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In this paper we present a hierarchical multi-rate control architecture for nonlinear autonomous systems operating in partially observable environments. Control objectives are expressed using syntactically co-safe Linear Temporal Logic (LTL) specifications and the nonlinear system is subject to state and input constraints. At the highest level of abstraction, we model the system-environment interaction using a discrete Mixed Observable Markov Decision Problem (MOMDP), where the environment states are partially observed. The high level control policy is used to update the constraint sets and cost function of a Model Predictive Controller (MPC) which plans a reference trajectory. Afterwards, the MPC planned trajectory is fed to a low-level high-frequency tracking controller, which leverages Control Barrier Functions (CBFs) to guarantee bounded tracking errors. Our strategy is based on model abstractions of increasing complexity and layers running at different frequencies. We show that the proposed hierarchical multi-rate control architecture maximizes the probability of satisfying the high-level specifications while guaranteeing state and input constraint satisfaction. Finally, we tested the proposed strategy in simulations and experiments on examples inspired by the Mars exploration mission, where only partial environment observations are available.