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

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.

The functions of leg muscles, structures and mechanisms in running

The actions of the major human leg muscles are well established; however, the functions of these muscle actions during steady running remain unclear. Here, leg structures and mechanisms are considered in terms of their functions in meeting the task of a vehicle acting as an effective machine, supporting body weight during translation with low mechanical work demand and in supplying mechanical work economically. Legs are modelled as a sequence of linkages that predict muscle actions and reveal the varying muscle functions within the integrated leg. Work avoidance is achieved with isometric muscles and linkages that promote a sliding of the hip over the ground contact, resulting in an approximately horizontal path of the centre of mass. Economical work supply requires, for muscle with constrained power, shortening over the entire stance duration; this function is achieved by the hamstrings without disrupting the linkages resulting in work avoidance. In late stance, the two functions occur through coactivation of antagonistic muscles, providing one answer to Lombard's paradox. Quadriceps and hamstring tensions result in opposing moments about both hip and knee joints, but by doing so perform the independent yet complementary roles of work avoidance during translating weight support and economical work supply.

Conserved mammalian muscle mechanics during eccentric contractions

Skeletal muscle has a broad range of biomechanical functions, including power generation and energy absorption. These roles are underpinned by the force-velocity relationship, which comprises two distinct components: a concentric and an eccentric force-velocity relationship. The concentric component has been extensively studied across a wide range of muscles with different muscle properties. However, to date, little progress has been made in accurately characterising the eccentric force-velocity relationship in mammalian muscle with varying muscle properties. Consequently, mathematical models of this muscle behaviour are based on a poorly understood phenomenon. Here, we present a comprehensive assessment of the concentric force-velocity and eccentric force-velocity relationships of four mammalian muscles (soleus, extensor digitorum longus, diaphragm and digastric) with varying biomechanical functions, spanning three orders of magnitude in body mass (mouse, rat and rabbits). The force-velocity relationship was characterised using a hyperbolic-linear equation for the concentric component a hyperbolic equation for the eccentric component, at the same time as measuring the rate of force development in the two phases of force development in relation to eccentric lengthening velocity. We demonstrate that, despite differences in the curvature and plateau height of the eccentric force-velocity relationship, the rates of relative force development were consistent for the two phases of the force-time response during isovelocity lengthening ramps, in relation to lengthening velocity, in the four muscles studied. Our data support the hypothesis that this relationship depends on cross-bridge and titin activation. Hill-type musculoskeletal models of the eccentric force-velocity relationship for mammalian muscles should incorporate this biphasic force response. KEY POINTS: The capacity of skeletal muscle to generate mechanical work and absorb energy is underpinned by the force-velocity relationship. Despite identification of the lengthening (eccentric) force-velocity relationship over 80 years ago, no comprehensive study has been undertaken to characterise this relationship in skeletal muscle. We show that the biphasic force response seen during active muscle lengthening is conserved over three orders of magnitude of mammalian skeletal muscle mass. Using mice with a small deletion in titin, we show that part of this biphasic force profile in response to muscle lengthening is reliant on normal titin activation. The rate of force development during muscle stretch may be a more reliable way to describe the forces experienced during eccentric muscle contractions compared to the traditional hyperbolic curve fitting, and functions as a novel predictor of force-velocity characteristics that may be used to better inform hill-type musculoskeletal models and assess pathophysiological remodelling.

The collisional geometry of economical walking predicts human leg and foot segment proportions

Human walking appears complicated, with many muscles and joints performing rapidly varying roles over the stride. However, the function of walking is simple: to support body weight as it translates economically. Here, a scenario is proposed for the sequence of joint and muscle actions that achieves this function, with the timing of muscle loading and unloading driven by simple changes in geometry over stance. In the scenario, joints of the legs and feet are sequentially locked, resulting in a vaulting stance phase and three or five rapid 'mini-vaults' over a series of 'virtual legs' during the step-to-step transition. Collision mechanics indicate that the mechanical work demand is minimized if the changes in the centre-of-mass trajectory over the step-to-step transition are evenly spaced, predicting an even spacing of the virtual legs. The scenario provides a simple account for the work-minimizing mechanisms of joints and muscles in walking, and collision geometry allows leg and foot proportions to be predicted, accounting for the location of the knee halfway down the leg, and the relatively stiff, plantigrade, asymmetric, short-toed human foot.

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.