ising techniques uses a chaser spacecraft that attaches itself to the debris by means of a robot arm. Subsequently, two sets of ‘tentacle’ arms will close around the target debris and four pushing rods will be deployed to keep the chaser firmly attached to the target. After de-tumbling the system and other in-orbit operations, a series of braking manoeuvres will take place to re-enter Earth’s atmosphere and eventually plunge into the South Pacific Ocean. The proximity operations, or docking, require an accurate attitude control system and a detailed knowledge of the system. These operations have been divided into three phases: an unconnected phase in which the chaser spacecraft synchronises its motion with that of the (spinning) target debris, a semi-connected phase where the robot arm and tentacles form a flexible connection between the chaser and the target, and a connected phase where the chaser and the debris are assumed to be rigidly attached to each other (and form a stack). An important question is whether the system can remain stable and controllable in all of these three phases. This article focuses on how that is done.

EVOLVING SYSTEMS Because the system changes its configuration during its mission, it can be classified as an Evolving System; a system with (actively controlled) components that "mate" to form a single connected system [1]. In an Evolving System, the connection between the components becomes stronger during the evolution and can be represented by compliant forces and moments. Originally, Evolving Systems were envisioned to be applied to the assembly of large space structures, such as space stations or large aperture telescopes. Now, it can also be applied to the formation flying of satellites or autonomous rendezvous and docking. However, up till now, Evolving Systems have just been a theoretical framework; it has not yet been used in practice. So let's try and apply Evolving Systems to the docking scenario. At first, the two satellites ENVISAT and the chaser - are unconnected, then as the robot arm attaches the tentacles close, and the pushing rods are deployed. Therefore the system slowly becomes less and less flexible, and finally the system is assumed to be rigid. To simplify the complex behaviour of all these flexible parts, the connection between the chaser and ENVISAT can be modelled as a spring and a damper, which become more rigid as the system evolves. Now, because attitude control is only concerned with rotational motion, the connection needs to be modelled as a three-dimensional torsional spring and damper. For the parameterisation of the spring, Euler's eigen axis was used as the instantaneous axis of rotation; a method that has not yet been found in literature. With the system modelled, a simple linear

stability analysis can be performed to get an indication of the stability of the system. Note that any nonlinear effects, such as gyroscopic coupling between the axes, disturbance torques, and nonlinear actuators, have not been taken into account. The analysis showed that the system remains stable during its evolution when the chaser is actively controlled. This is mostly because the motion of the chaser has little effect on the motion of ENVISAT due to the large size difference between the two spacecraft. ENVISAT has a mass of around 8,000kg, whereas the chaser weighs 1,500kg. But more importantly, the moments of inertia of ENVISAT are two orders of magnitude larger than those of the chaser. However, for a system with equally sized spacecraft, the linear stability analysis showed that instability could occur. Hence, future removal missions of smaller debris would require thorough stability analyses and even more advanced attitude control.

NONLINEAR ANALYSIS For a more accurate stability analysis, the nonlinear effects have to be considered as well. To this end, a representative simulator was built in MATLAB/Simulink. The simulator comprises two major parts; the attitude control algorithms and the control actuators. The latter consists of a Reaction Control System (RCS) and a control allocation algorithm. The attitude control algorithms that are investigated are a Linear Quadratic Regulator (LQR) and a Model Reference Adaptive Controller (MRAC). The objective of the attitude control algorithm in phases 1 and 2 of the docking is to reduce the angular error between the chaser and the target debris to zero. In phase 3, the objective is to de-tumble the complete stack. The LQR is a very simple control method, which multiplies the state error between the chaser and the target by a pre-computed gain. This gain is derived based on the characteristics of the system and tries to minimise a certain objective function [2]. One can imagine that the gains for the unconnected system will be quite different from the connected system. This means that the gains have to be scheduled between the various phases of the docking. The Model Reference Adaptive Controller aims to minimise the difference between the output of the system (in our case the state error), and a reference model output [3]. The reference model can have any desired form, as long as it is within the limits of the actual system. It was decided to use the LQR in combination with a linear representation of the chaser model as the reference model, because it was already shown that the LQR can stabilise the system. However, once the two satellites are connected, the chaser reference model is not an accurate representation of the actual system; it is much heavier! So the adaptive controller will have trouble matching the outputs. Therefore, the connected (stack) model

was also used as a reference model. Now the system is disconnected, this is an overestimate of the actual model, and as a result the response of the system will be a lot slower. The chaser is actuated by an RCS consisting of 24 thrusters placed in pods of three on the vertices of the chaser. After the control algorithm has computed a desired three-dimensional control moment, the thrusters have to try to deliver this moment. Therefore, the control moment has to be allocated to the correct thrusters. This problem can be translated into a Linear Programming optimisation problem, which can be solved with standard numerical tools. Furthermore, thrusters are discrete actuators and can only deliver full or zero thrust. Thus, some type of modulation technique has to be implemented to translate the continuous desired thrust vector into a discrete thrust vector. A common technique is PulseWidth Pulse-Frequency (PWPF) modulation [4]. It is evident that with all these techniques, the actual delivered control moment will not be exactly the same as the desired control vector. But will this result in instability?

RESULTS Nonlinear simulations of the model revealed that using the chaser reference model will lead to instabilities. Therefore, only the stack reference model was used for further analyses. Furthermore, for the unconnected phase, both the LQR and adaptive controller are able to synchronise the motion of the chaser with ENVISAT. Second, the effect of the connection between ENVISAT and the chaser on the stability of the system is very small and both the controllers are able to control the system. Third, the adaptive controller shows unacceptable performance during the control of the stack because of the large difference between the reference model for which the adaptive controller was tuned and the actual system. Using two different reference models for the unconnected and the connected phase could improve the performance of the adaptive controller. Currently, further research is being performed to better investigate the use of a second reference model and to see what the effect of uncertainties in the model are on the stability of the controller. References [1] Frost, S.A. and Balas, M., 2010, “Evolving Systems and Adaptive Key Component Control”. In: Aerospace Technologies Advancements. Ed. by T.T. Arif. Rijeka: InTech. [2] Anderson, B.D.O. and Moore, J.B., 1989. Optimal Control: Linear Quadratic Methods. 1st ed. Englewood Cliffs: Prentice-Hall. [3] Kaufman, H., Barkana, I., and Sobel, K., 1998. Direct Adaptive Control Algorithms. 2nd ed. New York: Springer. [4] Wie, B., 2008. Space Vehicle Dynamics and Control. 2nd ed. Reston, VA: American Institute of Aeronautics and Astronautics, Inc. LEONARDO TIMES N°2 2016

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