CoG Algorithms Overview

• Center-of-Gravity (CoG) algorithms are techniques that compute a representative centroid or weighted average from distributed measurements, crucial for precision applications.
• They address systematic errors such as sampling and truncation biases using methods like lookup-table and linear corrections to achieve near-optimal accuracy.
• These algorithms are applied in diverse fields including detector physics, robotic grasping, continuum arm dynamics, and supply chain optimization with advanced error modeling.

A Center-of-Gravity (CoG) algorithm calculates a representative point—typically the centroid or weighted average—of a distribution of samples, weights, or measurements. CoG algorithms are foundational in fields as diverse as precision position measurement in detector physics, industrial supply chain optimization, high-level vision and manipulation in robotics, compliant manipulator dynamics, fuzzy logic defuzzification, and various forms of sensor fusion. Advances in CoG methodology involve detailed analysis of their systematic biases, error distributions, robustification against outliers, computational performance, and unbiased estimation strategies close to theoretical statistical bounds.

1. Mathematical Formalism and Systematic Errors

CoG estimators nominally return the centroid of a distribution, e.g., $X_c = \frac{\sum_i x_i g_i}{\sum_i g_i}$ for sampled pixel locations $x_i$ and signal values $g_i$. For a Gaussian point-spread-function (PSF), sampling and truncation introduce two dominant sources of systematic error:

• Sampling error arises when the true, continuous PSF is discretized over pixels of finite pitch $\Delta$. The resulting periodic bias is given by

$$\delta_{x,\text{samp}}(x_0) = \frac{1}{\pi} \sum_{n=1}^\infty \sin\left(2\pi n \frac{x_0}{\Delta}\right) \exp\left[-2\pi^2 \left(n \frac{\sigma_g}{\Delta}\right)^2\right] (-1)^n,$$

with only the $n=1$ term usually significant.

• Truncation error occurs when the signal is windowed to a finite region of interest (ROI) of $N_s$ pixels. The linearized truncation bias is

$$\delta_{x,\text{cut}} = x_0 F_{\text{cut}}, \quad F_{\text{cut}} = \sqrt{\frac{2}{\pi}} \frac{N_s}{2\sigma_g} \frac{e^{-N_s^2/(8\sigma_g^2)}}{\operatorname{erf}\left(\frac{N_s}{2\sqrt{2}\sigma_g}\right)}.$$

Both effects combine: total systematic error is directly evaluated as

$$\delta_{x,\text{tot}}(x_0) = \frac{\sum_i x_i f(x_i;x_0)}{\sum_i f(x_i;x_0)} - x_0.$$

The variance over $x_0 \sim \mathrm{Uniform}[-\Delta/2, +\Delta/2]$ quantifies average bias, and an explicit PDF for errors can be constructed via the inverse mapping technique (Hechenblaikner, 2023, Landi, 2019).

2. Unbiased and Corrected CoG Algorithms

Two classes of bias correction have been established:

• Lookup-table correction: Numerically precompute the mapping $$f_\mathrm{lu}(x_0) = X_c(x_0) = x_0 + \delta_{x,\text{tot}}(x_0)$$ and invert to obtain $x_0 = f_\mathrm{lu}^{-1}(X_c)$. The resulting estimator $X_{c,\mathrm{ub}}$ is unbiased to within machine precision in the absence of noise, with variance

$$\sigma_{x,\mathrm{ub}}^2 = F_{\text{broad}} \left(\sigma_{\text{pix}}^2 + \sigma_{\text{phot}}^2\right),$$

where $F_{\text{broad}}$ quantifies noise broadening slightly exceeding unity (Hechenblaikner, 2023).

• Linear correction: When truncation dominates, the bias is nearly linear and corrected via $X_{c,\text{lin}} = X_c / (1 + F_\text{cut})$, yielding residual bias $O(x_0^3)$ and a total variance as function of $\sigma_{\text{pix}}^2, \sigma_{\text{phot}}^2$, and $\sigma_{\text{res}}^2$.

Monte Carlo simulation and analytical results agree to within a few percent, covering signal regimes from high to very low photon number (Hechenblaikner, 2023).

3. Error Distributions and Non-Gaussianity

In position-sensitive applications (e.g., silicon sensors, calorimetry), error PDFs of CoG algorithms are characteristically non-Gaussian:

• Two-strip and three-strip CoG: The distribution for $x = (x_1 - x_3)/(x_1 + x_2 + x_3)$ under Gaussian strip noise features a Gaussian-like core with Cauchy–Agnesi (long $1/x^2$) tails; variance is infinite for two-strip, finite for three-strip CoG. Extended forms (e.g., $x_{g3}, x_{g4}, x_{g5}$) in high granularity detectors yield multi-modal, gapped PDFs that are analytically tractable via CDF construction and differentiation (Landi et al., 2020, Landi et al., 2020, Landi et al., 2021).
• Impact for estimation: These heavy tails imply that least-squares minimization is statistically inconsistent (divergent variance), whereas maximum likelihood methods using exact PDFs yield robust, outlier-resistant fits and superior track/fitting resolution (Landi et al., 2020).

4. High-Dimensional and Application-Specific Extensions

4.1 Two-Dimensional and Arbitrary Tessellations

Generalization to 2D (pixels, calorimeter crystals, hexagonal/triangular arrays):

• Fourier analysis provides closed-form bias and resolution formulae for arbitrary lattice tessellations. Discretization errors are tied to the Fourier transform of the continuous signal at reciprocal lattice points (Landi, 2021).
• Elimination of bias is possible for tessellating signal shapes or by applying ideal crosstalk kernels that nullify both the value and derivatives of the transfer function at reciprocal lattice harmonics.

4.2 Robotics: Manipulation and Dynamics

Vision-Based CoG Estimation for Grasping

CoG localization models leveraging diffusion foundation models (DIFT) and vision-LLMs integrate scene segmentation, CLIP-based feature matching, diffusion-driven keypoint transfer, and commonsense pruning to estimate the balance point of physically asymmetric objects, enabling robust 6-DoF grasp selection (Xiangli et al., 25 Jul 2025). Quantitatively, this approach outperformed conventional baselines (76% success vs. 27% for keypoint methods, 65% for affordance-driven) and localized CoG on unseen objects at 76% pixel-level accuracy.

CoG-Based Continuum Arm Dynamics

For variable-length continuum robots, dynamic modeling at the per-section CoG yields recursive $O(n^2)$ complexity, enabling $>$kHz real-time control with sub-millimeter accuracy, validating against full-physics integral models. The CoG dynamics are extended with shaping coefficients to match full energy integrals, covering complex, compliant architectures (Godage et al., 2019).

CoG Trajectory Planning in Bipedal Gait

In ZMP-based trajectory planning for bipedal robots, the assumption of all mass at the hip CoG enables analytic trajectory synthesis along brachistochrone, circular, or linear height profiles, with quantifiable improvements in joint acceleration and jerk for brachistochrone descent compared to alternatives (Bhardwaj et al., 2020).

5. Optimization, Supply Chain, and Heuristic CoG Methods

The classical supply chain CoG problem determines effective facility placement as the weighted centroid over a demand network. State-of-the-art solutions embed this principle in MILP formulations, enabling integer site selection, explicit service-level constraints, and computationally accelerated variants through Candidate Location Selection (CLS) and “customer packet” aggregation. These heuristics achieve significant runtime reduction ($>90\%$) while retaining near-optimal location quality (average location shift $\sim18$ miles) (Houck et al., 2022).

Variant	Computational Approach	Application Domain
1D/2D Analytical CoG	Fourier, CDF, δ-function	Detectors, Imaging
CoG + Lookup Correction	Tabular inversion	Astrometry, Low-light tasks
MILP/Heuristics	Integer Linear Programming	Supply Chain
Vision-Driven CoG	CLIP, Diffusion Models	Robotics, Manipulation
CoG-Recursive Dynamics	Modal/Polynomial Recursion	Continuum Robots

6. Practical Implementation and Correction Strategies

Correcting for CoG bias and optimizing statistical performance involve several implementable strategies:

• Analytic inversion: Use explicit bias expansions (via Fourier or tabular form) and invert numerically or via root-finding to obtain unbiased position.
• Histogram inversion: Uniform scanning allows the empirical PDF of CoG to be derived, then invert the map to reconstruct the physical variable distribution (Landi, 2019).
• Ideal crosstalk engineering: Design pixel/strip readout and sharing kernels so the reconstructed CoG is exactly unbiased for any input shape.
• Finite cluster and border handling: Systematic corrections must be recalculated accounting for incomplete detector coverage, dead/noisy elements, and nonuniformity.
• Noise heteroscedasticity: Use per-channel error models in likelihood fitting to benefit from non-Gaussian error PDFs.

7. Broader Uses: Fuzzy Logic, Assessment, and Defuzzification

CoG defuzzification procedures, such as those in RFAM, GRFAM, TFAM, and TpFAM, map fuzzy membership functions to crisp values for assessment and decision-making. In educational or rating contexts, CoG-based models are affine in traditional GPA but add a second coordinate encoding grade compactness/tie-breaking, and admit model-appropriate tuning (rectangles, triangles, trapezoids) to address ambiguity and rank dispersion (Voskoglou, 2016).

• G. Landi & G.E. Landi, "Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms," (Landi et al., 2020).
• B. Fryer et al., "Unbiased centroiding of point targets close to the Cramer Rao limit," (Hechenblaikner, 2023).
• J.W. Hausler et al., "Properties of the Center of Gravity as an Algorithm for Position Measurements," (Landi, 2019).
• S.R. Kuindersma et al., "Planning Brachistochrone Hip Trajectory for a Toe-Foot Bipedal Robot going Downstairs," (Bhardwaj et al., 2020).
• Z. Zuo et al., "Center of Gravity-based Approach for Modeling Dynamics of Multisection Continuum Arms," (Godage et al., 2019).
• "Foundation Model-Driven Grasping of Unknown Objects via Center of Gravity Estimation," (Xiangli et al., 25 Jul 2025).
• SSG, "Advanced Quantitative Techniques to Solve Center of Gravity Problem in Supply Chain," (Houck et al., 2022).
• G. Landi & G.E. Landi, "Properties of the Center of Gravity as an Algorithm for Position Measurements: Two-Dimensional Geometry," (Landi, 2021).
• B. Ananiadou et al., "Comparison of the COG Defuzzification Technique and Its Variations to the GPA Index," (Voskoglou, 2016).