Influence of “J”-Curve Spring Stiffness on Running Speeds of Segmented Legs during High-Speed Locomotion

Both the linear leg spring model and the two-segment leg model with constant spring stiffness have been broadly used as template models to investigate bouncing gaits for legged robots with compliant legs. In addition to these two models, the other stiffness leg spring models developed using inspiration from biological characteristic have the potential to improve high-speed running capacity of spring-legged robots. In this paper, we investigate the effects of “J”-curve spring stiffness inspired by biological materials on running speeds of segmented legs during high-speed locomotion. Mathematical formulation of the relationship between the virtual leg force and the virtual leg compression is established. When the SLIP model and the two-segment leg model with constant spring stiffness and with “J”-curve spring stiffness have the same dimensionless reference stiffness, the two-segment leg model with “J”-curve spring stiffness reveals that (1) both the largest tolerated range of running speeds and the tolerated maximum running speed are found and (2) at fast running speed from 25 to 40/92 m s−1 both the tolerated range of landing angle and the stability region are the largest. It is suggested that the two-segment leg model with “J”-curve spring stiffness is more advantageous for high-speed running compared with the SLIP model and with constant spring stiffness

Compliant Leg Architectures and a Linear Control Strategy for the Stable Running of Planar Biped Robots

This paper investigates two fundamental structures for biped robots and a control strategy to achieve stable biped running. The first biped structure contains straight legs with telescopic springs, and the second one contains knees with compliant elements in parallel with the motors. With both configurations we can use a standard linear discrete-time state-feedback control strategy to achieve an active periodic stable biped running gait, using the Poincare map of one complete step to produce the discrete-time model. In this case, the Poincare map describes an open-loop system with an unstable equilibrium, requiring a closed loop control for tabilization. The discretization contains a stance phase, a flight phase and a touch-down. In the first approach, the control signals remain constant during each phase, while in the second approach both phases are discretized into a number of constant-torque intervals, so that its formulation can be applied easily to stabilize any active biped running gait. Simulation results with both the straight-legged and the kneed biped model demonstrate stable gaits on both horizontal and inclined surfaces

One-legged locomotion with a compliant passive joint

There is an increasing attention of exploiting compliant materials for the purpose of legged locomotion, because they provide significant advantages in locomotion performance with respect to energy efficiency and stability. Toward establishing a fundamental basis for this line of research, a minimalistic locomotion model of a single legged system is explored in this paper. By analyzing the dynamic behavior of the system in simulation and a physical robotic platform, it is shown that a stable locomotion process can be achieved without the necessity of sensory feedback. In addition, further analysis characterizes the relation between motor control and the natural body dynamics determined by morphological properties such as body mass and spring constant. © 2006 The authors

One-legged locomotion with a compliant passive joint

Abstract. There is an increasing attention of exploiting compliant materials for the purpose of legged locomotion, because they provide significant advantages in locomotion performance with respect to energy efficiency and stability. Toward establishing a fundamental basis for this line of research, a minimalistic locomotion model of a single legged system is explored in this paper. By analyzing the dynamic behavior of the system in simulation and a physical robotic platform, it is shown that a stable locomotion process can be achieved without the necessity of sensory feedback. In addition, further analysis characterizes the relation between motor control and the natural body dynamics determined by morphological properties such as body mass and spring constant

Tuning and Control of Human Locomotion

Mathematical modeling and analysis have been an integral part of legged locomotion research for many years. While models from the very simple inverted pendulum model of walking and the spring-mass model of running to multi-segmental models with numerous muscles across each joint have been used to explore the process of legged locomotion, they are not always sufficient to explain how human locomotion adapts to a changing environment. Using techniques from dynamical and control systems, I : 1) identify the time scales involved in metabolic minimization in running, 2) explore how stability differs between walking and running, 3) develop an algorithm for optimal control in discrete physical systems, and 4) examine the changes in leg mechanics involved in uphill and downhill running. For the first project, I use ideas from control systems to identify the processes involved in metabolic minimization in running. For the second project, I use ideas of orbital and local stability, measured using Floquet multipliers and finite time Lyapunov exponents, to try to quantify dynamic stability in walking and running at preferred and transition speeds. For the third project, I define a new method for the constrained optimization problem underlying Discrete Mechanics in Optimal Control. For the fourth project, I use a control systems approach to examine the changes in leg dynamics from level to uphill and downhill running. By exploring the adaptations that occur with changing environment, I hope to reveal the mechanisms, and possibly some of the strategies, that lead to stable locomotion

Engineering limit cycle systems:adaptive frequency oscillators and applications to adaptive locomotion control of compliant robots

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot