Solving Kepler's equation with CORDIC double iterations

TL;DR

This paper presents a novel shift-and-add algorithm for solving Kepler's equation using only bitshifts, additions, and one initial multiplication, enabling efficient hardware implementation.

Contribution

The paper introduces a CORDIC-based method that eliminates multiplications during iterations, improving efficiency for hardware solutions of Kepler's equation.

Findings

Requires 75% more iterations than previous methods

Provides eccentric anomaly and sine/cosine terms scaled by eccentricity

Applicable to hyperbolic Kepler's equation as well

Abstract

In a previous work, we developed the idea to solve Kepler's equation with a CORDIC-like algorithm, which does not require any division, but still multiplications in each iteration. Here we overcome this major shortcoming and solve Kepler's equation using only bitshifts, additions, and one initial multiplication. We prescale the initial vector with the eccentricity and the scale correction factor. The rotation direction is decided without correction for the changing scale. We find that double CORDIC iterations are self-correcting and compensate possible wrong rotations in subsequent iterations. The algorithm needs 75\% more iterations and delivers the eccentric anomaly and its sine and cosine terms times the eccentricity. The algorithm can be adopted for the hyperbolic case, too. The new shift-and-add algorithm brings Kepler's equation close to hardware and allows to solve it with cheap and simple hardware components.