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We consider the problem of designing controllers to steer mobile robots to the source (the minimizer) of a signal field. In addition to the mobility constraints, e.g., posed by the nonholonomic dynamics, we assume that the field is completely unknown to the robot and the robot has no knowledge of its own position. Furthermore, the unknown field is randomly switching. In the case where the information of the field (e.g., the gradient) is completely known, standard motion planning techniques for mobile robots would converge to the known source. In the absence of mobility constraints, convergence to the minimum of unknown fields can be pursued using the framework of numerical optimization. By considering these facts, this paper exploits an idea of the stochastic approximation for solving the problem mentioned in the beginning and proposes a source seeking controller which sequentially generates the next waypoints such that the resulting discrete trajectory converges to the unknown source and which steers the robot along the waypoints, under the assumption that the robot can move to any point in the body fixed coordinate frame. To this end, we develop a rotation-invariant and forward-sided version of the simultaneous-perturbation stochastic approximation algorithm as a method to generate the next waypoints. Based on this algorithm, we design source seeking controllers. Furthermore, it is proven that the robot converges to a small set including the source in a probabilistic sense if the signal field switches periodically and sufficiently fast. The proposed controllers are demonstrated by numerical simulations.