This paper addresses the autonomous robot ergodicity problem for efficient environment exploration. The spatial distribution as a reference distribution is given by a mixture of Gaussian and the mass generation of the robot is assumed to be skinny Gaussian. The main problem to solve is then to find out proper timing for the robot to visit as well as leave each component-wise Gaussian for the purpose of achieving the ergodicity. The novelty of the proposed method is that no approximation is required for the developed method. Given the definition of the ergodic function, a convergence condition is derived based on the timing analysis. Also, a formal algorithm to achieve the ergodicity is provided. To support the validity of the proposed algorithm, simulation results are provided.