Three-dimensional Nonlinear Path-following Guidance with Bounded Input Constraints

The paper rigorously proves fixed-time convergence of the range r and lead angles (θ_U, ψ_U) using composite Lyapunov functions V2 := r + |x|, W2 := |θ_U| + |z|, and W4 := |ψ_U| + |y|, together with a fixed-time lemma (Polyakov) and an inequality on sums of powers. See Theorem 2 and its design for U_c (Eqs. (7)–(9)), with the key identity r_dot = −x cosθ_U cosψ_U − (M1 r^{α1} + N1 r^{β1}) and the composite Lyapunov step that yields the bound with M̄1 = 2^{1−α1} M1 (a typeset “2(1−α1)” appears but is clearly intended as 2^{1−α1}) . The invariance of the saturated inputs is established for both the speed channel (Theorem 1) and the rate channel (Theorem 3) . For heading channels, fixed-time convergence is proven via W2 and W4 together with Lemma 2 and Lemma 1 (Theorems 4–5) . By contrast, the model’s solution (Phase 2) makes two critical missteps: (i) it claims r_dot = −cosσ_U[x + M1 r^{α1} + N1 r^{β1}] and then “ignores” −cosσ_U x to deduce D^+ r ≤ −M1 r^{α1} − N1 r^{β1}; this is not valid without first combining r and x in a composite Lyapunov function, precisely what the paper does; and (ii) it asserts sign(θ_U) z ≤ 0 and sign(ψ_U) y ≤ 0 almost everywhere based on the injected terms, instead of establishing negativity of the composite derivatives Ẇ2 and Ẇ4 as the paper proves. Hence, while the model’s final claims match the paper qualitatively, key proof steps are incorrect or missing.