scholarly journals The Murphy number: how pitch moment of inertia dictates quadrupedal walking and running energetics

2021 ◽  
pp. jeb.228296
Author(s):  
Delyle T. Polet
Keyword(s):  
Simple Model ◽  
Trajectory Optimization ◽  
Moment Of Inertia ◽  
Rotational Dynamics ◽  
Recent Analysis ◽  
Quadrupedal Locomotion ◽  
Central Factor ◽  
Small Pitch ◽  
Pitch Moment ◽  
Quadrupedal Walking

Many quadrupedal mammals transition from a four-beat walk to a two-beat run (e.g. trot), but some transition to a four-beat run (e.g. amble). Recent analysis shows that a two-beat run minimizes work only for animals with a small pitch moment of inertia (MOI), though empirical MOI were not reported. It also remains unclear whether MOI affects gait energetics at slow speeds. Here I show that a particular normalization of the pitch moment of inertia (the Murphy number) has opposite effects on walking and running energetics. During walking, simultaneous fore and hindlimb contacts dampen pitching energy, favouring a four-beat gait that can distribute expensive transfer of support. However, the required pitching of a four-beat walk becomes more expensive as Murphy number increases. Using trajectory optimization of a simple model, I show that both the walking and slow running strategies used by dogs, horses, giraffes and elephants can be explained by work optimization under their specific Murphy numbers. Rotational dynamics have been largely ignored in quadrupedal locomotion, but appear to be a central factor in gait selection.

scholarly journals The Murphy number: how pitch moment of inertia dictates quadrupedal walking and running energetics

2020 ◽  
Author(s):  
Delyle T. Polet
Keyword(s):  
Simple Model ◽  
Trajectory Optimization ◽  
Moment Of Inertia ◽  
Rotational Dynamics ◽  
Recent Analysis ◽  
Quadrupedal Locomotion ◽  
Small Pitch ◽  
Pitch Moment ◽  
Determining Factor ◽  
Quadrupedal Walking

AbstractMost quadrupedal mammals transition from a four-beat walk to a two-beat run (e.g. trot), but some transition to a four-beat run (e.g. amble). Recent analysis shows that a two-beat run minimizes work only for animals with a small pitch moment of inertia (MOI), though empirical MOI were not reported. It also remains unclear whether MOI affects gait energetics at slow speeds. Here I show that a particular normalization of the pitch moment of inertia (the Murphy number) has opposite effects on walking and running energetics. During walking, simultaneous fore and hindlimb contacts dampen pitching energy, favouring a four-beat gait that can distribute expensive transfer of support. However, the required pitching of a four-beat walk becomes more expensive as Murphy number increases. Using trajectory optimization of a simple model, I show that both the walking and slow running strategies used by dogs, horses, giraffes and elephants can be explained by work optimization under their specific Murphy numbers. Rotational dynamics have been largely ignored as a determining factor in quadrupedal locomotion, but appear to be a central factor in gait selection.

scholarly journals Energetically optimal running requires torques about the centre of mass

2012 ◽  
Vol 9 (73) ◽  
pp. 2011-2015 ◽  
Author(s):  
James R. Usherwood ◽  
Tatjana Y. Hubel
Keyword(s):  
Large Body ◽  
The Body ◽  
Moment Of Inertia ◽  
Mechanical Work ◽  
Vertical Force ◽  
Centre Of Mass ◽  
Pitch Moment ◽  
Tail Structure ◽  
Reaction Forces ◽  
Long Head

Bipedal animals experience ground reaction forces (GRFs) that pass close to the centre of mass (CoM) throughout stance, first decelerating the body, then re-accelerating it during the second half of stance. This results in fluctuations in kinetic energy, requiring mechanical work from the muscles. However, here we show analytically that, in extreme cases (with a very large body pitch moment of inertia), continuous alignment of the GRF through the CoM requires greater mechanical work than a maintained vertical force; we show numerically that GRFs passing between CoM and vertical throughout stance are energetically favourable under realistic conditions; and demonstrate that the magnitude, if not the precise form, of actual CoM-torque profiles in running is broadly consistent with simple mechanical work minimization for humans with appropriate pitch moment of inertia. While the potential energetic savings of CoM-torque support strategies are small (a few per cent) over the range of human running, their importance increases dramatically at high speeds and stance angles. Fast, compliant runners or hoppers would benefit considerably from GRFs more vertical than the zero-CoM-torque strategy, especially with bodies of high pitch moment of inertia—suggesting a novel advantage to kangaroos of their peculiar long-head/long-tail structure.

scholarly journals Effect of Pitch Moment of Inertia on Added Resistance of Container Ship in Waves

2021 ◽  
Vol 33 (0) ◽  
pp. 81-86
Author(s):  
Hironori Yasukawa ◽  
Kenji Enui
Keyword(s):  
Moment Of Inertia ◽  
Container Ship ◽  
Added Resistance ◽  
Pitch Moment

scholarly journals Moment of inertia: development of rotational dynamics KIT for physics students

2018 ◽  
Vol 434 ◽  
pp. 012014 ◽  
Author(s):  
D Mulhayatiah ◽  
H Y Suhendi ◽  
R Zakwandi ◽  
Y Dirgantara ◽  
M A Ramdani
Keyword(s):  
Moment Of Inertia ◽  
Rotational Dynamics ◽  
Physics Students

The Effect of Pitch Moment of Inertia in Body Axes on the Performance or a Yacht in Waves

1997 ◽  
Author(s):  
C. J. Sutcliffe ◽  
A. Millward
Keyword(s):  
Water Channel ◽  
Moment Of Inertia ◽  
Superior Performance ◽  
Correct Prediction ◽  
Damping Force ◽  
Head Waves ◽  
Pitch Moment ◽  
Experimental Project ◽  
Heave Motion ◽  
Phase Angles

Observation of full size yachts sailing upwind in a seaway has shown that, because of the presence of the sails, the yacht is constrained to move in body axes (parallel to the mast) rather than in earth axes (normal to the water). It is thought that this is due to the effect of the sails in the air and the keel and other appendages in the water providing a large damping force which resists any motion normal to the mast line. An experimental project has been carried out therefore to investigate the effect of this change in motion axes on the forces and motions induced by the seaway. The experiments were carried out on a model of an IACC class hull in regular head waves for a range of wave heights in both earth and body axes using a recirculating water channel. The magnitudes and phase angles of resistance, side force, pitch, heave and heel moment were measured. Comparisons between the results of the present work and previous experimentation, with motion in earth axes, showed similar trends. However the results from the experiments using motion in body axes showed marked changes in the measured motion and resistance characteristics when compared to the earth axes data. It is thought that this difference could well affect the order of merit when comparing the performance of two hulls and it was concluded therefore that the change to measurement in body axes is important for the correct prediction of the performance of a yacht in a seaway. The effect of the pitch moment of inertia on the motion and forces on the model was then studied, first using motion in earth axes and subsequently using motion in body axes. In general it was found that in both earth and body axes there was a strong cross coupling between the pitching motion and the heave motion and that a low gyradius resulted in smaller motions and a reduced added resistance; however the effect of any change in pitch moment of inertia was more significant in body axes. The results showed that a yacht optimised for low pitch moment of inertia would have superior performance in comparison with a yacht which has a high inertia under identical wave conditions.

Launch vehicle trajectory optimization including rotational dynamics

1976 ◽  
Vol 13 (1) ◽  
pp. 59-61 ◽  
Author(s):  
V. Adimurthy
Keyword(s):  
Trajectory Optimization ◽  
Launch Vehicle ◽  
Rotational Dynamics ◽  
Vehicle Trajectory

Quadrupedal locomotion using trajectory optimization and hierarchical whole body control

2017 ◽  
Author(s):  
Christian Gehring ◽  
C. Dario Bellicoso ◽  
Peter Fankhauser ◽  
Stelian Coros ◽  
Marco Hutter
Keyword(s):  
Trajectory Optimization ◽  
Whole Body ◽  
Quadrupedal Locomotion

scholarly journals Effect of passenger uncertainty on the inertial properties of a railway coach

2021 ◽  
Author(s):  
Xue-jun Gao ◽  
Yinghui Li
Keyword(s):  
Vehicle Dynamics ◽  
Transformation Matrix ◽  
Moment Of Inertia ◽  
Railway Vehicle ◽  
Chinese Adults ◽  
Pitch Moment ◽  
Before And After ◽  
Rotation Transformation ◽  
Highly Correlated ◽  
Yaw Moment

Abstract The rotation transformation matrix and translation transformation matrix are derived. They are combined to study the variation of inertial properties of the loaded coach with seating and standing passengers. After that, a CRH2 (China Railway Highspeed) motor coach and Chinese adults in statistical terms are illustrated for precise modelling. It is indicated that CG (Center of Gravity) positions and moments of inertia are all close to linear varying with passenger numbers but at different slopes before and after full-load. It is also found that yaw moment of inertia and pitch moment of inertia are highly correlated. The mass has larger correlation on CG z than CG x and CG y, and larger correlation on roll moment of inertia than yaw and pitch moment of inertia. It may offer some instructions and reference for more realistic simulation of railway vehicle dynamics and measure experiments.

scholarly journals A simple model of a quadruped discovers single-foot walking and trotting as energy optimal strategies

2019 ◽  
Author(s):  
Delyle T. Polet ◽  
John E. A. Bertram
Keyword(s):  
Simple Model ◽  
Trajectory Optimization ◽  
Animal Movement ◽  
Reaction Force ◽  
Energetic Cost ◽  
Global Optimality ◽  
Canis Lupus Familiaris ◽  
Cost Of Transport ◽  
Transition Speed ◽  
Movement Strategies

AbstractIt is widely held that quadrupeds choose steady gaits that minimize their energetic cost of transport, but it is difficult to explore the entire range of possible footfall sequences empirically. We present a simple model of a quadruped that can spontaneously produce any of the 2300+ planar footfall sequences available to quadrupeds. Through trajectory optimization of a work and force-rate cost, and a large sample of random initial guesses, we provide stronger evidence towards the global optimality of symmetrical four-beat walking at low speeds and two beat running (trotting) at intermediate speeds. Using input parameters based on measurements in dogs (Canis lupus familiaris), the model predicts the correct phase offset in walking and a realistic walk-trot transition speed. It also spontaneously reproduces the double-hump ground reaction force profile observed in walking, and the smooth single-hump profile observed in trotting, despite the model’s lack of springs. However, the model incorrectly predicts duty factor at the slowest speeds, and does not choose galloping as globally optimal at high speeds. These limitations might point to missing levels of complexity that could be added to future quadrupedal models, but the present results suggest that massive legs, elastic components, head-neck oscillations and even multi-linked legs are not critical determiners for the optimality of natural quadrupedal walking and trotting.Author summaryWhy do quadrupedal mammals move in consistent ways, when so many options are available? We tackled this problem by determining energetically-optimal gaits using a simple computational model of a four-legged animal. The model can use virtually any pattern of movement (physics-permitting!) but selects natural movement strategies as energetically optimal. The similarities between the computer-based predictions and natural animal movement are striking, and suggest mammals utilize movement strategies that optimize energy use when they move.

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