Abstract The task of position and velocity estimation of a body based on the Doppler effect, whether the body is transmitting or receiving a signal of known or unknown frequency, is a common problem arising in many different application domains. While prior works use nonlinear least-squares methods to solve the Doppler equations for positioning, such an approach requires an initial estimate to perform the initial linearization step. However, in certain scenarios (e.g., initial orbit determination), an initial estimate may not be readily available, and without a good initial estimate, the nonlinear least-squares solution may not converge. This work reformulates the Doppler-based positioning problem as a system of polynomial equations, allowing for a direct solution without any a priori state information. We then leverage homotopy continuation to obtain the global solution to the polynomial system. For an unknown transmitter (or receiver), we show that the data from six or seven receivers (or transmitters) is sufficient in the case of known or unknown frequency, respectively, to recover the unknown state up to finitely many possibilities. This technique provides a structured method for initializing nonlinear least-squares or sequential filters for further refinement of the estimated state, improving their convergence. After a brief development of the mathematics, two simple examples are provided: (1) initial orbit determination of a satellite emitting an electromagnetic signal and (2) position and velocity estimation of a vocalizing dolphin emitting an acoustic signal.
Mancini et al. (2026) studied this question.