We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope gamma. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < gamma < 0.015 rad. With increasing gamma, stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to gamma 1/3, and that the gravitational power used is proportional to v4 where v is the velocity along the slope.

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The simplest walking model: stability, complexity, and scaling.

Biped robots form a subclass of legged or walking robots. The study of mechanical legged motion has been motivated by its potential use as a means of locomotion in rough terrain, as well as its potential benefits to prothesis development and testing. The paper concentrates on issues related to the automatic control of biped robots. More precisely, its primary goal is to contribute a means to prove asymptotically-stable walking in planar, underactuated biped robot models. Since normal walking can be viewed as a periodic solution of the robot model, the method of Poincare sections is the natural means to study asymptotic stability of a walking cycle. However, due to the complexity of the associated dynamic models, this approach has had limited success. The principal contribution of the present work is to show that the control strategy can be designed in a way that greatly simplifies the application of the method of Poincare to a class of biped models, and, in fact, to reduce the stability assessment problem to the calculation of a continuous map from a subinterval of R to itself. The mapping in question is directly computable from a simulation model. The stability analysis is based on a careful formulation of the robot model as a system with impulse effects and the extension of the method of Poincare sections to this class of models.

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Asymptotically stable walking for biped robots: analysis via systems with impulse effects

Bipedal locomotion is among the most difficult challenges in control engineering. Most books treat the subject from a quasi-static perspective, overlooking the hybrid nature of bipedal mechanics. Feedback Control of Dynamic Bipedal Robot Locomotion is the first book to present a comprehensive and mathematically sound treatment of feedback design for achieving stable, agile, and efficient locomotion in bipedal robots.In this unique and groundbreaking treatise, expert authors lead you systematically through every step of the process, including:Mathematical modeling of walking and running gaits in planar robotsAnalysis of periodic orbits in hybrid systemsDesign and analysis of feedback systems for achieving stable periodic motionsAlgorithms for synthesizing feedback controllersDetailed simulation examplesExperimental implementations on two bipedal test bedsThe elegance of the authors' approach is evident in the marriage of control theory and mechanics, uniting control-based presentation and mathematical custom with a mechanics-based approach to the problem and computational rendering. Concrete examples and numerous illustrations complement and clarify the mathematical discussion. A supporting Web site offers links to videos of several experiments along with MATLAB® code for several of the models. This one-of-a-kind book builds a solid understanding of the theoretical and practical aspects of truly dynamic locomotion in planar bipedal robots.

/pdf/feedback-control-of-dynamic-bipedal-robot-locomotion-2pz7pbjgsi.pdf

Feedback Control of Dynamic Bipedal Robot Locomotion

Symmetry and limb dominance in able-bodied gait: a review.

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set - a two-dimensional zero dynamics submanifold of the full hybrid model $whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

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Hybrid zero dynamics of planar biped walkers

Model this paper, we developed a detailed mathematical model of dual action pneumatic actuators controlled with proportional spool valves. Effects of nonlinear flow through the valve, air compressibility in cylinder chambers, leakage between chambers, end of stroke inactive volume, and time delay and attenuation in the pneumatic lines were carefully considered. We performed system identification, numerical simulation, and model validation experiments for two types of air cylinders and different connecting tubes length. The mathematical model of the present article is used in a sequel article to develop high performance nonlinear pneumatic force controllers.@S0022-0434~00!00503-7# Modern force-reflecting teleoperation, haptic interfaces, and other applications in robotics require high performance force actuators, with high force output per unit weight. It is also important to have linear, fast and accurate response, as well as low friction and mechanical impedance. Traditional geared electrical motors cannot provide these characteristics. Few newly designed motors have special direct-drive actuators, with no intermediate mechanisms. Yet, applications such as teleoperation master arms with gravitational compensation, require long duration, static high force output. In these cases, direct drive electrical actuators necessitate special cooling systems to dissipate the excessive heat. We believe that pneumatic cylinders can offer a better alternative to electrical or hydraulic actuators for certain types of applications. Pneumatic actuators provide the previously enumerated qualities at low cost. They are also suitable for clean environments and safer and easier to work with. However, position and force control of these actuators in applications that require high bandwidth is somehow difficult. This is mainly due to compressibility of air and highly nonlinear flow through pneumatic system components. In addition, design and space considerations in many applications force the command valve to be positioned at relatively large distance from the pneumatic cylinder. Thus, the effects of time delay and attenuation caused by the connecting tubes becomes significant. These difficulties limited the early use of pneumatic actuators to simple applications that required only positioning at the two ends of the stroke. Subsequently, more complete mathematical models for the thermodynamic and flow equations in the charging-discharging processes were developed ~Shearer @1#!. As a result, more complex position controllers, based on the linearization around the mid stroke position were developed ~Burrows @2#, Liu and Bobrow @3#!. These simplified models provided only modest performance improvements. During the last decade, nonlinear control techniques were implemented using digital computers. Bobrow and Jabbari @4# and McDonell and Bobrow @5# used adaptive control for force actuation and trajectory tracking, applied to an air powered robot. Improved results were presented, but they were confined mainly for low frequencies ~approx. 1 Hz!. Sliding mode position controllers were also tested ~Arun et al. @6#, Tang and Walker @7#!, again with improved results at low frequencies. As a general characteristic, the mathematical models used in these controllers assumed no piston seals friction, linear flow through the valve, and neglected the valve dynamics. Ben-Dov and Salcudean @8# developed a forced-controlled pneumatic actuator that provided a force with an amplitude o f2Na t 16 Hz.Their model included the valve dynamics and the nonlinear characteristics of the compressible flow through the valve. A comparison between linear and nonlinear controllers applied to a rotary pneumatic actuator is presented by Richard and Scavarda @9#. The mathematical model accounted for the leakage between actuator’s chambers and the nonlinear variation of the valve effective area with the applied voltage. Their model depends heavily on curve fitting using experimental values, making it difficult to apply to even slightly different systems. The goal of this article is to provide an accurate model of a pneumatic actuator system controlled by a proportional spool valve. This model is targeted to develop force controllers that perform at significantly more demanding operating conditions. For this purpose, the model takes into consideration the friction in the piston seals, the difference in active areas of the piston due to the rod, inactive volume at the ends of the piston stroke, leakage between chambers, valve dynamics and flow nonlinearities through the valve orifice, and time delay and flow amplitude attenuation in valve-cylinder connecting tubes. Since the ultimate purpose of the modeling effort is for control applications, the proposed equations should be suitable for on-line implementation. We designed special experiments in order to identify the unknown characteristics of the pneumatic system, such as: valve discharge coefficient, valve spool viscous friction coefficient, and piston friction forces. The model was finally tested using two experiments that allowed the measurement of the actuator force output and piston displacement. The experimental results were compared with the results obtained by numerical simulation.

A High Performance Pneumatic Force Actuator System: Part I—Nonlinear Mathematical Model

This article deals with the rigid body collisions of planar, kinematic chains with an external surface while in contact with other surfaces. Two solution procedures that cast the impact equations in dierential and algebraic forms are developed to solve the general problem. The dierential formulation can be used to obtain three sets of solutions based on the kinematic, kinetic, and the energetic denitions of the coecient of restitution. Whereas the algebraic formulation can be used to obtain solutions based on the approaches presented in Whittaker (1904) and Brach (1990). A specic example of a planar, three-link chain with two contact points is studied to compare the outcomes predicted by each approach. A particular emphasis is placed on the energy loss that results from the application of each solution scheme. The circumstances where various methods lead to identical or distinct outcomes are investigated. Most importantly, the study elaborates on the rebounds at the non-colliding ends, a phenomenon that is observed only in multi-contact collisions. The interaction of the chain with the contact surfaces at the non-colliding contact points is examined and the dierences in the prediction of rebounds that arise from using various methods are investigated.

Rigid Body Collisions of Planar Kinematic Chains with Multiple Contact Points

https://hal.inria.fr/inria-00072297/document

Modeling, stability and control of biped robots-a general framework

In this article we present two nonlinear force controllers based on the sliding mode control theory. For this purpose we use the detailed mathematical model of the pneumatic system developed in the first part of the paper. The first controller is based on the complete model, and exhibits superior performance both in the numerical simulation and experiments, but requires complex online computations for the control law. The second controller neglects the valve dynamics and the time delay due to connecting tubes. The performance of this controller exhibits slight degradation for configurations with relatively short tubes, and at frequencies up to 20 Hz. At higher frequencies or when long connecting tubes are used, however, the performance exhibits significant degradation compared to the one provided by the full order controller.@S0022-0434~00!00703-6#

A High Performance Pneumatic Force Actuator System: Part II—Nonlinear Controller Design

The main focus of the present investigation is the development of quantitative measures to assess the dynamic stability of human locomotion. The analytical methodology is based on Floquet theory, which was developed to investigate the stability of nonlinear oscillators. Here the basic approach is modified such that it accommodates the study of the complex dynamics of human locomotion and differences among various individuals. A quantitative stability index has been developed to characterize the ability of humans to maintain steady gait patterns. Floquet multipliers of twenty normal subjects were computed from the kinematic data at Poincare sections taken at four instants of the gait cycle, namely heel strike, foot flat, heel off, and toe off. Then, an averaged stability index was computed for each subject. Statistical analysis was performed to demonstrate the utility of the stability indices as quantitative measures of dynamic stability of gait for the subject population tested during the present study.

Yildirim Hurmuzlu

Papers

A High Performance Pneumatic Force Actuator System: Part I—Nonlinear Mathematical Model

Rigid Body Collisions of Planar Kinematic Chains with Multiple Contact Points

Modeling, stability and control of biped robots-a general framework

A High Performance Pneumatic Force Actuator System: Part II—Nonlinear Controller Design

On the Measurement of Dynamic Stability of Human Locomotion