Cⁿ Trajectory Generation in 3D Under Arbitrary Bounds on Curvature and Higher-Order Curvature Derivatives

We propose a method to solve the general problem of generating a trajectory in 3D space that is Cⁿ continuous, and satisfies given bounds on the curvature κ_max, and its higher-order derivatives κ·_max, κ̈_max, up to κ⁽_maxⁿ−²⁾ where n > 2. The core idea is to first synthesize parametrically and geometrically continuous Cⁿ path segments each lying in a 2D hyperplane that satisfy the constraints on the curvature and its derivatives, then lift these planar segments to 3D via an affine anchor-plane transformation that preserves Cⁿ continuity and the κ-related constraints. The method supports local replanning, and is amenable to real-time implementation. We demonstrate this approach to C⁴ trajectory generation in 3D, of practical interest to robotics and aerospace applications, where fourth-order smoothness (snap continuity) together with explicit bounds on κ, κ·, κ̈ is required for dynamic feasibility. We apply the proposed trajectory-generation pipeline to solve a terrain-following problem for a fixed-wing aircraft.