Principle, Method, and Control
Which movements does a robot’s mechanics make possible — and how can we sustain them?
Doctoral manuscript · Mirado Mortel
Version 7
· 116 pages · PDF, 5.3 MB
Download V7 ↓Reading draft · English, with French and English abstracts. This is a working manuscript, not the final deposited thesis.
This version combines a chapter-by-chapter review with revised figures, clearer transitions and closer links between explanations and results. The latest visual pass recomposed eight figures and added one, making action and phase, energy exchange, the 2SEG theorem and the two control loops easier to follow.
| Version | Date | What changed | Read |
|---|---|---|---|
| V7 · latest | Chapter explanations and figure reading revised; eight figures recomposed and one added in the latest visual pass. Transitions and body–appendix organization refined. 116 pages. | PDF ↗ | |
| V5 · previous | Narrative and explanations revised; POE principle and 2SEG theorem foregrounded; 2SEG/3SEG evidence updated. 108 pages. | PDF ↗ | |
| V4 · archive | Previous scientific revision, before the V5 readability pass. Exact copy formerly available on this website. 119 pages. | PDF ↗ |
Each dated version link keeps its saved PDF unchanged. The latest-manuscript link always points to the newest available version.
The idea
01 · Natural motion
A pendulum has its own way of oscillating. Can an articulated robot also have its own way of moving forward? In an ideal model without losses or actuation, internal oscillations interact with environmental constraints to produce locomotion.
Joint angles, joint rates and admissible body velocities repeat, while the body advances. This is a natural locomotion cycle: the movement follows the dynamics, rather than a prescribed choreography.
02 · Energy exchange
Just as energy moves back and forth between a pendulum’s kinetic and potential stores, it can move between the robot’s effective oscillatory channel and its propulsive channel.
Ideal environmental constraints redirect this energy without supplying it. The principle requires cyclic propulsion–oscillator exchange (POE), with no net gain or loss in either channel. Both belong to the internal dynamics; they are a different partition from kinetic and potential energy.
03 · Sustain the motion
With dissipation, an actuator must replace lost energy. Like pushing a swing, timing matters. A phase-locked loop adjusts actuation frequency, while a slower amplitude loop regulates mean speed.
The simulated robot’s mechanics determines the detailed joint motion; the controller does not play back a prescribed gait.
Principle → Method → Control
Principle
The oscillatory and propulsive energy stores make POE explicit within the internal dynamics. Balanced exchange is necessary for a natural cycle; in general, an energy balance alone does not establish repetition of the whole internal state.
2SEG · theorem
Observe the joint crossing a fixed angle in the same direction: a Poincaré section. In the regular two-segment model, zero net POE is equivalent to recovery of the same joint rate. If the body velocities also repeat, the complete internal state repeats; nonzero reconstructed translation makes this a locomotion cycle.
The trajectory follows the original unforced conservative 2SEG equations in nonsingular admissible coordinates, with positive effective inertia and a finite next transverse crossing of the same oriented section. The energy–exchange identity holds along the trajectory. Comparing both body velocities is a separate requirement; energy balance does not supply it. A local NLM interpretation additionally requires a regular family and an embedded phase sweep.
Method · NLM
An auxiliary oscillation seeds a calculation that restores propulsive coupling and solves the full internal periodicity conditions. Continuation then follows neighbouring cycles as frequency and coordination change. A regular family together with its phases forms a local Natural Locomotion Manifold (NLM); mean speed can identify cycles locally, not drive passive migration between them.
3SEG · boundary
With three segments and two joints, scalar POE balance remains necessary but is insufficient. Every internal-state component must satisfy its periodicity condition. Numerical results distinguish in-phase and anti-phase organizations at a shared mechanical design, without turning these families into one global manifold.
Control · numerical
A local continuation connects one conservative cycle to maintained motion with the same mechanics. A separate, parameter-distinct forced-response study supports local Resonant Locomotion Manifold (RLM) descriptions. PLL timing and slower amplitude regulation maintain simulated motion: fundamental quadrature is a local resonance marker, not a universal rule or a guarantee of global nonlinear stability.
A reading route
Consult the seven appendices as needed for models, numerical protocols and geometric detail. The PDF’s linked contents provides chapter navigation.
Related publications
Publications before 2026 are indexed under Mirado Rajaomarosata. Full citations on the portfolio.