Irina Gavrilovich
email: irina.gavrilovich@lirmm.fr

Sébastien Krut

Marc Gouttefarde

François Pierrot

Laurent Dusseau
email: laurent.dusseau@um2.fr

-I. Gavrilovich 1 TEST BENCH for NANOSATELLITE ATTITUDE DETERMINATION and CONTROL SYSTEM GROUND TESTS

This paper presents recent progress in designing a three degree-of-freedom (3DoF) test bench for CubeSats. Such test benches offer many prospects for ground testing of CubeSats and their subsystems, and first of all Attitude Determination and Control System (ADCS). The difficulty of ground satellite tests is to create conditions simulating space environment with given accuracy while limiting the effects occurred on ground facilities. For nanosatellites, the accuracy of ground tests has to be very high due to reduced capability of actuators and high sensitivity to external forces. The main challenge of this research is to find a design solution allowing a nanosatellite to rotate freely around three axes with minimum disturbances from gravity and friction forces. We propose two ways to deal with these difficulties and avoid any limit in rotational motion. The first one is the active compensation of parasitic moments acting on CubeSat during the tests. The second one consists of an air bearing sphere structure providing a contactless suspension for all possible angular position of the CubeSat. In prospect, both approaches can be used together in one test bench to improve its efficiency. General aspects of the design and initial confirmation of system efficiency are presented.

INTRODUCTION

Since CubeSat-class of nanosatellites was started in 1999 it became widespread and a significant trend. Due to low cost and a number of off-the-shelf components, the development of such satellites became common, especially among school and universities. Building a CubeSat takes less time than needed to realize a nanosatellite from scratch. Hence, developers can focus on scientific payload integration and students can lead the project through all stages during their universities years. Despite small sizes and standardized construction, CubeSats are useful to solve wide range of tasks in different fields of space exploration -communication, earth and near-earth space observation, scientific missions. Passive ADCS can be enough for CubeSats that do not need accurate positioning on its orbit, but the more complex missions require precise ADCS. Traditional ADCS, designed for full-scale satellites, cannot be used for nanosatellite needs due to their sizes, price or power consumption. CubeSat developers have to build ADCS using standard hardware or design their own system. ADCS is a sophisticated system which needs validation testing to prove its operability and verify software compatibility with hardware and other subsystems. This procedure is widely used for fullsize satellites, but testing of nanosatellite comes with many difficulties. The efforts required to set up nanosatellite ADCS ground testing are generally comparable to the efforts needed for building a CubeSat. Although every element of the system can be individually verified before flight and computer simulations can be made, only satellite operation in space environment shows if all components work together correctly. Otherwise, one small mistake often leads to mission failure. Despite a CubeSat is relatively inexpensive, one failure on the orbit is costly in terms of money and time due to payload preparation and launch. After a number of CubeSats loss on the orbit, the necessity of ground testing system for nanosatellite-class ADCS became evident. Full-scale satellite developers have arrived at the same conclusion when first generation of satellites with complex orientation system was built some fifty years ago. Since then, the technology of ADCS ground testing evolved and became a common procedure for full-size satellites. For correct operation and valid results of the testing, precision of the ADCS test bench has to correlate with the precision of the satellite control system, the capabilities of its actuators and the magnitudes of disturbing torques acting on the satellite on its orbit. Unfortunately, it cannot be simply scaled down. Due to this limitation, test benches designed for full-scale satellites are not efficient for CubeSats.

Figure 1. Tabletop, umbrella and "dumbbell" types of air bearing platforms [START_REF] Schwartz | Historical Review of Air-Bearing Spacecraft Simulator[END_REF]: a sphere represents an air bearing; a disk represents a table for a payload

The simplest type of test bench use wire suspension. A wire is connected to a satellite (or a stack of ADCS components) through its center of gravity and provides rotation around one axis [START_REF] Karpenko | Laboratory Facility for Testing of Micro-and Nanosatellite Attitude Control Systems[END_REF]. Such a wire suspension is simple to build but only one DoF is usually not enough for correct test results.

The most widespread and reliable approach to ADCS test bench is using air bearing table, that allows motions with extremely small friction and let satellites freely rotate around at least one axis. Accordingly, three types of air bearing tables can be distinguished [START_REF] Schwartz | Historical Review of Air-Bearing Spacecraft Simulator[END_REF] tabletop, umbrella and "dumbbell", as shown in Fig. 1. All types provide full rotation in yaw axis. For tabletop and umbrella configurations, pitch and roll rotation ranges are typically constrained to less than ±90°.

The "dumbbell" table configuration increases free motion in pitch and roll, but limits the range of possible applications. Since dumbbell has two self-balanced ends, testing of separated components is more convenient than testing of an assembled satellite.

Figure 2. NanoSat Air Bearing Platform with only one vertical rotation axis [START_REF]NanoSat Air Bearing Flyer[END_REF] Nowadays, there are not many test benches suitable for CubeSats ADCS ground testing. As mentioned above, such test benches have some special features due to CubeSat sizes and have to be specifically designed. One example is an air bearing test bench designed in York University, Canada [START_REF] Ustrzycki | Spherical Air Bearing Attitude Control Simulator for Nanosatellites[END_REF]. This test bed provides 3 DoF (45° for roll and pitch rotations and 360° for yaw) and load capacity up to 9 kg. Another example is the NanoSat Air Bearing platform of Berlin Space Technologies [START_REF]NanoSat Air Bearing Flyer[END_REF]. It is a 1 DoF system suitable for 1-3U CubeSats and capable to simulate magnetic field and to test sun sensors (Fig. 2). The University Space Center of Montpellier and Nîme (CSU) is working for a few years on developing CubeSats for different purposes and needs a nanosatellite test bench for ADCS ground testing. Since both 1U and 3U CubeSats are designed at CSU, the test bench has to be suitable for 1-3U. According to present-day needs and willing to improve current test bench characteristics, testing platform has to provide 3 unlimited rotational DoF and challenging disturbing torques of less than 10 -5 Nm. In this paper, possible approaches satisfying those requirements are presented.

DESIGN STRATEGY

Requirements

The requirements to design a CubeSat ADCS test bench are determined by ADCS sensitivity and actuators capabilities. Those characteristics define the maximum value of the test bench disturbing torque which has no tangible influence on the test results. For a realistic simulation, this maximum disturbing torque has to be two orders of magnitude below the maximum control torque available in the spacecraft or one order of magnitude below the total expected external torque on the orbit [5].

According to calculations for the 3U CubeSat Robusta-3a designed at CSU, external torques on the orbit do not exceed 10 -4 Nm. Additionally, current off-the-shelf CubeSat ADCS system were reviewed to make the test bench suitable for any other satellite of this class. The control torque of the majority of 1-3U CubeSat reaction wheel blocks ranges from 0.6 to 1 mNm. Thus, the order of magnitude of the maximum uncompensated disturbing torque of the test bed shall be 10 ADCS is a combination of perfectly adjusted actuators, sensors and algorithms. Requirements mentioned above define environment for testing actuators and algorithms, but sensors have to be checked as a part of the system as well. For this reason the test bench has to support sun-, star-and magnetic field simulators, as well as measurement and telemetry systems for feedback and result collection.

Proposed concepts

Friction forces, gravitational torque and additional moment of inertia are the main factors which have undesirable influences on the results of ADCS ground testing. The most common way to deal with friction forces is using air bearings. Gravitational torque can be limited by precise balancing of the satellite on the test bench. The satellite's center of gravity (CG) and the test bench's center of rotation are brought together until the parasitic gravitational torque is lower than a maximum admissible value. The most evident way to minimize additional inertia is to design a test platform with minimum mass added to the satellite. These approaches to deal with the disturbing factors underlie the light test bench design based on spherical air bearing structure.

A fundamentally different approach to the minimization of parasitic torques is active compensation. We propose to eliminate the influence of the additional moment of inertia with help of an electric motor. This solution removes the restrictions on the moments of inertia of the test bench and therefore on its mass. In the future, this method of active compensation can be made more complicated and include the compensation of gravitational torque and friction forces.

TEST BENCH DESIGN with ACTIVE COMPENSATION of DISTURBING FORCES

Design overview

The proposed test bench structure is based on a gimbal suspension allowing 3DoF motion needed to meet the aforementioned requirements. A simplified gimbal with 3U CubeSat is shown in Fig. 3 (CubeSat model use here and below is taken from [START_REF]CubeSat 3D CAD Design[END_REF]) to better describe the concept. The gimbal consists of three holding rings which are connected with each other by air bearings providing contactless rotations.

Figure 3. 3U CubeSat on gimbal suspension

As can be seen, the moments of inertia of the holding rings are considerable with respect to moments of inertia of the CubeSat and cannot be neglected.

The principle of compensation of the additional moment of inertia proposed in this paper is based on a sensor and a motor. The sensor shall be able to detect the test bench angular position or acceleration. The motor shall be able to provide required direct and reverse accelerations to the platform. Fig. 4 shows the principle of compensation on the example of a one-axis device. There are two types of sensors which potentially can be used in this compensation schemea rotary encoder and a Ferraris sensor. A rotary encoder is a widely used and well-known sensor that is able to convert angular position or motion of an axle to digital code. Rotary encoders have wide range of operation characteristics and can be found in different design that helps to easily employ them in any configuration of the platform.

Ferraris sensors are able to directly measure angular acceleration of an axle which removes the need to differentiate an output signal when time derivatives of the position are required. It can usually increase the performance of a highly-dynamic system with control loop [START_REF] Baumer | [END_REF]. Nevertheless, Ferraris sensors have a disadvantage which could be critical for their use in systems with tight requirements to compact sizes. While Ferraris sensors are small, they require a conductive but non-magnetic disk connected to the rotating object to induce eddy currents. The diameter of this disk influences the accuracy of the measurements. Linear tangential acceleration detected by the sensor increases with distance from the center of rotation. Accordingly, the sensor needs smaller gain to obtain measurements and provides less noise. Diameter of a disk is a result of trade-off between efficiency and size of the sensor. The active compensation system for ADCS test bench requires good accuracy. Noise, as a result of double differentiation of the position signal, may harmfully affect the efficiency of the system. Simulations, presented below, are made to study the feasibility of the active compensation of the additional moment of inertia and to determine the performances of the different types of sensors (rotary encoders and Ferraris sensors).

SIMULINK model

For preliminary verifications, the active compensation principle was simplified to only one motion axis. This simplification allows checking the approach in the case of a simple system modeling. In prospect, the model shall be improved and extended to three axes with mutual interferences.

The main idea of ground ADCS tests is to simulate behavior of the satellite as if it were in space. In this condition, the ADCS performance can be evaluated realistically. Angular motion of the satellite in space can be described by Eq.1.

S S e a I θ ( t ) Γ Γ    (1)
where S I is the moment of inertia of the satellite, S θ is the angular acceleration of the satellite, e Γ and rw Γ are the external torque and the torque produced by the actuators of the satellite ADCS, respectively. The equation of motion of the satellite and test bench platform can be written as:

S P p e a p ( I I ) θ ( t ) Γ Γ Γ      (2)
In Eq. 2, P I is the moment of inertia of the test bench platform, p θ is the angular acceleration of the set composed of the satellite and the platform, p Γ is a compensation torque produced by the motor on the platform. The influence of the platform moment of inertia is compensated if the satellite motion is close to its motion in space. Feedback is needed to define the influence of the platform on the motion of the system and compensate it. It can be done if p Γ is equal to the platform's contribution to the left hand side of Eq. 2. However, since there is a feedback loop in the system, the time delay cause by the loop cannot be ignored. Eq. 3 is then used for the compensation torque.

p P p Γ I θ ( t τ )    (3) p θ ( t τ )
 is the angular acceleration of the satellite and the platform at time t-τ, τ is the time delay.

The total equation modeling the system behavior is found as a combination of Eqs. 1-3:

S P p P p S S ( I I ) θ ( t ) I θ ( t τ ) I θ ( t )        (4)
Figure 6 shows the three main parts composing the SIMULINK model of the systemthe satellite, the test bench platform and the computer. The satellite behavior is described by Eq. 1 and represented with blue block. The block for motion of the test bench platform with the satellite is based on Eq. 2 and it includes a negative feedback that means the output signal is sent back to change the original signal by a given gain. The output of this block is arranged in two waysa Ferraris sensor (output called Acceleration in Fig. 6) or a rotary encoder (output Position in Fig. 6).

Only one type of sensor can be used in modeling at one time. The computer and drive block includes a switch allowing choosing the type of sensor. In case of the encoder, double differentiation is required. It is made in the computer block. The computer is a controller which takes the signal from the sensor, transforms it according to Eq. 3 and sends the resulting compensation torque back to the test bench platform. Moreover, the computer block emulates time delay in the system which is due to the negative feedback. According to Eq. 4, the behavior of the system depends on the time delay value and on the value of the moment of inertia of the platform. These values were varied in modeling to define how they affect the performance of the system. The time delay depends on the capabilities of the computer and drive. Up-to-date technologies of robotic systems make possible operation frequency of up to 20 kHz which means delay of 5•10 -5 s [START_REF]ETEL motion technologies[END_REF].

Results and discussion

In the first steps of the system study, when some assumptions are made, results have to be parameterized. General character of the system behavior is more important for early analysis. Bode plots help to analyze stability of the system without reference to input signal and to find the range of frequencies where the compensation principle can be efficient. Furthermore, the inertia moment of the platform can be characterized with respect to the satellite to make the results independent from particular numerical values. The model includes the following parameters:

-Ratio of moments of inertia of the platform and the satellite ( PS I / I ) from 0.5 to 2; -Time delay values ranging from 5•10 -5 to 10 -4 s; Fig. 7 and Fig. 8 illustrate results obtained with the SIMULINK modeling. The system is considered stable and the compensation is considered efficient when the gain of the system and its phase shift stay equal to zero. As can be seen in Fig. 7 and Fig. 8, the maximum of magnitude and phase shift increase and plots move slightly to the left when the compensated moment of inertia grows (first three lines of the plots). The time delay in the system has an influence only on the frequency of stability loss. Thus, the zone of effective compensation shortens when inertia of the platform and time delay (or only one of them) increase. It is important that the zone of effective compensation includes all frequencies of CubeSat ADSC. Otherwise vibrations of the satellite seen by the sensor can lead to malfunction of the compensation system. Average frequency of ADCS operation is 10 Hz. Additionally, the reaction wheels have to be taken into account. The maximum speed of reaction wheels for CubeSats is 6500 rpm that is approximately equal to 110 Hz. Figures 7 and8 show that this compensation system, with a Ferraris sensor or with an encoder, cannot provide stable operation at frequencies higher than 40 Hz at delay 5•10 -5 s. This result is already acceptable and it shows feasibility of the compensation principle. But the compensation system has to be improved to perfectly satisfy all requirements. The results obtained from the modeling shall be validated by practical experiment to prove correctness. The design of an experimental setup for validation of the compensation system is shown in Fig. 9. This setup is a 1 DoF equivalent of the satellite test bench with compensation system. It provides a contactless rotation around axle and motion can be corrected by direct drive motor mounted together with air bearing in the base rotation stage [START_REF]PI miCos Ultra Precision Rotation Stage[END_REF]. The CubeSat is replaced with the simple shape rotating body with known mass and moments of inertia. Fig. 9 shows a Ferraris sensor, but a rotary encoder also can be used. Mass and moments of inertia of the axle and all elements connected to it shall be known with required accuracy before experimental tests. The moment of inertia of the rotating body shall be comparable with the moment of inertia of other rotating parts or smaller. The value of the initial impulse is also important for a correct analysis of the obtained results and shall be known. Hence, the first impulse to the system shall be given by the motor to know its value accurately. Once the axle is rotating, the sensor gets information about motion and sends it to the computer. The computer calculates the required compensation torque according to the algorithm presented above. The motor of the rotary stage applies the torque to the axle with respect to data received from the computer. Information about motion, obtained from the sensor, shall be compared with theoretical computation of free motion of the rotating body without influence of the other test bed elements. Equal results tell that the compensation system operates correctly and efficiently. As a double check, measurement of motion with compensation can be also compared with measurement of motion without compensation. Summarizing results of the work presented above, the next steps to improve the compensation system are clearly:

-Include filters to reduce frequencies of vibration detected by sensors; -Design a control scheme for the compensation system; -Build the experimental test bed and check efficiency of the compensation system; -Include in the system a component of the compensation torque, which is able to deal with a friction force, to avoid using air bearing.

LIGHT TEST BENCH DESIGN with NO COMPENSATION

Concept

Air bearings are well-known means to create contactless suspension for satellite ADCS tests. Such bearings deal well with friction force and do not need any particular operating conditions while their counterpartsmagnetic bearingshave some special needs. Passive magnetic bearings operate at low temperature because of the superconductors included in their structure. Active

Ferraris sensor

Axel Air bearing rotation stage with a direct drive motor Rotating body magnetic bearings work at standard conditions, but they are usually massive, bulky and require a complicated and heavy controller. Moreover, the generated magnetic field is an issue since it can interact with the satellite body and ADCS actuators and obstructs the magnetic field simulation. This effect can be prevented, but it demands additional efforts and expenses. Unfortunately, air bearing have restrictions in freedom of motion. The fluid film of the bearing is achieved by supplying an air flow through the bearing face and into the bearing gap [10]. This fluid film holds the load and provides contactless motion. The obtained motion depends on the shape of bearing face. Flat bearings offer planar motions while spherical bearings allow rotational motions. Motion ranges are limited by the area of the fluid film. A spherical air bearing is usually a puck with a spherical cap cut out and the sliding part is a hemisphere. In this case, contactless performance is available only within the area of the spherical cap which means pitch and roll rotation limitations around ±45°. These limitations can be slightly pushed if the sliding part is almost a full sphere and a payload is placed on a table above (umbrella air bearing platform type mentioned above -Fig. 1). But this configuration leads to difficulties with stability of the motion because the center of gravity of such a structure is usually above its center of rotation. It yields to limited tilt angles or inconvenient requirements to placing the payload. As a result, developers of the air bearing platforms have to choose between restrictions of motion and problematic payload placing. A solution could be found if the payload is placed at the center of the sphere gliding on air bearing. This concept was tried at Los Angeles/California Institute of Technology where a hollow spherical bearing with all hardware mounted internally was designed [START_REF] Schwartz | Historical Review of Air-Bearing Spacecraft Simulator[END_REF]. This system provides ±180° but reported tests include only single-axis rotations [START_REF] Wang | Synchronized Rotation of Multiple Autonomous Spacecraft with Rule-Based Controls: Experimental Study[END_REF]. The system shown on Fig. 10 was developed for experimental work with control of multiple spacecrafts. Designed for this purpose, the air-levitated spheres contain only a microcontroller, sensors, batteries and a flywheel, but not a full satellite. The sphere able to hold a CubeSat inside would be bulky and heavy. The design presented below in the present paper is based on the advantage of hollow air bearing but without a massive structure. We propose a design where few spots form spherical surface and all unnecessary segments of the sphere are eliminated (Fig. 11).

Figure 11. Drawing of hallow air bearing and spherical structure of the same diameter formed by several small air bearings One of the main requirements to the test bench is 3 DoF unconstrained rotations. Hollow air bearings fit this requirement due to the continuous surface of the ball which allows "contact" with bearing puck in every configuration (Fig. 11, left). But the same spherical surface also can be formed with several small air bearings with one common center of rotation (Fig. 11, right). In the latter case, it can be easily seen that the moving part of the structure is much smaller and lighter than that of full sphere. Size and mass of this part is critical for the ADCS test bench because it leads to unwanted moment of inertia.

When sliding spherical caps are placed opposite bearing pucks, dynamics of the structure is similar to a hollow bearing. Nevertheless, the frame with bearing pucks needs to be movable to follow a trajectory of the fictitious inner sphere. Although it yields additional mass, the external part of the structure is independent from the CubeSat mounted on the test bench and therefore makes no influence on results of the tests.

Design

The external frame with air bearing pucks has to spin freely and follow inner part with the CubeSat mounted on it. Motion of the satellite is a key point of the tests and shall not be predicted but only observed. The positioning of the bearing pucks has to be corrected rapidly based on the current position of the satellite. For these purposes, a tracking system is required. Optical sensors offer speed and accurate coordinates of the object within a small delay. Delay in the feedback control of the tracking system is not a critical point. Indeed, narrow relative motions of the inner and external parts of the test bench without correction is available due to different sizes of spherical caps and bearing pucks, as shown in the right side of Fig. 11. Suspension of the external frame is realized by gimbals as shown in Fig. 12. Thin and rigid ring sections in the gimbal provide 4 DoF that helps to avoid possible constraints of the motion. Moreover, the slim design of the gimbals leaves space to place the simulators needed for full imitation of space environment. This design of the test bench presupposes precise balancing of the CubeSat with the holding frame and the spherical caps attached to it. The geometrical center of the inner sphere defines the center of rotation and it has to be perfectly coincidental with its CG. Inaccuracy in the balancing yields a disturbing gravitational torque acting on the satellite. Value of the acceptable inaccuracy will be defined such that the resulting torque does not exceed 10 -5 Nm. Precise balancing is a challenging task, which has to be solved in a next step of this work. The realization of the light test bench with no compensation and infinite rotation angles is a complicated work that was never made before. In this paper, the concept of such a test bench is proposed and the critical points of its design are identified. The balancing of the inner part of the test bench with the payload has to be done. The efficiency and accuracy of the test bench will depend on the: -Coincidence of center of rotation and CG; -Stiffness and thermal stability of the holding frame connected to the CubeSat. The design of the holding structure has to be approved with strength analysis. This structure design is expected to result from a tradeoff between mass, strength and stiffness. Many indefinite characteristics of the air bearings have to be found from tests. Usual applications of spherical air bearings do not involve their use in positions different from the vertical, when the puck is under the sphere. But, in our use, the bearings can be located with every possible tilt angle. Air bearing capability, friction and stiffness could highly change with tilt angle and the corresponding relationships need to be defined from tests. The tracking system for the external frame has to be designed. It shall include sensors measuring the spherical cap positions, simple motors to actuate the gimbal and algorithms to control the tracking. As mentioned above, this system does not need high accuracy or speed. Verification of the test bench is as important as its development. The CubeSat developers have to know the conditions of the testing and influences on the nanosatellite. All disturbing torques, produced by the test bench, have to be estimated in theory and checked by verification tests:

-Gravitational torque; -Residual friction in air bearings; -Aerodynamic friction; -Average parasitic torque of the test bench.

CONCLUSION

In this paper, two approaches to the problem of nanosatellite test bench design are proposed. This work was started recently and this paper introduces the first results as well as the intended next steps. The results show that the test bench with compensation is a promising way to design CubeSat test beds. But the light test bench design looks more feasible on short time. It is a completely new approach to use air bearings for satellite ground testing. In our future works, the mathematical model of the compensation will be expanded to 3 DoF and both designs could be integrated together into a single system.

Figure 4 .

 4 Figure 4. Principle of compensation of the additional moment of inertia of a one-axis platform

Figure 5 .

 5 Figure 5. Ferraris sensors with a 500mm diameter eddy current disc

Figure 6 .

 6 Figure 6. Block diagram of the SIMULINK model of the test bench with active compensation

Figure 7 .Figure 8 .

 78 Figure 7. Bode plots of the model with Ferraris sensor

Figure 9 .

 9 Figure 9. Validation experimental setup for the compensation system

Figure 10 .

 10 Figure 10. Experimental hollow air bearing platform in University of California, LosAngeles/California Institute of Technology model spacecraft spheres[START_REF] Wang | Synchronized Rotation of Multiple Autonomous Spacecraft with Rule-Based Controls: Experimental Study[END_REF]

Figure 12 . 4 . 3

 1243 Figure 12. Overall view of the test bench based on spherical air bearing structure

ACKNOWLEDGMENTS

This research work was funded by The Van Allen Foundation of University of Montpellier 2 and Labex NUMEV.


Van Allen

I Garilovich

Sébastien Krut
email: sebastien.krut@lirmm.fr

Marc Gouttefarde
email: marc.gouttefarde@lirmm.fr

François Pierrot
email: francois.pierrot@lirmm.fr

Irina Gavrilovich
email: irina.gavrilovich@lirmm.fr

Laurent Dusseau
email: laurent.dusseau@ies.univ-montp2.fr

Robotic Test Bench for CubeSat Ground Testing: Concept and Satellite Dynamic Parameter Identification*

This paper introduces a novel concept of an air bearing test bench for CubeSat ground testing together with the corresponding dynamic parameter identification method. Contrary to existing air bearing test benches, the proposed concept allows three degree-of-freedom unlimited rotations and minimizes the influence of the test bench on the tested CubeSat. These advantages are made possible by the use of a robotic wrist which rotates air bearings in order to make them follow the CubeSat motion. Another keystone of the test bench is an accurate balancing of the tested CubeSat. Indeed, disturbing factors acting on the satellite shall be minimized, the most significant one being the gravity torque. An efficient balancing requires the CubeSat center of mass position to be accurately known. Usual techniques of dynamic parameter identification cannot be directly applied because of the frictionless suspension of the CubeSat in the test bench and, accordingly, due to the lack of external actuation. In this paper, a new identification method is proposed. This method does not require any external actuation and is based on the sampling of free oscillating motions of the CubeSat mounted on the test bench.

I. INTRODUCTION

One of the widespread and demanded branches of space technologies is CubeSat-class satellites. CubeSats are miniaturized satellites with standardized sizes of one-unit (1U, 10x10x10 mm), or a multiple of one unit (e.g. 3U). The development of such satellites is common in schools and universities due to the relatively short development time, the wide range of components available off-the-shelf, and compatibility with standardized launch ejectors. In addition to educational purposes, nowadays, CubeSats are employed in full scale scientific and technological missions, and every year the number of professional CubeSat programs increases [START_REF] Swartwout | The first one hundred CubeSats: a statistical look[END_REF]. This tendency leads to the necessity of complete CubeSat ground testing. One of the main issues of the pre-launch examination is the inspection of the Attitude Determination and Control System (ADCS) once it is assembled and integrated in the satellite. ADCS testing, unlike tests of other subsystems, requires the satellite to move. Each constraint impairing the satellite free rotation yields cases where the ADCS cannot be checked and might thus cause problems during its on-orbit operation. Results obtained for the available range of angles are often extrapolated to other The development of an ADCS test bench is a challenging task, but some examples can be found in the literature. Most of the developed test benches are designed for satellites with a mass greater than 50 kg [START_REF] Kim | Automatic mass balancing of air-bearingbased three-axis rotational spacecraft simulator[END_REF]- [START_REF] Meissner | A three degrees of freedom test bed for nanosatellite and CubeSat attitude dynamics, determination, and control[END_REF]. They cannot be used for CubeSat ADCS tests, because inertia parameters of the test bench are critical for a reliable checkout and shall be comparable with those of the tested satellite. Some test benches have been designed specifically for CubeSats [START_REF] Gallardo | Six degrees of freedom experimental platform for testing autonomous satellites operations[END_REF]- [START_REF] Schwartz | The distributed spacecraft attitude control system simulator: from design concept to decentralized control[END_REF], but their maximum rotation angle is approximately 90° or less.

In this paper, a novel approach to the design of 1..3U CubeSat test benches is proposed. The corresponding design eliminates the limitations on the satellite attitude (orientation) during the testing since it allows full three degree-of-freedom (DoF) satellite rotations. The test bench consists of two frames, which have spherical mating surfaces conventionally called Spheres, and of a 4 DoF redundant robotic wrist. Moreover, four air bearings are employed to allow frictionless motion of the satellite mounted on the test bench.

Reliable and trustworthy results of ADCS testing can be obtained only if the effects of all disturbing torques caused by gravity, aerodynamic drag, and friction, are minimized. The total disturbing torque shall be one order of magnitude smaller than the total expected external torque on the orbit [START_REF] Kim | Designing a lowcost spacecraft simulator[END_REF]. While other sources of disturbances can be reduced up to negligible values by optimizing environmental conditions at the test facility, the torque due to gravity always influences the satellite when there is an offset between the satellite center of mass (CM) and the test bench center of rotation (CR). A balancing procedure, aiming at eliminating this offset, requires an accurate identification of the satellite CM position together with its inertial parameters. In the sequel, the inertial parameters and the CM position are referred to as the dynamic parameters.

The identification of dynamic parameters is a well-known technique in robotics and related fields. Several approaches have been proposed in the literature to solve the dynamic parameter identification problem [START_REF] Kim | Automatic mass balancing of air-bearingbased three-axis rotational spacecraft simulator[END_REF], [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]- [START_REF] Gautier | Dynamic parameter identification of a 6 DOF industrial robot using power model[END_REF]. Some common features of these approaches can be found in [START_REF] Gautier | New method for global identification of the joint driven gains of robots using a known payload mass[END_REF], [START_REF] Gautier | Dynamic parameter identification of a 6 DOF industrial robot using power model[END_REF], namely:

• The use of inverse dynamic or energy model to form the identification equations; • The use of an optimal exciting trajectory for efficient model sampling; • The use of an over-determinate linear system of equations resulting from the model sampling; • Solving the linear system by the Least Squares method to estimate the parameters. In all the aforementioned approaches, the identification is done for systems subjected to external actuation (usually, joint forces/torques). For example in [START_REF] Kim | Automatic mass balancing of air-bearingbased three-axis rotational spacecraft simulator[END_REF], where the identification and balancing of an 800 kg satellite is done, the test bench uses reaction wheels for actuation. The identification equations can be written in the following general form

( ) = y Γ W x (1)
where Γ is an external torque, W is an observation matrix, and x is the vector of the dynamic parameters to be identified. In the case of a passive system (with no actuation), the right-hand side of ( 1) is always equal to zero. The latter case is the one dealt with in this paper where the identification problem concerns the payload of the test bench which is passive, i.e., it is not subjected to external influences apart from the torque due to gravity. A method of dynamic parameter identification suitable to this passive case and based on the sampling of free oscillating rotations is presented in this paper. In summary, the first contribution of this paper is a novel design of an air bearing test bench for CubeSat ADCS testing, which includes a 4-DoF redundant robotic wrist. The second contribution is a dynamic parameter identification technique for a passive system. This technique is based on the observation of its oscillating rotational motions.

The paper is organized as follows: Section 2 introduces the details of the test bench design, Sections 3 and 4 are devoted to the identification method. Section 5 presents simulation results while section 6 concludes the paper.

II. THE TEST BENCH

The most popular and advanced technique to obtain frictionless 3-DoF rotational motions of a satellite on a test bench is the use of air bearings. Different types of air bearing test benches are discussed in [START_REF] Schwartz | Historical review of airbearing spacecraft simulators[END_REF], but all of these designs have limitations on the available pitch and roll rotations of the mounted payload, the maximum rotation amplitudes being around ±45°. These limitations could be eliminated if the payload is placed at the center of a hollow sphere gliding on an air bearing puck. Such an approach was tried for satellite disassembled hardware, the system provided rotations of ±180° but the reported tests included single-axis rotations only [START_REF] Wang | Synchronised rotation of multiple autonomous spacesrafts with rule-based control: experimental study[END_REF]. In [START_REF] Meissner | A three degrees of freedom test bed for nanosatellite and CubeSat attitude dynamics, determination, and control[END_REF], an air-levitated sphere with a 1U CubeSat placed inside is described, but it was not realized because of technological issues. A hollow sphere able to hold 3U CubeSats would be bulky and heavy, which are undesirable characteristics for a CubeSat test bench. Another disadvantage is technological complexity: The hollow sphere shall be built in several sections to be able to place a payload inside the sphere, while the surface shall be accurately machined to provide the roughness required by the air bearings. The design proposed in this paper has the advantage of the air-levitated sphere (satellite unconstrained rotations) without its aforementioned weaknesses.

A. Concept

The key advantage of the hollow sphere is the continuous surface that allows friction-free contact with the bearing puck at every possible angular position. The idea proposed here is that the same property can be achieved by means of several small air bearings rotating around a common CR placed at the spherical surface center (Fig. 1). The inner part of the structure, which consists of small hemispheres (spherical segments) as shown in the right part of Fig. 1, becomes much smaller and lighter than the air-levitated spheres suggested before. It is relevant to minimize the size and mass of this part of the test bench because of its unwanted moment of inertia. The frame on which the bearing pucks are attached, called the "External Sphere", needs to follow the motion of the fictitious "Inner Sphere" wherein the satellite is mounted. The External Sphere is independent from the CubeSat mounted on the test bench and, therefore, has no influence on results of the tests.

B. Design overview

The proposed CubeSat test bench mainly consists of three parts: (i) the Inner Sphere composed of the sliding hemispheres attached to the CubeSat; (ii) the External Sphere built of the air bearing pucks; and (iii) a 4 DoF redundant robotic wrist, able to rotate the External Sphere around the CR (Figure 2). The External Sphere shall be rotated around the CR in order to follow the motion of the Inner Sphere so that each hemisphere remains aligned with its air bearing puck. The CR is thus common to both the Inner and External Spheres and is coincident with the common intersection point of the four robotic wrist revolute joint axes. While the whole system needs only 3 rotational DoF, the additional DoF of the wrist aims at eliminating kinematic singularities which would limit the possible rotations.

The CubeSat is fixed to the Inner Sphere by means of an adjusting mechanism, which allows limited modifications of the satellite position with respect to the CR. This feature is needed for sampling the payload motion with different initial offsets between the CR and the CM and for the final balancing of the CubeSat. The maximum displacements of the satellite inside the Inner Sphere are ±20 mm along the X and Y directions and ±35 mm in the Z direction. These values have been chosen according to CubeSat Design Specification [START_REF]The CubeSat Program, CubeSat Design Specification Rev[END_REF] that defines allowable CM locations for 1...3U CubeSats.

The motion of the satellite is a key point of ADCS tests and shall not be predicted but only observed. Moreover, the positioning of the External Sphere has to be corrected rapidly based on the current position of the satellite. For these purposes, a tracking system is required. Two possible means to observe the Inner Sphere motion are: 3D motion capture cameras or distance sensors. The sensors measure a relative position of the hemispheres and pucks. In order to obtain an absolute estimation of the Inner Sphere attitude, the encoders of the robotic wrist joints are then required.

III. THE DYNAMICS OF THE TEST BENCH PAYLOAD

The payload of the test bench is the CubeSat together with the Inner Sphere and the adjusting mechanism, which are rigidly connected and move as one body. Before dynamic parameter identification, the dynamics of the payload shall be described. In the sequel, "body" refers to the payload.

A. Frames

The inertial fixed frame I R is defined by the basis I B , denoted ( ) , ,

I I I
x y z , and its origin centered at the point CR . Vector I z is aligned with the local vertical (Fig. 3). Let the basis bf B , denoted ( ) As shown in Fig. 3, it can be written as the sum of the vectors

,
O ρ and r expressed in bf B = + O ρ r ρ (2)
It should be noted that the vector r is related to the position of the body in BF R and the vector O ρ is constant for the given body (Fig. 3).

B. Kinematics

The orientation of BF R with respect to I R is given by the rotation matrix I BF A . In I B , the angular velocity vector of the body I ω can be found as follows [ ]

T ´= I I I BF BF ω A A & (3)
where [ ] Íω is the skew-symmetric matrix associated to the vector I ω . The following operation can be used to express the angular velocity vector ω in bf B

= BF I I ω A ω (4)
In the sequel, all vectors without a left superscript are expressed in bf B .

C. Dynamics

The Euler's equations of motion traditionally describes the rotational dynamics of a body with respect to a coordinate frame whose origin is the body's CM. The test bench payload is subjected to the action of a gravity torque (when the points CR and CM are not perfectly coincident) and to the reaction forces at the air bearings. The reaction forces are pointing towards CR . They can be represented by a resulting force passing through CR . The magnitude and direction of this force are unknown.

However, the equations of motion does not include this unknown resulting force if the Euler's equations of motion are expressed in BF R + ´= where vector x contains the body dynamic parameters to be identified.

IV. IDENTIFICATION OF THE DYNAMIC PARAMETERS

The goal of the identification process is to find the dynamic parameters of the body. The dynamic model [START_REF] Bisiacchi | Three-axis air-bearing based platform for small satellite attitude determination and control simulation[END_REF] in the current formulation is a function of these parameters taken with respect to CR ( CR I and ρ ) so that the vector b in ( 8) is always equal to zero. Moreover, CR I and ρ are related to the position of the body in BF R , hence the result of the identification depends on the initial position of the body with respect to CR . Such an objectionable situation can be avoided if the dynamic parameters are taken at a point attached to the body. The points CM and O (Fig. 3) are two candidate points. Choosing CM yields nonlinearities in the dynamic parameters. On the contrary, choosing O leads to a linear system. CR I and ρ shall thus be expressed with respect to O and substituted into [START_REF] Bisiacchi | Three-axis air-bearing based platform for small satellite attitude determination and control simulation[END_REF].

A. The Inertia Matrix

The inertia matrix

Then, CR I can be found by applying the Huygens-Steiner theorem:

( )

T T m = + - CR CM I I ρ ρ 1 ρ ρ ( 10
)
where 1 is the 3 3 ´identity matrix. Substituting (2) into

where ( )

T T m = + - O CM O O O O I I ρ ρ 1 ρ ρ (12)
is the inertia matrix of the body taken at O and expressed in bf B and ( ) ( )

T T m = - C r r r 1 r r (13)
is a value depending only on the position of the body in BF R and on the body's mass m. The decomposition of the inertia matrix presented in [START_REF] Ustrzycki | Spherical air bearing attitude control simulator for nanosatellites[END_REF] is convenient for further simplification and transformation of [START_REF] Bisiacchi | Three-axis air-bearing based platform for small satellite attitude determination and control simulation[END_REF].

B. The Identification Equations

Taking into account the relations obtained in the previous section, [START_REF] Bisiacchi | Three-axis air-bearing based platform for small satellite attitude determination and control simulation[END_REF] can be rewritten as a linear equation in the dynamic parameters. The product of the inertia matrix and the angular velocity becomes ( )

( ) , m = + + CR O O I ω I ω C r ω B ω r ρ (14)
where matrix ( ) , B ω r is defined as ( )

2 2 3 3 2 1 1 2 3 1 1 3 1 2 2 1 1 1 3 3 3 2 2 3 1 3 3 1 2 3 3 2 1 1 2 2 ω ω 2 ω ω 2 ω ω , 2 ω ω ω ω 2 ω ω 2 ω ω 2 ω ω ω ω r r r r r r r r r r r r r r r r r r - - - - é ù = - - - - ê ú - - - - ê ú ë û B ω r (15)
Equation ( 14) can be modified to highlight the fact that it is linear in the unknown values O I and O ρ . The first term on the right-hand side can be written ( )

O O I ω = Ω ω j (16)
where O j is a 6 1

´ vector composed of the elements of the ( )

1 2 3 2 1 3 3 1 2 ω 0 0 ω ω 0 0 ω 0 ω 0 ω 0 0 ω ω ω 0 é ù = ê ú ê ú ë û Ω ω (18)
Accordingly, the expression of CR I ω & can be written as

( ) ( ) ( ) , m = + + CR O O ( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ( )

C. The Least Squares Solution

In this work, the identification of the components of the inertia matrix and of the position of CM is based on the knowledge of the angular position of the body-fixed frame with respect to I R . A Least Squares (LS) solution requires data obtained from p experiments each having a duration of i N sec or i n time steps. i N and p as well as the initial conditions of the experiments should be optimized in order to obtain the best results. During the experiments, the body is moving freely under the influence of the gravity torque created by the body weight when CM is not coincident with CR . In each experiment, the initial conditions, i.e. the offset between CR and O , described by the vector i r , the initial orientation of the body in I R and its initial angular velocity could be different. For the th j measurement of the th i experiment, ( 20) is where Φ is a 3 9 np ´ observation matrix, x is the 9 1

( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ] , ) ( ( ( , ) ) ( ) ij i ij ij ij ij ij ij i ij i i i i j j i ij m m ´´´é ù é ù + + + ë ûê ú ë û =- - - O O j Ω ω ω Ω ω B ω B g ρ g r ω C ω C ω r ω r r r ω &
´ vector of dynamic parameters and b is a 3 1 np óbservation vector. Since there are more equations than unknowns, this problem can be solved computing the LS solution

+ x = Φ b (23)
where + Φ is the pseudo-inverse of the observation matrix. This LS solution is an estimation of the dynamic parameters

x . The Huygens-Steiner theorem can finally be applied to find the inertia matrix of the body at its CM ( )

T T m = - - CG O O O O O I I ρ ρ 1 ρ ρ (24)

D. Iterative Approach for Multi-Body Payload

The equations in the previous subsections are given for a single rigid body. Practically, in the test bench studied in this paper, the "body" consists of the CubeSat, the Inner Sphere and the adjusting mechanism. The identification shall be done only for the CubeSat's dynamic parameters, which are constant since the CubeSat does not change a shape during the experiments. The parameters of the other elements are assumed to be well known. While they are not a subject of interest here, they influence the motion and shall thus be taken into account. Accordingly, (2) becomes

The resulting identification equations can be obtained by substituting (25) and ( 26) into [START_REF]The CubeSat Program, CubeSat Design Specification Rev[END_REF].

Before a direct application of (21) to the identification of dynamic parameters, it should be mentioned that the vector x is not homogeneous. The components of x have different units (unit of O ρ is m and unit of O j is kg•m 2 ). For CubeSats, the values of O ρ can be up to 2 orders of magnitude larger than the diagonal components of the inertia matrix, depending on the position of the CM and on the choice of the point O on the body. An accurate identification can be reached when all the components of x are of the same order of magnitude. To this end, an iterative approach aiming to minimize the magnitude of O ρ can be considered. The following sequence of operations is suggested: • Update r based on given scaling coefficients;

• Initialization: Choose the origin O of bf R
• Repeat the operations until the required accuracy is reached This iterative approach results in decreasing the magnitude of O ρ at every step. It can be seen in ( 12) that the contribution of 0 CG I in O I is larger than the addition, which depends on O ρ , because O ρ is small and squared. While O ρ decreases fast, O j changes slowly. Accordingly, after a few iterations, the values of O ρ and O j will be of the same order of magnitude and the identification error should be minimized. Besides, when O ρ is decreasing, the oscillation period increases and the experiment duration i N might not be large enough to capture the character of the motion. Consequently, a scaling of i N after each iteration is recommended to avoid this issue.

MATLAB SIMULATION

The identification algorithm proposed above shall be tested and, if required, improved before its application to the test bench. Simulations of the identification process in MATLAB are thus used to verify the algorithm efficiency. The MATLAB model consists of two parts: The simulation of the body dynamics and the identification of the dynamic parameters (Fig. 4).

The rotating body dynamics is described by [START_REF] Bisiacchi | Three-axis air-bearing based platform for small satellite attitude determination and control simulation[END_REF]. The simulation of the body motion requires to solve this equation to determine the angular position of the body, described by the rotation matrix I BF A . Finding I BF A emulates data that will be obtained from the sensors of the test bench during the real experiments.

The part of the MATLAB model, dedicated to the identification of the inertial parameters, is similar to the one that will be used during the real experiments. The accuracy of the identification process can be estimated by comparing the "real" values CR I and O ρ , chosen by the user for the current model run, and the values CR I $ and $ O ρ obtained by the identification process. E. Simulation of the body motion Equation ( 7) shall be written in the following form to be solved by the MATLAB Ordinary Differential Equation (ODE) solver ( )

f = y y & , ( 27
)
where y is the vector of unknowns. Equation ( 7) contains first and second order derivatives of the angular position of the body. Moreover, the angular velocity is not linearly dependent on the sought-for rotation matrix I BF A . The following solution is proposed to deal with this issue.

Let y to be the vector composed of the components of

Thereby, the equations of motion can be expressed in the form (27) which is required by the ODE solver.

F. Results

MATLAB simulations were run for CAD models of 1U and 3U CubeSats with different dynamic characteristics and initial conditions, represented in Table I The results of the identification and the corresponding number of iterations, required to obtain O ρ with an accuracy of 1µm (that is needed to meet disturbing torque requirements) are given in Table 3.

These simulations provide good results in the identification of the CM location (largest error <0.1%) and of the moments of inertia (largest error <1%), but for the accurate identification of the products of inertia some additional measures might be needed. The latter can be explained by the small magnitudes of the products of inertia with respect to other identified values. However, the products of inertia should always be small due to the parallelepiped shape of the satellite. Furthermore, they are not important to the CubeSat balancing.

Based on the conducted simulations of the identification process, better results are obtained when at least 4 experiments are made such that, in each experiment, the geometric center of the CubeSat is located in a different octant of I R . However, further research on the optimization of the identification process shall be done to improve its efficiency.

5452

V. CONCLUSION A novel concept of an air bearing test bench for CubeSat ground testing was proposed in this paper. In contrast to existing air-bearing test benches, it allows unlimited 3 DoF rotations of the tested CubeSat while reducing the undesired influences caused by the other mobile elements attached to the CubeSat. Moreover, a dynamic parameter identification method, adapted to the proposed test bench, was presented. This method is mainly based on the sampling of free oscillating motions. Simulation results of dynamic parameter identification for CAD models were presented. Future work will aim at validating the identification method on the assembled test bench with a CubeSat prototype.

Figure 2 .

 2 Figure 2. The test bench.

Figure 1 .

 1 Figure 1. An air-levitated sphere and a spherical structure of the same diameter formed by several air bearings.

Figure 3 .

 3 Figure 3. The frames involved in the identification process.

  , x y z , be attached to the body, and let the body-fixed reference frame bf R consist of bf B centered at a given point O of the body. The choice of O will be discussed in Section 4. The frame BF R shown in Fig. 3 has the same orientation as bf B but its origin is CR . As shown in the sequel, the frame BF Ris introduced to simplify the writing of the equations of motion, suitable for identification. The position of the point CM in BF R is defined by the column vector ρ .

  matrix of the body at CR , CR T is the torque induced at CR by the weight m g of the body describes the dynamics of the test bench payload in bf B . As detailed in Section 4, this equation can be used to obtain the identification equations in the following form

ρ r ρ r 1 ρ r ρ r I C r ρ r 1 ρ r r ρ

Figure 4 .

 4 Figure 4. Simulation scheme of the identification of dynamic parameters.

  where m and O ρ refer now to the CubeSat while k m and k O ρ refer to the elements attached to the satellite,

  at the geometric center of the CubeSat (as a first approximation of the CM position) and select the initial position of the CubeSat in I R (i.e. vector r ) for each one of the p experiments; • Run p experiments in order to form an overdeterminate linear system of equations (22); • Derive x using the LS method; • Update the position of O in I R according to the identified value O ρ ;

3 components

 3 can be solved for I ω & , which are the last

  and TableIII. At the beginning of each experiment, the CubeSat is at rest and the axes of bf R and I R are perfectly aligned.The scaling coefficients for r and the duration of each experiment, which depends on the magnitude of O ρ derived during the previous iteration, were found empirically and are given in TableII. The sampling time for all iterations is 0.1 sec.

TABLE I .

 I INITIAL CONDITIONS

	Experiment number, p		1			2			3			4
	Initial offset, r (m)	é ê ê ë	0.02 0.02 0.02	ù ú ú û	é ê ê ë	0.02 0.02 0.02 --	ù ú ú û	é ê ê ë	0.02 0.02 0.02 -	ù ú ú û	é ê ê ë	0.02 0.02 0.02 ---	ù ú ú û

TABLE II .

 II SCALING COEFFICIENTS

	O ρ at the previous	Scaling coefficient	Duration of each
	iteration (m)		for r	experiment, Ni (sec)
	≥ 0.01		1	5
	≥ 0.001		0.5	10
	≥ 0.0001		0.1	20
	< 0.0001		0.05	25
	TABLE III. RESULTS OF SIMULATIONS
		1U CubeSat	3U CubeSat
		CAD	Identified	CAD	Identified
		values	values	values	values
	O 1 ρ (m 10 ) 3 -×	-0.2560	-0.2558	-1.6393	-1.6398
	O 2 ρ (m 10 ) 3 -×	-0.9320	-0.9317	-1.2807	-1.2814
	O 3 ρ (m 10 ) 3 -×	-9.9570	-9.9572	17.1741	17.1749
	CM xx I (kgm 10 ) 2 3 -×	1.5460	1.5325	30.6915	30.4716
	CM yy I (kgm 10 ) 2 3 -×	1.5910	1.5797	29.6998	29.4381
	CM zz I (kgm 10 ) 2 3 -×	1.3840	1.3817	4.5775	4.5399
	CM xy I (kgm 10 ) 2 3 -×	0.0090	-0.0087	0.0250	-0.2039
	CM xz I (kgm 10 ) 2 3 -×	-0.0070	0.0030	-0.1459	-0.0052
	CM yz I (kgm 10 ) 2 3 -×	0.0060	0.0082	0.0030	0.0021
	Number of
	iterations	4		4
	(Section IV-D)

& & & &(19)Collecting all dynamic parameters on the left-hand side, (7) is finally written in matrix form as

& &(21)

Foundation and FEDER.