A Universal Formulation for Path-Parametric Planning and Control

Path-parametric methods have gained popularity in the formulation of navigation algorithms, such as high-level planners [1, 2, 3], reinforcement learning (RL) policies [4, 5, 6] or low-level model predictive controllers (MPC) [7, 8, 9]. The fundamental concept behind these parametric methods is to either introduce the path parameter as an additional degree of freedom, enabling the system to regulate its progress along the path [10, 11], or to conduct a change of coordinates that project the Euclidean states to the spatial states, i.e., the progress along the path and the orthogonal distance to it [12, 13, 14]. These parametric formulations have proven successful for three primary reasons: firstly, they inherently capture the notion of advancement along the path, secondly they allow for embedding the path's geometric features, such as the curvature and the torsion, into the system dynamics, and thirdly, spatial bounds manifest as convex constraints in the orthogonal terms of the spatial states. Given the broad range of problems encompassing path-parametric approaches, existing methods remain detached from each other and are frequently presented as independent work. This has resulted in a disjointed body of literature, where these techniques are viewed as distinct methods. Consequently, the reader is left with a fragmented view of the path-parametric problem, making it difficult to understand the interplay between the different techniques. To close this gap, in this paper we show how all these approaches are interconnected by presenting a universal formulation for path-parametric planning and control. To this end, the path-parametric problem is analyzed from three different yet interconnected perspectives (i-iii): First, in Section 2, we study the (i) interplay of existing parametric techniques and show how they can be unified under a single framework consisting of two ingredients: (ii) a path-parameterization technique and (iii) a spatial representation of the system dynamics. These are discussed in-depth across the subsequent Sections 3 and 4, respectively.