Justin Seipel

Running in Three Dimensions: Analysis of a Point-mass Sprung-leg Model

We analyze a simple model for running: a three-dimensional spring-loaded inverted pendulum carrying a point mass (3D-SLIP). Our formulation reduces to the sagittal plane SLIP and horizontal plane lateral leg spring (LLS) models in the appropriate limits. Using the intrinsic geometry and symmetries and appealing to the case of stiff springs, in which gravity may be neglected during stance, we derive an explicit approximate mapping describing stride-to-stride behavior. We thereby show that all left-right symmetric periodic gaits are unstable, deriving a particularly simple mapping for sagittal plane dynamics. Continuation to fixed points for the “exact” mapping confirms instability of these gaits, and we describe a simple feedback stabilization scheme for leg placement at touchdown.

Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motions

We develop a simple hexapedal model for the dynamics of insect locomotion in the horizontal plane. Each leg is a linear spring endowed with two inputs, controlling force-free length and “hip” position, in a stereotypical feedforward pattern. These represent, in a simplified manner, the effects of neurally activated muscles in the animal and are determined from measured foot force and kinematic body data for cockroaches. We solve the three-degree-of-freedom Newtonian equations for coupled translation-yawing motions in response to the inputs and determine branches of periodic gaits over the animal’s typical speed range. We demonstrate a close quantitative match to experiments and find both stable and unstable motions, depending upon input protocols.Our hexapedal model highlights the importance of stability in evaluating effective locomotor performance and in particular suggests that sprawled-posture runners with large lateral and opposing leg forces can be stable in the horizontal plane over a range of speeds, with minimalsensory feedback from the environment. Fore–aft force patterns characteristic of upright-posture runners can cause instability in the model. We find that stability can constrain fundamental gait parameters: our model is stable only when stride length and frequency match the patterns measured in the animal. Stability is not compromised by large joint moments during running because ground reaction forces tend to align along the leg and be directed toward the center of mass. Legs radiating in all directions and capable of generating large moments may allow very rapid turning and extraordinary maneuvers. Our results further weaken the hypothesis that polypedal, sprawled-posture locomotion with large lateral and opposing leg forces is less effective than upright posture running with fewer legs.

Insects running on elastic surfaces

SUMMARY In nature, cockroaches run rapidly over complex terrain such as leaf litter. These substrates are rarely rigid, and are frequently very compliant. Whether and how compliant surfaces change the dynamics of rapid insect locomotion has not been investigated to date largely due to experimental limitations. We tested the hypothesis that a running insect can maintain average forward speed over an extremely soft elastic surface (10 N m−1) equal to 2/3 of its virtual leg stiffness (15 N m−1). Cockroaches Blaberus discoidalis were able to maintain forward speed (mean ± s.e.m., 37.2±0.6 cm s−1 rigid surface versus 38.0±0.7 cm s−1 elastic surface; repeated-measures ANOVA, P=0.45). Step frequency was unchanged (24.5±0.6 steps s−1 rigid surface versus 24.7±0.4 steps s−1 elastic surface; P=0.54). To uncover the mechanism, we measured the animals centre of mass (COM) dynamics using a novel accelerometer backpack, attached very near the COM. Vertical acceleration of the COM on the elastic surface had a smaller peak-to-peak amplitude (11.50±0.33 m s−2, rigid versus 7.7±0.14 m s−2, elastic; P=0.04). The observed change in COM acceleration over an elastic surface required no change in effective stiffness when duty factor and ground stiffness were taken into account. Lowering of the COM towards the elastic surface caused the swing legs to land earlier, increasing the period of double support. A feedforward control model was consistent with the experimental results and provided one plausible, simple explanation of the mechanism.

Energy Efficiency of Legged Robot Locomotion With Elastically Suspended Loads

Elasticity is an essential property of legged locomotion. Elastically suspending a load can increase the efficiency of locomotion and load carrying in biological systems and for human applications. Similarly, elastically suspended loads have the potential to increase the energy efficiency of legged robot locomotion. External loads and the inherent mass of a legged robot, such as batteries, electronics, motors, and fuel, can be elastically suspended from the robot with compliant springs, passively reducing the energetic cost of locomotion. An experimental elastic load suspension mechanism was developed and utilized on a hexapod robot to test the energetic cost of legged robot locomotion over a range of suspension stiffness values. While running at the same speed, the robot with an elastically suspended load consumed up to 24% less power than with a rigidly attached load. Thus, elastically suspended loads could increase the operation time, load-carrying capacity, or top speed of legged robots, enhancing their utility in many roles.

Conformable actuation and sensing with robotic fabric

Future generations of wearable robots will include systems constructed from conformable materials that do not constrain the natural motions of the wearer. Fabrics represent a class of highly conformable materials that have the potential for embedded function and are highly integrated into our daily lives. In this work, we present a robotic fabric with embedded actuation and sensing. Attaching the same robotic fabric to a soft body in different ways leads to unique motions and sensor modalities with many different applications for robotics. In one mode, the robotic fabric acts around the circumference of the body, and compression of the body is achieved. Attaching the robotic fabric in another way, along one surface of a body for example, bending is achieved. We use thread-like actuators and sensors to functionalize fabric via a standard textile manufacturing process (sewing). The actuated fabric presented herein yields a contractile force of 9.6N and changes in length by approximately 60% when unconstrained. The integrated strain sensor is evaluated and found to have an RMS error of 14.6%, and qualitatively differentiates between the compressive and bending motions demonstrated.

Animals prefer leg stiffness values that may reduce the energetic cost of locomotion.

Despite the neuromechanical complexity and wide diversity of running animals, most run with a center-of-mass motion that is similar to a simple mass bouncing on a spring. Further, when animals׳ effective leg stiffness is measured and normalized for size and weight, the resulting relative leg stiffness that most animals prefer lies in a narrow range between 7 and 27. Understanding why this nearly universal preference exists could shed light on how whole animal behaviors are organized. Here we show that the biologically preferred values of relative leg stiffness coincide with a theoretical minimal energetic cost of locomotion. This result strongly implies that animals select and regulate leg stiffness in order to reduce the energy required to move, thus providing animals an energetic advantage. This result also helps explain how high level control targets such as energy efficiency might influence overall physiological parameters and the underlying neuromechanics that produce it. Overall, the theory presented here provides an explanation for the existence of a nearly universal preferred leg stiffness. Also, the results of this work are beneficial for understanding the principles underlying human and animal locomotion, as well as for the development of prosthetic, orthotic and robotic devices.

Energetics of bio-inspired legged robot locomotion with elastically-suspended loads

Elasticity is an essential property of locomotion. In biology, elastically-suspended loads increase the efficiency of locomotion and load carrying. Similarly, elastically-suspended loads have the potential to increase the energy efficiency of legged robot locomotion. External loads and the inherent mass of a legged robot, such as batteries, electronics, and fuel, can be elastically-suspended from the robot with compliant springs, passively reducing the energetic cost of locomotion. An experimental prototype hexapod robot with a novel elastic load suspension mechanism based on the Christie suspension system is developed and utilized to test the energy efficiency of legged robot locomotion with elastically-suspended loads versus rigidly-attached loads. Elastically-suspended loads are shown to reduce the energetic cost of locomotion compared to rigidly-attached loads. Thus, the speed, operation time, or load carrying capacity could be increased for robots that utilize elastically-suspended loads.

A model of human walking energetics with an elastically-suspended load

Elastically-suspended loads have been shown to reduce the peak forces acting on the body while walking with a load when the suspension stiffness and damping are minimized. However, it is not well understood how elastically-suspended loads can affect the energetic cost of walking. Prior work shows that elastically suspending a load can yield either an increase or decrease in the energetic cost of human walking, depending primarily on the suspension stiffness, load, and walking speed. It would be useful to have a simple explanation that reconciles apparent differences in existing data. The objective of this paper is to help explain different energetic outcomes found with experimental load suspension backpacks and to systematically investigate the effect of load suspension parameters on the energetic cost of human walking. A simple two-degree-of-freedom model is used to approximate the energetic cost of human walking with a suspended load. The energetic predictions of the model are consistent with existing experimental data and show how the suspension parameters, load mass, and walking speed can affect the energetic cost of walking. In general, the energetic cost of walking with a load is decreased compared to that of a stiffly-attached load when the natural frequency of a load suspension is tuned significantly below the resonant walking frequency. The model also shows that a compliant load suspension is more effective in reducing the energetic cost of walking with low suspension damping, high load mass, and fast walking speed. This simple model could improve our understanding of how elastic load-carrying devices affect the energetic cost of walking with a load.

Analytic-Holistic Two-Segment Model of Quadruped Back-Bending in the Sagittal Plane

Back-bending in the sagittal plane is common in many animals during legged locomotion and could be useful for robots. However, to our knowledge, there exists no analytical mechanistic model of sagittal-plane back bending legged locomotion of quadrupeds. Such a mechanistic model and knowledge derived from it is expected to enable direct analysis and insight into back bending locomotion and can be applied to the study of biomechanics or the design of robots. Here a whole-body mechanistic model is developed and governing equations of motion are derived to provide insight into the mathematical structure of the system dynamics. The model is energy conserving, consisting of massless elastic legs pinned to two body segments. The two body segments are pin-joined together with a rotational spring. The massless legs are returned to a resting angle relative to the body during swing phase. We discover: 1) Whole-body configuration variables simplify the resulting equations of motion. 2) The sagittal-plane back-bending two-segment model of legged locomotion yields stable periodic gaits.Copyright

The leg stiffnesses animals use may improve the stability of locomotion.

Despite a wide diversity of running animals, their leg stiffness normalized by animal size and weight (a relative leg stiffness) resides in a narrow range between 7 and 27. Here we determine if the stability of locomotion could be a driving factor for the tight distribution of animal leg stiffness. We simulated an established physics-based model (the actuated Spring-Loaded Inverted Pendulum model) of animal running and found that, with the same energetic cost, perturbations to locomotion are optimally corrected when relative leg stiffness is within the biologically observed range. Here we show that the stability of locomotion, in combination with energetic cost, could be a significant factor influencing the nearly universally observed animal relative leg stiffness range. The energetic cost of locomotion has been widely acknowledged as influencing the evolution of physiology and locomotion behaviors. Specifically, its potential importance for relative leg stiffness has been demonstrated. Here, we demonstrate that stability of locomotion may also be a significant factor influencing relative leg stiffness.