# Natural efficient gaits from Nonholonomic Locomotion Nonlinear Normal Mode (NL-NNM): The Pendrivencar case

Mirado Rajaomarosata, Luc Jaulin, Lionel Lapierre, Simon Rohou

► To cite this version:

Mirado Rajaomarosata, Luc Jaulin, Lionel Lapierre, Simon Rohou. Natural efficient gaits from Nonholonomic Locomotion Nonlinear Normal Mode (NL-NNM): The Pendrivencar case. Mechatronics, 2025, 110, pp.103366. 10.1016/j.mechatronics.2025.103366. hal-05306055

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Submitted on 9 Oct 2025

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# Natural efficient gaits from Nonholonomic Locomotion Nonlinear Normal Mode (NL-NNM): The Pendrivencar case☆

Mirado Rajaomarosata ID*, Luc Jaulin ID, Lionel Lapierre ID, Simon Rohou ID

ENSTA, Lab-STICC - UMR CNRS 6285, Brest, France

## ARTICLE INFO

**Keywords:**

Energy-efficient locomotion

Bio-inspired robot

Nonholonomic locomotion

Nonlinear normal modes

Invariant manifold

Passive dynamics

Natural oscillation

## ABSTRACT

Bio-inspired robots remain far less energy-efficient than animals because conventional controllers impose trajectories that fight passive dynamics, whereas animals exploit resonance through natural nonlinear normal modes (NNM), whose periodic internal motions form a smooth 2D invariant surface; We ask how to define and compute the natural motions of a conservative locomotion system: propulsion arises only from no-slip constraints, and once initiated, a gait persists without actuation—like a frictionless pendulum. We tackle non-holonomic constraints on the Pendrivencar, a vehicle driven by a motorised pendulum with a cubic torsional spring; We introduce the Nonholonomic Locomotion - NNM (NL-NNM): extract a high-speed spectral seed – where chassis oscillations vanish and the pendulum is neutrally stable – refine the periodic orbit, and continue the resulting 2D invariant manifold via pseudo-arclength across three slow centre manifolds (stable for positive speed, neutral at zero, unstable for negative) from non-isolated rectilinear equilibria; We demonstrate the first NL-NNM for a moving non-holonomic robot: internal orbits produce a pendulum–chassis choreography whose energy-dependent frequency shifts and harmonic richness exceed linear predictions. Via geometric phase, each orbit yields undulatory straight-line motion. A dual-loop control simulation confirms autonomous path tracking with only the pendulum; Extending to dissipative regimes via non-linear resonant modes offers a path to high-efficiency locomotion in aquatic, aerial, legged, soft-bodied, and other robots.

## 1. Introduction

### Motivation

Robots remain markedly less energy-efficient than animals: fish, birds and humans exhibit costs of transport (COT) 4–5 × lower than their best robotic counterparts [1–5]. A major culprit is that conventional controllers impose joint trajectories, forcing actuators to fight the machine’s passive dynamics [6]. By contrast, animals exploit resonance: fish tune body stiffness to match tail-beat frequency [7], moths align wing-beat to thorax elasticity [8], and humans walk near the pendular frequency of their limbs [9].

### Linear resonance is not enough

For a pendulum of length $l$ the small-angle period is $T_0 = 2\pi\sqrt{l/g}$, but the exact period increases with amplitude as

$$T(\theta_0) = 4\sqrt{l/g} K\left(\sin\frac{\theta_0}{2}\right),$$

where $K$ is the complete elliptic integral [10]. Hence energy changes the natural frequency. Linearising an n-DOF mechanism yields n un-coupled harmonic oscillators whose constant-energy trajectories are ellipses confined to two-dimensional linear normal-mode (LNM) planes [11]. Because all oscillations in one LNM share the same frequency, linear resonance cannot explain the frequency-amplitude and frequency-speed variations observed in flapping and swimming [12, 13].

### Non-linear normal modes (NNMs)

A non-linear normal mode (NNM) is a two-dimensional invariant manifold on which any orbit remains trapped; period and shape now vary smoothly with energy [14,15]. For the pendulum, the LNM ellipses inflate into such a curved surface, reproducing the exact energy-period relation. NNMs are well established in structural dynamics, but their translation to robotics is still in its early stages. To our knowledge, the first robotics-focused NNM studies are by Albu-Schäffer and colleagues

<footnote>☆ This paper was recommended for publication by Associate Editor Toru Namerikawa.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: mirado.ajaomarosata@ensta.fr (M. Rajaomarosata), luc.jaulin@ensta.fr (L. Jaulin), lionel.lapierre@ensta.fr (L. Lapierre), simon.rohou@ensta.fr (S. Rohou).</footnote>
at DLR—e.g. surveys of conservative systems [16], energy-efficient soft-robot motions [17], and eigenmodes for elastic legs [18]—but a locomotion-centred treatment with non-holonomic constraints is still missing [19].

## Problem statement and platform

Apart from Rand's fixed Chaplygin sleigh study [20], NNMs have never been computed for a moving non-holonomic system. We close this gap by introducing the non-holonomic locomotion NNM (NL-NNM) and computing it on a deliberately minimalist two-link Chaplygin car, the Pendrivencar. A no-side-slip chassis is coupled to an internal pendulum via a cubic torsional spring; a single actuator drives the relative angle, producing both propulsion and steering—an architecture reminiscent of Tallapragada's one-motor swimmer [21–23].

## High-speed spectral seed

As the forward speed $v \to \infty$ the chassis motion freezes while the pendulum continues to wobble. The resulting one-parameter line of equilibria—the asymptotic centre—has two purely imaginary eigenvalues, all others collapsing towards zero. Centre-manifold theory therefore provides a two-dimensional neutral subspace that serves as an ideal seed for NL-NNM continuation [24,25].

## Computational pipeline

For the conservative plant we combine (i) spectral seeding at the asymptotic centre, (ii) Newton refinement to periodicity, and (iii) pseudo-arclength continuation. Dissipation and forced resonant modes are left for future work.

## Contributions

* First NL-NNM of a mobile non-holonomic robot.

* Linear–nonlinear bridge via a high-speed spectral seed.

* Generic extraction of constrained linear modes.

## Organisation

Section 2 formulates the mechanical model; Section 3 analyses LNMs and the asymptotic centre; Section 4 details NL-NNM construction; Section 5 presents numerical backbones; Sections 6–9 address energetics, control, discussion and conclusions.

## 2. Mechanical model of the pendrivencar

Fig. 1 shows the vehicle: a no-side-slip chassis ($l_1, m_1$) carrying an internal pendulum ($l_2, m_2$) linked by a cubic spring and driven by a single motor.

### 2.1. Geometry and spring

* Total length $l$, mass $M$; split with $0 < \epsilon < 1$: $l_1 = (1 - \epsilon)l$, $l_2 = \epsilon l$, $m_1 = \epsilon M$, $m_2 = (1 - \epsilon)M$.

* Inertias $I_i = \gamma m_i l_i^2$.

* Cubic torsional spring: $\tau_s = -k(\delta + 10\delta^3)$.

### 2.2. Coordinates and constraint

State splits as $q = (G, \delta)$ with $G = [x \ y \ \theta]^\top \in SE(2)$. Side-slip is forbidden through

$$A(\theta)\dot{q} = 0, \quad A(\theta) = \begin{bmatrix} -\sin\theta & \cos\theta & 0 & 0 \end{bmatrix}.$$

A right-null basis $H(\theta) = \begin{bmatrix} \cos\theta & 0 & 0 \\ \sin\theta & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ yields Pendrivencar kinematics

$$\dot{q} = Hv \text{ with admissible velocities } [26] \quad v = [v \ \theta \ \delta]^\top.$$

<figure><img src="image_4.png" /><figcaption>Fig. 1. Pendrivencar. Top: CAD; bottom: schematic with pose $G = [x \ y \ \theta]^\top$, speed $v$ and pendulum angle $\delta$.</figcaption></figure>

### 2.3. Scaling and auxiliary terms

Scale time by the elastic period $t_c = \sqrt{Ml^2/k}$, and non-dimensionalise velocities via $\{v, \dot{\theta}, \delta\} \mapsto \{v t_c / l, \dot{\theta} t_c, \delta t_c\}$. Set $\Delta = \epsilon - 1$ and $D = \dot{\delta} + \dot{\theta}$. The kinetic-energy cross-terms involve

$$A = -\Delta^3(\gamma + 1),$$

$$B = \Delta^2[-\epsilon\gamma + \epsilon(\cos\delta - 1) + \gamma + 1],$$

$$C = \epsilon^3(\gamma + 1) - \gamma\Delta^3 - \Delta[-2\epsilon^2(\cos\delta - 1) + 2\epsilon(\cos\delta - 1) + 1].$$

### 2.4. Conservative dynamics

With the Lagrangian $\mathcal{L} = \mathcal{K} - \mathcal{V}$,

$$\mathcal{K} = \frac{1}{2}[A\delta^2 + 2B\dot{\delta}\theta + C\dot{\theta}^2] - \Delta^2 D v \sin\delta + \frac{1}{2}v^2,$$

$$\mathcal{V} = \frac{1}{2}\delta^2 + \frac{5}{2}\delta^4.$$

The Euler–Lagrange equations [27] with multiplier $\lambda$ are

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = A(\theta)^\top \lambda.$$

Projecting onto admissible velocities by pre-multiplying with $H(\theta)^\top$ eliminates the constraint force [28], $A(\theta)^\top \lambda$ and yields

$$M_H(\delta)\dot{v} + C_H(\delta)v + g_H(\delta) = 0,$$

where the reduced mass matrix

$$M_H = \begin{bmatrix} 1 & -\Delta^2 \sin \delta & -\Delta^2 \sin \delta \\ -\Delta^2 \sin \delta & M_{22} & M_{23} \\ -\Delta^2 \sin \delta & M_{23} & -\Delta^3(\gamma + 1) \end{bmatrix},$$

$$M_{22} = \Delta(2\epsilon\Delta\cos\delta - \epsilon^2) + (3\epsilon\Delta + 1)(\gamma + 1),$$

$$M_{23} = \Delta^2(\epsilon \cos \delta - \Delta(\gamma + 1)),$$
and the *generalised forces*

$$F_H = \begin{bmatrix} \epsilon \dot{\theta}^2 + \Delta^2 D^2 \cos \delta \\ \Delta^2 \epsilon (D^2 - \dot{\theta}^2) \sin \delta - \dot{\theta} v (\epsilon + \Delta^2 \cos \delta) \\ -\Delta^2 [\epsilon \dot{\theta}^2 \sin \delta + \dot{\theta} v \cos \delta] - \delta - 10 \delta^3 \end{bmatrix}$$

Defining $F_H = C_H v + g_H$ and the internal state $\xi = [\delta v^T]^T$ gives the first-order form

$$\dot{\xi} = f(\xi) := [\delta \mathbf{v} - \mathbf{M}_H^{-1} \mathbf{F}_H]^T, \quad (1)$$

used as the seed for the spectral study of Section 3.

## 2.5. Single-joint actuation

A motor torque $\tau_a$ acts only at the joint [21]. Setting $\tau_a = 0$ recovers the conservative cycles that compose the NL-NNM; Section 7 later re-activates $\tau_a$ for guided locomotion.

## 3. Linear spectral structure and slow manifolds

Small deviations from straight-line motion reveal the spectral skeleton that later supports the NL-NNM. After linearisation the chassis speed $\bar{v}$ is a neutral coordinate (i.e. in the linearised dynamics one may set $v = \bar{v}$, constant to first order), so we analyse a 3 × 3 Jacobian that depends smoothly on $\bar{v}$.

### 3.1. Internal equilibrium families

The internal state $\xi = (\delta, v, \dot{\theta}, \dot{\delta})^T$ admits two one-parameter equilibrium curves [22]:

$$\bar{\xi}_{\text{rect}}(\bar{v}) = (0, \bar{v}, 0, 0)^{\top}, \quad \bar{\xi}_{\text{circ}}(\dot{\theta}) = (\delta_{\dot{\theta}}, v_{\dot{\theta}}, \dot{\theta}, 0)^{\top}, \\ \delta_{\dot{\theta}} = \arccos\left(-\frac{\epsilon}{(\epsilon-1)^2}\right), \quad v_{\dot{\theta}} = -\frac{\delta_{\dot{\theta}}}{\epsilon \dot{\theta}} + \dot{\theta} \sqrt{(\epsilon-1)^4 - \epsilon^2}.$$

This paper focuses on the rectilinear branch; the circular family underlies steady-turn gaits and is left for future work.

### 3.2. Reduced Jacobian at speed $\bar{v}$

Linearising (1) about $\bar{\xi}_{\text{rect}}(\bar{v})$ and removing the neutral $\bar{v}$-row/ column yields

$$J_r(\bar{v}) = \begin{bmatrix} 0 & 0 & 1 \\ \alpha & -a_1 \bar{v} & 0 \\ \beta & b \bar{v} & 0 \end{bmatrix}, \quad (2)$$

with geometry-dependent coefficients

$$\begin{align*} a_1 &= \frac{\epsilon + \gamma}{\epsilon F}, & b &= \frac{\epsilon^2 \gamma + \epsilon^2 + \epsilon \gamma - \epsilon - \gamma}{\epsilon (\epsilon - 1) F}, \\ \alpha &= \frac{\epsilon \gamma - \gamma - 1}{\epsilon^2 (\epsilon - 1) F}, & \beta &= \frac{\epsilon^3 + 3\epsilon^2 \gamma - 3\epsilon \gamma - \epsilon + \gamma + 1}{\epsilon^2 (\epsilon - 1)^3 F}, \\ F &= \epsilon(\gamma^2 + \gamma + 1) + \gamma > 0. \end{align*}$$

### 3.3. Eigenvalues and Routh argument

The characteristic polynomial

$$P(\lambda; \bar{v}) = \lambda^3 + a_1 \bar{v} \lambda^2 - \beta \lambda - \bar{v} (a_1 \beta + b \alpha)$$

has one real root $\lambda_1(\bar{v})$ and a conjugate pair $\lambda_{\pm}(\bar{v}) = \pm i \omega(\bar{v})$. Using the Routh-Hurwitz criterion [29]:

* $\bar{v} > 0: \lambda_1(\bar{v}) < 0; \omega(\bar{v}) > 0$.

* $\bar{v} = 0: \lambda_1(0) = 0; \omega(0) = \sqrt{-\beta}$.

* $\bar{v} < 0: \lambda_1(\bar{v}) > 0; \omega(\bar{v})$ unchanged.

Asymptotics needed later:

* Low-speed: $\lambda_1(\bar{v}) \approx \pm c |\bar{v}|$ with $c = (a_1 \beta + b \alpha)/\beta > 0$.

* High-speed: $\lambda_1(\bar{v}) \sim -a_1 \bar{v}$, $\omega(\bar{v}) \to \omega_{\infty}$, $\omega_{\infty} = \sqrt{\frac{a_1 \beta + b \alpha}{a_1}}$. Yaw oscillations decay as $\exp(-a_1 |\bar{v}| T_{\infty})$, with $T_{\infty} = 2\pi/\omega_{\infty}$.

### 3.4. Three-dimensional slow manifolds

Re-inserting the neutral direction adds $\lambda_4 = 0$. According to the sign of $\lambda_1(\bar{v})$, the slow [30] subspace $(\delta, \delta, \bar{v})$ extends to three local manifolds which we index by $\sigma = \text{sign}(\bar{v}) \in \{-, 0, +\}$, and denote $\{M_{\sigma}\}$:

<table><thead><tr><th>σ</th><th>λ₁(ν̄)</th><th>manifold</th></tr></thead><tbody><tr><td>+</td><td>&lt; 0</td><td>centre-stable M<sub>+</sub></td></tr><tr><td>0</td><td>0</td><td>centre M<sub>0</sub></td></tr><tr><td>-</td><td>&gt; 0</td><td>centre-unstable M<sub>-</sub></td></tr></tbody></table>

$M_+$ is $C^r$ and exponentially attracting by Fenichel's theorem [24]; $M_0$ is only $C^1$ and algebraically attracting; $M_-$ repels. We collect them as

$$\hat{M} := M_+ \cup M_0 \cup M_-,$$

which is tangential to the linear modal plane $(\delta, \delta)$ along $(\delta, \delta) = (0, 0)$; its curvature appears at order $O(\|(\delta, \delta)\|^2)$. This tangency provides the natural bridge towards the nonlinear continuation in Section 4.

## 4. NL-NNM — Non-holonomic locomotion non-linear normal mode

### 4.1. Classical idea, non-holonomic twist

For conservative holonomic mechanisms a nonlinear normal mode (NNM) is a two-dimensional invariant manifold tangent to a linear mode and foliated by periodic orbits indexed by an energy-phase pair [14,31]. Non-holonomic robots such as the Pendrivencar differ in two crucial respects:

(i) the constraint introduces an neutral direction (the cycle-mean forward speed $\bar{v}$), so the linear modal plane $(\delta, \delta)$ sits inside a 3-D slow manifold; (ii) a locomotion mode should convert internal oscillation into net motion of the chassis. These observations motivate the extended definition below.

**Definition 1 (Non-holonomic Locomotion NNM).** A 2-D manifold $\mathcal{N}$ is an NL-NNM if it satisfies

(1) Internal invariance — every point of $\mathcal{N}$ belongs to a periodic orbit of the conservative internal dynamics, and

(2) Geometric drift — each such orbit generates a nonzero geometric phase $\Delta x_b > 0$ of the chassis over one period.

**Property (2)** is irrelevant for fixed structures but essential for autonomous robots: it distinguishes “oscillations that move” from those that remain confined to fixed frame.

**Remark.** The total translation over one cycle decomposes as

$$X_{\text{total}} = \underbrace{\bar{v}T}_{\text{kinematic drift}} + \underbrace{\Delta x_b}_{\text{geometric phase}}.$$

For a locomotion NNM, it is the geometric phase $\Delta x_b$ — independent of the trivial $\bar{v}T$ — that produces net propulsion and heading control through the internal actuator [32].

### 4.2. Internal Hamiltonian and periodic seeds

Let $s = (\delta, \bar{v}, \delta)$; the mean speed $\bar{v}$ remains a slow coordinate, not a frozen equilibrium. On each slow manifold $\{M_{\sigma}\}$, the yaw dynamics satisfy the graph $\dot{\theta} = h_{\sigma}(s)$ with $C^r$ regularity for $\bar{v} \neq 0$ and $C^1$ at $\bar{v} = 0$. Substituting into the Lagrangian energies yields the conserved Hamiltonian

$$H(s) = \mathcal{K}(\delta, \bar{v}, h_{\sigma}(s), \delta) + \mathcal{V}(\delta).$$
**Local periodic seeds at $\bar{v} = 0$.** At $\bar{v} = 0$, the rectilinear equilibrium $\bar{\xi}_{\text{rect}}(0)$ has spectrum $\{\lambda_{\pm}(0) = \pm i\omega_0, \lambda_1(0) = 0\}$ with $\omega_0 = \omega(0) = \sqrt{-\bar{\beta}}$ (Section 3.3). Because no non-zero integer $k$ satisfies $k\omega_0 = 0$, the Lyapunov Center Theorem yields a one-parameter family of small-amplitude periodic orbits $\{\mathcal{O}_{E,0}\}_{0<E\ll 1} \subset \mathcal{M}_0$, each oscillating at $\omega(E) = \omega_0 + O(E)$ and depending $C^r$-smoothly on $E$ [33].

**Continuation to $\bar{v} \neq 0$.** For small but nonzero $\bar{v}$, Fenichel's theorem carries each seed orbit $\mathcal{O}_{E,0}$ off $\mathcal{M}_0$ into two invariant leaves $\mathcal{O}_{E,\pm} \subset \mathcal{M}_{\pm}$, where $\sigma = \pm 1$ labels the centre-stable (+) and centre-unstable (-) manifolds. Under the same integer-resonance exclusion, Spectral Submanifold (SSM) theory guarantees each $\mathcal{O}_{E,\pm}$ is unique, analytic in $(E, \bar{v})$, and tangent to its $\bar{v} \to 0$ seed [34]. Together these two foliations produce a global $C^r$ 2D NL-NNM across $\tilde{\mathcal{M}}$.

### 4.3. Candidate manifold and immediate properties

$$ \mathcal{N} := \bigcup_{\sigma} \bigcup_{E>0} \mathcal{O}_{E,\sigma} \subset \tilde{\mathcal{M}} $$

By construction, $\mathcal{N}$ is the union of all periodic orbits $\mathcal{O}_{E,\sigma}$ on the slow manifolds. Its key features are:

* **Linear limit.** As $E \to 0$, each $\mathcal{O}_{E,\sigma}$ coalesces with the planar linear normal-mode ellipse of angular frequency $\omega_0$.

* **Regularity.** The two-parameter family $\{\mathcal{O}_{E,\sigma}\}$ is $C^r$ in $(E, \sigma)$, reducing to $C^1$ at $E = 0$, in alignment with the slow-manifold smoothness.

* **Yaw enslaved.** Since $\dot{\theta} = h_\sigma(s) = O(1/|\bar{v}|)$ for $\sigma = \pm 1$, yaw oscillations vanish at high speed (recovering the spectral seed).

### 4.4. Yaw symmetry and branch dichotomy

**Reversibility.** The internal vector field is reversible [35] under $\mathcal{R}: (\delta, \dot{\delta}, \bar{v}, \dot{\theta}) \mapsto (-\delta, -\dot{\delta}, \bar{v}, -\dot{\theta})$. Hence every rectilinear cycle satisfies

$$ \langle \dot{\theta} \rangle = 0, \quad \Delta\theta = 0, $$

so the mean heading is preserved and the chassis can only drift straight ahead.

**Circular branch (contrast).** At circular equilibria $(\dot{\theta}^* \neq 0)$ reversibility is lost: each cycle accumulates an orientation change $\Delta\theta = 2\pi k (k \neq 0)$. Although the cycle still produces a body-frame displacement $\Delta\mathbf{p}_b \neq 0$, the successive cycles are rotated by multiples of $2\pi/|k|$; after $|k|$ cycles they cancel out and the centre of mass closes into a rosette, so the net inertial drift averaged over many cycles is zero.

### 4.5. Geometric phase, curvature and straight-line drift

We now relate an *internal* pendulum cycle to an *external* translation of the chassis via the geometry of non-holonomic connections.

**Body-frame twist ($V_b$).** For a planar rigid body the twist is the spatial velocity expressed in the moving body frame:

$$ V_b = g^{-1} \dot{g} = (v_{x_b}, v_{y_b}, \dot{\theta})^T \in \mathcal{S}\mathcal{E}(2), $$

where $g = (x, y, \theta) \in \text{SE}(2)$ is the chassis pose and $(x_b, y_b, z_b)$ is the forward-left-up triad rigidly attached to it. Throughout the paper the scalar $v$ used in Section 3 is precisely $v_{x_b}$, the forward component of $V_b$.

**Ehresmann connection $A(s)$.** Because of the no-slip constraint, $V_b$ is not free but generated by the *internal* variables $s = (\delta, \bar{v}, \dot{\delta})$ through the affine map

$$ V_b = -A(s)\dot{s}, \quad A_{x_b}(s) = a_1\dot{\delta} + a_2\dot{\delta}\dot{\delta} + \mathcal{O}(|s|^3), $$

with $a_1 = b\bar{v}/(-\lambda_1) > 0$. This $A$ is the classical Ehresmann *local connection* of geometric mechanics: it converts shape-space velocities $\dot{s}$ into body-frame motions [36].

**Translational curvature $\kappa_i$.** The curvature of $A$ measures how a loop in shape space produces a rigid-body displacement. We only need its forward component:

$$ \kappa_i := \partial_{\delta} A_{x_b} - \partial_{\delta} A_{x_b} = a_1 + \mathcal{O}(|s|), $$

which is strictly positive inside the ball $|s| \le \rho$. Intuitively $\kappa_i$ is the “area-to-distance” conversion factor: a larger positive curvature means a larger forward drift for the same internal cycle.

**Geometric phase formula.** Given a periodic orbit $\mathcal{O}_{E,\sigma}$, the net displacement in body coordinates is the geometric phase

$$ \Delta \mathbf{p}_b = - \oint_{\mathcal{O}_{E,\sigma}} A \, ds, $$

an integral that involves only the internal trajectory. Using Stokes's theorem we convert the line integral into an area integral over the disc $D_E$ bounded by the loop's projection on the $(\delta, \dot{\delta})$ plane:

$$ \Delta x_b(E) = - \iint_{D_E} \kappa_i d\delta d\dot{\delta}, \quad \Delta y_b(E) = O(E^{5/4}). $$

Because $|D_E| \sim E^{3/4}$ in all energy regimes and $\kappa_i \ge a_1/4$, we obtain the boxed drift law

$$ \Delta x_b(E) \ge C E^{3/4} > 0, \quad C = \frac{a_1}{4} $$

valid from infinitesimal oscillations to wildly non-linear kicks. Heading preservation (Section 4.4) turns this body-frame drift into a global straight-line translation, proving part (2) of Definition 1.

**Theorem 2 (Passive Straight-Line Locomotion).** For every mean speed $\bar{v}$ and energy $E$, each rectilinear cycle $\mathcal{O}_{E,\sigma}$ satisfies $\|\Delta\mathbf{p}\| > 0$; circular-branch cycles translate by 0.

(i) Forward cycles ($\sigma = +1$) live on the centre-stable manifold and are self-stabilising; reverse cycles require yaw control. (ii) The sign of $a_1 \propto (\epsilon - 1)$ fixes the travel direction; swapping link lengths flips the drift.

### Corollary 3. Theorem 2 plus property (P1) yield

$\mathcal{N}$ is an NL-NNM (Definition 1)

which completes the conceptual part of the paper.

### F. Numerical construction of NL-NNM, $\mathcal{N}$

**Seed at large positive speed.** When $\bar{v} \gg 1$, yaw oscillations scale as $\mathcal{O}(1/\bar{v})$; the linear pair $\lambda_{\pm} = \pm i\omega_{\infty}$ therefore supplies an accurate harmonic guess

$$ \xi^{(0)}(t) = \bar{\xi}_{\text{rect}}(\bar{v}_{+}^{\text{max}}) + \epsilon [\phi_r \cos \omega_{\infty} t - \phi_l \sin \omega_{\infty} t]. $$

**Residual and phase locking.** For $(\xi_0, T)$ set

$$ R(z) = \begin{bmatrix} \Phi_T(\xi_0) - \xi_0 \\ \langle \dot{\xi}_0, \phi_r \rangle \end{bmatrix}, \quad z := (\xi_0, T) $$

where the second row fixes the phase to $\phi_r$ so that the Jacobian is full rank.

**Newton phase shooting.** Newton iterations on $R(z) = 0$ refine the seed into the first exact cycle $\mathcal{O}_{E_0,+}$.

**Pseudo-arclength continuation** [37] ($C^1$ crossing). Augment $R$ with $\langle z - z^{\text{prev}}, t^{\text{prev}} \rangle = \Delta s$; perform an *iteration* composed of (i) a *prediction* using the tangent (the null-space of the augmented Jacobian, or a higher-order secant), then (ii) a Newton *correction*. Because the slow
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Period *T* versus mean speed $\bar{v}$ (NL/NNM backbone). Colour coded curve: NL-NNM data; dashed curve: extrapolation of NL-NNM; dotted curve: extrapolation asymptote.</figcaption></figure>

manifold is still $C^1$ at $\bar{v} = 0$ (centre case), the solver passes through that point smoothly and returns the complete backbone {$O_{E,\sigma}$}, whose period, mean speed and holonomy feed Section 5. Our implementation uses PyDSTool-PyCont [38], Python interfaces to AUTO-07P [39], although other numerical continuation tools can be adapted similarly.

## 5. Numerical NL-NNM results

Numerical continuation uses the non-dimensional model of Section 2 with $\epsilon = 0.681$, $\gamma = 2.504$. Choosing $M = l = k = 1$ fixes the scaling time $\tau_c = 1$ s. Colour encodes the orbit energy $E$ on a log$_{10}$ scale.

### 5.1. Hardening of the period

*Hardening effect.* The linear spectrum of Section 3.3 predicts a nearly constant period $T_0 \approx T_\infty \approx 1.9$ s. Fig. 2 instead exhibits a pronounced **hardening**: all nonlinear orbits fall in $0.67 < T < 0.81$ s. The cubic torque $\tau \propto \delta + 10\delta^3$ adds a quartic term in the potential, raising the effective stiffness as the oscillation grows.

*Energy-speed link.* The NL-NNM emerges from the *non-isolated rectilinear* equilibrium family $\bar{\xi}_{\text{rect}}(v)$ (Section 3.1). Since $\bar{v}$ is the neutral direction, moving along that family simultaneously raises the total energy; hence the smooth colour→speed→energy gradient.

### 5.2. Projections of the rectilinear NL-NNM (Fig. 3)

**Panel (a)** illustrates the 2-D invariant surface in $(\dot{\theta}, v, \delta)$. Its funnel reflects the eigen-law $\dot{\theta} = O(1/v)$ (Section 3.3). **Panel (b)** — Quartic energy contour. For large amplitudes $|\delta| \gtrsim 0.63$ rad the cubic spring term dominates, the torsion cubic simplifies to a quartic potential $\frac{5}{2}\delta^4$. A constant-energy slice satisfies $\dot{\theta}^2 + \frac{5}{2}\delta^4 = E$, i.e. an implicit quartic curve, matching the plot.

**Panel (c)** confirms the yaw-collapse $\text{amp}\theta \sim 1/|v|$ predicted in Section 3.3.

**Panel (d)** — Why $v$ vibrates at $2\omega$. Dynamics of $v$ shows only order 2 coupling term; an harmonic-balance thus places the main component of $v(t)$ at twice the fundamental frequency of $\delta$ and $\theta$, hence the double loop.

*Amplitude of δ.* With increasing $E$ a larger fraction is stored in the quartic potential, so less is available for pendulum excursion; the

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Four projections of the 2D NL/NNM orbits: (top) projection in $(\dot{\theta}, v, \delta)$ space; (second) projection in $(\dot{\theta}, \delta)$ space; (third) projection in $(v, \theta)$ space; (bottom) zoom in $(v, \theta)$ space.</figcaption></figure>

amplitude decreases and plateaus once the high-speed asymptote is reached.

### 5.3. Phase lag and harmonic distortion

Fig. 4(top) shows the phase lag $\varphi = \arg\delta - \arg\theta$. It transits smoothly from $\pi/2$ on the centre-stable branch to $-\pi/2$ on the centre-unstable branch as $\bar{v}$ crosses 0, in accordance with the sign change of $\lambda_1$ (Section 4.4). The linear formula $\tan\varphi = -\lambda_1/\omega$ captures the trend well.

The total harmonic distortion of $\theta$ (bottom) is minimum ($\approx 2\%$) near the origin and rises to $\approx 7\%$ at high energy. Although the swing of $\delta$ decays, the cubic torque flattens the extrema of $\theta(t)$, enriching its harmonic content.
<figure><img src="image_7.png" /><figcaption>Fig. 4. Phase lag δ − θ vs. mean speed v̄ (top) and total harmonic distortion of θ vs. v̄ (bottom).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Forward (bottom) and reverse (top) NL-NNM cycles at | v̄ | ≈ 10 m/s.</figcaption></figure>

5.4. Summary of numerical evidence

(1) Slow-manifold bridge. Continuation starts on the centre-stable manifold M $_{+}$, crosses the C 1 centre M 0 and finishes on the centre-unstable M $_{−}$, stitching them into the 2-D surface N de- fined in Section 4.

(2) Cartesian translation Each cycle in Fig. 5 produces a net trans- lation of 8–9 m, confirming the locomotion capability. Although only one cycle is shown here as an example, all other cycles of the NL-NNM exhibit the same behaviour.

(3) Spectral and nonlinear fingerprints. Period hardening arises from the cubic stiffness nonlinearity of the torsion spring (Duffing effect); yaw decay stem from the linear eigen-structure of Sec- tion 3 and the 2 ω component in v to the absence of linear term in v dynamics; the phase reversal (phase lag) provides insight into the optimal choreography of the gait, i.e. the timing relationship between the segment angles.

(4) Passive persistence. NL-NNM orbits are exact solutions of the conservative dynamics; once the initial state lies on N the motion persists indefinitely without internal actuation (see the energy balance discussion in Section 6).

Note: We display relatively large speeds because we have assumed the absence of dissipation, which allows arbitrarily high speeds to be reached.

6. Energetic viewpoint

In a purely conservative setting ( M = l = k = 1, neglecting drag and electrical losses), the positive work per cycle W + provides a realistic

measure of actuator effort when energy recuperation is limited [40]. We compare two strategies for executing oscillations about the torsion spring:

Positive Work Metric. Define

$$
W^{+} = \int_{0}^{T} [\tau_{a} \dot{\delta}]_{+} dt, \quad [x]_{+} = \max\{x, 0\},
$$

where τ$_{a}$ is the actuator torque and δ the pendulum angle. Even though the net work over a cycle vanishes on a conservative plant, W + captures the unavoidable energy input when negative power phases cannot be fully recovered [41].

**Linear-Mode Stabilisation.** Enforcing a small-amplitude sinusoid δ ( t ) = A sin( ω ∞ t ) (Section 3) requires an actuator torque that cancels both linear and cubic spring terms. Substitution into the positive work integral yields

$$
W_{\text{lin}}^{+} = \frac{5\pi}{2} A^{4} + O(A^{6}),
$$

so that effort grows quartically with amplitude. The quadratic term drops out as torque reversals balance over each half-cycle.

**Passive NL-NNM Execution.** When the system is launched precisely onto the invariant NL-NNM manifold N (Section 4), the natural nonlin- ear dynamics sustain oscillations without further actuator intervention ( τ$_{a}$ ≡ 0). Consequently,

$$
W_{\text{NL-NNM}}^{+} = 0,
$$

and the actuator remains dormant aside from an initial impulse.

**Comparative Insight.** For equal amplitude A , the energetic gap

$$
\Delta W^{+} = W_{\text{lin}}^{+} - W_{\text{NL-NNM}}^{+} \propto A^{4}
$$

rapidly widens, underscoring the benefits of hugging the passive non- linear normal mode when regeneration is impractical.

7. Dual-loop autonomous control demonstration

Two loosely coupled loops steer the Pendrivencar along a Cartesian ‘squirrel’ while keeping its internal state on the NL-NNM invariant manifold. Fig. 6 summarises the architecture and the symbols used below.

7.1. Control architecture

Outer Vector-Field-Guided (VFG) [42] loop. The target squirrel is the zero-level set φ $^{∗}$( x, y ) = x 4 + y 4 − a $^{4}$. Its gradient furnishes the unit normal N = ∇ φ $^{∗}$/ ||∇ φ $^{∗}$|| , while the Frenet frame yields the unit tangent T . Combining them gives the analytic guidance field

$$
F = T - h(\phi^*) N, \quad h(s) = \tanh(|s|/\delta) \operatorname{sgn}(s),
$$

so that F is tangent on the curve ( h = 0) and points inward/outward elsewhere. The desired heading is θ ∗ = arctan 2( F$_{y}$, F$_{x}$ ); a PID on the wrapped error generates the yaw torque u$_{YAW}$ .

Inner NL-NNM loop. Given the forward speed selected by the GVF, a look-up table returns the periodic reference v $^{∗}$( t ) = ( v $^{∗}$, ˙ θ $^{∗}$, ˙ δ $^{∗}$) on the NL-NNM manifold. A pseudo-inverse feedback linearisation u$_{NNM}$ = g $^{†}$( δ ) [ K$_{NNM}$ ( v $^{∗}$− v ) + ˙ v $^{∗}$− f ( ξ )] with g is M$_{H}$ third column assists the internal state to converge towards v $^{∗}$( t ) [43]. The total command is u = u$_{YAW}$ + u$_{NNM}$ .

7.2. Simulation set-up

**Plant.** Parameters of Table 1 ( ε ≈ 0 . 681 , γ ≈ 2 . 504) plus a small linear drag c$_{d}$ = 2 × 10 − $^{3}$. **Reference path.** Squirrel of half-size a = 30 m; initial pose ( a, 0 , π/ 2). **Integration.** ODE45, step 1 ms, horizon 200 s.
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Dual-loop autonomous control architecture. The outer GVF block provides the desired heading $\dot{\theta}^*$, while the inner NL-NNM guidance selects the periodic orbit $v^*(t)$ on the invariant manifold; the feedback-linearised law $u_{\text{NNM}}$ assists convergence.</figcaption></figure>

### 7.3. Results

**Internal convergence.** Despite having only one actuator, the pseudo-inverse law drives the three coordinates $(v, \dot{\theta}, \ddot{\delta})$ simultaneously towards the periodic NL-NNM orbit (black curve matching the red dashed reference in Fig. 7).

**Cartesian tracking.** The outer GVF keeps the chassis on the squirrel while the inner NL-NNM oscillations proceed. Blue markers (end of each internal cycle) stay evenly spaced along the path for one squirrel loops (Fig. 8).

This proof-of-concept shows that the pendulum actuation, guided by the NL-NNM backbone, can both propel the vehicle and steer its pose along a prescribed planar path, opening the door to fully autonomous non-holonomic robots that exploit their natural dynamics.

## 8. Discussion and perspectives

Our results demonstrate that NL-NNMs truly bridge the gap between classical nonlinear structural dynamics and the emerging paradigm of energy-aware locomotion introduced in the Introduction. In line with the idealised, conservative setting of our *Pendrivencar*, three key insights emerge:

**Resonant motions beat imposed sinusoids.** As anticipated from the pendular example in Section 6, enforcing a linear sinusoidal mode incurs a positive work cost scaling as $A^4$ with amplitude, whereas the NL-NNM—being a genuine invariant orbit—requires zero net actuation under the conservative hypothesis. This stark contrast confirms that exploiting the machine's passive dynamics via its natural nonlinear modes yields an unrivalled energetic advantage.

**Linear modes as a springboard, not a solution.** The LNM captures the dominant phase locking between chassis yaw and pendulum swing and provides the high-speed “spectral seed” at the *asymptotic centre* $(v \to \infty)$. Through pseudo-arclength continuation this seed smoothly deforms into the full NL-NNM. However, the linear orbit itself is not natural to the nonlinear plant: maintaining it demands sustained work

(hence the $A^4$ scaling), underscoring that the LNM's descriptive power must be weighed against its energetic cost.

**Non-holonomic constraints enrich modal design.** By extending NNMs to velocity-constrained systems, we add locomotion to the classic applications of vibration mitigation and structural safety. Beyond Rand's fixed Chaplygin sleigh, our mobile variant exposes a phase-dependent exchange of momentum between chassis and pendulum—an interaction impossible to reproduce with holonomic models.

### Limitations

Our conservative analysis yields a baseline of zero-work gaits, but real robots experience drag, damping, and friction. NL-NNM implies unlimited speed; real-world losses cap it. When dissipation becomes significant, the conservative NL-NNM ceases to be optimal and morphs into a non-linear resonant mode (NLRM): the invariant gait maximising forward speed per unit actuation. Adapting our spectral-continuation pipeline to compute NLRMs under damping is essential for translating these concepts to aquatic, aerial or soft robots, and will be validated next on the *Pendufish*.

### Synergy with data-driven methods

Model-based manifolds – NL-NNMs today, NLRMs tomorrow – establish a guaranteed lower bound on mechanical work. Model-free optimisers (e.g. RL, evolutionary search) can compensate unmodelled effects but lack global optimality guarantees. Seeding or regularising learning with the invariant manifold narrows the search space, accelerates convergence and flags policies that exceed the analytic energy bound, making modal analysis and data-driven control highly complementary.

### Future directions

Building on our three main axes:

(1) Dissipative extensions—develop NLRM continuation in fluids by incorporating added mass and drag;
<figure><img src="image_10.png" /><figcaption>Fig. 7. Convergence of the internal state (green → black) towards the NL-NNM reference (red dashed).</figcaption></figure>

(2) *Complex body plans*—generalise to multi-segment and soft-body swimmers to uncover natural choreographies beyond two-link systems;

(3) *Hardware validation*—finish the Pendrivencar prototype to empirically confirm the predicted energy savings.

## 9. Conclusion

We have introduced a *modal* framework for non-holonomic locomotion, culminating in the first conservative **Nonholonomic Locomotion Nonlinear Normal Mode NL-NNM** of a moving robotic car. Starting from the high-speed *asymptotic centre*, we leveraged pseudo-arclength continuation to deform a linear spectral seed into a fully nonlinear invariant manifold. This 2D surface not only sets an energy-optimal benchmark for the Pendrivencar but also serves as:

* A design target for resonance-centred robots;

<figure><img src="image_11.png" /><figcaption>Fig. 8. One autonomous laps of the squircle: reference path in red, realised Pendrivencar trajectory in black.</figcaption></figure>

* A control reference enabling dual-loop path tracking with minimal actuation;

* A quantitative yardstick against which any learning-based gait can be measured.

Looking forward, computing dissipative *non-linear resonant modes* and exploring richer body geometries will pave the way from minimalist theory to practical, high-efficiency machines that resonate with their physics rather than fight it.

## CRediT authorship contribution statement

**Mirado Rajaomarosata:** Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Luc Jaulin:** Writing – review & editing, Validation, Supervision, Project administration. **Lionel Lapierre:** Writing – review & editing, Validation, Supervision, Funding acquisition. **Simon Rohou:** Validation, Supervision.

## Declaration of Generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the authors used generative AI-based tools (e.g., ChatGPT) to assist in language editing and to provide coding suggestions. After using these tools, the authors carefully reviewed and edited the content as needed and take full responsibility for the content of this publication.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors acknowledge funding from the French National Research Agency, France http://dx.doi.org/10.13039/501100001665 under the Electro-Karst project https://anr.fr/Projet-ANR-22-CE33-0005 for robotic underwater exploration of karsts.
# Data availability

Data will be made available on request.

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<figure><img src="image_12.png" /></figure>

Mirado Rajaomarosata earned a degree in Mechatronics from the École normale supérieure de Rennes. In the same year, 2023, he also obtained an engineering degree in Autonomous Robotics from ENSTA Bretagne and a Master's degree in Dynamical Systems from the University of Angers. Currently, Mirado is pursuing a Ph.D. in Robotics at ENSTA Bretagne - Lab-STICC (Brest, France), focusing on robotic locomotion for karst exploration.

<figure><img src="image_13.png" /></figure>

Luc Jaulin was born in Nevers, France in 1967. He received the Ph.D. degree in automatic control from the University of Orsay, France in 1993. He is currently Professor of Robotics at the ENSTA Bretagne engineering school Brest, France from 2004. He does his research on underwater robotics using interval methods and constraint propagation.
<figure><img src="image_14.png" /></figure>

**Lionel Lapierre** earned his Ph.D. in robotics from the University of Montpellier 2, France, in 1999. He joined Prof. A. Pascoal's team on the European FreeSub project for three years before affiliating with the Underwater Robotics Division at LIRMM in 2003. In 2015, he obtained his habilitation to direct research, focusing on autonomous underwater vehicles. He is currently a professor at ENSTA Bretagne and a member of the ROBEX team at Lab-STICC, CNRS.

<figure><img src="image_15.png" /></figure>

**Simon Rohou** received a M.Sc. in Information Technology at INSA de Rouen, France, in 2014. He started a Franco-British Ph.D. thesis in 2014 between ENSTA Bretagne – Lab-STICC (Brest, France) and the University of Sheffield (Sheffield, UK). His work focuses on robotics and especially set-membership localisation methods for autonomous underwater robots.