Performance Analysis of an Attitude Control System for Solar Sails Using Sliding Masses

Performance analysis of an attitude control system for solar sails using sliding masses

Christina Scholz1,Daniele Romagnoli²,Bernd Dachwald1

1University of Applied Science Aachen, Germany

²German AerospaceCenter – Institute of Aerospace Systems, Germany

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Abstract

This paper deals with the attitude control performance analysis of a square solar sail with two sliding masses moving along the mast lanyards for pitch and yaw control. A robust nonlinear controller with a feedback and feed forward part is used to control the attitude of the sail. Numerical simulations have been carried out to investigate the system’s ability of performing precise and near-time-optimal reorientation maneuvers as well as the controller’s sensitivity with respect to the sail parameters, like the center of pressure to the center of mass offset or the sail’s geometry. Our simulation results are finally discussed and compared to previous results that have been obtained by others and are reported in the literature.

INTRODUCTION

The possibility of using only solar radiation pressure as thrust, as well as their unlimited propulsion capability, makes solar sails very attractive for high demanding missions [1].

In order to control a solar sail spacecraft in three axes, various solutions for the attitude control system have been designed. Most of them cause an offset between the center of pressure and the center of mass. For example, it is possible to control a solar sail with one gimbaled boom with two degrees of freedom, mounted at the center of the sail. Alternatively the change of attitude can be performed by articulated reflecting vanes at the masts end or by sail panel translation/warping [1,2].

In this paper another kind of attitude control actuator is considered. The so-called sliding masses, also known as trim control masses or ballast masses in the literature, are two small masses, which move inside and along the masts via lanyard lines and electric motors [3,4]. The sliding masses also provide a center of pressure to center of mass displacement. Assuming that the center of mass and the center of pressure are in the same plane, this displacement can be used for pitch and yaw attitude control. To control the movement around the roll axis a second controller is needed, for example, pulsed plasma thrusters mounted at the top of the booms could be used [4,5], or a rotating boom around the roll axis [6]. This paper, however, will not go into details about the control of the roll axis.

Let us consider a 40by40m square sail with four carbon fiber-reinforced plastics booms, four triangular sail segments and two small sliding masses of 1kg each [3]. The sail area is assumed to be 1200m² and the satellites bus has a mass of 150kg. The total mass of the configuration is 185kg, taking into account the sail mass, the mass of the masts, the sliding masses, the central assembly and the payload mass. For more detailed information about the properties of the sail see Table 1 [3].

The used coordinate system is a spacecraft-centered system, called body frame b, as shown in Fig.1. The origin is located at the geometric center of the sail. The +X-direction is taken normal to the illuminated sail surface, whereas the Y- and Z-axes are along the masts.

The satellite is located at 1AU from the Sun, so that the solar radiation pressure can be assumed to be equal to 4.563x10-6 N/m².

1. Problem description

1.1The Controller

The attitude control strategy used in this paper is based on a two-degrees-of-freedom controller design.

The feed forward part is based on a kind of inversion of the systems dynamics (see equations 1a-c). In case of changing the operating point, that means, changing from one attitude to another, the feed forward part determines the design of the desired trajectory, while the feed back part takes care of rejecting the effects of the disturbances. The design of a desired change of the attitude is specially based on the transition time T, the unique Eigenaxis of the related rotational maneuver and the progress of the rotation angle around the Eigenaxis over time. Notice that the Eigenaxis of the rotation and the angle completely determines the states as well as the control inputs of the system for all times t0≤t≤T. Assuming that the underling model of the feed forward controller describes the system, the latter follows the desired trajectory, using the control signal transmitted to the system. In this context it is emphasized that the torques due to the displaced center of pressure with respect to the center of mass is considered by the feed forward controller.

The feed back controller is designed under the assumption that the linearized model is close enough to the world system. This part controls the errors between the desired trajectory and the trajectory of the system. Possible errors can be caused, e.g. by external or internal disturbance torques, differences between the model and the real world system or parameter uncertainties.

Under the previously mentioned assumptions the feed forward part contributes the major part of the control signal. [2,7]

2.1Equations of Motion

The attitude kinematics is propagated using the well known quaternion representation according to the vector first notation [8]. In addition, the Gibbs-vector and the Euler axis/angle representations are also used to design both the feed forward and the feed back controllers, so that maneuvers around the three axes can be performed at the same time [9].

The attitude dynamics equations for a solar sail with sliding masses have been developed by Wie and Murphy [1] and can be written as:

Where ω is the angular velocity, is the angular acceleration, FS,x. is the x-component of the solar radiation pressure force (FS,y and FS,z, respectively), is the so called reduced mass, M is the total mass of the satellite bus, ∑T is the sum of the external torques Text and the offset torque Toff, which is computed as the cross product of the position of the center of pressure vector and the solar radiation pressure force vector. Jx, Jy, Jz, the principal moments of inertia of the sail considering the position of the masses, can be expressed by the relations:

1. Test Results

Our simulations have been run in a Matlab/Simulink environment, using dedicated software that has been internally developed. The goal of this simulation campaign was to find a near-optimal time to perform a desired maneuver and to follow the commanded trajectory provided by the feed forward part. The controller and the integration of the equations of motions have been integrated in a Simulink model, whereas the near-optimal time is computed in a Matlab script before launching the simulation.

3.1Single Maneuver Around one Axis

The controller is initially tested using a 35° maneuver around the yaw axis. This maneuver is performed with sliding masses of 1kg. In this test case, the Z-component of the offset vector has been increased over the simulations, to study the sensitivity of the performance of the controller with the mass of the sliding masses and different values of the offset vector.

Figure 2 shows the position, velocity and acceleration of the sliding masses over time during the simulated maneuver, while Fig. 3 describes the desired control torques over time as given by the feed forward controller. The predicted control torque isaround the pitch axis, as expected, whereas the roll and yaw torques are zero over time. In this special case, an offset of 0.1m in +Z-direction is introduced, so that the near-optimal maneuver time is 72.5min.

The sliding mass which is moving on the Y-direction clearly remains at the center of the sail, which is equal to the center of mass, and is not moving over time, as expected. The other mass, on the contrary, moves along the Z-direction to perform the maneuver and compensate the center of pressure to the center of mass offset with an initial/final displacement of about 15m. By changing the offset, for example in the +Z-direction, the equilibrium position of the sliding mass is also changing.

The first plot of Fig. 2 shows that the control mass moves at the beginning of the maneuver to the end of the boom (28m) and reaches this position after about 22min. Doing so, the reorientation maneuver is started. After that the sliding mass has to slow the sail down, so that the sail has no residual angular velocity at the end of the maneuver. The mass slows the sail down by moving the composite center of mass to the opposite position with respect to the center of pressure.

4.1Influence of the Offset

Under the assumption that the center of pressure lies on the plane of the sail, the offset vector expressed in the body frame has only components in Y- and Z-direction.

The simulation results presented in Fig. 4 show the big influence of the offset vector for a single 35° yaw maneuver. With no offset, this reorientation of the sail can be performed in less than 1h. Whereas the same maneuver with an offset of 0.14m needs nearly 100min to be performed. As it can be seen in this figure, the near-optimal maneuver time is in the case of +0.02m and +0.04m smaller than with no offset. That means that the offset has a positive effect on the near-optimal maneuver time. The reason is that the reorientation of the sail is in positivedirection. The steady state position is already in the proper direction to perform the reorientation. For an offset of 0.06m and bigger, this condition is not dominating anymore, so that the near-optimal maneuver time increases.

A sliding mass of 1kg can not handle an offset of 0.19m or larger. In this case the equilibrium position of the sliding mass is located outside of the sail and is, therefore, not feasible. Other simulations, not presented here, showed, as expected, that by increasing the mass of the sliding masses the manageable offset increases and the reorientation can be performed in less time.

Table 2 shows the influence of the offset vector for a two-axes maneuver. The reorientations are 40° around the yaw axis and 10° around the pitch axis. The bigger reorientation of the two axes maneuver determines the near-optimal maneuver time.

5.1Two Axes Maneuver with Offset and Environmental Torques as well as Torques Due to a Disturbed Inertia Matrix

Besides the torques due to the offset between the center of pressure and the center of mass, also other disturbances can be integrated in the simulation, such as environmental torques in all three axes or an inertia matrix with components different from zero at the non-principal elements.

The next plots present a two-axes maneuver, whereas a reorientation of 10° around the yaw axis and a 40° reorientation around the pitch axis is introduced. The maneuver is performed with two sliding masses of 1kg each. The reorientation is achieved with an offset of 0.1m in +Y and +Z direction so that the offset vector is m. An environmental torque with avalueof[0.001, 0, 0.0001]T Nm is applied to the simulation. An inertia matrix with off-diagonal elements is introduced as well. The non-diagonal elements of the inertia matrix are filled with random values of about 10% of the principal elements.

In this special case the disturbed inertia matrix is:

kg m²

Figure 5 shows the necessary control torques to perform the maneuver. As being visible, the control torque has also a component in the X-direction, even if no rotation around that axis has been commanded. The main reason is that the Eigenaxis of the rotation has components in all three axes of the body frame. For this example, the Eigenaxis turns out to be: [0.08476, 0.96881, 0.23287]T. In addition, the disturbance torque around the roll axis determines a roll rotation. As a consequence, to perform the desired reorientation, it is necessary to rotate also around the X-axis.

Figure 6 shows the real positions of the sliding masses compared with the feed forward prediction of the positions. The first plot shows the mass that is moving along the Y-direction to perform the 10° reorientation. Due to the small maneuver, the mass is not moving a lot and stays close to the center of the sail over the transition time. On the other hand, the second mass uses the whole length of the sail booms. After ~25min the sliding mass reaches the end of the sail, so that the position of this mass is determining the near-optimal maneuver time.

At certain points, it can be seen that the real states of the masses are not equal to the feed forward prediction. The explanation for these variations is that the feed forward part takes only the offset torques into account. Therefore, the feed back part of the controller has to compensate the disturbed inertia matrix as well as the environmental torques with a dedicated control signal. In the first plot, it can be seen that at the end of the maneuver the sliding mass is not located at the predicted equilibrium position. That can be explained by the fact that the environmental torques are acting constantly over the whole simulation time and causes a new steady state position.

Due to the fact that the real positions are most of the time really close to the predicted ones, it can be said that the feed forward part of the controller is doing most of the work.

conclusions

A means for solar sailing attitude control using sliding masses has been developed. The attitude control strategy developed here is based on an attitude controller that tracks a given trajectory while rejecting disturbances, such as torques due to the center of pressure to center of mass offset, environmental torques or torques due to a non-diagonal inertia matrix.

The developed controller is able to perform a single 35° yaw maneuver with a center of mass to center of pressure offset of 0.1m, in 72min, which is less time as it can be found in the literature for similar simulations [3]. It has also been shown that a maneuver around two axes can be performed simultaneously, reducing the overall time for the maneuver.

A sensitivity study has been performed to show how the controller performance is related to design parameters, like the center of pressure to center of mass displacement or asymmetries in the structure, expressed in a non-diagonal inertia matrix.

As a future research objective, it will be very useful to establish a better maneuver-time optimization. Evolutionary neurocontrol (ENC) as described in [10], can be used for near-globally optimal computations. In addition, new controllers for bigger sails (like a 80m or 160m sail) have to be developed and integrated into the simulation model, in order to compare the reorientation performance of sails with different sizes and inertia matrices.

Acknowledgments

This work was carried out at the DLR (GermanAerospaceCenter) as a partial fulfillment of a diploma thesis. The authors wish to thank Mr. Thimo Oehlschlägel for his precious help in designing the controller.

References

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