Task-constrained path planning on implicit manifolds via successive projections

This paper proposes a projection-based layer for constrained path planning, where task requirements are encoded as equality constraints defining an implicit manifold. The method is compatible with geometric planners that optimise a path inside a corridor represented as a sequence of convex safety regions, and returns a collision-free B-spline path, with user-prescribed degree and continuity, that is uniformly 𝜀-adherent to the implicit task constraints. Corridor construction is biased toward task feasibility by projecting exploration seeds onto the manifold, so that each corridor element contains at least one feasible configuration. After corridor optimisation, the task constraints are enforced at the B-spline control points within a prescribed projection tolerance through a constrained closest-point problem that preserves corridor containment. Continuity is then recovered by a smoothing procedure based on alternating projections while maintaining the prescribed controlpoint tolerance. Uniform 𝜀-adherence along the whole path is finally enforced by an outer loop that recomputes the knot vector with progressively more knots and repeats optimisation and projection until the prescribed tolerance is satisfied. The approach is validated through simulations and real-world experiments.