Center-of-Mass Frame Fixing

• Center-of-Mass Frame Fixing is a procedure that reforms physical descriptions to eliminate bulk translational motion and coordinate artifacts.
• It is applied across various domains such as nuclear density functional theory, numerical relativity, sensor calibration, and robotic grasp planning to improve intrinsic observables.
• Methodologies range from projection techniques and charge-based adjustments to Kalman filtering and closed-form corrections, yielding reduced systematic errors and enhanced model fidelity.

Searching arXiv for recent and directly relevant papers on center-of-mass frame fixing across domains. Center-of-mass (CoM) frame fixing denotes the class of procedures by which a physical description is reformulated so that bulk translational motion, frame drift, or off-center coordinate choices do not contaminate intrinsic observables. The operation takes different technical forms in different fields: projection onto zero total momentum in nuclear density functional theory, estimation of an offset vector between a sensor and a spacecraft’s true center of mass, BMS-frame transformations of numerical-relativity waveforms, laboratory-to-CoM kinematic corrections in associated particle imaging, and explicit alignment of a manipulator pose to an object CoM in grasp planning. Despite these differences, the common aim is to separate intrinsic dynamics from coordinate artifacts or mechanically induced bias (Kafker et al., 12 Mar 2025, Mitman et al., 2021, Wei et al., 2023).

1. Formal scope and common mathematical structure

At the most basic level, CoM frame fixing begins with the decomposition of motion into collective and relative components. For a two-body system with positions $\mathbf x_1,\mathbf x_2$, masses $m_1,m_2$, CoM coordinate $\mathbf R$, and relative coordinate $\mathbf r$,

$$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$

and the equations of motion separate into

$(m_1+m_2)\ddot{\mathbf R}=0$

together with the relative dynamics. In an exact CoM frame, $\mathbf R^{(\mathrm{cm})}\equiv 0$, so only intrinsic motion remains (Žugec et al., 2020, Gómez et al., 2014).

In many-body and field-theoretic settings the same idea is expressed through symmetry generators rather than elementary coordinates. In nuclear DFT the issue is that the trial Slater determinant or Bogoliubov vacuum $|\Phi\rangle$ is not an eigenstate of the total momentum operator $\hat P=\sum_{k=1}^A \hat p_k$, so translational invariance is broken. In numerical relativity at future null infinity, the relevant object is the CoM charge

$G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$

whose nonzero intercept or slope signals residual translation or boost freedom in the waveform’s BMS frame (Kafker et al., 12 Mar 2025, Mitman et al., 2022).

A concise cross-domain summary is:

Domain	Quantity being fixed	Representative operation
Nuclear EDF/DFT	Total momentum of the many-body state	Projection with $m_1,m_2$0
Numerical relativity	CoM charge $m_1,m_2$1 at $m_1,m_2$2	Boost/translation fit and BMS transform
Space instrumentation	Offset $m_1,m_2$3 between sensor and spacecraft COM	EKF–RTS smoothing with $m_1,m_2$4 outlier rejection
API reconstruction	Lab-frame effect of nonzero $m_1,m_2$5	Kinematic correction using average CoM velocity
Grasp planning	Misalignment between gripper origin and object CoM proxy	Closed-form translation $m_1,m_2$6

These examples suggest that “frame fixing” is not a single algorithm but a structural requirement: one must either impose the correct CoM frame, estimate it, or transform to it before interpreting observables.

2. Translational symmetry restoration in nuclear many-body theory

In nuclear energy density functional theory, translational symmetry breaking is universal: unlike rotational or gauge symmetry breaking, it occurs for all nuclei. The isolated nucleus must have zero total momentum, yet the standard mean-field energy

$m_1,m_2$7

contains spurious CoM kinetic energy because $m_1,m_2$8 is not an eigenstate of $m_1,m_2$9. The intrinsic state is obtained by projection,

$\mathbf R$0

with $\mathbf R$1. In coordinate representation,

$\mathbf R$2

so the projected wave function is translationally invariant by construction, with $\mathbf R$3 and $\mathbf R$4 (Kafker et al., 12 Mar 2025).

The associated energy correction is

$\mathbf R$5

Using the generator-coordinate form,

$\mathbf R$6

A Gaussian overlap approximation, valid up to a few fm of shift, recovers the usual renormalized kinetic-energy prescriptions at lowest order, but the full GCM integral captures both kinetic and interaction contributions to $\mathbf R$7 (Kafker et al., 12 Mar 2025).

Using the SeaLL1 functional and exact CoM projection, the correction is large across the nuclear chart:

Nucleus	$\mathbf R$8 (MeV)
$\mathbf R$9	$\mathbf r$0
$\mathbf r$1	$\mathbf r$2
$\mathbf r$3	$\mathbf r$4
$\mathbf r$5	$\mathbf r$6

These values exceed the Bethe–Weizsäcker mass-formula RMS error of about $\mathbf r$7 MeV and also exceed the typical $\mathbf r$8–$\mathbf r$9 MeV RMS deviations of uncorrected DFT mass fits. The same work reports slight but systematic changes in rms radii, of order $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$0–$$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$1 fm, comparable to experimental uncertainties. Implementation requires overlap and Hamiltonian kernels over $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$2–$$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$3 shifts and increases cost by a modest factor of a few relative to a single mean-field run, provided box size, mesh handling, and FFTs are treated carefully (Kafker et al., 12 Mar 2025).

Historically, the method is traced to the CoM projection suggested by Peierls in 1957. In current EDF practice its importance is methodological rather than optional: the spurious translational contribution is numerically larger than the accuracy targets of modern mass models.

3. BMS-frame fixing of numerical-relativity waveforms

For gravitational-wave calculations, CoM frame fixing is a problem in asymptotic symmetry rather than ordinary mechanics. Numerical waveforms at future null infinity carry the freedom of the Bondi–van der Burg–Metzner–Sachs group, so comparison among post-Newtonian, numerical-relativity, and perturbative waveforms requires a common BMS frame. Early practice often used a Newtonian CoM trajectory built from horizon coordinates,

$$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$4

followed by a least-squares fit for a translation $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$5 and boost $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$6. That approach was shown to be less effective than previously thought, because coordinate black-hole motion need not track the true asymptotic CoM motion, especially for high mass ratio or precessing runs (Mitman et al., 2021).

Charge-based methods replace bulk gauge-dependent coordinates with asymptotic quantities extracted from the waveform itself. In the Moreschi–Boyle convention one defines the mass aspect $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$7, Lorentz aspect $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$8, and energy-moment aspect $$\mathbf R=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{m_1+m_2}, \qquad \mathbf r=\mathbf x_1-\mathbf x_2,$$9, from which the Poincaré charges $(m_1+m_2)\ddot{\mathbf R}=0$0, $(m_1+m_2)\ddot{\mathbf R}=0$1, $(m_1+m_2)\ddot{\mathbf R}=0$2, and $(m_1+m_2)\ddot{\mathbf R}=0$3 are constructed by sphere integrals. The CoM charge

$(m_1+m_2)\ddot{\mathbf R}=0$4

is then fit over an inspiral window. In the linearized method one writes

$(m_1+m_2)\ddot{\mathbf R}=0$5

takes the boost parameter as $(m_1+m_2)\ddot{\mathbf R}=0$6, and the translation as $(m_1+m_2)\ddot{\mathbf R}=0$7, iterating until residuals are small. The same framework is integrated with supertranslation fixing through the Moreschi supermomentum, enabling a complete BMS-frame specification (Mitman et al., 2022).

On a set of 13 binary black-hole systems, the asymptotic-charge CoM fix reduced residual CoM charge from typical $(m_1+m_2)\ddot{\mathbf R}=0$8–$(m_1+m_2)\ddot{\mathbf R}=0$9 to $\mathbf R^{(\mathrm{cm})}\equiv 0$0. It also reduced leakage at twice the $\mathbf R^{(\mathrm{cm})}\equiv 0$1 frequency in the $\mathbf R^{(\mathrm{cm})}\equiv 0$2 mode by a factor of $\mathbf R^{(\mathrm{cm})}\equiv 0$3–$\mathbf R^{(\mathrm{cm})}\equiv 0$4, lowered mode-by-mode mismatches from $\mathbf R^{(\mathrm{cm})}\equiv 0$5 to $\mathbf R^{(\mathrm{cm})}\equiv 0$6 or below, improved NR–PN $\mathbf R^{(\mathrm{cm})}\equiv 0$7 alignment errors over a four-orbit window typically by factors $\mathbf R^{(\mathrm{cm})}\equiv 0$8–$\mathbf R^{(\mathrm{cm})}\equiv 0$9, and restored the correct memory offset in the $|\Phi\rangle$0 mode (Mitman et al., 2021). A separate charge-based BMS-fixing framework reported a method that is $|\Phi\rangle$1 times faster than previous optimization-based approaches when mapping to the superrest frame (Mitman et al., 2022).

For quasicircular, nonprecessing binaries, a later refinement replaced the pure linear ansatz with a post-Newtonian model of the boosted CoM charge that captures physical out-spiraling oscillations. Across 20 SXS simulations, the largest improvement in robustness to fitting-window choice was by a factor of $|\Phi\rangle$2 for the boost vector and $|\Phi\rangle$3 for the translation vector, with the maximum robustness obtained when the window is centered in the inspiral. That method was incorporated into the scri frame-fixing workflow for waveforms produced with Cauchy-characteristic evolution (Khairnar et al., 25 Mar 2026).

A persistent misconception in this area is that Poincaré-only alignment or Newtonian CoM tracking is sufficient. The charge-based results indicate otherwise: waveform-intrinsic asymptotic charges are needed to suppress physically spurious mode mixing and to make hybridization with PN data reliable.

4. Estimation and correction in sensing and measurement systems

In experimental systems, CoM frame fixing often appears as an estimation problem. For TaiJi-1, the gravitational reference sensor requires the test-mass center to coincide with the satellite’s center of gravity in order to avoid disturbances from angular acceleration and gradient. The relevant frames are the body frame $|\Phi\rangle$4, the test-mass frame $|\Phi\rangle$5, and the center-of-mass frame $|\Phi\rangle$6, all with identical orientation, while the offset is

$|\Phi\rangle$7

from spacecraft COM to test-mass COM, expressed in $|\Phi\rangle$8. Over the calibration interval the state is treated as constant,

$|\Phi\rangle$9

and the measured GRS output obeys

$\hat P=\sum_{k=1}^A \hat p_k$0

The estimation pipeline combines an Extended Kalman Filter with a Rauch–Tung–Striebel smoother, uses a $\hat P=\sum_{k=1}^A \hat p_k$1 residual test with $\hat P=\sum_{k=1}^A \hat p_k$2 and $\hat P=\sum_{k=1}^A \hat p_k$3 for outlier rejection, and cross-checks the result with nonlinear least squares solved by Levenberg–Marquardt (Wei et al., 2023).

The final TaiJi-1 offsets were reported as $\hat P=\sum_{k=1}^A \hat p_k$4, $\hat P=\sum_{k=1}^A \hat p_k$5, and $\hat P=\sum_{k=1}^A \hat p_k$6, equivalently $\hat P=\sum_{k=1}^A \hat p_k$7 mm, $\hat P=\sum_{k=1}^A \hat p_k$8 mm, and $\hat P=\sum_{k=1}^A \hat p_k$9 mm. After in-orbit CoM calibration, the modulation peak in the GRS ASD at the applied torque frequency was suppressed, and the residual acceleration noise in the $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$0–$G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$1 Hz band was reduced by a factor $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$2–$G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$3 (Wei et al., 2023).

Associated Particle Imaging presents a different measurement problem. In the DT fusion reaction used by API, the $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$4 particle and neutron are exactly back-to-back in the CoM frame, but in the laboratory frame the opening angle is slightly less than $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$5 because the reacting ion has nonzero $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$6. The CoM velocity is

$G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$7

and in a thick Ti target it varies with depth through the stopping-power equation $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$8. The reconstruction therefore uses a cross-section-weighted average CoM speed obtained from stopping powers and DT fusion cross sections (Egan et al., 2022).

When the CoM effect is included in API reconstruction, the mean of reconstructed locations becomes a correctable systematic shift or tilt, but the distribution retains an irreducible spread because the CoM velocity fluctuates event by event. Reported consequences include a systematic tilt or shift of up to $G^i(u)\equiv \frac{K^i(u)+uP^i(u)}{P^t(u)},$9–$m_1,m_2$00 cm at $m_1,m_2$01 m and an irreducible broadening in the beam-axis direction with a 90% interval of order $m_1,m_2$02–$m_1,m_2$03 cm at $m_1,m_2$04 m (Egan et al., 2022). This distinction between correctable bias and non-correctable spread is central to experimental CoM correction.

5. Off-center actuation, control, and manipulation

In control problems, CoM frame fixing may require either moving the reference frame away from the CoM or aligning the controlled body to the CoM of another object. For a UAV with an off-center slung load, Lv et al. formulate the dynamics about the suspension point rather than the UAV CoM. The frames are the inertial frame $m_1,m_2$05, quadrotor body frame $m_1,m_2$06 at the quadrotor CoM, suspension-point frame $m_1,m_2$07, and payload frame $m_1,m_2$08. With $m_1,m_2$09 the vector from suspension point to quadrotor CoM and $m_1,m_2$10 the vector from suspension point to payload CoM, the inertial positions are

$m_1,m_2$11

The resulting equations of motion show explicit coupling terms induced by the offset $m_1,m_2$12, including the $m_1,m_2$13 term in the attitude dynamics (Lv et al., 6 Jan 2026).

The control design is cascaded. In the middle loop, the virtual input $m_1,m_2$14 regulates the swing angle through a locally exponentially stable subsystem. In the inner loop, the torque

$m_1,m_2$15

cancels the coupling exactly, without neglecting the $m_1,m_2$16 term or treating it as a disturbance. The paper proves local exponential stability by Lyapunov analysis and argues that fixing the frame at the suspension point simplifies swing-angle control while preserving exact treatment of inertial coupling (Lv et al., 6 Jan 2026). This suggests that CoM frame fixing is not always equivalent to “placing the origin at the CoM”; in some systems, the technically correct move is to choose a frame in which the physically relevant coupling becomes controllable.

In robotic grasp planning, DISF uses CoM alignment explicitly as a contact-stability term. Because the true volumetric CoM is unavailable from point clouds, both object and gripper CoM are approximated by centroids,

$m_1,m_2$17

with $m_1,m_2$18 and $m_1,m_2$19. The translation-refinement stage minimizes

$m_1,m_2$20

whose closed-form solution is

$m_1,m_2$21

This step is embedded between rotation optimization and finger-aperture optimization in the sequence RotOpt $m_1,m_2$22 TransRefine $m_1,m_2$23 FingerOpt (Yamanokuchi et al., 31 Dec 2025).

The reported effect is substantial. In the Known-shape regime, VISF achieved median $m_1,m_2$24 m and DISF reduced it to about $m_1,m_2$25 m. In the Observed-shape regime, VISF had median misalignment $m_1,m_2$26 m and DISF reduced it to $m_1,m_2$27 m. Across three robots, average success rose from $m_1,m_2$28 to $m_1,m_2$29 in Known-shape and from $m_1,m_2$30 to $m_1,m_2$31 in Observed-shape; on a real UR3e with observed point clouds, DISF achieved $m_1,m_2$32 successes versus $m_1,m_2$33 for VISF (Yamanokuchi et al., 31 Dec 2025). Here CoM fixing functions as a stability prior rather than a symmetry restoration.

6. Artificial fixing, geometric interpretation, and broader implications

A recurring failure mode is to hold a body fixed in a frame that is only approximately inertial. Gómez et al. examined the Milky Way–Large Magellanic Cloud system and showed that when the LMC is massive, artificially fixing the Milky Way center of mass biases both the LMC orbit and the phase-space structure of other tracers. For $m_1,m_2$34 in the range $m_1,m_2$35 to $m_1,m_2$36, the Milky Way CoM within $m_1,m_2$37 kpc can be displaced by $m_1,m_2$38–$m_1,m_2$39 kpc and $m_1,m_2$40–$m_1,m_2$41 km/s over the last $m_1,m_2$42–$m_1,m_2$43 Gyr (Gómez et al., 2014).

The orbital consequences are large. In a representative model with $m_1,m_2$44 and $m_1,m_2$45, the fixed-MW treatment yields orbital period $m_1,m_2$46 Gyr and apocenter $m_1,m_2$47, whereas the free-MW treatment gives $m_1,m_2$48 Gyr and $m_1,m_2$49. For the Sagittarius stream, including MW recoil as well as LMC torque reduces the angular separation between apocenters by up to $m_1,m_2$50, tilts the orbital plane by $m_1,m_2$51, and changes debris predictions by tens of degrees on the sky and dozens of km/s in velocity (Gómez et al., 2014). In this context, CoM frame fixing is a requirement for dynamical fidelity, not a bookkeeping preference.

The geometric value of the CoM frame is especially transparent in repulsive Rutherford scattering. In the fixed-target frame the shadow caustic has the universal paraboloidal form

$m_1,m_2$52

in scaled units, with the target at the focus. In the CoM frame the projectile and target each cast their own paraboloidal shadow,

$m_1,m_2$53

and the focal points of the two shadows coincide at the CoM itself (Žugec et al., 2020). This example gives a precise geometrical meaning to “fixing the CoM frame”: the intrinsic symmetry of the two-body interaction becomes manifest only after the collective coordinate is removed.

Across these domains, a common misconception is that CoM fixing is merely a postprocessing convenience. The evidence points in the opposite direction. In nuclei it removes a $m_1,m_2$54–$m_1,m_2$55 MeV spurious energy bias; in waveform modeling it suppresses mode mixing and improves PN–NR compatibility; in precision instrumentation it lowers acceleration noise; in Galactic dynamics it changes inferred orbital histories; and in manipulation it alters grasp stability (Kafker et al., 12 Mar 2025, Mitman et al., 2021, Wei et al., 2023, Gómez et al., 2014, Yamanokuchi et al., 31 Dec 2025). A plausible implication is that CoM frame fixing should be regarded as part of model definition whenever translational degrees of freedom are not directly observable but still enter the computation.