Inertial Shape Estimation: Algorithms & Applications

• Inertial shape estimation schemes are a class of algorithms that infer object configuration using inertial sensor data across applications like nuclear physics, robotics, and motion capture.
• The methodology employs variational principles, geometric observers, and convex optimization to deliver robust estimates under dynamic conditions and measurement noise.
• Key approaches include energy minimization, probabilistic sensor fusion, and data-driven models that enhance observability and accuracy in complex, multi-domain systems.

Inertial shape estimation scheme encompasses a diverse class of algorithms and physical models whose aim is to accurately infer the shape or configuration of an object, structure, or system by leveraging measurements derived from inertial sensors or the underlying inertial properties. These mechanisms can pertain to nuclear collective motion (microscopic many-body physics), rigid body dynamics (robotics, satellite platforms), rolling shutter camera motion (computer vision), wearable sensor-based human motion tracking, tensegrity robotics, and collaborative robot manipulation. The following sections present a comprehensive overview of key methodologies, mathematical formalisms, computational strategies, and experimental validations associated with inertial shape estimation schemes, with emphasis on schemes employing constrained dynamics, variational principles, geometric observers, energy minimization, probabilistic sensor fusion, and data-driven shape-aware algorithms.

1. Microscopic Inertial Shape Estimation in Nuclear Collective Motion

Microscopically, inertial shape estimation arises in the modeling of large-amplitude collective dynamics of atomic nuclei, especially those exhibiting multipolar deformation and shape mixing. The key mathematical formalism is the five-dimensional quadrupole collective Hamiltonian:

$$H_\mathrm{coll} = T_\mathrm{vib} + T_\mathrm{rot} + V(\beta, \gamma)$$

where vibrational kinetic energy and rotational kinetic energy depend on the inertial tensor components $D_{\beta\beta}$, $D_{\beta\gamma}$, $D_{\gamma\gamma}$ (vibrational masses), and $J_k$ (rotational moments of inertia), with $(\beta, \gamma)$ representing the shape degrees of freedom.

The constrained Hartree–Fock–Bogoliubov plus local QRPA (CHFB+LQRPA) scheme enables self-consistent derivation of these inertial functions directly from the underlying microscopic Hamiltonian, as detailed in equations (1)–(5) in (Hinohara et al., 2011). This approach recovers time-odd mean-field dynamical contributions absent from traditional cranking formulas (Inglis-Belyaev), yielding improved fidelity when modeling shape coexistence and mixing, notably in phenomena such as the "island of inversion" in neutron-rich Mg isotopes. The method operationalizes local normal mode analysis on constrained mean-field states, applying minimal metric selection to obtain the most effective vibrational subspace at each point in the deformation plane.

2. Geometric and Variational Principles in Rigid Body and Robotics Estimation

Many inertial shape estimation schemes for rigid bodies, vehicles, and robotic platforms employ geometric observer frameworks and variational mechanics. The Lagrange-d'Alembert principle is a pivotal theoretical foundation: state estimation is cast as minimizing the action functional associated with an artificial Lagrangian on $\mathrm{SE}(3)$ (or its higher-dimensional variants). The Lagrangian comprises a kinetic energy-like term quadratic in velocity estimation error and potential functions penalizing attitude and translation estimation errors (generalized Wahba's cost, quadratic position residuals), plus explicit dissipation terms to guarantee asymptotic stability via Lyapunov constructions (Izadi et al., 2015, Izadi et al., 2015).

Implementation strategies include both continuous-time estimators (energy-based observers driven by sensor residuals and dissipative feedback) and discrete-time Lie group variational integrators that maintain geometric consistency during numerical simulation. Sensor requirements, observability analysis, and robustness to measurement noise and dynamic model uncertainties are analyzed in detail within this class.

A geometric variant introduces an SE$_5$ embedding for INS-type models, enabling observer design with decoupled error dynamics: rotational corrections are computed using innovation terms proportional to cross-product residuals, while translational corrections follow Kalman filter-like Riccati equation driven by uniform observability conditions (Benahmed et al., 2 Apr 2025).

3. Convex Optimization and Energy-Minimization-based Schemes

For non-conventional structures such as tensegrity robots (lacking well-defined joint encoders), inertial shape estimation is formulated as a real-time optimization problem minimizing physical energy subject to inclination data from onboard IMUs. Strut inclination is measured, and missing yaw (rotational) degrees of freedom are estimated via gradient descent minimizing stored elastic energy in the cable network:

$$e = \frac{1}{2} (C_s A^T p + C_s B^T q)^T K (C_s A^T p + C_s B^T q)$$

where $p$ are strut center positions, $q$ orientation vectors computed from inclination and estimated yaw, $K$ cable stiffness, $A$ and $B$ mapping matrices, and $C_s$ the connectivity matrix (Bhat et al., 16 Apr 2025). Fast convergence and mm-scale accuracy are demonstrated in controlled experiments.

Similarly, collaborative robot object identification leverages mass discretization: the object's volume is partitioned into point masses, and convex optimization is solved subject to force-torque and kinematic constraints, with a weighting term adapting between gravity-dominated and full dynamic identification depending on motion "dynamism" (Nadeau et al., 2022).

4. Sensor Fusion, Uncertainty, and Data-driven Shape Estimation

Emerging schemes for inertial shape and motion capture in humans rely on integrating inertial sensor signals (IMUs) with additional distance constraints (e.g., ultra-wideband, UWB) and anthropometric priors, using probabilistic state estimators such as the unscented Kalman filter (UKF) to fuse uncertain measurements and enforce SMPL-model constraints (Liu et al., 14 May 2025). The state vector includes relative positions, velocities, and bias terms, propagated via strapdown models and corrected using fused inter-node distance and pose uncertainties. Pose reconstruction uses LSTM architectures and per-joint uncertainty modeling to constrain the SMPL mesh parameters.

Shape-aware algorithms explicitly decouple pose and shape: multi-layer perceptrons (MLPs) are trained to infer SMPL shape parameters from IMU signals, pose, and subject height, over sliding windows, compensating for the influence of varying limb morphology on accelerometer and velocity data. Retargeting networks map real-body sensor signals to a fixed template, run standard pose estimation, and then map the reconstructed velocities back, followed by shape-aware physical optimization utilizing voxelized mesh mass, inertia, and center-of-mass computations (Yin et al., 20 Oct 2025).

5. Observability, Experimental Validation, and Performance Analysis

Accurate inertial shape estimation depends critically on the observability of the system, i.e., sufficient excitation in input/measurements to infer parameters such as mass, inertia tensor, and dynamic shape. For satellite module manipulation, analysis of sensitivity partials $\partial x/\partial \theta$ and corresponding output derivatives determines which thruster input sequences best stimulate overall observability (Parikh et al., 15 Sep 2024). In practice, phase-shifted sine wave inputs lead to joint excitation of mass and inertia parameters, confirmed by sub-percent RMSE in both simulation and experimental (VICON-measured) data.

Validation across domains typically juxtaposes estimator outputs against high-resolution ground truth, using optical tracking (MoCap) for pose verification, synthetic datasets for simulation benchmarking, and real-world deployments. Key metrics include orientation errors (SIP, global angular error), joint/mesh position errors, energy convergence rates, and robustness under noise, occlusion, or dynamic stress.

6. Domain-Specific Extensions and Future Directions

Inertial shape estimation schemes continue to evolve across application domains:

• Nuclear physics integrates QRPA-based inertial functions into collective Hamiltonians to address quantum fluctuations and shape coexistence in complex isotopes (Hinohara et al., 2011).
• Robotics leverages variational mechanics and geometric observers for pose/velocity estimation under sensor-limited, dynamics-free contexts (Izadi et al., 2015, Benahmed et al., 2 Apr 2025).
• Real-time shape estimation in tensegrity structures eschews unreliable yaw sensors in favor of energy-based inference (Bhat et al., 16 Apr 2025).
• Human motion tracking exploits shape-conditioned IMU-pose correlations, retargeting IMU data using MLP networks to address population-wide body variation (Yin et al., 20 Oct 2025, Liu et al., 14 May 2025).
• Visual-inertial SLAM fuses inertial guidance with image patch errors and probabilistic depth modeling to improve correspondences and shape reconstruction under degraded vision (Yoon et al., 2023).

Prospective directions include improved fusion algorithms for multi-modal sensing, expanded datasets incorporating greater diversity (shape, motion type), physically-constrained global trajectory inference, and batch polynomial optimization for robust windowed attitude estimation (Zhu et al., 2023).

In summary, inertial shape estimation encompasses a spectrum of models and algorithms, unified by their exploitation of inertial information (physical, sensor-derived, or dynamical) to infer configuration, pose, or deformation with high precision under uncertainty and experimental validation, across diverse engineering and physical science domains.