Zhuohua Shen

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.

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.

A Nonlinear Leg Damping Model for the Prediction of Running Forces and Stability

Despite the neuromechanical complexity underlying animal locomotion, the steady-state center-of-mass motions and ground reaction forces of animal running can be predicted by simple spring-mass models such as the canonical spring-loaded inverted pendulum (SLIP) model. Such SLIP models have been useful for the fields of biomechanics and robotics in part because ground reaction forces are commonly measured and readily available for comparing with model predictions. To better predict the stability of running, beyond the canonical conservative SLIP model, more recent extensions have been proposed and investigated with hip actuation and linear leg damping (e.g., hip-actuated SLIP). So far, these attempts have gained improved prediction of the stability of locomotion but have led to a loss of the ability to accurately predict ground reaction forces. Unfortunately, the linear damping utilized in current models leads to an unrealistic prediction of damping force and ground reaction force with a large nonzero magnitude at touchdown (TD). Here, we develop a leg damping model that is bilinear in leg length and velocity in order to yield improved damping force and ground reaction force prediction. We compare the running ground reaction forces, small and large perturbation stability, parameter sensitivity, and energetic cost resulting from both the linear and bilinear damping models. We found that bilinear damping helps to produce more realistic, smooth vertical ground reaction forces, thus fixing the current problem with the linear damping model. Despite large changes in the damping force and power loss profile during the stance phase, the overall dynamics and energetics on a stride-to-stride basis of the two models are largely the same, implying that the integrated effect of damping over a stride is what matters most to the stability and energetics of running. Overall, this new model, an actuated SLIP model with bilinear damping, can provide significantly improved prediction of ground reaction forces as well as stability and energetics of locomotion.

A Simple Model for Body Pitching Stabilization

Pitching dynamics are an important component of the dynamics of legged locomotion. We develop an energy-open pitching stabilization model to achieve full asymptotic stability of locomotion. This model is called the pitched-actuated Spring-Loaded Inverted Pendulum (pitched-actuated SLIP) model. It extends the conservative SLIP model to include a trunk as well as net nonzero hip torque and leg damping. The hip torque is governed by a proportional and derivative controller which uses only the angle between body and leg as feedback during stance. During the swing phase of the leg, inertial frame feedback is used to reset the leg to a fixed angle in space. The use of body-frame feedback in stance is thought to be relevant to biology and robotic control, as time delays and uncertainty in inertial frame feedback could be challenging in stance. Further, this method of control during stance could be implemented in a neural feedforward manner using antagonistic pairs of muscle or muscle-like actuators around the hip joint. This model of pitching dynamics exhibits full asymptotic stability over a range of model parameters. Further, derivative control significantly impacts disturbance mitigation. Periodic locomotion solutions of the model with large energy cost tend to be unstable. Whereas, the most energy efficient locomotion solutions found tend to be within the stable region of the parameter space. The correlation between energy efficiency and stability found in this model may have significant implications for locomotion with pitching.Copyright

Towards the Understanding of Hip Torque and Leg Damping Effects on Model Stability

A new class of actuated Spring-Loaded Inverted Pendulum (SLIP) models with hip actuation and leg damping has been developed, and is found to be more stable than the canonical version. However, it is not known how the addition of hip torque and leg damping change locomotion stability. In this paper, we study the effects of leg damping and hip torque on locomotion stability of actuated-SLIP models. All other modeling assumptions of SLIP are conserved. And hip torque is turned off during the flight phase. We show that for a given set of nondimensional SLIP parameters the hip torque and leg damping required for a periodic solution are not independent. Further, we show that adding hip torque and damping changes the dynamics of actuated-SLIP solutions in a non-intuitive way. When a very small amount of torque and damping are added the affect is initially to destabilize SLIP solutions. As torque and damping are added further, it eventually improves the stability properties of solutions.Copyright

Comparing Legged Locomotion With a Sprung-Knee and Telescoping-Spring When Hip Torque is Applied

Legged locomotion has been a subject of study for many years. However, the role of the knee in whole-body dynamics of locomotion is not well understood, especially for non-conservative dynamics. Based upon a hip actuated Spring-Loaded Inverted Pendulum (Hip-actuated SLIP) model, we develop a more human-like, two-segment leg model with a pin-jointed springy knee, to see what effects a knee has in the context of an applied hip torque. Overall, we find that the governing equations for the two-segment (knee) version have a distinct structure when compared to the telescoping version of SLIP. The two-segment model with a knee spring influences forces acting on the mass center in a more complex way than a telescoping spring. While a wide variation of behavior is possible for the two-segment model, here we focus on comparing the dynamics for a special case when the knee spring resting angle is 90°. For this particular choice of resting knee angle we find that the knee version of actuated SLIP can have similar locomotion dynamics to the telescoping version of actuated SLIP. This result provides one explanation for how animals and robots with multi-segmented legs could produce overall center-of-mass dynamics that are similar to models with telescoping legs. Nonetheless, despite overall similarities for this special case, small differences in the stability of locomotion are still observed. In particular, we find that the knee version tends to be slightly more stable than the telescoping SLIP in terms of the allowable size of perturbations, while requiring higher input power.Copyright

Comparison of Hip Torque and Radial Forcing Effects on Locomotion Stability and Energetics

Hip torque and radial forcing along the leg are two common actuation methods for legged robots. However, hip torque and radial forcing have not been compared as potential alternative strategies of actuation. The respective advantages and disadvantages of hip torque and radial forcing are not well known. In this paper, we compare hip torque and radial forcing actuation through the simulation of two models: a Rotary-forced Spring-Loaded Inverted Pendulum and a Radially Forced Spring-Loaded Inverted Pendulum. Both actuation methods can produce fully asymptotically stable locomotion. Interestingly, it is found that they improve locomotion stability in different ways: hip torque first destabilizes locomotion when initially introduced but greatly stabilizes locomotion when it keeps increasing; radial forcing always stabilizes locomotion, but in a moderate way.© 2013 ASME

Design of Elastic Element and Controller Algorithm

Recent development of series elastic actuators have revealed a capability to mimic muscle-like properties and achieve accurate force control. Series elastic actuators have also been widely used in humanoid and surgical robotic devices. The design of the elastic elements are critical and complex. This tends to increase costs and complexity of designing and controlling series elastic actuators. Here, we present a novel low cost and easy-to-fabricate design for a series elastic element that also has adjustable stiffness. The design consists of simple shaft couplers and spring steel plates. During the test, the stiffness of the designed elastic elements is very close to linear (R2 = 0.999) when the clamped spring-steel strip length is sufficiently long. As the clamped strip length shortens, the resulting torque deflection curve becomes slightly quadratic but remains largely linear. Also, the designed elastic element exhibits little hysteresis during loading and unloading. The stiffness of the designed elastic element can be tuned to achieve a range of stiffness values, thus it is suitable for different applications with different stiffness requirements. We also design a simple control algorithm and develop a simulation based on the dynamic properties of the designed elastic element. In simulation, the controller is able to accurately track the commanded torque values. Overall, this design could help reduce the cost and development time required for series elastic actuators.Copyright

A Simple Analytical Tool for Legged Robot Design

Although legged locomotion is better at tackling complicated terrains compared with wheeled locomotion, legged robots are rare, in part, because of the lack of simple design tools. The dynamics governing legged locomotion are generally nonlinear and hybrid (piecewise-continuous) and so require numerical simulation for analysis and are not easily applied to robot designs. During the past decade, a few approximated analytical solutions of Spring-Loaded Inverted Pendulum (SLIP), a canonical model in legged locomotion, have been developed. However, SLIP is energy conserving and cannot predict the dynamical stability of real-world legged locomotion. To develop new analytical tools for legged robot designs, we first analytically solved SLIP in a new way. Then based on SLIP solution, we developed an analytical solution of a hip-actuated Spring-Loaded Inverted Pendulum (hip-actuated-SLIP) model, which is more biologically relevant and stable than the canonical energy conserving SLIP model. The analytical approximations offered here for SLIP and the hip actuated-SLIP solutions compare well with the numerical simulations of each. The analytical solutions presented here are simpler in form than those resulting from existing analytical approximations. The analytical solutions of SLIP and the hip actuated-SLIP can be used as tools for robot design or for generating biological hypotheses.Copyright

A Spring-Mass Model of Locomotion With Full Asymptotic Stability

The concept of passive dynamic walking and running [5] has demonstrated that a simple passive model can represent the dynamics of whole-body human locomotion. Since then, many passive models were developed and studied: [3,1,2,11]. The later developed Spring-Loaded Inverted Pendulum (SLIP) [1, 4, 11, 2] exhibits stable center of mass (CoM) motions just by resetting the landing angle at each touch down. Also, compared to SLIP, a SLIP-like model with simple flight leg control is better at resisting perturbations of the angle of velocity but not the magnitude [11, 2, 7]. Energy conserving models explain much about whole-body locomotion. Recently, there has been investigations of modified spring-mass models capable of greater stability, like that of animals and robots [9, 10, 8, 12]. Inspired by RHex [6], the Clock-Torqued Spring-Loaded Inverted Pendulum (CT-SLIP) model [9] was developed, and has been used to explain the robust stability of animal locomotion [12]. Here we present a model (mechanism) simpler than CT-SLIP called Forced-Damped SLIP (FD-SLIP) that can attain full asymptotically stability of the CoM during locomotion, and is capable of both walking and running motions. The FD-SLIP model, having fewer parameters, is more accessible and easier to analyze for the exploration and discovery of principles of legged locomotion.© 2011 ASME