Motion planning has received much attention over the past 40 years. More than 15 years have passed since the introduction of the successful samplingbased approach known as the Probabilistic RoadMap Method (PRM). PRM and its many variants have demonstrated great success for some high-dimensional problems, but they all have some level of difficulty in the presence of narrow passages. Recently, an approach called Toggle PRM has been introduced whose performance does not degrade for 2-dimensional problemswith narrowpassages. In Toggle PRM, a simultaneous, coordinatedmapping of both ℂfree and ℂobst is performed and every connection attempt augments one of the maps – either validating an edge in the current space or adding a configuration’witnessing’ the connection failure to the other space. In this paper, we generalize Toggle PRM to d-dimensions and show that the benefits of mapping both ℂfree and ℂobst continue to hold in higher dimensions. In particular, we introduce a new narrow passage characterization, α-ε -separable narrow passages, which describes the types of passages that can be successfully mapped by Toggle PRM. Intuitively, α-ε -separable narrow passages are arbitrarily narrow regions of ℂfree that separate regions of ℂobst, at least locally, such as hallways in an office building. We experimentally compare Toggle PRM with other methods in a variety of scenarios with different types of narrow passages and robots with up to 16 DOF.