A linear mode is only a seed
At finite amplitude, period, waveform and inter-coordinate phase all vary along the family.
PhD research · Robotics · 2023–2026
Natural Locomotion Families
Let the mechanism organize the motion before controlling it.
Rather than prescribing an arbitrary periodic joint choreography, this thesis asks which repeatable motions are admitted by the locomotor's own dynamics. A natural- locomotion cycle solves the declared unforced conservative equations, returns its reduced mechanical state exactly, and leaves the reconstructed body pose free to move.
Here natural means dynamically compatible with the stated ideal model. It does not mean stable, optimal, biologically inevitable or already validated on hardware.
Current V3 working manuscript · automatically compiled from the thesis source
- Status
- Complete working manuscript
- Affiliation
- ENSTA · Institut Polytechnique de Paris
- PhD supervisor
- Luc Jaulin
- Co-supervisors
- Lionel Lapierre · Simon Rohou
The question
Four familiar ideas are each insufficient on their own.
The thesis combines nonlinear dynamics, relative periodicity, reconstruction and forced response without asking any one of them to certify more than it can.
Full periodicity suppresses transport
If body pose closes with the mechanical state, the cycle has no pose increment. Locomotion therefore requires reduced return with open pose.
A transporting loop need not be natural
Reconstruction tells where a prescribed loop goes; it does not prove that the timed loop solves the unforced dynamics.
Loss changes the object
A conservative cycle does not survive positive dissipation unchanged. A maintained response must return under forcing and replace its lost work.
The target is an exact reduced return admitted by the mechanism, with pose left open — not one preferred sinusoid.
Qualification
A cycle qualifies before it is organized.
Continuation may connect solutions and modal metrics may classify them, but the locomotor object itself must first pass three model-level checks.
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01
Mechanical compatibility
The complete time-parameterized reduced trajectory solves the declared unforced conservative field.
Solve
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02
Nontrivial exact reduced return
After a declared phase or minimal-period convention, the mechanical state returns: z(T) = z(0).
Return
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03
Declared locomotor output
Pose reconstruction gives the required nonidentity increment Δg ≠ eG. This establishes transport, not by itself an oscillation-only cause of propulsion.
Move
Qualified cycles can then be continued into branches and described by modal or harmonic observables. The historical acronym NLM names the research programme; the mathematical results are called families or branches because no global invariant-manifold theorem is claimed.
Finite-amplitude family
A family is not one sinusoid at different amplitudes.
The corrected two-segment archive contains 379 displayed cycles. Their clock, carrier participation, phase relation and harmonic content change with mean speed.
What control should accompany
A controller aligned with the mechanics must allow frequency and coordination to change along the branch instead of imposing one fixed harmonic choreography.
What transport does — and does not — show
A matched comparison on SE(2) shows that, away from near-zero drift, straight motion usually dominates the archived translation. The oscillation modulates that moving baseline; the comparison is not a causal energy decomposition.
Mechanical return
Exchange certifies return only when the missing information is scalar.
The carrier-closed / carrier-open construction is a field decomposition after carrier elimination, not a physical clamp. It exposes a transverse storage and the exchange induced when the original coupled field is restored.
Contributions and evidence
Each result is stated at the level its evidence supports.
The theorem, numerical constructions, audits and control demonstration are deliberately kept distinct.
Finite-amplitude relative-periodic locomotion
A spectral seed is corrected and continued into a 379-cycle moving family whose period and choreography vary. A matched-straight SE(2) audit then separates total transport from modulation relative to straight drift.
Scalar exchange–return equivalence
On an oriented finite-speed scalar section with injective storage, and once the other return rows close, zero integrated exchange is equivalent to return of the remaining crossing speed. The theorem is realized on 29 accepted cycles.
Coexisting returned in-phase and anti-phase sectors
At one fixed parameter vector, 2,278 AP and 1,358 IP opened cycle files pass the audited return, transport and modal-classification rows. Full vector return qualifies each cycle; modal metrics classify it afterwards.
One conservative cycle continued into maintained motion
Keeping geometry, inertia, constraint and elasticity fixed, one 2SEG cycle is followed through 19 damping-and-forcing homotopy points. This is one synchronized fixed-speed branch, not persistence or stability of the whole family.
A response map and local phase–speed navigation
A parameter-distinct archive contains 13,782 period-matched responses with folds and changing waveforms. Work–loss balance is exact; quadrature is a local marker only on the lower-amplitude segment. A fast phase loop and slower mean-speed update traverse that chart locally from two named starts.
Scientific scope and current limits
The mechanical results concern smooth ideal nonholonomic models with fixed nondimensional parameters. Conservative orbital stability, basins, robustness and hardware identity are not established.
The 3SEG population audit covers the stated return, modal, exchange and pose rows, not a complete genealogy or exhaustive atlas. The same-model bridge contains one anchored branch. The forced-response and controller archives belong to a different parameter set and have no experimental validation.
All 42 archived controller stages were accepted, so rejection, rollback and recovery remain untested. The demonstrated claim is local navigation of a forced-response chart, not stabilization of conservative natural-locomotion families.
Manuscript
Read the current working manuscript.
The concise V3 is designed as a self-contained three-day reading path: the necessary foundations are introduced where they become useful, and every main contribution is paired with its evidence boundary.
Download the latest manuscriptStable URL · refreshed automatically after each successful build of the V3 source branch · not yet the final institutional deposit.
Question and foundations
Action and phase, nonlinear modes, relative periodicity, reconstruction, return maps and losses.
Conservative families and return
The 2SEG family, scalar exchange theorem, 3SEG vector boundary and matched-pose audits.
Maintained response and navigation
The same-mechanics local bridge, parameter-distinct Resonant Locomotion Map and local phase–speed controller.
Synthesis, limits and protocols
A direct answer, evidence topology, research programme and technical appendices for reproduction.
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