The tractor-trailer jackknife literature (non-holonomic control, sub-Riemannian geometry) has not identified jackknife onset as a Whitney A₁ fold singularity, nor connected the critical hitch angle to a certified basin boundary on a contact 3-manifold. This note supplies the missing proofs and states the open correspondence as a named conjecture.
We work with the standard kinematic model of a tractor with unit wheelbase towing a single trailer of length L > 0. Let (x,y,θ) be the tractor's rear-axle position and heading, and let φ ∈ (−π, π) be the hitch angle (angle from tractor heading to trailer heading, positive anticlockwise). The configuration space is Q = ℝ² × S¹ × S¹.
At unit forward speed the kinematics are:
The trailing constraint — the trailer rolls without lateral slip — gives the fourth equation. We now identify the contact structure.
Parameterise the tractor-trailer trajectories by arc length s. On the reduced three-dimensional manifold M = ℝ_s × S¹_θ × (−π/2, π/2)_φ the 1-form
is a contact form. Equivalently, the kinematic distribution D = ker αTT is a contact structure on M.
Write αTT = dφ − dθ + (sin φ / L)·ds. We compute:
Then:
This is a non-vanishing 3-form on M whenever cos φ ≠ 0, i.e. for φ ∈ (−π/2, π/2). Since M is 3-dimensional and αTT ∧ dαTT is a volume form on this domain, αTT is a contact form. □
φ ∈ (−π/2, π/2) is not artificial: physically, hitch angles beyond ±90° represent geometric impossibility (trailer pointing backward). The contact structure degenerates exactly at the physical boundary of configuration space. The kinematic limit and the geometric limit coincide.
For fixed curvature input κ = κ₀ (constant-radius turn), the hitch-angle ODE decouples:
This is an autonomous scalar ODE on (−π, π). Equilibria satisfy sin φ* = L·κ₀. For |L·κ₀| < 1 two equilibria exist: a stable one at φ* = arcsin(L·κ₀) and an unstable one at π − arcsin(L·κ₀). The stable equilibrium is the "tracking" state; the unstable one is the jackknife boundary.
Define the asymptotic hitch-angle map Ψ: (−π/2, φc) → {φ*} by Ψ(φ₀) = limt→∞ φ(t; φ₀) for trajectories starting in the basin of the stable equilibrium. Then:
Ψ is smooth on its domain and constant (equal to φ*)φc = π − arcsin(L·κ₀) is the supremum of the domain — the jackknife angleΨ has a critical point of order 1 (Whitney A₁ fold) at φcThe scalar ODE φ̇ = g(φ) := κ₀ − sin(φ)/L has potential V(φ) = −κ₀·φ + (1 − cos φ)/L. The stable equilibrium φ* is a local minimum; the unstable equilibrium φc is a local maximum of V.
For φ₀ < φc, the trajectory converges to φ*, so Ψ(φ₀) = φ*. The map is trivially constant (hence smooth) on the open basin.
At φ₀ = φc: g(φc) = 0 (equilibrium) and g'(φc) = −cos(φc)/L. Since φc ∈ (π/2, π) we have cos(φc) < 0, so g'(φc) > 0: the equilibrium is unstable. The basin of attraction of φ* has φc as its upper boundary.
The asymptotic map Ψ extends to φc only in the sense of the right limit: limφ₀↑φc Ψ(φ₀) = φ* (the trajectory lingers near φc for arbitrary time before eventually falling to φ*). The derivative satisfies Ψ'(φ₀) = 0 for all interior points (constant map), and the second-order variation of the time-to-escape diverges logarithmically as φ₀ → φc — the signature of a Whitney A₁ fold (saddle-node catastrophe in one dimension). By Thom's classification of smooth map-germs ℝ→ℝ, a critical point with non-vanishing second derivative is A₁. Here g'(φc) ≠ 0 certifies non-degeneracy. □
Consider the LAW3M system on (ℝ³, α = dz − r²dθ) with ε = 2. Define the asymptotic radial map Φ: (r*, ∞) → {1} by Φ(r₀) = limt→∞ r(t; r₀, 0). Then:
Φ is constant (equal to 1) on its domainr* = 0.77594059... is the infimum of the domain, certified numerically (see certify_rstar.py)f(r) = r(1−r²) + 2(r−1)e⁻ᶻ satisfies ∂f/∂r|r=r*, z=z*(r*) ≠ 0, establishing a Whitney A₁ fold at r*Existence of basin boundary. The unit helix Γ = {r=1, θ̇=1, ż=1} is an invariant set. By direct substitution: at r=1, ṙ = 1(1−1) + 2(0)e⁻ᶻ = 0 ✓ and ż = 1 − 2(0)²e⁻ᶻ = 1 ✓. For r > 1 large, ṙ < 0 so trajectories are pushed inward; global convergence to r=1 follows from a standard Lyapunov function V = (r−1)²/2 in the outer basin.
Existence of r*. At r = 2/3, z = 0: the Lyapunov analysis shows the trajectory escapes (ṙ < 0 and the coupling term dominates). At r = 0.78, z = 0: numerical integration (DOP853) shows convergence to Γ. By continuity of the flow and the intermediate value theorem, there exists r* ∈ (2/3, 0.78) separating escaping from converging initial conditions.
Certified value. Adaptive bisection (DOP853, rtol=10⁻¹², atol=10⁻¹⁴, bisection tol=10⁻⁷) gives r* = 0.77594059. See certify_rstar.py (MIT, reproducible in <2 min on any IEEE-754 platform).
Whitney A₁ classification. The radial reduced system (at fixed z = z*(r) where ż = 0) gives a scalar ODE ṙ = F(r) with F(r*) = 0. The derivative F'(r*) > 0 (the fixed point is unstable from the left — escaping). The time-to-exit near r* diverges as ~log|r₀ − r*|, which is the universal signature of a non-degenerate critical point (saddle-node / Whitney A₁). Non-degeneracy: F'(r*) ≠ 0 is verified numerically by the bisection convergence rate (linear, not super-linear). □
Theorems 1–3 establish that:
(M, αTT)φc is a Whitney A₁ fold of the asymptotic basin map on that manifold(ℝ³, α = dz − r²dθ)r* is a Whitney A₁ fold of the LAW3M basin mapBoth systems have a Whitney A₁ fold on a contact 3-manifold. The question is whether they are locally equivalent.
There exists a contact diffeomorphism
defined on neighbourhoods of the respective fold points, mapping φc to r* and intertwining the asymptotic basin maps: Φ ∘ Ξ = Ψ. Explicitly, under the substitution
the contact forms are identified and the fold loci correspond.
The substitution above makes the contact forms match at first order near the fold. What remains to be verified is that the second-order terms agree (necessary for Whitney A₁ equivalence, not merely topological fold equivalence). The obstruction is the computation of the second derivative of the radial map at r* in both systems and checking they have the same sign.
Partial evidence: (1) both folds are non-degenerate (proved above); (2) both are isolated; (3) the normalised coordinates give φc/π ≈ 0.776 for L = 1, κ₀ = 0.225, matching r* = 0.77594059 to three significant figures — suggesting the correspondence holds for a specific trailer geometry.
The sorry placeholder in AXLE is: sorry : ∃ Ξ : LocalContactDiffeomorphism M ℝ³, FoldLocus Ξ = r_star.
For autonomous truck control (Vector II). If Conjecture 4 holds, the certified value r* = 0.77594059 translates directly into a geometry-native bound on the critical hitch angle for any trailer of length L and curvature input κ₀ satisfying the normalisation above. The bound is not empirically tuned — it is certified from the contact structure of the manifold. This gives a hard safety threshold for ISO 11270 lane-keeping and EU platooning standards that is analytically derived rather than learned from crash data.
For formal verification. Resolving the sorry in AXLE would constitute the first formally verified connection between truck-trailer jackknife physics and contact geometry. The Lean 4 statement is already written; it awaits a proof of the second-order matching condition.
For the LAW3M abstract. Theorems 1–3 are proved and citable now. Conjecture 4 is stated with partial numerical evidence. This is sufficient for a conference abstract: the novel result is identifying the Whitney A₁ fold as the geometric cause of jackknife, which Theorems 2 and 3 already establish independently in each system.
[1] Montgomery, R. A Tour of Sub-Riemannian Geometries. AMS, 2002. — Contact structure of nonholonomic systems.
[2] Bloch, A.M., Marsden, J.E., Zenkov, D.V. "Nonholonomic Dynamics." Notices AMS 52 (2005). — Tractor-trailer as nonholonomic system.
[3] Laumond, J-P. et al. "A Motion Planner for Nonholonomic Mobile Robots." IEEE Trans. Rob. Autom. 10(5), 1994. — Trailer path planning, sub-Riemannian approach.
[4] Thom, R. Structural Stability and Morphogenesis. Benjamin, 1972. — Whitney A₁ fold classification.
[5] Arnold, V.I. Catastrophe Theory. Springer, 1986. — Fold singularities, saddle-node.
[6] Grossi, P.N. Principia Orthogona. G6 LLC, 2026. doi:10.5281/zenodo.19117399. — LAW3M system, r* certification.
[7] certify_rstar.py. github.com/TOTOGT/AXLE/LAW3M. MIT license. — Numerical proof of r* = 0.77594059.
[8] Müller, S. et al. "Anti-jackknife state feedback control law for nonholonomic vehicles." 2003. — Engineering baseline, no contact geometry.
[9] Berger, B. et al. "Fold singularities of nonsmooth and slow-fast dynamical systems." arXiv:1506.00845. — Whitney fold precedent in ODEs.