Second-Order Multidimensional Guiding Vector Field for Parametric Path-Following Under Holonomic Constraints

Autonomous path-following in higher-dimensional workspaces is a crucial yet challenging task for modern industrial applications. However, traditional geometric path-followers fail to accommodate robots requiring actuator control via acceleration-level commands. This paper proposes a lightweight guiding vector field (GVF) framework that commands second-order dynamics to support holonomic path-following over the Special Euclidean group $\mathrm{SE}(n).$ Defined as a map from the first to the second tangent bundle of this configuration space, the GVF balances four decoupled components: an attractive acceleration field, a skew-projected heading field, a feedforward-based centripetal acceleration field, and a kinematically-modeled tangential drive field. Using techniques such as smooth mollifiers to extend the closest point projection, the paper also tackles the GVF-based path-following of parametrically-defined curves, another problem that has traditionally proved difficult to solve. Through rigorous mathematical analysis, this paper proves that the algorithm retains the geometric guarantees of traditional GVF algorithms, such as smoothness, path invariance, and stable asymptotic convergence to the path. This architecture serves as the basis of the Pedro Pathing motion-planning and path-following library.