Pushing-Pulling Method

• Pushing-Pulling Method is a framework that distinguishes between pushing, pulling, and sign-changing regimes across various fields, including robotics, photonics, acoustics, active matter, and federated learning.
• In robotics and optical applications, the method employs geometry-aware sampling, phase control, and interference to achieve precise manipulation and force reversal, with successful trials reported using energy-based metrics and directional filters.
• In federated learning and active matter, the approach implements asymmetric communication and activity-induced propulsion, enabling performance gains through local compensation and effective control over model updates or cellular forces.

The expression Pushing-Pulling Method is used for several technically distinct procedures in which transport, manipulation, or control is obtained by selecting between a pushing regime, a pulling regime, or a sign-changing transition between them. In robotics, it denotes nonprehensile repositioning by contact; in photonics and acoustics, it denotes force reversal under structured fields; in active matter and cell mechanics, it denotes orientation- or load-dependent propulsion; and in federated learning, it denotes asymmetric communication in which clients push gradients and intermittently pull global parameters (Li et al., 22 Sep 2025, Lee et al., 2019, Rajabi et al., 2021, Wang et al., 2020).

1. Conceptual scope and formal structure

Across the reported formulations, the central object is a signed actuation quantity. In dexterous manipulation, the sign is encoded geometrically by the relation between a pre-contact hand pose and a desired object-translation direction. In optical and acoustic methods, the sign is the sign of the axial or lateral radiation force, such as $F_z$, $\vec{P}_\Sigma$, or $\langle F_z\rangle$. In algorithmic communication systems, “push” and “pull” refer to directional information flow: clients push local updates and pull synchronized global state (Li et al., 22 Sep 2025, Lee et al., 2021, Rajabi et al., 2021, Wang et al., 2020).

Domain	Representative state or quantity	Switching variable
Dexterous nonprehensile manipulation	$H=(\theta,T)$, $u_{dir}$, $V(H)$	geometry, contact pose, planner ranking
Optical or acoustic actuation	$F_z$, $F_x$, $\vec{P}_\Sigma$, $\langle F_z\rangle$	beam angle, phase, polarization, gain, bubble geometry
Active matter and cell mechanics	orientation $\vec{P}_\Sigma$0, mobility $\vec{P}_\Sigma$1, force–velocity relation $\vec{P}_\Sigma$2	linkage orientation, load dipole, protrusion–contraction balance
Federated learning	local model $\vec{P}_\Sigma$3, global model $\vec{P}_\Sigma$4	pull ratio $\vec{P}_\Sigma$5, local compensation

The recurrent mathematical pattern is a decomposition into a direct term and an interference, gradient, or compensation term. Optical pulling from a forward beam is written as

$\vec{P}_\Sigma$6

so the sign changes when forward scattering is sufficiently enhanced. In dexterous robotics, candidate poses are ranked by

$\vec{P}_\Sigma$7

and execution selects the lowest-cost feasible action. In acoustic pulling with active spherical carriers, the sign of the spatially averaged acoustic radiation force is controlled by the phase difference $\vec{P}_\Sigma$8 between two excited modes. In PRLC for federated learning, the communication asymmetry is governed by a Bernoulli pull indicator with ratio $\vec{P}_\Sigma$9 (Chen et al., 2011, Li et al., 22 Sep 2025, Rajabi et al., 2021, Wang et al., 2020).

2. Robotics: nonprehensile pushing and pulling by contact planning

In robotic manipulation, the most explicit pushing-pulling formulation in the cited material is Geometry-aware Dexterous Pushing and Pulling (GD2P). The task is to synthesize pre-contact hand poses $\langle F_z\rangle$0 for an object $\langle F_z\rangle$1 on a flat surface, with object geometry represented by a point cloud $\langle F_z\rangle$2 and a Basis Point Set encoding $\langle F_z\rangle$3. A trial specifies a direction $\langle F_z\rangle$4 with zero table-normal component and a target $\langle F_z\rangle$5. Success is defined after a 20 cm translation along $\langle F_z\rangle$6: the object center must lie within 3 cm of $\langle F_z\rangle$7 and yaw, pitch, and roll deviation must be at most $\langle F_z\rangle$8. GD2P generates diverse hand poses by contact-guided sampling, filters them in IsaacGym, and trains a conditional 1D U-Net DDPM conditioned on BPS geometry. The training set contains 1.3 million successful hand poses across 2.3k objects. On 840 real-world Allegro trials, aggregate success rates across three directions were 61.4%, 47.1%, and 57.1% for GD2P, compared with 50.0%, 40.0%, and 51.4% without ranking; 32.9%, 28.6%, and 32.9% for nearest-neighbor retrieval; and 27.1%, 31.4%, and 37.1% for a pre-trained grasp-pose baseline. On a LEAP Hand, 210 trials yielded a 68.1% success rate (Li et al., 22 Sep 2025).

The geometric core of GD2P is an energy-based pose synthesis stage,

$\langle F_z\rangle$9

with

$H=(\theta,T)$0

This makes directional alignment explicit and couples it to collision, joint-limit, and penetration penalties. The learned DDPM then minimizes

$H=(\theta,T)$1

while test-time action selection combines cuRobo feasibility with the ranking metric $H=(\theta,T)$2. This formulation treats pushing and pulling not as separate motion primitives, but as outcomes of geometry-conditioned whole-hand support and directionally aligned contact.

The same data block also contains adjacent robotic methods in which only the pushing half is implemented. Push-MOG computes “fork pushing” actions using a parallel-jaw gripper to consolidate convex polygonal objects for multi-object grasping. Its objective metric is $H=(\theta,T)$3, and in physical decluttering experiments it achieved $H=(\theta,T)$4, compared with $H=(\theta,T)$5 for frictional single-object grasping and $H=(\theta,T)$6 for MOG-Net without pushing. Pulling is explicitly out of scope in that method (Aeron et al., 2023).

A second boundary case is stable pushing with a nonholonomic mobile base. There the essential result is that single-point sticking is not achievable for general motions, whereas line contact is necessary. The stable pushing condition and the robot’s nonholonomic constraint reduce to the linear motion constraint

$H=(\theta,T)$7

which is embedded in an NMPC planner. Real-world experiments with Husky and Boxer reported 100% stiff-contact success, with average robot distance reduced from 8.53 m to 6.53 m and time reduced from 58.4 s to 13.2 s relative to a reactive strategy. Pulling is only discussed conceptually as a possible extension (Tang et al., 2023).

3. Optical methods: radiation-pressure reversal, directional recoil, and field-engineered sign control

Optical pushing-pulling methods are defined by sign control over the optical force, usually along the beam axis but sometimes transversely. A general criterion is the backward-scattering-force condition

$H=(\theta,T)$8

which shows that pulling requires the average scattered momentum to be more forward-directed than the incident axial momentum budget. This is not achievable with a plane wave alone; the necessary condition reported in the cited work is the simultaneous excitation of multiple radiation multipoles together with a small projection of the incident photon momentum along the propagation direction, as in propagation-invariant beams such as Bessel beams (Chen et al., 2011).

One experimentally simple realization uses a single Gaussian beam acting on plasmonic Au nanoparticles that generate a nanobubble at surface plasmon resonance. Under 800 nm femtosecond illumination, the particle can become encapsulated by a vapor cavity, creating supercavitation and drastically suppressing viscous drag. The sign of the axial optical force $H=(\theta,T)$9 is then set by NP–bubble geometry. Negative $u_{dir}$0 occurs when a sufficiently large bubble, with radius $u_{dir}$1 nm, forms at the “back” of the nanoparticle relative to the incident wavevector, with polar angle $u_{dir}$2; otherwise $u_{dir}$3 is positive. Reported peak speeds were 336,000 $u_{dir}$4m s$u_{dir}$5 for positive motion and 245,000 $u_{dir}$6m s$u_{dir}$7 for negative motion, with effective viscosities near $u_{dir}$8 kg m$u_{dir}$9 s$V(H)$0, close to steam viscosity (Lee et al., 2019).

A second optical class uses surface-wave recoil rather than bubble-mediated field reshaping. In a one-dimensional photonic crystal supporting Bloch surface waves, the lateral force on a dipolar bead above the interface consists of positive radiation-pressure terms and a recoil term proportional to $V(H)$1. The sign is controlled by the phase of the p-polarized reflection coefficient: $V(H)$2 yields pulling, and $V(H)$3 yields pushing. Angle- and wavelength-assisted switching is therefore intrinsic to the photonic-crystal design (Kostina et al., 2021).

A third class uses resonant interference inside active nanostructures. In a gain-assisted plasmonic core-shell sphere, a narrow quadrupole mode couples to a broad dipole mode and produces a Fano resonance. Near $V(H)$4 nm, the internal and external electric fields reverse phase, the Lorentz force on bound currents and charges in the metallic shell flips sign, and the reported negative force becomes about two orders of magnitude larger than that on a pure gain sphere of the same size. Pump rates around $V(H)$5 s$V(H)$6 mark the force-sign transition in the reported geometry (Gao et al., 2017).

The same sign-switching logic appears in PT-symmetric photonics, but there it is organized in a generalized transfer-matrix parameter space. For single-beam normal incidence, the pressure from the left is

$V(H)$7

and from the right

$V(H)$8

The exact PT-symmetry phase yields symmetric pushing, exceptional points yield uni-directional null force, the line $V(H)$9 yields bi-directional null force, and the broken-symmetry phase yields pulling-pushing flipped forces. Under two counter-propagating plane waves, the resultant force becomes phase-tunable through the relative phase $F_z$0 (Lee et al., 2021).

Recent optical nanomotor work separates longitudinal and transverse control. A dielectric glass cylinder with anti-reflection coatings is driven by an azimuthally polarized Bessel beam to generate longitudinal pulling, while embedded asymmetric plasmonic dimers supply lateral motion under plane-wave illumination. Reported peak pulling forces were about $F_z$1 pN for a design with $F_z$2 nm, $F_z$3, $F_z$4, $F_z$5m, and $F_z$6m, and about $F_z$7 pN for a shorter design at $F_z$8 nm, $F_z$9, and $F_x$0. The same structure also supports plane-wave pushing and lateral forces of about 0.1 pN (Serrera et al., 20 Jun 2025).

Optical pulling on deformable bodies introduces an additional constraint: the force-to-stress ratio can be much smaller than in conventional pushing. For water microdroplets in a Bessel beam, the local force density is strongly peaked near the poles, and the resulting linearized surface dynamics,

$F_x$1

predict shape oscillations with lifetimes of order microseconds for droplet radii of a few micrometers. The dominant deformations are jet-like protrusions along the beam axis (Ellingsen, 2011).

4. Acoustic and ponderomotive implementations

In acoustics, the pushing-pulling method can be realized with an active spherical carrier in a progressive plane wave. A single activated mode, either monopole or dipole, always has a positive spatially averaged acoustic radiation force, equal to the passive-sphere value, so net pulling is impossible in that case. The reported method therefore excites the $F_x$2 and $F_x$3 modes synchronously with a controlled phase difference $F_x$4. The time-averaged force is written as

$F_x$5

and the spatial average over one wavelength gives $F_x$6 as an explicit function of $F_x$7 and $F_x$8. At $F_x$9, wide intervals of $\vec{P}_\Sigma$0 produce negative averaged force, and the reported $\vec{P}_\Sigma$1 curves include four regimes: always negative, negative average with sign changes, positive average with sign changes, and always positive (Rajabi et al., 2021).

Wave-packet and beat-wave methods generalize this logic beyond resonant carriers. For a small particle in a quasi-monochromatic propagating packet, the slow drift obeys

$\vec{P}_\Sigma$2

which, for a traveling envelope $\vec{P}_\Sigma$3, becomes

$\vec{P}_\Sigma$4

Forward waves, $\vec{P}_\Sigma$5, are always pushing, whereas backward waves, $\vec{P}_\Sigma$6, pull when $\vec{P}_\Sigma$7. Beat waves generated by two forward-propagating waves with slightly different frequencies emulate periodic sequences of forward- or backward-propagating sub-pulses. For simple particles with backward sub-pulses, pulling occurs above a critical amplitude

$\vec{P}_\Sigma$8

while trapping sets in above

$\vec{P}_\Sigma$9

For polarizable particles, the sign structure is inverted: forward sub-pulses pull for $\langle F_z\rangle$0 and push for $\langle F_z\rangle$1 (Bliokh, 3 Feb 2026).

These acoustic and ponderomotive formulations make the sign reversal explicitly phase- and dispersion-controlled. The relevant control variables are no longer object geometry or photonic scattering modes alone, but modal phase difference, effective group velocity, and amplitude thresholds.

5. Active matter, catalytic carriers, and biological asymmetry

At microscale and cellular scales, pushing and pulling are determined by internal activity and by the geometry of the active element relative to cargo or external load. In a catalytically active carrier connected by a rigid rod to an inert cargo, the carrier generates solute gradients and an interfacial slip $\langle F_z\rangle$2. The composite propulsion speed follows

$\langle F_z\rangle$3

For repulsive solute–surface interactions, $\langle F_z\rangle$4 corresponds to a pull configuration and $\langle F_z\rangle$5 to a push configuration. As the rod length increases, the puller speed asymptotes to $\langle F_z\rangle$6 and the pusher speed to $\langle F_z\rangle$7. A fully catalyst-covered sphere, which is motionless when isolated, becomes motile once attached to cargo because the cargo breaks spherical symmetry (Popescu et al., 2011).

A distinct biological use of the same terminology appears in the one-dimensional active-gel model of crawling cells. External boundary tractions define a resultant load

$\langle F_z\rangle$8

with $\langle F_z\rangle$9 corresponding to pushing and $\vec{P}_\Sigma$00 to pulling. In the traveling-wave regime, the force–velocity relation is

$\vec{P}_\Sigma$01

For pushing, the curve is concave and protrusion-dominated. For pulling, protrusion dominates only at small $\vec{P}_\Sigma$02; above the threshold

$\vec{P}_\Sigma$03

contraction dominates, producing a convex branch and a negative-mobility interval. The regular branch terminates at

$\vec{P}_\Sigma$04

The reported biologically relevant negative-mobility regime lies around 1–1.7 nN (Recho et al., 2013).

These active-matter examples show that pushing-pulling behavior need not depend on an external field. It may instead be selected by linkage orientation, catalytic symmetry breaking, or by the reallocation of internal force-generating mechanisms between protrusion and contraction.

6. Algorithmic push–pull communication in federated learning

In federated learning, the same terminology is used in a non-mechanical sense. PRLC, or Pulling Reduction with Local Compensation, treats pushing as transmitting local stochastic gradients to the server and pulling as synchronizing a client with the server’s global model. The server update is

$\vec{P}_\Sigma$05

while each client independently pulls with probability $\vec{P}_\Sigma$06. If client $\vec{P}_\Sigma$07 does not pull, it performs a local compensation step

$\vec{P}_\Sigma$08

The gap $\vec{P}_\Sigma$09 is thus corrected locally rather than by immediate synchronization (Wang et al., 2020).

The theoretical result is that PRLC preserves the same order of convergence as classical synchronous SGD for both strongly convex and non-convex cases and exhibits linear speedup with respect to the number of training nodes. The paper further shows that PRLC admits lower pulling frequency than a pulling-reduction method without local compensation. In experiments on CIFAR-10 with logistic regression and ResNet-18, PRLC with $\vec{P}_\Sigma$10 required about half of the pulling operations of synchronous SGD and, as reported, only half of the pulling operations of LAG. In an edge-like environment, it achieved about 30% faster time-to-convergence than ASGD.

This usage extends the term beyond physical force. Here the push–pull asymmetry is a resource-allocation mechanism: communication is reduced by making pulls intermittent, while local compensation preserves optimization stability. The structural analogy with physical pushing-pulling methods lies in directional asymmetry plus a compensating correction term, not in mechanical transport.