Central pivot point control of a body-attached spring-mass runner: periodic gaits, stabilization, template embedding and approximate analytic return maps
Abstract
The spring-loaded inverted pendulum (SLIP) describes the center-of-mass (COM) motion of runners but has no trunk. In trunk models, the hip torque couples posture and COM motion and destroys the central-force structure on which SLIP analysis relies. As a result, trunk models have been studied almost only numerically. We show that this structure can be restored. The central pivot point (CPP), a hip-torque law that directs the ground reaction force through the COM, decouples pitch exactly and reduces stance to a planar central-force problem. At upright posture, time-reversal symmetry produces one-parameter families of periodic gaits. A stance-phase pitch law and a stride-level regulator stabilize all computed gaits, also with bounded hip torque and friction. Under CPP, any radial force law in virtual-leg coordinates can be embedded independently of posture. For a linear radial template, the stance radius and the constant-momentum sweep have exact closed forms. With a new gravity correction, the resulting analytic return map, to our knowledge the first for a trunk model, predicts liftoff velocity and apex position with mean errors of 1.6% and 0.9%. The natural gaits of the embedded runner are symmetric and need no force step at touchdown. A near-reversibility argument explains their unit multiplier, and a scalar equation predicts them within 0.8%. A deadbeat stride controller based on the analytic map tracks changes of speed, height and posture. CPP thus makes trunk models accessible to the analytic tools developed for SLIP.
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BibTeX
@misc{sever2026central,
title = {{Central pivot point control of a body-attached spring-mass runner: periodic gaits, stabilization, template embedding and approximate analytic return maps}},
author = {İzel Sever and O. Kaan Karagöz and M. Mert Ankaralı},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02564},
eprint = {2611.02564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02564},
}
RIS
TY - UNPB
TI - Central pivot point control of a body-attached spring-mass runner: periodic gaits, stabilization, template embedding and approximate analytic return maps
AU - İzel Sever
AU - O. Kaan Karagöz
AU - M. Mert Ankaralı
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02564
UR - https://arxiv.org/abs/2611.02564
AB - The spring-loaded inverted pendulum (SLIP) describes the center-of-mass (COM) motion of runners but has no trunk. In trunk models, the hip torque couples posture and COM motion and destroys the central-force structure on which SLIP analysis relies. As a result, trunk models have been studied almost only numerically. We show that this structure can be restored. The central pivot point (CPP), a hip-torque law that directs the ground reaction force through the COM, decouples pitch exactly and reduces stance to a planar central-force problem. At upright posture, time-reversal symmetry produces one-parameter families of periodic gaits. A stance-phase pitch law and a stride-level regulator stabilize all computed gaits, also with bounded hip torque and friction. Under CPP, any radial force law in virtual-leg coordinates can be embedded independently of posture. For a linear radial template, the stance radius and the constant-momentum sweep have exact closed forms. With a new gravity correction, the resulting analytic return map, to our knowledge the first for a trunk model, predicts liftoff velocity and apex position with mean errors of 1.6% and 0.9%. The natural gaits of the embedded runner are symmetric and need no force step at touchdown. A near-reversibility argument explains their unit multiplier, and a scalar equation predicts them within 0.8%. A deadbeat stride controller based on the analytic map tracks changes of speed, height and posture. CPP thus makes trunk models accessible to the analytic tools developed for SLIP.
ER -