This study proposes a technique to estimate the output state of twisted string actuators (TSAs) based on payload's acceleration measurements. We outline differential kinematics relationships of the actuator, re-formulate these into a nonlinear parameter identification problem and then apply linearization techniques to efficiently solve it as a quadratic program. Using accurate estimates of string parameters obtained with the proposed method, we can predict TSA position with sub-millimeter accuracy via conventional kinematic relationships. In addition, the proposed method supports accurate estimation under varying operating conditions, unpredictable perturbations, and poorly-excited trajectories. This technique can be employed to improve the accuracy of trajectory tracking when the use of direct position measurements is challenging, with the list of potential applications including flexible and soft robots, long-span cable robots, multi-DOF joints and others.