This paper presents iterative forward kinematics algorithms for cable-driven parallel robots (CDPRs) that are based on Halley’s method. In contrast to other iterative forward kinematics methods that use a linearization of the CDPRs loop-closure equations, such as Newton’s method or the Levenberg-Marquardt method, Halley’s method uses a second-order Taylor series approximation of these equations. A hybrid method is also proposed that performs a Halley update for the first few iterations and then switches to a Levenberg-Marquardt update. The proposed algorithms are applied to a six degree-of-freedom suspended CDPR and are shown to reduce the number of iterations and increase the rate of successful convergence to the truth pose compared to the Levenberg-Marquardt method. The proposed hybrid method reduces the computation time required for convergence compared to the Levenberg-Marquardt method in the presence of large initial estimation errors.