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About this book

Small satellite technology is opening up a new era in space exploration offering reduced cost of launch and maintenance, operational flexibility with on-orbit reconfiguration, redundancy etc. The true power of such missions can be harnessed only from close and precise formation flying of satellites. Formation flying missions support diverse application areas such as reconnaissance, remote sensing, solar observatory, deep space observatories, etc. A key component involved in formation flying is the guidance algorithm that should account for system nonlinearities and unknown disturbances. The main focus of this book is to present various nonlinear optimal control and adaptive guidance ideas to ensure precise close formation flying in presence of such difficulties. In addition to in-depth discussion of the relevant topics, MATLAB program files for the results included are also provided for the benefit of the readers. Since this book has concise information about the various guidance techniques, it will be useful reference for researchers and practising engineers in the space field.

Table of Contents

Frontmatter

Chapter 1. Introduction and Motivation

Abstract
Over the past few decades, there has been immense advancement in the miniaturization of onboard computer, sensor, actuator, and battery technologies. These developments have helped in miniaturing the satellites. Despite this advantage, however, due to their limited size and weight, no meaningful practical mission is possible using small satellites in stand-alone mode. However, in many missions, in principle, one can achieve similar or better performance as compared to a larger satellite using multiple small satellites flying in formation. In view of this, and also because certain missions can only be realized using multiple satellites with some minimum physical separation, an emerging trend across the globe is to have missions involving multiple small, distributed, and inexpensive satellites flying in formation to achieve common objectives.
S. Mathavaraj, Radhakant Padhi

Chapter 2. Satellite Orbital Dynamics

Abstract
Orbital dynamics is primarily concerned with the motion of orbiting celestial and man-made bodies. A well-studied specific orbital dynamics problem is the classic two-body problem, where two celestial bodies keep moving under the gravitational influence of each other. This chapter presents an overview of the two-body orbital mechanics first. Satellite orbital dynamics is presented next, which is a special case of the two-body problem, where the mass of one celestial body (e.g., the satellite) is negligible as compared to the body around which it orbits. Subsequently, the relative motion of two satellites is presented, which is used to synthesize the guidance schemes for formation flying.
S. Mathavaraj, Radhakant Padhi

Chapter 3. Infinite-Time LQR and SDRE for Satellite Formation Flying

Abstract
In this chapter, we demonstrate the applicability of the standard linear quadratic regulator (LQR) and state-dependent Riccati Equation (SDRE) based on linear and nonlinear optimal controllers to achieve the objective of satellite formation flying. Infinite-time formulations of LQR and SDRE offer the advantage of simplicity naturally and hence they are widely preferred in various applications. The utility of these methods for satellite formation flying is demonstrated in their respective validity regions. Both methods are effective when the chief satellite is in the circular orbit and the relative separation distance is small. On the other hand, the SDRE-based formulation is found to be effective even for eccentric orbits, but only with small eccentricity.
S. Mathavaraj, Radhakant Padhi

Chapter 4. Adaptive LQR for Satellite Formation Flying

Abstract
As shown in Chapter 3, a linearized relative dynamics based LQR control theory does not lead to satisfactory results when the formation is desired in an elliptic orbit and/or when the relative separation is high. It was also demonstrated in Chapter 6 that the SDRE approach is a relatively better to address such issues to a reasonable extent. Unfortunately, however, practicing engineers are typically not comfortable to switch over to nonlinear control theory completely. In view of this, an alternate approach is presented here. This approach uses the standard infinite-time LQR control as the nominal controller. However, it is augmented with online learning based adaptive optimal controller that accounts for the effects of the neglected system dynamics and external state-dependent disturbances together. This additional adaptive component in fact is computed from the identified system. The overall controller is termed as ‘Adaptive LQR’ controller for obvious reasons. As expected, this Adaptive LQR controller performs much better than the nominal LQR controller. It is worth mentioning here that this technique is capable enough to handle small eccentricity as well as large desired relative distance scenarios. Details of the process are discussed next.
S. Mathavaraj, Radhakant Padhi

Chapter 5. Adaptive Dynamic Inversion for Satellite Formation Flying

Abstract
The benefit of satellite formation flying can truly be realized with greater mission flexibility such as higher inter-satellite separation, formation in elliptic orbits, etc. However, under the above-enhanced conditions, linear system dynamics based control design approaches fail to achieve the desired objectives. Even though the LQR philosophy inspired the SDRE approach discussed in Chapter 3 offers a limited solution, it suffers from the drawback that the success of the approach largely depends on the typical state-dependent coefficient form one adopts (which remains as ‘art’). Moreover, if the eccentricity deviates significantly from circular orbit or separation distance requirement becomes large significantly, even SDRE can fail. The Adaptive LQR offers a fairly good solution to this issue, but introduces neural network learning concepts even for the system dynamics that is fairly known which can be handled directly. This brings in additional transients at the beginning of learning as well, which should preferably be avoided. In view of these observations, this chapter presents an alternate approach that need not be optimal, but can be successful under such realistic conditions as well.
S. Mathavaraj, Radhakant Padhi

Chapter 6. Finite-Time LQR and SDRE for Satellite Formation Flying

Abstract
Even though the results of the infinite-time LQR, SDRE, DI approaches are seemingly easier to understand and implement and can lead to acceptable results with appropriate tuning, it is very important to understand that such approaches are not strongly recommended for satellite formation flying problems in general. This is because in infinite-time formulations, the error is usually driven to zero asymptotically. However, as one can see from Chapter 2, the satellite formation flying problem is fundamentally a problem where one should ensure formation flying in two neighboring ‘orbits’ (unlike the formation flying of aerial vehicles). Hence, the right problem formulation should ensure that the relative desired position and velocity vectors are achieved at a ‘particular time’ (not earlier, not later). Once that is achieved, from that time onward, the deputy satellite remains in the desired orbit with respect to the chief satellite. Hence, such a problem formulation should ideally be done under the ‘finite-time’ optimal control paradigm instead. To address such finite-time terminal constraints problems, fortunately a few advanced techniques are also available in the literature. The present and next chapters deal with a few of these techniques, and demonstrate the suitability and applicability of these methods for satellite formation flying problems. In this chapter, an overview of the finite-time LQR and SDRE techniques is presented, followed by their usage for the satellite formation flying application.
S. Mathavaraj, Radhakant Padhi

Chapter 7. Model Predictive Static Programming

Abstract
In Chapters 3 and 6, a nonlinear suboptimal control design technique known as the SDRE technique has been successfully employed (both in infinite-time and finite-time frameworks) for the satellite formation flying problem. However, the method requires that the plant model to be expressed in the linear-looking state-dependent coefficient form, which is a nontrivial task in general. Moreover, because of the usage of linear optimal control theory, the resulting solution turns out to be suboptimal. In order to overcome such issues, an alternate optimal control design approach, known as the model predictive static programming (MPSP), is proposed in this chapter. This fairly recent approach solves the original nonlinear optimal control problem in a computationally efficient manner without introducing any transformation or approximation. Because of its fast convergence with less computational demand, it is suitable for implementation in the onboard processors as well. The generic technique is discussed first, followed by its usage for controlling the problem of satellite formation flying problem. Both discrete and continuous time versions of the MPSP technique are presented here for completeness.
S. Mathavaraj, Radhakant Padhi

Chapter 8. Performance Comparison

Abstract
By now, it must be obvious to the reader that the major goal for this book is to present various modern optimal control and adaptive guidance design techniques that are best suited in application to the satellite formation flying problems. Namely, the following design techniques have been discussed in detail in various preceding chapters: (i) Infinite-time linear quadratic regulator (LQR), (ii) Infinite-time state-dependent Riccati Equation (I-SDRE), (iii) Adaptive LQR, (iv) Adaptive dynamic inversion (DI) (v) Finite-time linear quadratic regulator (F-LQR), (vi) Finite-time state-dependent Riccati Equation (F-SDRE), and (vii) both discrete and continuous versions of model predictive static programming (MPSP). It is apparent, a curious reader would definitely want a comparative study to figure out the best performing method for the problem in hand. This chapter is aimed at to bring out the performance aspects of various methods presented in the earlier chapters.
S. Mathavaraj, Radhakant Padhi

Chapter 9. Conclusion

Abstract
An emerging trend across the globe is to have missions involving many small, distributed, and largely inexpensive satellites flying in formation to achieve a common objective. The advantages of satellite formation flying can be summarized as (i) higher redundancy and improved fault tolerance, (ii) on-orbit reconfiguration within the formation (which offers multi-mission capability and design flexibility), (iii) lower individual launch mass in case of small satellite missions, typically leading to reduced launch cost and increased launch flexibility, and (iv) minimal financial loss in case of failures both during launch as well as operation. Satellite formation flying enables new application areas such as spar antenna arrays for remote sensing, distributed sensing for solar and extra-terrestrial observatories, interferometry synthetic aperture radar, and many more. Also a variety of interesting and useful applications are possible in satellite formation flying, both as a substitute to single large satellite missions as well as the ones unique to formation flying missions.
S. Mathavaraj, Radhakant Padhi

Backmatter

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