## 2 Constellation Dynamics (Walker)  
The mathematical module plays a major role in aligning the right orbit and orientation of the satellites in a plane, as required by the mission. Hence, detailed research was carried out to ensure that each element is finely analyzed and implemented through simulations. Small satellites have specific conditions and ways to operate due to their compact sizes, weight constraints, and restricted applications with given payloads. So, a well-defined system engineering approach needs to be adopted to orient satellites in such a way that they align to achieve optimal coverage and reach the desired locations in the shortest time possible in their particular plane. Walker constellation is one such significant pattern which is used globally for various applications, but for this paper, a further delicate approach was made to expand the use of walker type constellation and maintain better optimality for a dedicated period of operation. The major objective of this paper was to maintain and optimize the travel period of each orbit significantly to reach the end-user as quickly as possible while choosing the best solution through optimization technique. To do so, the walker dynamics had to be potentially robust for each satellite in every plane of the constellation. There has been certain research carried out to determine a reasonably optimal walker constellation for several real-time applications, but for disaster management the scope still remains open. There can be significant improvisation in maintaining and optimizing trajectories with an autonomous alignment approach to make the data delivery faster as needed. Figure 5: Lunar mission strategy. Hence, detailed research was carried out to study the mathematical models which are most suited for this kind of application using autonomous technology and algorithms. This is not only to enhance the trajectories, but also to significantly reduce the number of satellites to be operated to achieve a particular coverage objective. From the literature, it has been observed that a variety of modules have been used for understanding several scenarios of constellation operation through walker patterns. For this paper, a few of the most significant approaches were followed and implemented as suitable to the major objective (disaster management) for the optimization and maintenance of a global small satellite (using COTS) constellation understanding and undertaking all the constraints and situations in real-time conditions, respectively. The main issue of coverage was addressed using the phenomena of nadir-pointing coverage circle area \(A\) presented by Ref. [4] to be given as follows:  
\[A=\frac{2\Pi R^{2}h}{R+h} \tag{1}\]  
where \(R\) is the radius of Earth and \(h\) is the altitude of the satellite. The potential of the walker pattern can be properly utilized with its parameters aligning as required/suitable by their appropriate mathematical modeling considering the initial real-time conditions. The important integers mentioned as N (satellites), P (planes), and F (phasing parameter) define the relative movement of the satellites in a plane and autonomous control is also implemented to secure their movement with respect to the satellites in other planes. The specialty of the walker constellation is that the satellites remain uniformly distributed along the plane, giving enough opportunity to plan and implement the autonomous formation among them in a plane they are operating. Additionally, one interesting fact about this is that the only two parameters, which vary and have significant impacts on the constellation, have semi-major axes and inclination of the orbits with respect to the travel trajectory optimization. The right ascension of the ascending nodes (RAAN) also varies when aligning the orbits in different positions without interfering with one another. The correlation among the RAAN and the mean anomaly was also set using Ref. [4] as follows:  
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