MATLAB Implementation of Classical Orbital Elements (Keplerian Elements)

Classical orbital elements (also known as Keplerian elements) are fundamental parameters describing spacecraft orbital motion, consisting of six key parameters: semi-major axis, eccentricity, orbital inclination, right ascension of ascending node, argument of perigee, and true anomaly. In space mission analysis, conversions between position/velocity vectors and orbital elements are frequently required. In MATLAB implementations, orbital element conversions typically involve two main directions: Position/Velocity to Orbital Elements: This conversion calculates six orbital parameters from given position and velocity vectors. Key algorithmic steps include computing the angular momentum vector using MATLAB's `cross()` function, determining the orbital plane, solving for eccentricity using vector operations, and calculating the direction to perigee. The implementation involves careful handling of trigonometric functions to resolve quadrant ambiguities. Orbital Elements to Position/Velocity: This conversion predicts spacecraft position and velocity at any given time based on specified orbital elements. The algorithm requires iterative solution of Kepler's equation for eccentric anomaly using Newton-Raphson methods, followed by coordinate transformation to Cartesian system. MATLAB's `fzero()` or custom iteration functions can be employed for solving transcendental equations. Important implementation considerations include: Handling singularities for special orbits (equatorial or circular orbits) through conditional checks and alternative formulations. Utilizing MATLAB's built-in functions like `norm()` for vector magnitudes and `cross()` for vector products to simplify mathematical operations. For high-precision requirements, implementing perturbation models accounting for Earth's non-spherical gravity effects using additional dynamical equations. Extension possibilities: Validation of results through integration with STK (Systems Tool Kit) Development of reusable function modules for space mission analysis toolboxes with error handling and unit conversion capabilities