Figure 4 - available via license: Creative Commons Attribution 4.0 International
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The boom of a sailing boat demonstrates that vertical forces can be opposed with passive structures while allowing free horizontal rotations about a vertical-axis joint. The boom is free to rotate about the mast (double headed arrow) while loaded in compression; vertical forces are supported with tension (dashed arrows) upwards (sail) and downwards (boom vang or kicking strap).
Source publication
![Figure 1. Watt's linkage (A, [2]) provides an analogue for the sprawled...](publication/341937655/figure/fig1/AS:899048974344192@1591361387540/Watts-linkage-A-2-provides-an-analogue-for-the-sprawled-crocodilian-B-3-with_Q320.jpg)
Animal legs are diverse, complex and perform many roles. One defining requirement of legs is to facilitate terrestrial travel with some degree of economy. This could, theoretically, be achieved without loss of mechanical energy if the body could take a continuous horizontal path supported by vertical forces only – effectively a wheel-like translati...
Context in source publication
Context 1
... is horizontal. The analogy of the boom of a sailing boat demonstrates that this could feasibly be achieved with biological structures without unrealistic mechanisms: the boom, analogous to femur or humerus bones, resists compression and achieves lowresistance rotations about the mast supported by tension along the sail (up) or boom vang (down) (Fig. 4), analogous to passive tensile tissues acting across shoulder or hip. The general principle is simple: horizontal translation need not demand mechanical work if weight can be supported with frictionless vertical-axis joints. The screen support and the tortoise: the potential for true zero limb ...
Citations
... There may also be some ambiguity about the appropriate limb "length" to use, due to the complexities of limb attachment (e.g. as through the muscular sling of the forelimbs). Finally, it also neglects the work of forces not aligned with the leg axis-either using the limb as a lever (Gray, 1944) or taking advantage of hypothesized leg linkages (Usherwood, 2020b(Usherwood, , 2022. However, the axial force assumption remains a useful approximation and simplification (at least for large parasagittal mammals; Fischer and Blickhan, 2006), especially in a modelling context. ...
... The cost of locomotion is not exactly proportional to axial limb work during locomotion. Ground reaction forces are not entirely leg-axial during quadrupedal walking in general Usherwood, 2020b), and various antagonist muscles co-contract (Fischer and Lilhe, 2011). Isometric contraction and muscle (de) activation (Kushmerick and Paul, 1977;van der Zee and Kuo, 2021) are metabolically costly, and passive dissipation contributes to the work of locomotion (Zelik and Kuo, 2010). ...
The walking gaits of cursorial quadrupedal mammals tend to be highly stereotyped as a four-beat pattern with interspersed periods of double and triple stance, often with double-hump ground reaction force profiles. This pattern has long been associated with high energetic economy, due to low apparent work. However, there are differing ways of approximating the work performed during walking and, consequently, different interpretations of the primary mechanism leading to high economy. A focus on Net Center of Mass (COM) Work led to the claim that quadrupedal walking is efficient because it effectively trades potential and kinetic energy of the COM. Individual Limbs COM Work instead focuses on the ability of the limbs to manage the trajectory of the COM to limit energetic losses to the ground (“collisions”). By focusing on the COM, both these metrics effectively dismiss the importance of rotation of the elongate quadrupedal body. Limb Extension Work considers work required to extend and contract each limb like a strut, and accounts for the work of body pitching. We tested the prescriptive ability of these approximations of work by optimizing them within a quadrupedal model with two approximations of the body as a point-mass or a rigid distributed mass. Perfect potential-kinetic energy exchange of the COM was possible when optimizing Net COM Work, resulting in highly compliant gaits with duty factors close to one, far different than observed mammalian gaits. Optimizing Individual Limbs COM Work resulted in alternating periods of single limb stance. Only the distributed mass model, with Limb Extension Work as the cost, resulted in a solution similar to the stereotypical mammalian gait. These results suggest that maintaining a near-constant limb length, with distributed contacts, are more important mechanisms of economy than either transduction of potential-kinetic energy or COM collision mitigation for quadrupedal walking.
... Theoretical force profiles and foot contact timings that result in zero vertical displacements of the centre of mass, zero horizontal forces and zero roll or pitch accelerations have been described [5], and are broadly consistent with observed timings and forces of tortoises [5,6]. While the original intent of such modelling was in exploring strategies that prevent excessive roll and pitch despite strides of relatively very long duration, it also demonstrates a strategy that allows weight support and progression while approaching absolutely zero limb work [6]. ...
... Theoretical force profiles and foot contact timings that result in zero vertical displacements of the centre of mass, zero horizontal forces and zero roll or pitch accelerations have been described [5], and are broadly consistent with observed timings and forces of tortoises [5,6]. While the original intent of such modelling was in exploring strategies that prevent excessive roll and pitch despite strides of relatively very long duration, it also demonstrates a strategy that allows weight support and progression while approaching absolutely zero limb work [6]. The term 'limb work' here is the mechanical work associated with forces and deflections from ground contact (the foot), all the way to the centre of mass, for each leg-it does not include work associated with accelerating the limb mass. ...
... Hips and shoulders could, therefore, translate the centre of mass perfectly horizontally and, if horizontal forces are avoided, no limb power is ever required. Force profiles of walking alligators and tortoises appear highly consistent with this wheel-like or 'sliding' strategy [6] (this is not typical for mammals, particularly the familiar, larger species-see below). Note that zero limb work as defined here, while providing one option for demanding zero work from the muscle, is not the only one. ...
Quadrupedal animal locomotion is energetically costly. We explore two forms of mechanical work that may be relevant in imposing these physiological demands. Limb work, due to the forces and velocities between the stance foot and the centre of mass, could theoretically be zero given vertical limb forces and horizontal centre of mass path. To prevent pitching, skewed vertical force profiles would then be required, with forelimb forces high in late stance and hindlimb forces high in early stance. By contrast, joint work—the positive mechanical work performed by the limb joints—would be reduced with forces directed through the hip or shoulder joints. Measured quadruped kinetics show features consistent with compromised reduction of both forms of work, suggesting some degree of, but not perfect, inter-joint energy transfer. The elbows-back, knees-forward design reduces the joint work demand of a low limb-work, skewed, vertical force profile. This geometry allows periods of high force to be supported when the distal segment is near vertical, imposing low moments about the elbow or knee, while the shoulder or hip avoids high joint power despite high moments because the proximal segment barely rotates—translation over this period is due to rotation of the distal segment.
... The second mechanism, gyroscopic backspin, was first discovered in one plant species with explosive seed dispersal (Ruellia ciliatiflora) (Cooper et al. 2018), which led to the re-evaluation of a previous study on a different plant, the Jabillo or Sandbox tree (Hura crepitans) (Swaine and Beer 1977;Ribera et al. 2020). The third example is linkages, such as four-bar linkages, Watt's linkages, and Peaucellier linkages, which have been found in animal skulls, legs, and wings (Muller 1987;Westneat 1990;Wissa et al. 2015;Hu et al. 2017;Usherwood 2020). Linkages are used in engineering and biology to convert one type of motion into another, and they help reduce mechanical work or inertia and provide leverage to manage the force or speed of a movement. ...
Synopsis
Plants and animals have evolved solutions for a wide range of mechanical problems, such as adhesion and dispersal. Several of these solutions have been sources for bio-inspiration, like the Lotus Effect for self-cleaning surfaces or Velcro for adhesion. This symposium brought together plant and animal biomechanics researchers who tackle similar problems in different systems under the unifying theme of structure–function relations with relevance to bio-inspiration. For both communities it holds true that the structural systems, which have evolved in the respective organisms to address the mechanical challenges mentioned above, are often highly complex. This requires interdisciplinary research involving “classical” experimental biology approaches in combination with advanced imaging methods and computational modeling. The transfer of such systems into biomimetic technical materials and structures comes with even more challenges, like scalability issues and applicability. Having brought all these topics under one umbrella, this symposium presented the forefront of biophysical basic and application-oriented international research with the goal of facilitation knowledge transfer across systems and disciplines.
Considerable attention has been given to the spring-like behaviour of stretching and recoiling tendons, and how this can reduce the work demanded from muscle for a given loss–return cycling of mechanical energy during high-speed locomotion. However, even completely isometric muscle–tendon units have the potential to act as tension struts, forming links in linkages that avoid the demand for mechanical work-cycling in the first place. Here, forelimb and hindlimb structures and geometries of quadrupeds are considered in terms of linkages that avoid mechanical work at the level of both the whole limb and the individual muscles. The scapula, isometric serratus muscles and forelimb can be viewed as a modified Roberts' straight-line mechanism that supports an approximately horizontal path of the body with vertically orientated forces, resulting in low work demand at the level of both limb and muscle. Modelled isometric triceps brachii inserting to the olecranon form part of a series of four-bar linkages (forelimb) and isometric biceps femoris cranial, rectus femoris and tensor fascia latae form part of a series of six-bar linkages (hindlimb), in both cases potentially resulting in straight-line horizontal motion, generating appropriate moments about shoulder and hip to maintain vertical ground reaction forces and again low mechanical work demand from the limb. Analysing part of the complexity of animal limb structure as linkages that avoid work at the level of both the whole limb and the supporting muscles suggests a new paradigm with which to appreciate the role of isometric muscle–tendon units and multiple muscle origins.
