Quadrupedal Running

Investigating the biomechanics of quadrupedal running provides a framework for understanding aspects from control to energetics of running.

A model of scale effects in mammalian quadrupedal running -- Although the effects of body size on mammalian locomotion are well documented, the underlying mechanisms are not fully understood. The Biomechatronics group has developed a computational model of animal mechanics, control, and energetics that unifies some well-known scale effects in running quadrupeds. The model consists of dynamic, physics-based simulations of six running mammals ranging in size from a chipmunk to a horse (0.115-676 kg). The ‘virtual animals’ are made up of rigid segments (head, trunk, and four legs), linked by joints and similar in morphology to particular species. In the model, each stance limb acts as a spring operating within a narrow range of stiffness, forward motion is powered and controlled by active hip and shoulder torques, and metabolic cost is predicted from the time course of supporting body weight. Model parameters important for stability (joint stiffnesses, limb-retraction times, and target positions and velocities of the limbs) are selected such that (i) running kinematics (aerial height, forward speed, and body pitch) are smooth and periodic; and (ii) overall leg stiffness is in agreement with published data. Both trotting and galloping gaits are modeled, and comparisons across size are made at speeds that are physiologically similar among species. Model predictions are in agreement with data on vertical stiffness, limb angles, metabolic cost of transport, stride frequency, peak force, and duty factor. This work supports the idea that a single, integrative model can predict important features of running across size by employing simple strategies to control overall leg stiffness. More broadly, the model provides a quantitative framework for testing hypotheses that relate limb control, stability, and metabolic cost.

A galloping horse model -- A two-dimensional numerical model of a horse is presented that predicts the locomotory behaviors of galloping horses, including how stride frequency, stride length, and metabolic rate change from a slow canter to a fast gallop. In galloping, each limb strikes the ground sequentially, one after the other, with distinct time lags separating hind and forelimb footfalls. In the model, each stance limb is represented as an ideal linear spring, and both feed-forward and feedback control strategies determine when each limb should strike the ground. In a feed-forward strategy, the first hindlimb and the first forelimb to strike the ground are phase-locked such that the time separating their adjacent footfalls is held constant by the controller. In distinction, in a feedback strategy, the footfalls of the second hindlimb and the second forelimb begin when the first hindlimb and the first forelimb are perpendicular to the model’s trunk, respectively. While any limb is in contact with the ground, the controller also employs a feedback control to move each stance foot at a constant tangential velocity relative to the model’s trunk. With these control schemes, the galloping model remains balanced without sensory knowledge of its postural orientation relative to vertical. This work suggests that a robot will exhibit behavior that is mechanically similar to that of a galloping horse if it employs spring-like limbs and simple feed-forward and feedback control strategies for which postural stabilization is an emergent property of the system.

A trotting horse model -- A new control strategy is used to stabilize numerical simulations of a horse model in the trotting quadrupedal gait. Several well-established experimental findings are predicted by the model, including how stride frequency and stride length change with forward running speed. Mass is distributed throughout the model’s legs, trunk, and head in a realistic manner. Leg and trunk flexion is modeled using four flexible legs, a back joint, and a neck joint. In the control model, pitch stabilization is achieved without directly controlling body pitch, but rather by controlling both the aerial time and the foot speed of each stance leg. The legs behave as ideal springs while in contact with the ground, enabling the model to rebound from the ground with each trotting step. Numerical experiments are conducted to test the model’s capacity to overcome a change in ground impedance. Model stability is maximized and the metabolic cost of trotting is minimized within a narrow range of leg stiffness where trotting horses of similar body size have been observed to operate. This work suggests that a horselike robot will exhibit behavior that is mechanically similar to that of a trotting horse if it operates in a narrow range of leg stiffness and employs simple control strategies where postural stabilization is an emergent property of the system.