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Defense Information, Communication, and Space Technology
Published in Anna M. Doro-on, Handbook of Systems Engineering and Risk Management in Control Systems, Communication, Space Technology, Missile, Security and Defense Operations, 2023
Eccentric anomaly, E—the angle subtended at the center of the orbit between the perigee and the projection of the satellite onto a circle of radius a. The true anomaly is converted to the eccentric anomaly using the following relationship (Kaplan and Hegarty 2006): E=2tan−1[1−e1+etan(12)v]
Attitude Determination Using Two Vector Measurements – TRIAD Method
Published in Chingiz Hajiyev, Halil Ersin Soken, Fault Tolerant Attitude Estimation for Small Satellites, 2020
Chingiz Hajiyev, Halil Ersin Soken
In the analyses, the satellite’s orbital parameters are assumed as: inclination i = 97°; right ascension of the ascending node Ω = 15°; eccentricity e = 0 and orbital altitude of h = 550 km. The Earth radius is R = 6378.14 km; the Earth angular velocity is; the Earth magnetic field moment is M = 7.86 × 1015 Wb ⋅ m; the magnetic tilt angle is ε = 11.4°. The accuracy of the orbital parameters i, Ω and u are 5e-6 rad, 1e-5 rad and 1.5e-4 rad, respectively. The attitude sensors’ accuracy is ~1° for magnetometer, 0.1° for sun sensor and 0.36° for horizon sensor (horizon sensor determines the roll and pitch angles). It is assumed that eccentric anomaly is equal to the mean anomaly. Only one orbital period was simulated. In Figure 5.4, the change of the satellite attitude accuracy throughout the orbit is shown when the first algorithm (SUN-MAG) is used (required accuracy is 1°). As accuracy characteristics pitch (θ), yaw (ψ) and roll (φ) angles’ variances are taken. The results for the second and third algorithms are given in Figures 5.5 and 5.6, respectively.
Orbit Dynamics and Properties
Published in Yaguang Yang, Spacecraft Modeling, Attitude Determination, and Control Quaternion-based Approach, 2019
The relation between true anomaly and eccentric anomaly is derived as follows. Note () x+y=ae=c () x=acos(ψ) () y=rcos(180−θ)=−rcos(θ)
A problem-based learning proposal to teach numerical and analytical nonlinear root searching methods
Published in International Journal of Mathematical Education in Science and Technology, 2022
Juan Luis González-Santander, Fernando Sánchez-Lasheras
Nonlinear equations are found in many branches of Applied Mathematics, Physics, and Engineering, but very few of them can be solved analytically. The impact of this fact on teaching at undergraduate level has been highly significant. For instance, in the calculation of planetary elliptic orbits, we need to evaluate the roots of Kepler's equation, where E is the eccentric anomaly, ϵ is the eccentricity and M the mean anomaly. The solution of Kepler’s equation can be formulated analytically as where Jn denotes the Bessel function of the first kind, and the sum in (1) converges for ϵ < 1 like a geometric series with ratio (Colwell, 1993):
Adaptive kriging-assisted optimization of low-thrust many-revolution transfers to geostationary Earth orbit
Published in Engineering Optimization, 2021
Renhe Shi, Teng Long, Hexi Baoyin, Nianhui Ye, Zhao Wei
The minimum-time low-thrust GEO transfer optimization problem can be generally formulated in Equation (1): where x and u are the state and control variables, respectively; J is the performance index (i.e. the total transfer time ); are the motion equations; and presents the terminal state constraints. The motion equations are established in terms of classical orbital elements combined with the mass–flow rate equation, as shown in Equation (2) (Yang 2001): where n, E and f are the orbit mean motion, eccentric anomaly and true anomaly, respectively; p = a(1− e2) is the latus rectum; u = + f is the argument of latitude; Isp is the specific impulse of the EP system; g0 is the gravitational acceleration at sea level; m is the mass of the spacecraft; T0 is the maximum thrust of the EP system; and F = [FR, FT, FN] are the acceleration components expressed in the RTN coordinate system that is defined in the Appendix. F consists of two parts, i.e. the thrust accelerations provided by the thrusters and the Earth’s oblateness perturbations. In this work, the disturbances caused by the first four zonal harmonics of non-spherical gravitational potential are taken into account (Ghosh et al. 2015). Since the EP system is incapable of operating during eclipses because the solar arrays cannot generate power in the shadow of the Earth, an Earth conical shadow model (Yang 2001) is employed to address the eclipse problem during orbital transfer. It is assumed that T0 = 0 when the satellite enters the shadow of the Earth.