Passive Limit Cycles Research Articles

Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincaré map

The compass-gait biped robot is a two-DoF legged mechanical system that has been known by its passive dynamic walking. This kind of passive biped robot is modeled by an impulsive hybrid nonlinear system that exhibits complex behaviors. This paper is concerned with the stabilization of the passive dynamic walking of the compass-gait biped robot by designing an analytical expression of the controlled Poincare map. The suggested analytical method starts with the time-piecewise linearization of the impulsive system augmented with the controller, around a desired one-periodic passive hybrid limit cycle. By virtue of the first-order Taylor series, we design an explicit expression of the controlled Poincare map. We present also a simplified expression of the controlled Poincare map having a reduced dimension. In order to accomplish our goal concerned with the stabilization of the passive walking dynamics of the compass-gait biped robot, we develop first the linearized Poincare map around the period-1 fixed point of the Poincare map and we adopt a state-feedback control law to stabilize it. Furthermore, in order to enhance the effectiveness and robustness of the stabilization process, we adopt an LMI-based optimization approach by considering the controlled Poincare map to design the feedback gain. In the end of this work, we provide some numerical and graphical simulation results, to show the validity of the designed control law by means of the controlled Poincare map in the stabilization of the passive dynamic walking of the compass-gait biped robot.

Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

This paper aims at providing a numerical calculation method of the spectrum of Lyapunov exponents in a four-dimensional impulsive hybrid nonlinear dynamics of a passive compass-gait model under the OGY control approach by means of a controlled hybrid Poincaré map. We present a four-dimensional simplified analytical expression of such hybrid map obtained by linearizing the uncontrolled impulsive hybrid nonlinear dynamics around a desired one-periodic passive hybrid limit cycle. In order to compute the spectrum of Lyapunov exponents, a dimension reduction of the controlled hybrid Poincaré map is realized. The numerical calculation of the spectrum of Lyapunov exponents using the reduced-dimension controlled hybrid Poincaré map is given in detail. In order to show the effectiveness of the developed method, the spectrum of Lyapunov exponents is calculated as the slope (bifurcation) parameter varies and hence used to predict the walking dynamics behavior of the compass-gait model under the OGY control.

Gait Generation and Stabilization for Nearly Passive Dynamic Walking Using Auto‐distributed Impulses

AbstractWe propose a state feedback control design via linearization for flexible walking on flat ground. First, we generate nearly passive limit cycles, being stable or not, using impulsive toe‐off actuations. The term ‘nearly passive’ means that the dynamics is completely passive almost everywhere except at the toe‐off moment. A feature of our gait generation method is that walking gaits are characterized only by amounts of supplied energy, and we observe that other variables, including input torques, are auto‐balanced via our method. After gait generation, we design a feedback controller considering robustness and input saturation. As a result, each limit cycle can be matched with its respective controller classified only by energy levels. We have verified that walking speeds monotonically increase by adding more energy, and the ankle joint plays a significant role in compass‐gait walking. Finally, instead of applying impulsive torques, we discuss a practical issue regarding realistic control inputs that ensure stable gait transitions as energy levels are elevated.

Determination of the initial conditions by solving boundary value problem method for period-one walking of a passive biped walking robots

SUMMARYWith regard to the small basin of attraction of the passive limit cycles, it is important to start from a proper initial condition for stable walking. The present study investigates the passive dynamic behaviors of two-dimensional bipedal walkers of a compass gait model with different foot shapes. In order to find proper initial conditions for stable and unstable period-one gait limit cycles, a method based on solving the nonlinear equations of motion is presented as a boundary value problem (BVP). An initial guess is required to solve the related BVP that is obtained by solving an initial value problem (IVP). For parametric analysis purposes, a continuation method is applied. Simulation results reveal two, period-one gait cycles and the effects of parameters variation for all models.

Analysis of period-1 passive limit cycles for flexible walking of a biped with knees and point feet

SUMMARYIn this paper, we investigate dynamic walking as a convergence to the system's own limit cycles, not to artificially generated trajectories, which is one of the lessons in the concept of passive dynamic walking. For flexible walking, gait transitions can be performed by moving from one limit cycle to another one, and thus, the flexibility depends on the range in which limit cycles exist. To design a bipedal walker based on this approach, we explore period-1 passive limit cycles formed by natural dynamics and analyze them. We use a biped model with knees and point feet to perform numerical simulations by changing the center of mass locations of the legs. As a result, we obtain mass distributions for the maximum flexibility, which can be attained from very limited location sets. We discuss the effect of parameter variations on passive dynamic walking and how to improve robot design by analyzing walking performance. Finally, we present a practical application to a real bipedal walker, designed to exhibit more flexible walking based on this study.

Finite difference method to find period-one gait cycles of simple passive walkers

Passive dynamic walking refers to a class of bipedal robots that can walk down an incline with no actuation or control input. These bipeds are sensitive to initial conditions due to their style of walking. According to small basin of attraction of passive limit cycles, it is important to start with an initial condition in the basin of attraction of stable walking (limit cycle). This paper presents a study of the simplest passive walker with point and curved feet. A new approach is proposed to find proper initial conditions for a pair of stable and unstable period-one gait limit cycles. This methodology is based on finite difference method which can solve the nonlinear differential equations of motion on a discrete time. Also, to investigate the physical configurations of the walkers and the environmental influence such as the slope angle, the parameter analysis is applied. Numerical simulations reveal the performance of the presented method in finding two stable and unstable gait patterns.

Chaos control in passive walking dynamics of a compass-gait model

The compass-gait walker is a two-degree-of-freedom biped that can walk passively and steadily down an incline without any actuation. The mathematical model of the walking dynamics is represented by an impulsive hybrid nonlinear model. It is capable of displaying cyclic motions and chaos. In this paper, we propose a new approach to controlling chaos cropped up from the passive dynamic walking of the compass-gait model. The proposed technique is to linearize the nonlinear model around a desired passive hybrid limit cycle. Then, we show that the nonlinear model is transformed to an impulsive hybrid linear model with a controlled jump. Basing on the linearized model, we derive an analytical expression of a constrained controlled Poincaré map. We present a method for the numerical simulation of this constrained map where bifurcation diagrams are plotted. Relying on these diagrams, we show that the linear model is fairly close to the nonlinear one. Using the linearized controlled Poincaré map, we design a state feedback controller in order to stabilize the fixed point of the Poincaré map. We show that this controller is very efficient for the control of chaos for the original nonlinear model.

Parameter Optimization of Passive Dynamic Walking Biped with Knees

A method to find the optimized parameter values of passive dynamic walking biped is presented. The effects of biped physical parameters on the stability property of passive gaits are studied by simulation experiments. The chosen parameters include the mass distribution, length of leg and slope angle. The stability property of passive walking limit cycles is used as criterion of optimization calculation, including the orbital stability described by eigen-values of linearized Poincaré map and the global property described by size of attraction region. The simulation results show how the stability of limit cycle varies when physical parameters of the passive biped change. The work is useful to explore the inherent property of passive dynamic walking and can be used as an important instruction in the mechanical design of biped robots based on principle of passive dynamic walking.

Gait generation and control for biped robots with underactuation degree one

The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait.

Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.

Stable running of a planar underactuated biped robot

SUMMARYThis paper investigates the stable-running problem of a planar underactuated biped robot, which has two springy telescopic legs and one actuated joint in the hip. After modeling the robot as a hybrid system with multiple continuous state spaces, a natural passive limit cycle, which preserves the system energy at touchdown, is found using the method of Poincaré shooting. It is then checked that the passive limit cycle is not stable. To stabilize the passive limit cycle, an event-based feedback control law is proposed, and also to enlarge the basin of attraction, an additive passivity-based control term is introduced only in the stance phase. The validity of our control strategies is illustrated by a series of numerical simulations.

Passivity-Based Control of Bipedal Locomotion

In this article, we have shown how to design energy-based and passivity-based control laws that exploit the existence of passive walking gaits to achieve walking on different ground slopes, to increase the size of the basin of attraction and robustness properties of stable limit cycles, and to regulate walking speed. Many of the results presented in this are the compass gait are equally applicable to bipeds with knees and a torso. Practical considerations such as actuator saturation, ground reaction forces, and ground friction need to be addressed. The problem of foot rotation introduces an underactuated phase into the walking gait, which greatly challenges the application of energy shaping ideas. For walking in 3D, finding purely passive limit cycles, which is the first step in applying our energy control results, may be difficult. It was shown how ideas of geometric reduction can be used to generate 3D stable gaits given only 2D passive limit cycles.

STABLE PERIODIC GAITS OF N-LINK BIPED ROBOT IN THREE DIMENSIONAL SPACE

In passive walking, dissipation due to impacts or damping is offset by the use of potential energy supplied by walking down a slope. In this paper, we develop an analytical procedure to prove the existence and find active limit cycles of a rigid biped in 3-dimensional space. From an existing passive limit cycle, we use the theoretical framework of dynamic geometry and energy shaping, to develop a nonlinear feedback control law wich allows the robot to reach stable gaits corresponding to various velocities. Finally, an example that treat a biped robot with knees is presented to illustrate the theoretical results.

CONTROLLED SYMMETRIES AND PASSIVE WALKING

It was shown in Spong [1999b] that the passive gaits for a planar 2-DOF biped walking on a shallow slope can be made slope invariant by a passivity based control that compensates only the gravitational torques acting on the biped. In this paper we extend these results to the general case of a 3-D, n-DOF robot. We show that if there exists a passive walking gait, i.e. a stable limit cycle, then there exists a passivity-based nonlinear control law that renders the limit cycle slope invariant. The result is constructive in the sense that we generate the resulting control law and initial conditions from the initial conditions of the passive biped and the ground slope. This intuitively simple result relies on some well-known symmetries in the dynamics of mechanical systems with respect to the group action of SO(3) on solution trajectories of the system. We also discuss the design of an additional passivity based control term designed to increase the basin of attraction of the passive limit cycle.

Passivity based control of the compass gait biped

In this paper we discuss the passivity based control of the two-link robot known as the Compass Gait Biped. Starting from a narrow region of initial conditions, the compass gait biped is capable of locomotion down shallow inclines without actuation or feedback control of any kind. We will discuss some feedback control strategies that can exploit these passive dynamics by shaping the energy of the system. Using potential energy shaping we can make the passive limit cycle slope invariant. We also discuss the use of switching control to address the sensitivity to initial conditions and robustness to disturbances.

Limit Cycles in a Passive Compass GaitBiped and Passivity-Mimicking Control Laws

It is well-known that a suitably designed unpowered mechanical biped robot can “walk” down an inclined plane with a steady periodic gait. The energy required to maintain the motion comes from the conversion of the biped‘s gravitational potential energy as it descends. Investigation of such passive natural motions may potentially lead us to strategies useful for controlling active walking machines as well as to understand human locomotion. In this paper we demonstrate the existence and the stability of symmetric and asymmetric passive gaits using a simple nonlinear biped model. Kinematically the robot is identical to a double pendulum (similar to the Acrobot and the Pendubot) and is able to walk with the so-called compass gait. Using the passive behavior as a reference we also investigate the performance of several active control schemes. Active control can enlarge the basin of attraction of passive limit cycles and can create new gaits.