We present an approach to motion planning under motion and sensing uncertainty, formally described as a continuous partially-observable Markov decision process (POMDP). Our approach is designed for non-linear dynamics and observation models, and follows the general POMDP solution framework in which we represent beliefs by Gaussian distributions, approximate the belief dynamics using an extended Kalman filter (EKF), and represent the value function by a quadratic function that is valid in the vicinity of a nominal trajectory through belief space. Using a variant of differential dynamic programming, our approach iterates with second-order convergence towards a linear control policy over the belief space that is locally-optimal with respect to a user-defined cost function. Unlike previous work, our approach does not assume maximum-likelihood observations, does not assume fixed estimator or control gains, takes into account obstacles in the environment, and does not require discretization of the belief space. The running time of the algorithm is polynomial in the dimension of the state space. We demonstrate the potential of our approach in several continuous partially-observable planning domains with obstacles for robots with non-linear dynamics and observation models.