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Bibliography & Sources

Reaction Mass Pendulum & centroidal dynamics​

Lee, S.-H. & Goswami, A. (2007). "Reaction Mass Pendulum (RMP): An explicit model for centroidal angular momentum of humanoid robots." IEEE ICRA 2007, pp. 4667–4672. PDF Β· DOI: 10.1109/ROBOT.2007.364198 β€” The foundational RMP paper. The estimation original: how to build the reaction-mass ellipsoid from a body's mass distribution. Closest published thing to what we implement.

Lee, S.-H. & Goswami, A. (2009). "The Reaction Mass Pendulum (RMP) Model for Humanoid Robot Gait and Balance Control." In Humanoid Robots (InTech). PDF β€” The Lie-group (SE(3)) construction of the RMP-of-a-humanoid; composite/locked inertia (Eq. 12), parallel-axis spatial inertia (Eq. 9), and the reaction-mass ellipsoid figures we reproduce in the viewport.

Sanyal, A. K. & Goswami, A. (2013). "Dynamics and Balance Control of the Reaction Mass Pendulum: A Three-Dimensional Multibody Pendulum With Variable Body Inertia." ASME J. of Dynamic Systems, Measurement, and Control, 136(2), 021002. PDF Β· DOI: 10.1115/1.4025607 β€” Full 3D RMP dynamics and a Morse-Lyapunov controller. We use its representation of the variable body inertia; the control half is out of scope (we measure, not actuate).

Orin, D. E., Goswami, A. & Lee, S.-H. (2013). "Centroidal dynamics of a humanoid robot." Autonomous Robots, 35(2–3), 161–176. PDF Β· DOI: 10.1007/s10514-013-9341-4 β€” The computational reference: the Centroidal Composite Rigid Body Inertia (I_G, Eq. 22), the average spatial velocity Ο‰ = I_G⁻¹ H_G (Eq. 24), and an efficient O(N) algorithm. Reports full centroidal dynamics at ~3 kHz on 2013 hardware β†’ our subset is trivially realtime.

Balance metrics & ground-reference points​

Popovic, M. B., Goswami, A. & Herr, H. (2005). "Ground reference points in legged locomotion: Definitions, biomechanics, and applications." Int. J. of Robotics Research, 24(12), 1013–1032. PDF Β· DOI: 10.1177/0278364905058363 β€” Definitions of ZMP, FRI, and the Centroidal Moment Pivot (CMP). Source for the CoP/CMP relationship (CMPβˆ’CoP = angular-momentum-rate readout) and for the kinematics-only ZMP reconstruction (Eq. 5) we note as a future force-plate-free CoP upgrade.

Koolen, T., de Boer, T., Rebula, J., Goswami, A. & Pratt, J. (2012). "Capturability-based analysis and control of legged locomotion, Part 1: Theory and application to three simple gait models." Int. J. of Robotics Research, 31(9), 1094–1113. PDF Β· DOI: 10.1177/0278364912452673 β€” The N-step capturability framework. States explicitly that the instantaneous capture point equals Hof's extrapolated CoM, and shows how angular momentum (flywheel / RMP territory) extends the capture region beyond the single XCoM point.

Pratt, J., Carff, J., Drakunov, S. & Goswami, A. (2006). "Capture point: A step toward humanoid push recovery." IEEE-RAS Humanoids 2006. PDF Β· DOI: 10.1109/ICHR.2006.321385 β€” Introduces the capture point; the robotics-side origin of the quantity Hof named the XCoM.

Hof, A. L. (2008). "The 'extrapolated center of mass' concept suggests a simple control of balance in walking." Human Movement Science, 27(1), 112–125. PDF Β· DOI: 10.1016/j.humov.2007.08.003 β€” The XCoM and the "margin of stability." This is the formula freemocap already implements; the biomechanics-side origin of the capture point.

Anthropometric segment-inertia data (Phase 2)​

de Leva, P. (1996). "Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters." J. of Biomechanics, 29(9), 1223–1230. PDF Β· DOI: 10.1016/0021-9290(95)00178-6 β€” Per-segment mass, CoM location, and radii of gyration (Table 4, female + male). Transcribed into freemocap/core/kinematics/inertial/anthropometry.py β€” both sexes recorded, mean used by default; mass fractions sum to 1.0 (verified by test).

Dumas, R., ChΓ¨ze, L. & Verriest, J.-P. (2007). "Adjustments to McConville et al. and Young et al. body segment inertial parameters." J. of Biomechanics, 40(3), 543–553. PDF Β· DOI: 10.1016/j.jbiomech.2006.02.013 β€” Full 3D segment inertia tensors (including off-diagonal terms). Alternative to de Leva for higher-fidelity ellipsoids.

Winter, D. A. (2009). Biomechanics and Motor Control of Human Movement (4th ed.). Wiley. β€” freemocap's existing source for segment mass fractions and CoM locations; its full tables also include radii of gyration.

Internal references​