An inelastic quadrupedal model discovers four-beat walking, two-beat running, and pseudo-elastic actuation as energetically optimal

Muscle-Driven Predictive Physics Simulations of Quadrupedal Locomotion in the Horse

Musculoskeletal simulations can provide insights into the underlying mechanisms that govern animal locomotion. In this study, we describe the development of a new musculoskeletal model of the horse, and to our knowledge present the first fully muscle-driven, predictive simulations of equine locomotion. Our goal was to simulate a model that captures only the gross musculoskeletal structure of a horse, without specialized morphological features. We mostly present simulations acquired using feedforward control, without state feedback ("top-down control"). Without using kinematics or motion capture data as an input, we have simulated a variety of gaits that are commonly used by horses (walk, pace, trot, tölt, and collected gallop). We also found a selection of gaits that are not normally seen in horses (half bound, extended gallop, ambling). Due to the clinical relevance of the trot, we performed a tracking simulation that included empirical joint angle deviations in the cost function. To further demonstrate the flexibility of our model, we also present a simulation acquired using spinal feedback control, where muscle control signals are wholly determined by gait kinematics. Despite simplifications to the musculature, simulated footfalls and ground reaction forces followed empirical patterns. In the tracking simulation, kinematics improved with respect to the fully predictive simulations, and muscle activations showed a reasonable correspondence to electromyographic signals, although we did not predict any anticipatory firing of muscles. When sequentially increasing the target speed, our simulations spontaneously predicted walk-to-run transitions at the empirically determined speed. However, predicted stride lengths were too short over nearly the entire speed range unless explicitly prescribed in the controller, and we also did not recover spontaneous transitions to asymmetric gaits such as galloping. Taken together, our model performed adequately when simulating individual gaits, but our simulation workflow was not able to capture all aspects of gait selection. We point out certain aspects of our workflow that may have caused this, including anatomical simplifications and the use of massless Hill-type actuators. Our model is an extensible, generalized horse model, with considerable scope for adding anatomical complexity. This project is intended as a starting point for continual development of the model and code that we make available in extensible open-source formats.

Muscle-controlled physics simulations of bird locomotion resolve the grounded running paradox

Humans and birds use very different running styles. Unlike humans, birds adopt "grounded running" at intermediate speeds-a running gait where at least one foot always maintains ground contact. Avian grounded running is a paradox: Animals usually minimize locomotor energy expenditure, but birds prefer grounded running despite incurring higher energy costs. Using predictive gait simulations of the emu (Dromaius novaehollandiae), we resolve this paradox by demonstrating that grounded running represents an optimal gait for birds, from both energetics and muscle excitations perspectives. Our virtual experiments decoupled effects of posture and tendon elasticity, biomechanically relevant anatomical features that cannot be isolated in real birds. The avian body plan prevents (near) vertical leg postures, making the running style used by humans impossible. Under this anatomical constraint, grounded running is optimal if the muscles produce the highest forces in crouched postures, as is true in most birds. Shared anatomical features suggest that, as a behavior, avian grounded running first evolved within non-avian dinosaurs.

Optimal Gearing of Musculoskeletal Systems

Movement is integral to animal life, and most animal movement is actuated by the same engine: striated muscle. Muscle input is typically mediated by skeletal elements, resulting in musculoskeletal systems that are geared: at any instant, the muscle force and velocity are related to the output force and velocity only via a proportionality constant G, the "mechanical advantage". The functional analysis of such "simple machines" has traditionally centered around this instantaneous interpretation, such that a small vs large G is thought to reflect a fast vs forceful system, respectively. But evidence is mounting that a comprehensive analysis ought to also consider the mechanical energy output of a complete contraction. Here, we approach this task systematically, and deploy the theory of physiological similarity to study how gearing affects the flow of mechanical energy in a minimalist model of a musculoskeletal system. Gearing influences the flow of mechanical energy in two key ways: it can curtail muscle work output, because it determines the ratio between the characteristic muscle kinetic energy and work capacity; and it defines how each unit of muscle work is partitioned into different system energies, that is, into kinetic vs "parasitic" energy such as heat. As a consequence of both effects, delivering maximum work in minimum time and with maximum output speed generally requires a mechanical advantage of intermediate magnitude. This optimality condition can be expressed in terms of two dimensionless numbers that reflect the key geometric, physiological, and physical properties of the interrogated musculoskeletal system, and the environment in which the contraction takes place. Illustrative application to exemplar musculoskeletal systems predicts plausible mechanical advantages in disparate biomechanical scenarios, yields a speculative explanation for why gearing is typically used to attenuate the instantaneous force output ($G_{\text{opt}} \lt 1)$, and predicts how G needs to vary systematically with animal size to optimize the delivery of mechanical energy, in superficial agreement with empirical observations. A many-to-one mapping from musculoskeletal geometry to mechanical performance is identified, such that differences in G alone do not provide a reliable indicator for specialization for force vs speed-neither instantaneously, nor in terms of mechanical energy output. The energy framework presented here can be used to estimate an optimal mechanical advantage across variable muscle physiology, anatomy, mechanical environment, and animal size, and so facilitates investigation of the extent to which selection has made efficient use of gearing as a degree of freedom in musculoskeletal "design."

Viscoelastic materials are most energy efficient when loaded and unloaded at equal rates

Biological springs can be used in nature for energy conservation and ultra-fast motion. The loading and unloading rates of elastic materials can play an important role in determining how the properties of these springs affect movements. We investigate the mechanical energy efficiency of biological springs (American bullfrog plantaris tendons and guinea fowl lateral gastrocnemius tendons) and synthetic elastomers. We measure these materials under symmetric rates (equal loading and unloading durations) and asymmetric rates (unequal loading and unloading durations) using novel dynamic mechanical analysis measurements. We find that mechanical efficiency is highest at symmetric rates and significantly decreases with a larger degree of asymmetry. A generalized one-dimensional Maxwell model with no fitting parameters captures the experimental results based on the independently characterized linear viscoelastic properties of the materials. The model further shows that a broader viscoelastic relaxation spectrum enhances the effect of rate-asymmetry on efficiency. Overall, our study provides valuable insights into the interplay between material properties and unloading dynamics in both biological and synthetic elastic systems.

Optimization Design and Performance Analysis of a Bionic Knee Joint Based on the Geared Five-Bar Mechanism

Animal joint motion is a combination of rotation and translational motion, which brings high stability, high energy utilization, and other advantages. At present, the hinge joint is widely used in the legged robot. The simple motion characteristic of the hinge joint rotating around the fixed axis limits the improvement of the robot's motion performance. In this paper, by imitating the knee joint of a kangaroo, we propose a new bionic geared five-bar knee joint mechanism to improve the energy utilization rate of the legged robot and reduce the required driving power. Firstly, based on image processing technology, the trajectory curve of the instantaneous center of rotation (ICR) of the kangaroo knee joint was quickly obtained. Then, the bionic knee joint was designed by the single-degree-of-freedom geared five-bar mechanism and the parameters for each part of the mechanism were optimized. Finally, based on the inverted pendulum model and the Newton-Euler recursive method, the dynamics model of the single leg of the robot in the landing stage was established, and the influence of the designed bionic knee joint and hinge joint on the robot's motion performance was compared and analyzed. The proposed bionic geared five-bar knee joint mechanism can more closely track the given trajectory of the total center of mass motion, has abundant motion characteristics, and can effectively reduce the power demand and energy consumption of the robot knee actuators under the high-speed running and jumping gait.

Competing Models of Work in Quadrupedal Walking: Center of Mass Work is Insufficient to Explain Stereotypical Gait

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.

Legs as linkages: an alternative paradigm for the role of tendons and isometric muscles in facilitating economical gait

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.

Center of Mass Offset Enhances the Selection of Transverse Gallop in High-Speed Running by Horses: A Modeling Study

Horses use the transverse gallop in high-speed running. However, different animals use different gaits, and the gait preference of horses remains largely unclear. Horses have fore-aft asymmetry in their body structure and their center of mass (CoM) is anteriorly located far from the center of the body. Since such a CoM offset affects the running dynamics, we hypothesize that the CoM offset of horses is important in gait selection. In order to verify our hypothesis and clarify the gait selection mechanisms by horses from a dynamic viewpoint, we developed a simple model with CoM offset and investigated its effects on running. Specifically, we numerically obtained periodic solutions and classified these solutions into six types of gaits, including the transverse gallop, based on the footfall pattern. Our results show that the transverse gallop is optimal when the CoM offset is located at the position estimated in horses. Our findings provide useful insight into the gait selection mechanisms in high-speed running of horses.

Estimating Gaits of an Ancient Crocodile-Line Archosaur Through Trajectory Optimization, With Comparison to Fossil Trackways

Fossil trackways provide a glimpse into the behavior of extinct animals. However, while providing information of the trackmaker size, stride, and even speed, the actual gait of the organism can be ambiguous. This is especially true of quadrupedal animals, where disparate gaits can have similar trackway patterns. Here, predictive simulation using trajectory optimization can help distinguish gaits used by trackmakers. First, we demonstrated that a planar, five-link quadrupedal biomechanical model can generate the qualitative trackway patterns made by domestic dogs, although a systematic error emerges in the track phase (relative distance between ipsilateral pes and manus prints). Next, we used trackway dimensions as inputs to a model of Batrachotomus kupferzellensis, a long-limbed, crocodile-line archosaur (clade Pseudosuchia) from the Middle Triassic of Germany. We found energetically optimal gaits and compared their predicted track phases to those of fossil trackways of Isochirotherium and Brachychirotherium. The optimal results agree with trackways at slow speeds but differ at faster speeds. However, all simulations point to a gait transition around a non-dimensional speed of 0.4 and another at 1.0. The trackways likewise exhibit stark differences in the track phase at these speeds. In all cases, including when simulations are constrained to the fossil track phase, the optimal simulations after the first gait transition do not correspond to a trot, as often used by living crocodiles. Instead, they are a diagonal sequence gait similar to the slow tölt of Icelandic horses. This is the first evidence that extinct pseudosuchians may have exhibited different gaits than their modern relatives and of a gait transition in an extinct pseudosuchian. The results of this analysis highlight areas where the models can be improved to generate more reliable predictions for fossil data while also showcasing how simple models can generate insights about the behavior of extinct animals.

Whole-body kinematic and dynamic modeling for quadruped robot under different gaits and mechanism topologies

Dynamic locomotion plays a crucial role for legged robots to fulfill tasks in unstructured environments. This paper proposes whole-body kinematic and dynamic modeling method s based on screw theory for a quadruped robot using different gaits and mechanism topologies. Unlike simplified models such as centroid or inverse pendulum models, the methods proposed here can handle 10-dimensional mass and inertia for each part. The only simplification is that foot contact models are treated as spherical joints. Models of three different mechanism topologies are formulated: (1) Standing phase: a system consisting of one end-effector, the body, and four limbs, the legs; (2) Walking phase: a system consisting of one or two lifting legs (depending on the chosen gait), two or three supporting legs; (3) Floating phase: a system in which all legs detach from the ground. Control strategies based on our models are also introduced, which includes walk and trot gait plans. In our control system, two additional types of information are provided: (1) contacting forces are given by force sensors installed under feet; (2) body poses are determined by an inertial measurement unit (IMU). Combined with the sensor data and calibrated mass, inertia, and friction, the joint torque can be estimated accurately in simulation and experiment. Our prototype, the "XiLing" robot, is built to verify the methods proposed in this paper, and the results show that the models can be solved quickly and leads to steady locomotions.

Elastic energy savings and active energy cost in a simple model of running

The energetic economy of running benefits from tendon and other tissues that store and return elastic energy, thus saving muscles from costly mechanical work. The classic "Spring-mass" computational model successfully explains the forces, displacements and mechanical power of running, as the outcome of dynamical interactions between the body center of mass and a purely elastic spring for the leg. However, the Spring-mass model does not include active muscles and cannot explain the metabolic energy cost of running, whether on level ground or on a slope. Here we add explicit actuation and dissipation to the Spring-mass model, and show how they explain substantial active (and thus costly) work during human running, and much of the associated energetic cost. Dissipation is modeled as modest energy losses (5% of total mechanical energy for running at 3 m s-1) from hysteresis and foot-ground collisions, that must be restored by active work each step. Even with substantial elastic energy return (59% of positive work, comparable to empirical observations), the active work could account for most of the metabolic cost of human running (about 68%, assuming human-like muscle efficiency). We also introduce a previously unappreciated energetic cost for rapid production of force, that helps explain the relatively smooth ground reaction forces of running, and why muscles might also actively perform negative work. With both work and rapid force costs, the model reproduces the energetics of human running at a range of speeds on level ground and on slopes. Although elastic return is key to energy savings, there are still losses that require restorative muscle work, which can cost substantial energy during running.

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

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 was also 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 forelimb 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.

Limb work and joint work minimization reveal an energetic benefit to the elbows-back, knees-forward limb design in parasagittal quadrupeds

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 Possibility of Zero Limb-Work Gaits in Sprawled and Parasagittal Quadrupeds: Insights from Linkages of the Industrial Revolution

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 translation, and a condition closely approximated by walking tortoises. If this is a potential strategy for zero mechanical work cost among quadrupeds, how might the structure, posture, and diversity of both sprawled and parasagittal legs be interpreted? In order to approach this question, various linkages described during the industrial revolution are considered. Watt's linkage provides an analogue for sprawled vertebrates that uses diagonal limb support and shows how vertical-axis joints could enable approximately straight-line horizontal translation while demanding minimal mechanical power. An additional vertical-axis joint per leg results in the wall-mounted pull-out monitor arm and would enable translation with zero mechanical work due to weight support, without tipping or toppling. This is consistent with force profiles observed in tortoises. The Peaucellier linkage demonstrates that parasagittal limbs with lateral-axis joints could also achieve the zero-work strategy. Suitably tuned four-bar linkages indicate this is feasibly approximated for flexed, biologically realistic limbs. Where "walking" gaits typically show out of phase fluctuation in center of mass kinetic and gravitational potential energy, and running, hopping or trotting gaits are characterized by in-phase energy fluctuations, the zero limb-work strategy approximated by tortoises would show zero fluctuations in kinetic or potential energy. This highlights that some gaits, perhaps particularly those of animals with sprawled or crouched limbs, do not fit current kinetic gait definitions; an additional gait paradigm, the "zero limb-work strategy" is proposed.