startup errors.
In addition to the phase cancellation, the actual global trajectory of the follower satellite contains more noise than the leader satellite. This is due to the fact that the noise of the leader satellites actual position is added to the noise of the follower satellite’s actual position. It is important to notice that the addition of an acceptable level of trajectory error in the follower relative position (shown in Figure 5) and an acceptable level of trajectory error in the leader global position (shown in Figure 6) result in an acceptable level of trajectory error in the follower global position (shown in Figure 7). In the experimental results shown in the next section, this will not be the case.
Figure 5: Simulated Follower Relative Position Figure 6 shows the leader satellite’s global desired and actual trajectory. In Figure 4, these are the orange and
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Figure 7: Simulated Follower Global Position Figure 8: Experimental Follower Relative Position EXPERIMENTAL RESULTS The plot in Figure 9 shows the global trajectory that was followed by the leader satellite. Although the trajectory has significant tracking errors it is interesting to observe how these errors were translated to the follower satellite.
These navigation and control algorithms were implemented on the SPHERES hardware and tested on the MIT Space Systems Laboratory’s air table. The trajectory the satellites were programmed to follow was a circle followed by a figure eight over the course of seven minutes. This required the SPHERES to travel with a constant speed of approximately two cm/s, which is a significantly fast rate for the satellites, and increases the magnitude of dynamic frictional forces. The MIT SSL air table provides a low friction testing environment where SPHERES satellites are capable of generating accurate trajectory tracking results. Previous tests [5] have shown between two SPHERES have demonstrated precision control. However high precision tests typically require a large effort to level the air table, minimize the effects of the curvature and maintain the flat surface smooth. The tests presented in this paper did not account for such details. As such the global position of the satellites was not expected to be precise; however, one should recall that fractionated spacecraft do not require global control. Figure 8 shows the actual and desired relative trajectory of the follower satellite with respect to the leader body frame. The desired trajectory was closely tracked with small errors that were primarily due to disturbances from the air table.
Figure 9: Experimental Leader Global Position The global trajectory of the follower satellite is shown in Figure 10. It is clear that the follower satellite has significant difficulties in following the global trajectory. These results show that the disturbances and noise are
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compounded in the follower satellite to the point where the actual trajectory is even close to the desired trajectory. Additionally, the effect of a phase difference of 120 degrees completely cancels the circle in the X-Axis. This destructive interference produces an almost perfectly straight horizontal line near the bottom of the plot.
In the simulation, all of the results were shown to have an acceptable level of trajectory tracking error. However, the experimental results show that the addition of an acceptable level of trajectory error in the follower relative position (shown in Figure 8) and an acceptable level of trajectory error in the leader global position (shown in Figure 9) result in an unacceptable level of trajectory error in the follower global position (shown in Figure 10). This is different from the simulation results, and although is likely due to the effects of the disturbances, this discrepancy warrants further investigation.
It is interesting to observe that the relative trajectory shown in Figure 8 is tracked much more closely than the global position of the leader satellite shown in Figure 9. It seems intuitive that the errors should be equal in both cases since they are performed on the same air table. Small scratches in the air table cause many of the tracking errors observed. These will typically slow the satellite down to a stop, where static friction becomes the primary force that opposes motion. When this occurs the direction of the leader’s error vector changes very slowly in direction, and the satellite continues to be stuck. However, the follower satellite’s error vector changes direction much more quickly and causes the satellite to fire its thrusters in a different direction. Since the static friction force caused by the small scratches are typically very directional, the follower satellite can typically free itself in a much shorter time frame than the leader satellite. While the air table is not applicable to space, the results do show that the way potential errors add from a leader to a follower may result in unacceptable results. Even if the leader trajectory is acceptable, and the following error is acceptable, it can result in an unacceptable global follower trajectory. At the same time, if one is only concerned with relative control between the satellites, the error can be acceptable.
CONCLUSION This paper introduces an approach for controlling global trajectories using a relative navigation system. A leaderfollower approach applicable to space missions that require a fractionated formation of spacecraft is presented. Simulation and hardware tests demonstrated the effects of error propagation from the leader satellite to the follower satellite. From the investigation of phase cancellation, noise addition and disturbance propagation it can be concluded that any satellite systems that utilize this leader-follower architecture should be designed to minimize the Leader’s trajectory errors. This can be done in a number of ways. For instance, the bandwidth of the desired trajectory can be reduced, the control authority can be increased, disturbances can be minimized or highly tuned control algorithms can be employed. FUTURE WORK Future work will include a number of different areas. Future research will investigate error performance experimentally by characterizing the addition of noise over a series of follower satellites and demonstrating accurate global tracking in the microgravity environment on the ISS. Other possibilities include transmitting the global error of the leader satellite, which allows determination of the “actual” relative distance (expand this idea, its important). Further, consideration will be given to use the relative sensors hardware available on the SPHERES satellites (rather than differencing the global positions). An additional area of future research is the investigation of global attitude navigation and control using only relative sensors. ACKNOWLEDGMENTS The authors wish to acknowledge the complete SPHERES team at MIT and Aurora Flight Sciences, as well as the DoD and its Space Test Programs office for their funding, launch opportunities, and continued operational support.
Figure 10: Experimental Follower Global Position
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REFERENCES [1] J. R. O´Donnell, Jr., M. Concha, et al, Space Technology 5 Launch and Operations, Paper AAS 07091, 2007 AAS Guidance, Navigation and Controls Conference, Breckenridge, CO (February 2007). [2] W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, (2003) [3] S. Nolet, A. Saenz-Otero, D. W. Miller, A. Fejzic, SPHERES Operations aboard the ISS: Maturation of GN&C Algorithms in Microgravity, Paper AAS 07-042, 2007 AAS Guidance, Navigation and Controls Conference, Breckenridge, CO (February 2007) [4] S. Nolet, The SPHERES Navigation System: from Early Development to On-Orbit Testing, AIAA Guidance, Navigation & Control, Hilton Head, SC (August 2007) [5] C. Mandy, A. Saenz-Otero, D. W. Miller, Satellite Formation Flight and Realignment Maneuver Demonstration aboard the International Space Station, SPIE Optics & Photonics, San Diego, CA (August 2007) [6] S. Nolet, D. W. Miller, Autonomous Docking Experiments Using the SPHERES testbed inside the ISS, SPIE Sensors and Systems for Space Applications, edited by R. T. Howard, R. D. Richards, SPIE Vol 6555 65550P1 (August 2007)
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