Flatness-Based Geometric Path Following via Guiding Vector Fields

I. INTRODUCTION

• GVFs provide geometric path convergence without a timing law, but do not compensate for internal dynamics; DFBC handles nonlinear drift yet depends on a time-parameterized reference.Timing mismatch can introduce avoidable tracking error, particularly for infeasible or aggressive paths.Source: §I · p. 1
• The motivation includes agile flight, where missions may be specified geometrically but high-speed performance still calls for dynamics-informed control.Examples include aggressive quadrotor maneuvering and aerobatics.Source: §I · p. 1
• Prior flatness-based field tracking follows the field’s current value rather than anticipating its changes, and provides no bound on resulting tracking error.The introduction contrasts this with the proposed use of the GVF’s higher-order time derivatives [17].Source: §I · p. 1
• FGC uses a GVF and its higher-order time derivatives to generate DFBC references, avoiding a pre-planned timing law while retaining dynamics-informed control.A cascade Lyapunov argument establishes exponential convergence; the singularity-free extension removes the topological obstruction and guarantees the condition for global exponential convergence.Source: §I · pp. 1–2

A. Notation

• The notation distinguishes agent position ξ ∈ R^n, full state x ∈ R^nx, and flat output z = h(x) ∈ R^m.The letter also defines derivative notation, partial Bell polynomials, and the Hadamard product and power.Source: A. Notation · p. 2

B. Differential flatness-based control

• A differentially flat system admits a flat output from which its state and input can be recovered algebraically using finitely many output derivatives.The flat output has the same dimension as the input and differentially independent components; sufficiently smooth output trajectories generate admissible system trajectories when the parametrization is well-defined.Source: B. Differential flatness-based control · p. 2
• Inverting the input-output form and prescribing the desired flat-output derivative yields a DFBC law with positive-definite gains that assign the tracking-error poles.The input-output representation separates nonlinear drift from the decoupling matrix.Source: B. Differential flatness-based control · p. 2
• When relative degree does not span the full state, the internal dynamics must be input-to-state stable with respect to the flat-output derivatives for the law to stabilize the full state.Source: B. Differential flatness-based control · p. 2
• Model uncertainty and disturbances can make cancellation of the nonlinear drift imperfect; a sensor-based inner loop such as INDI is often added for model-independent disturbance rejection.Source: B. Differential flatness-based control · p. 2

C. Guiding Vector Fields for Path Following

• A GVF’s integral curves converge to and traverse a path without imposing a timing law.Source: C. Guiding Vector Fields for Path… · p. 2
• The path is described by n − 1 independent constraints, and the GVF combines tangential propagation with a converging term that vanishes on the path.The vanishing term makes the path invariant; the weighted constraint error has nonincreasing dynamics under the field.Source: C. Guiding Vector Fields for Path… · p. 2
• The constraint error decreases according to the squared norm of the converging term, while positive rescaling preserves the integral curves up to time reparameterization.A normalized GVF can traverse the path at constant speed where the field is nonzero.Source: C. Guiding Vector Fields for Path… · p. 2
• For a single closed path in R^2, at least one equilibrium inside the enclosed region is unavoidable for a standard GVF.This follows from the Poincaré–Bendixson theorem [19].Source: C. Guiding Vector Fields for Path… · pp. 2–3
• Adding a virtual coordinate lifts a parameterized path into an unbounded, non-self-intersecting path, removing the topological obstruction; the resulting field is nonzero everywhere and guarantees global convergence to the original path.Source: C. Guiding Vector Fields for Path… · p. 3

• FGC generates the reference velocity from the GVF at the agent’s current position, combining spatial path following with DFBC’s model-inversion structure.This avoids a pre-specified time-parameterized reference while retaining dynamics-informed control.Source: §III · p. 3
• Successive differentiation of the GVF along system trajectories supplies the higher-order references required by DFBC, and each derivative is computable from the current state.The relative degree ensures that the control input does not appear in the state derivatives used to compute these references.Source: §III · p. 3
• The GVF derivatives are formed by the multivariate chain rule using Faà di Bruno’s formula, with each order adding one field tensor and one state derivative.For piecewise or nondifferentiable paths, mollification can produce a C∞ approximation; the formula requires χ ∈ C^ρ and ϕ ∈ C^(ρ+1).Source: §III · p. 3

• Choosing gains so that A_e is Hurwitz makes the tracking-error system globally exponentially stable.The tracking error is a cascade of derivative errors, with e_1 = χ − ξ̇ representing velocity error.Source: A. Stability and Convergence Analysis · p. 3
• The path error converges exponentially when the tracking error decays, because its dynamics reduce to a stabilizing path-error law plus an exponentially vanishing perturbation.The convergence proof proceeds from tracking-error convergence to convergence of the physical trajectory to the path.Source: A. Stability and Convergence Analysis · pp. 3–4
• The convergence proposition assumes Hurwitz tracking gains, ISS internal dynamics when mρ < n_x, a valid input–output representation, nonsingular G(x), and a positive lower bound on the path-error gradient.These conditions delimit the stated convergence guarantee.Source: A. Stability and Convergence Analysis · p. 4
• The path-error convergence rate is min(β/2, λ), and the argument extends to n−1 constraints with a common gain when the path-error Jacobian has a positive minimum singular value.The gradient condition is a key geometric hypothesis and can fail where the gradient vanishes, as occurs at singular points of standard GVFs on closed paths.Source: A. Stability and Convergence Analysis · p. 4

• A virtual coordinate w lifts the agent state to ζ = (ξ, w), and its evolution follows the w-component of the singularity-free GVF.The GVF derivatives are then computed with respect to the augmented state.Source: B. Extension to the Singularity-Free GVF · p. 4
• The lifted path-error Jacobian satisfies σ_min(Ñ) ≥ 1 everywhere, so the gradient condition holds globally with c_min = 1.This lets Proposition 1 apply in the augmented space, yielding exponential convergence of the lifted path error and physical trajectory to the path.Source: B. Extension to the Singularity-Free GVF · p. 4
• The lifted reference derivatives preserve the linear tracking-error cascade by differentiating the singularity-free GVF through w, and path parameterizations must satisfy f_i ∈ C^(ρ+1).Higher-order terms are generated through the extended Jacobian and Bell polynomials.Source: B. Extension to the Singularity-Free GVF · p. 4

• MATLAB simulations test the convergence bound, internal dynamics, and the singularity-free extension.Source: §IV · p. 4

A. Circular Path With Radial Drift

• The example uses a double integrator in R2 with radial drift f(ξ) = ξ/∥ξ∥ and a circular path of radius R = 5.The flat output is z = ξ with relative degree ρ = 2; the drift map is F(ξ) = f(ξ) and G(ξ) = I2.Source: A. Circular Path With Radial Drift · pp. 4–5
• Because χ depends only on ξ, its first reference derivative is exact: χ^(1)(ξ, ν) = Jχ(ξ)ν.The source states that the ρ = 2 instance of the control law then reduces accordingly.Source: A. Circular Path With Radial Drift · p. 5
• The path error converges exponentially to the circle despite radial drift, while remaining within the predicted envelope.The velocity tracking error decays exponentially at a constant rate; the path error is nonlinear because the cascade combines its stabilizing dynamics with the tracking-error perturbation.Source: A. Circular Path With Radial Drift · p. 5, Figure 1 · p. 5

B. Helical Path with Internal Dynamics

• The R3 system uses ξ as its flat output with relative degree ρ = 2 and full state x = (ξ, ν, η) ∈ R9.The internal state evolves as η-dot = −η + ξ◦2, while the agent dynamics include the coupled drift −η◦ξ.Source: B. Helical Path with Internal Dynamics · p. 5
• The helix is constrained by a cylinder-radius condition and a separate angular-phase condition that sets its pitch.The guiding vector field’s propagation direction is τ(ξ) = g2 × g1.Source: B. Helical Path with Internal Dynamics · p. 5
• The agent follows the helix despite drift induced by the internal state η.The helix has radius R = 5 and pitch c = 1; its path error decays nearly linearly on the logarithmic scale, with the ascending constraint setting the effective rate.Source: B. Helical Path with Internal Dynamics · p. 5, Figure 2 · p. 5
• The internal-state subsystem satisfies the input-to-state stability condition, so the law applies when relative degree does not span the full state.On ξ = 0, ν = 0, the internal dynamics reduce to η-dot = −η, which is globally exponentially stable and confirms the cascade is minimum phase.Source: B. Helical Path with Internal Dynamics · p. 5

C. Singularity-Free FGC

• The figure-eight has a singularity under the standard GVF, so the example uses the singularity-free construction with virtual coordinate w.The curve is parameterized by f1(w) = R sin w and f2(w) = R sin w cos w, then lifted to ζ = (ξ, w) using implicit constraints.Source: C. Singularity-Free FGC · p. 5
• The leading-order reference derivative is computed in the lifted state, alongside the ρ = 2 control-law instance and virtual-coordinate update.The simulation uses k1 = k2 = ke = 4 and K1 = 12.Source: C. Singularity-Free FGC · p. 6
• Exponential convergence is achieved on the self-intersecting figure-eight, which the standard GVF construction cannot deliver.The agent traverses the origin; in the extended space, the lifted path is non-self-intersecting and the virtual coordinate w resolves the crossing.Source: C. Singularity-Free FGC · p. 6, Figure 3 · p. 6

V. CONCLUSION

• FGC generates the DFBC reference hierarchy directly from a guiding vector field rather than a pre-planned trajectory.The references are generated from the field rather than supplied as a time-parameterized trajectory.Source: §V · p. 6
• A cascade Lyapunov argument establishes exponential convergence to the desired path, with the gradient condition made global through singularity-free GVFs.Source: §V · p. 6
• Simulations confirm the results on helical, circular, and self-intersecting figure-eight paths.Source: §V · p. 6