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1. Relative hindlimb length and hindlimb segmental proportions as indicators of locomotor category in primates, rodents, and tree shrews

   Yapuncich GS, Hegarty E (March 2022, American journal of biological anthropology)

   Leaping is an important locomotor behavior for arboreal taxa such as primates, providing means to cross discontinuous substrates, escape predation, and/or capture prey. Primates that leap frequently have relatively longer hindlimbs than those taxa that leap less often. However, it is unknown if this pattern holds across a broader phylogenetic sample that includes non-primate arboreal taxa and non-primate specialized leapers. Here, we examine if relative hindlimb length and segmental proportions correlate with locomotor category across a sample of small-bodied (800g) mammals. Lengths of six hindlimb elements (summing to total hindlimb length) were measured on micro-computed tomography scans. Total hindlimb length was regressed against body mass to calculate relative hindlimb length. Segmental proportions were calculated as the ratio of femoral, tibial, and pedal (the sum of calcaneal, cuboidal, metatarsal, and phalangeal lengths) lengths to total hindlimb length. We found that while three arboreal/scansorial taxa (common marmosets, greater dwarf lemurs, and palm squirrels) exhibit short hindlimbs relative to their body mass, all other arboreal and scansorial taxa have relatively long hindlimbs. Most arboreal, scansorial, terrestrial, and fossorial taxa distribute length evenly across segments (femur, tibia, and pes each comprise 33% of total hindlimb length). Saltatorialists (e.g., jerboas and kangaroo rats) were the only locomotor group with exceptional proportions, with pedal segments contributing 38% of total hindlimb length. These results suggest to us that segmental proportions may distinguish specialized ricochetal hoppers from taxa that leap sporadically, while relative hindlimb length may predict general leaping ability across mammals. 

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2. A Two-DOF Bipedal Robot Utilizing the Reuleaux Triangle Drive Mechanism

   https://doi.org/10.1109/IROS40897.2019.8967952  

   Yang, Jiteng; Saab, Wael; Ben-Tzvi, Pinhas (November 2019, 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS))

   This paper presents the design, modeling, analysis, and experimental results of a bipedal robotic system that utilizes two interconnected single degree-of-freedom leg mechanisms to produce stable forward locomotion and steering. The legs are composed of double four-bar mechanism connected in series that maintain a parallel orientation of a flat foot, relative to the biped body, that is actuated via a Reuleaux triangle cam-follower system to produce a desirable foot trajectory. The mechanical design of the leg mechanism is presented followed by kinematic analysis of the cam-follower system to select the optimal foot trajectory and synthesize the mechanism dimensions and produce a desired step height and step length. The concept of leg sequencing is then presented to maintain a constant body height above the ground and a constant forward walking velocity. Experimental results using an integrated prototype indicate that the proposed biped robot is capable of maintaining quasi-static stability during locomotion, maintaining a constant robot body height, maintaining a constant body orientation, move forward with a constant maximum velocity of 27.4 cm/s, and steer. 

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3. On the generalized Fitting height and insoluble length of finite groups

   https://doi.org/doi.org/10.1112/blms.12372  

   Guralnick, Robert; Tracey, Gareth (January 2020, The bulletin of the London Mathematical Society)

   null (Ed.)

   We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by‐product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism. 

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4. Stair Climbing Using the Angular Momentum Linear Inverted Pendulum Model and Model Predictive Control

   https://doi.org/10.1109/IROS55552.2023.10342369  

   Dosunmu-Ogunbi, Oluwami; Shrivastava, Aayushi; Gibson, Grant; Grizzle, Jessy W (October 2023, IEEE)

   A new control paradigm using angular momentum and foot placement as state variables in the linear inverted pendulum model has expanded the realm of possibilities for the control of bipedal robots. This new paradigm, known as the ALIP model, has shown effectiveness in cases where a robot's center of mass height can be assumed to be constant or near constant as well as in cases where there are no non-kinematic restrictions on foot placement. Walking up and down stairs violates both of these assumptions, where center of mass height varies significantly within a step and the geometry of the stairs restrict the effectiveness of foot placement. In this paper, we explore a variation of the ALIP model that allows the length of the virtual pendulum formed by the robot's stance foot and center of mass to follow smooth trajectories during a step. We couple this model with a control strategy constructed from a novel combination of virtual constraint-based control and a model predictive control algorithm to stabilize a stair climbing gait that does not soley rely on foot placement. Simulations on a 20-degree of freedom model of the Cassie biped in the SimMechanics simulation environment show that the controller is able to achieve periodic gait. 

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5. On the essential minimum of Faltings' height

   https://doi.org/https://doi.org/10.1090/mcom/3286  

   Burgos Gil, Jose; Menares, Ricardo; Rivera-Letelier, Juan (January 2018, Mathematics of computation)

   We study the essential minimum of the (stable) Faltings' height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files. 

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