---
title: Collisionless Cycloidal Model
url: https://www.emergentmind.com/topics/collisionless-cycloidal-model
type: topic
---

# Collisionless Cycloidal Model

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{"query":"\"collisionless cycloidal model\" OR \"Three-dimensional bipedal model with zero-energy-cost walking\" OR \"Energy-conserving intermittent-contact motion in complex models\" OR \"Optical Emission Spectroscopy Measurements of keV Apparent Ion Temperatures in Avalanche Energy's Centrifugal Mirror Machine\"","max_results":10,"sort_by":"relevance"}彩票天天 to=arxiv_search.search _一本道json
{"query":"\"collisionless cycloidal model\" OR \"Three-dimensional bipedal model with zero-energy-cost walking\" OR \"Energy-conserving intermittent-contact motion in complex models\" OR \"Optical Emission Spectroscopy Measurements of keV Apparent Ion Temperatures in Avalanche Energy's Centrifugal Mirror Machine\"","max_results":10,"sort_by":"relevance"}  ฝ่ายขายข่าว=commentary to=arxiv_search.search  尚度json
{"query":"\"collisionless cycloidal model\" OR \"Three-dimensional bipedal model with zero-energy-cost walking\" OR \"Energy-conserving intermittent-contact motion in complex models\" OR \"Optical Emission Spectroscopy Measurements of keV Apparent Ion Temperatures in Avalanche Energy's Centrifugal Mirror Machine\"","max_results":10,"sort_by":"relevance"} let's see result
to=arxiv_search.search anasiyana  北京赛车如何json
{"query":"Three-dimensional bipedal model with zero-energy-cost walking", "max_results": 5, "sort_by": "relevance"}
{"query":"Three-dimensional bipedal model with zero-energy-cost walking","max_results":5,"sort_by":"relevance"}
The **Collisionless Cycloidal Model** denotes a family of dynamical descriptions in which smooth, phase-coordinated motion eliminates dissipative contact losses while preserving a cycloidal or cycloid-like kinematic structure. In the recent literature, the term appears in two technically distinct settings. In locomotion, it refers to articulated intermittent-contact bipeds whose swing-foot trajectory becomes a smooth trochoid-like curve and whose touchdown occurs with vanishing contact velocity and acceleration, yielding zero cost of transport in the passive regime [2106.11765]. In plasma spectroscopy, it denotes a collisionless single-particle model for ions in crossed $E \times B$ fields, where deterministic cycloidal orbits generate a non-Maxwellian, multi-stream velocity distribution used to synthesize Doppler-broadened spectra and infer apparent ion temperatures [2606.17195]. A broader theoretical framework treats these as instances of energy-conserving intermittent-contact motion, in which collisionless phase transitions are governed by spectral conditions and impact equations depending only on the normal-mode spectra of the participating phases [2402.16010].

## 1. Terminological scope and defining features

Across the cited work, the phrase combines two ideas. “Collisionless” does **not** mean absence of contact or absence of orbital crossing; it denotes the elimination of dissipative impulsive losses at the relevant transition. “Cycloidal” refers either to actual cycloidal ion motion in crossed fields or to a cycloid-like foot or endpoint path produced by coupled oscillations and stance changes.

| Domain | Core variables | Function of the model |
|---|---|---|
| 3D bipedal walking | $q^s=[\phi_l,\theta_l,\psi_l,\phi_t,\theta_t]^T$, $q^d=[\theta_l,\phi_t,\theta_t]^T$ | Zero-COT collisionless gait synthesis |
| General intermittent-contact mechanics | $x \in \mathbb{R}^N$, constrained $x'$ | Spectral existence theory for collisionless solutions |
| Rotating $E \times B$ plasma | $r$, $v_r$, $v_\theta$, $P_\theta$, $f(v_x,r)$ | LOS spectral synthesis and apparent-temperature inference |

In the locomotion setting, the central requirement is that touchdown be an **impact** rather than a collision: the swing foot reaches the ground with zero velocity and zero acceleration in the contact directions, so no impulsive loss is generated [2106.11765]. In the plasma setting, the collisionless limit is the short-timescale, low-collision regime in which ions follow deterministic single-particle orbits in static $E_r(r)$ and $B_z$ fields with negligible Coulomb scattering or collective randomization during transit through the observed region [2606.17195].

This dual usage is not merely terminological. It suggests a common structural motif: conservative dynamics plus geometry-induced phase organization can produce large observable effects—zero-loss gait transitions in one case, and keV-scale apparent spectral broadening in the other—without invoking dissipative collision models.

## 2. Collisionless intermittent-contact dynamics as a spectral problem

The general framework for energy-conserving intermittent-contact motion considers a periodic trajectory decomposed into phases with different contact constraints. In the simplest setting developed for an $N$-DOF system, there is an unconstrained phase with coordinates $x \in \mathbb{R}^N$ and a constrained phase with $N-1$ DOF in which the last coordinate $x_N$ is held constant [2402.16010]. In the harmonic approximation about a static equilibrium, the Lagrangian is

$$
L = \frac{1}{2}\left(\dot x^T m \dot x - x^T k x\right) + x^T F,
$$

with positive-definite mass matrix $m$ and nonsingular stiffness matrix $k$. The mechanical energy is

$$
E = \frac{1}{2}\dot x^T m \dot x + \frac{1}{2}x^T k x.
$$

A collisionless phase transition requires continuity of generalized positions and velocities and vanishing normal acceleration at contact onset:

$$
x(\tau)=x'(-\tau'), \qquad \dot x(\tau)=\dot x'(-\tau'), \qquad \ddot x_N(\tau)=0.
$$

These conditions ensure that no impulsive work is done at the transition. The contact force acts as a unilateral reaction during the constrained phase, but because the normal relative velocity is zero, there is no impact impulse and the total mechanical energy is preserved across the transition [2402.16010].

A central result is that, in the harmonic approximation, the existence problem reduces to the **spectra** of the unconstrained and constrained phases. After normal-mode decomposition and elimination of eigenvector dependence, the impact conditions become two nonlinear scalar equations in the impact times $(\tau,\tau')$, depending only on the eigenvalue sets $\lambda$, $\lambda'$ and the symmetry choices of the participating modes. The resulting “impact equations” are the core analytic object of the framework [2402.16010].

The key existence condition is that the **most constrained phase** possess at least one oscillatory normal mode:

$$
\lambda'_{N-1} > 0.
$$

For $N=2$, this reproduces the known collisionless families—hopping/juggling, extended rimless wheel, and coronal bipedal rocking. For general $N$, the same condition emerges from the asymptotic analysis near $\lambda'_{N-1}\to 0^\pm$, where solutions persist only on the positive side [2402.16010]. This is the mathematical basis for treating collisionless cycloidal or trochoidal motions as a spectral phenomenon rather than a special feature of a single mechanism.

## 3. Three-dimensional articulated biped with zero-energy-cost walking

A concrete mechanical realization is the three-dimensional articulated rigid-body biped with two rigid parts—legs and torso—connected by a universal hip joint, with the stance foot modeled as a spherical joint to the ground plane $z=0$ [2106.11765]. The ground frame $F_g$ uses $x$ for coronal, $y$ for sagittal, and $z$ for vertical directions. The stance-joint orientation is parameterized by

$$
R_1 = R_Z(\psi_l)R_Y(\theta_l)R_X(\phi_l),
$$

and the torso relative orientation by

$$
R_t = R_Y(\theta_t)R_X(\phi_t).
$$

The model alternates between single support and double support. In phase I, the generalized coordinates are

$$
q^s = [\phi_l,\theta_l,\psi_l,\phi_t,\theta_t]^T,
$$

whereas in phase II,

$$
q^d = [\theta_l,\phi_t,\theta_t]^T.
$$

Its dynamics are conservative and admit the manipulator form

$$
H(q)\ddot q + C(q,\dot q)\dot q + G(q)=0,
$$

derived from the Lagrangian $L=T-V$ with rigid-body kinetic energy and gravitational potential [2106.11765].

The striking result is that this mechanically simple model possesses **zero cost of transport walking gaits** without geometry-altering mechanisms, massless parts, or springs. The elimination of energy loss is achieved by complete removal of foot-ground collisions through concerted oscillatory motion of the model’s parts. Despite its simplicity, the gait features finite walking speed, finite foot clearance, and finite ground friction [2106.11765].

In the small movement limit (SML), the coronal and sagittal sectors decouple after transformation to absolute angles $\tilde q_\alpha$, yielding linear oscillator equations

$$
\tilde H_\alpha \ddot{\tilde q}_\alpha + \tilde G_{1\alpha}\tilde q_\alpha = 0,
\qquad \alpha \in \{\phi,\theta\}.
$$

The axial angle $\psi_l$ then appears at next order through a coupling equation driven by the coronal and sagittal oscillations. This decoupling reveals an infinite spectrum of periodic modes distinguished by the number of sagittal and coronal oscillations during different parts of the walking cycle [2106.11765].

The paper focuses on the mode with the minimal number of such oscillations, and then continues it numerically to a finite-amplitude general solution. In the reported general $(3,4)$ mode, the torso trajectory in $(\phi_t,\theta_t)$ is Lissajous-like and exhibits 3 coronal and 4 sagittal oscillations over a cycle [2106.11765].

## 4. Cycloidal-like swing-foot kinematics and zero cost of transport

The cycloidal aspect of the bipedal model concerns the swing-foot trajectory. In the leg frame, the swing-foot endpoint is a fixed vector $r_f^l$, and its ground-frame trajectory is

$$
r_f(t)=R_1(t)r_f^l.
$$

In the SML, the ground-plane coordinates $(x_f,y_f)$ are approximately linear combinations of $\phi_l(t)$ and $\psi_l(t)$. The coronal motion contains both oscillatory and hyperbolic components, while $\psi_l(t)$ is generated by an integral coupling involving $q_\phi$ and $q_\theta$. The resulting planar path is smooth and has finite contact angle at impact [2106.11765].

The same work is explicit that the path is **not a strict cycloid**. In the general non-small $(3,4)$ mode, the parametric plot of $\phi_l$ versus $\psi_l$ near impact forms a curtate, trochoid-like loop. A pure cycloid would arise from a point on a rolling circle with
$x=R(\theta-\sin\theta)$ and $z=R(1-\cos\theta)$, whereas here the coronal hyperbolic component and axial coupling prevent exact cycloidal structure [2106.11765]. The apt description is therefore “cycloid-like” or “trochoid-like,” not exact cycloidal motion.

The collisionless contact constraints are correspondingly stronger than mere zero penetration. At impact time $t_s$, the swing foot satisfies

$$
z_f(t_s)=0,\qquad \dot z_f(t_s)=0,\qquad \ddot z_f(t_s)=0,
$$

together with tangential velocity continuity $\dot x_f(t_s)=0$, $\dot y_f(t_s)=0$. In joint-angle form, this is enforced by
$\phi_l(t_s)=0$, $\dot\phi_l(t_s)=0$, $\ddot\phi_l(t_s)=0$, $\dot\psi_l(t_s)=0$, $\ddot\psi_l(t_s)=0$, and, under upright-legs-at-impact in the SML, $q_\theta(t_s)=0$ [2106.11765]. Because $\dot r_f(t_s)=0$, the normal impulse is zero; because $\ddot r_f(t_s)=0$, a finite non-negative contact force can be supported without suction. This is the precise reason impulsive energy losses are eliminated.

The cost of transport is defined by

$$
\mathrm{COT} = \frac{\int_0^T P(t)\,dt}{mgD}.
$$

For this passive conservative gait, there are no generalized non-conservative forces, no Rayleigh dissipation, and the ideal no-slip constraints do no work. With collisionless impacts, the total mechanical energy is conserved over a cycle, so $\int_0^T P(t)\,dt = 0$ and therefore $\mathrm{COT}=0$ [2106.11765].

A representative finite-amplitude solution, in dimensionless units $g=m_2=l_t=1$, uses
$l_h=1.0941669$, $l_1=0.16$, $d=0.15$, $m_1=0.19$,
$I_{1\phi}=2\times 10^{-5}$,
$I_{1\theta}=I_{1\psi}=3.2\times 10^{-4}$,
$I_{2\phi}=9.2\times 10^{-6}$,
$I_{2\theta}=I_{2\psi}=0.019$.
For this case, the ground reaction remains non-negative throughout the cycle, the maximal numerically determined friction demand is approximately $0.161$, and the step length is approximately $0.085$, about $7.7\%$ of model height and $27.4\%$ of feet separation [2106.11765].

## 5. Collisionless cycloidal model in rotating $E \times B$ plasmas

In Avalanche Energy’s centrifugal mirror machine, the same phrase refers to a collisionless, deterministic, single-particle model for ions born nearly at rest in a strong radial electric field and an axial magnetic field [2606.17195]. The device consists of a negatively biased central conductor at radius $a$ and a grounded anode at $b \approx 6.6\,\mathrm{cm}$, producing the coaxial-capacitor potential

$$
\Phi(r) = -V \frac{\ln(r/a)}{\ln(b/a)},
\qquad
E_r(r)=\frac{V}{r\ln(b/a)},
$$

with approximately uniform $B_z$ over the OES region. The analysis assumes low density, neglects space charge, and treats hydrogen ions.

In time-independent axisymmetric fields, both energy and canonical angular momentum are conserved:

$$
E = \frac{1}{2}m(v_r^2+v_\theta^2)+q\Phi(r),
\qquad
P_\theta = mr v_\theta + \frac{qB_z}{2}r^2.
$$

For ions born at rest at radius $r_0$, one has

$$
P_\theta = \frac{qB_z}{2}r_0^2,
$$

which gives the azimuthal speed along the orbit:

$$
v_\theta(r,r_0)=\frac{qB_z}{2m}\left(\frac{r_0^2}{r}-r\right).
$$

Energy conservation then yields

$$
v_r^2 = F(r,r_0)=\frac{2q}{m}\big[\Phi(r_0)-\Phi(r)\big]-v_\theta^2(r,r_0).
$$

Rather than an explicit $r(t),\theta(t)$ parametric solution, the model uses these invariants and the radial period
$T_r(r_0)=2\int_{r_-}^{r_+} dr/v_r$ to construct orbit-averaged distributions [2606.17195].

The physical consequence is **orbit crossing**. Ions born at different radii experience different potential drops and acquire different canonical momenta, so at a given observation radius $r$ they arrive with markedly different $\{v_r,v_\theta\}$. The resulting distribution is strongly non-Maxwellian and multi-stream. The orbit-averaged LOS distribution is built from residence-time weighting:

$$
f(v_r,r)=\int_a^b dr_0
\frac{r_0 n_0(r_0)}{r\,T_r(r_0)\,u}
\big[\delta(v_r-u)+\delta(v_r+u)\big],
\qquad
u=\sqrt{F(r,r_0)},
$$

and for a point $(x,y)$ with $r=\sqrt{x^2+y^2}$ the LOS velocity is

$$
v_x=\frac{x}{r}v_r-\frac{y}{r}v_\theta.
$$

This mapping yields non-Gaussian, multi-peaked $f(v_x,r)$ and hence broad spectral wings [2606.17195].

The optical forward model convolves the local emission with the LOS geometry and the measured Gaussian instrument response, with $\sigma_{\mathrm{inst}}\approx 0.04\,\mathrm{nm}$ and $\mathrm{FWHM}\approx 0.1\,\mathrm{nm}$. Five chords at
$y_{\mathrm{LOS}}=(-37.5,-22.5,0.0,15.0,30.0)\,\mathrm{mm}$ are fit simultaneously by discretizing the unknown birth profile $n_0(r_0)$ into radial shells and minimizing reduced $\chi^2$ with a smoothness regularization on the shell weights [2606.17195].

The apparent ion temperature is defined through the velocity variance,

$$
\frac{1}{2}k_B T_{\mathrm{app}}(r)=\frac{1}{2}m\sigma_v^2(r),
\qquad
\sigma_v^2=\langle v^2\rangle-\langle v\rangle^2,
$$

and the scalar discharge value is the density-weighted radial average

$$
\langle T_{\mathrm{app}}\rangle
=
\frac{\int_a^b r n(r)T_{\mathrm{app}}(r)\,dr}
{\int_a^b r n(r)\,dr}.
$$

The best-fit collisionless cycloidal model reproduces the observed wing widths and chord-to-chord asymmetries and yields

$$
\langle T_{\mathrm{app}}\rangle = 1.55 \pm 0.24\ \mathrm{keV}.
$$

A complementary fully relaxed rotating-Gaussian limit yields
$1.40\pm0.43\,\mathrm{keV}$, so the two models bracket the degree of velocity-space relaxation while both supporting keV-scale apparent ion energies in a device only a few centimeters in size [2606.17195].

## 6. Universality, parameter tuning, and misconceptions

The bipedal collisionless construction exhibits a notable degree of universality. In the SML, the minimal-oscillation mode obeys impact-phase relations

$$
o_+ \coth(o_+) = o_- \cot(o_-) = o \tan(o),
$$

together with

$$
\left(o^{-2}-o_-^{-2}\right)\left(o^{-2}+o_+^{-2}\right)=\bar\tau^{-4},
$$

where the reduced impact time is $\bar\tau=\tau/\sqrt{\gamma}$ [2106.11765]. These relations are independent of the detailed inertial or geometric parameterization, provided the signs of the relevant eigenvalues remain as in the SML. The same paper further argues, by a topology-based counting rule
$n=d_i+n_e-2$, that the three-dimensional biped with a two-dimensional impact should require only one tuned parameter to achieve collisionlessness; numerically, a single geometric parameter, $l_h$, is indeed sufficient to tune the general finite-amplitude $(3,4)$ mode [2106.11765].

The general intermittent-contact theory places this observation in a broader class. It shows that collisionless solutions are fixed primarily by spectral structure, symmetry, and impact timing, not by ad hoc restitution laws or special compliant elements [2402.16010]. This suggests that the bipedal construction is not an isolated curiosity but an instance of a wider family of conservative intermittent-contact motions.

Several misconceptions are directly corrected by the cited work. First, a collisionless gait is **not** a no-contact gait; it is a gait whose contact onset occurs with vanishing relative velocity and normal acceleration, so no impulsive loss occurs [2106.11765][2402.16010]. Second, the bipedal swing-foot trajectory is **not exactly cycloidal**; it is cycloid-like or trochoid-like because the coronal dynamics contain hyperbolic components and the axial angle is generated by coupled-sector integrals rather than a single sinusoid [2106.11765]. Third, a keV-scale apparent ion temperature in the plasma model does **not** imply a fully thermalized Maxwellian ion population; it is a variance-based quantity inferred from the LOS-projected, emission-weighted distribution, and the sharper-than-observed spectral features in the collisionless model are consistent with neglected collisional or collective smoothing [2606.17195].

The main limitations are equally clear. The intermittent-contact theory is developed in the linear harmonic approximation, assumes one-dimensional impact, distinct eigenvalues, and a symmetry-restricted solution class [2402.16010]. The bipedal analysis derives its spectrum analytically only in the small-movement regime and treats Floquet stability qualitatively rather than computing it explicitly [2106.11765]. The plasma model assumes axisymmetric static vacuum fields, neglects space charge, ion-ion and ion-neutral collisions, and collective effects, treats $B_z$ as uniform in the analysis region, neglects explicit Zeeman/Stark/natural broadening beyond the measured instrument response, and assumes that excited charge-exchange neutrals inherit the ion velocity distribution without $\sigma_{cx}(v)$ weighting [2606.17195].

Taken together, these results establish the Collisionless Cycloidal Model as a technically precise concept rather than a metaphor. In mechanics, it denotes conservative intermittent-contact motion in which a cycloid-like endpoint path suppresses impact loss and can produce zero-COT walking. In plasma physics, it denotes invariant-based cycloidal orbit synthesis for ions in crossed fields, yielding non-Maxwellian LOS spectra and keV apparent temperatures. The unifying principle is that coordinated conservative dynamics can replace dissipative collision models while remaining quantitatively predictive in both locomotion and spectroscopy [2106.11765][2606.17195][2402.16010].

Source: https://www.emergentmind.com/topics/collisionless-cycloidal-model