Compass-gait Biped Robot Research Articles

Occurrence of Complex Behaviors in the Uncontrolled Passive Compass Biped Model

It is widely known that an appropriately built unpowered bipedal robot can walk down an inclined surface with a passive steady gait. The features of such gait are determined by the robot's geometry and inertial properties, as well as the slope angle. The energy needed to keep the biped moving steadily comes from the gravitational potential energy as it descends the inclined surface. The study of such passive natural motions could lead to ideas for managing active walking devices and a better understanding of the human locomotion. The major goal of this study is to further investigate order, chaos and bifurcations and then to demonstrate the complexity of the passive bipedal walk of the compass-gait biped robot by examining different bifurcation diagrams and also by studying the variation of the eigenvalues of the Poincaré map's Jacobian matrix and the variation of the Lyapunov exponents. We reveal also the exhibition of some additional results by changing the inertial and geometrical parameters of the bipedal robot model.

Modeling, Analysis and Control of the Compass Gait Biped Robot and Extensions: A Review

Abstract This paper gives a review of recent results in the field of modeling and control of hybrid walking systems with a particular focus on the Compass Gait model. Since dynamic models of biped robots are hybrid in nature, the paper presents an overview of modeling the continuous and discrete dynamics of a hybrid system representing a walking robot. The paper briefly describes the design evolution in underactuated bipedal robots and presents several prominent prototypes, explains a number of related concepts and lists a variety of suitable control techniques.

A new Poincaré map for investigating the complex walking behavior of the compass-gait biped robot

The planar compass-gait biped robot is recognized by its simple passive morphological structure and by a complex dynamic walking system modeled by an impulsive hybrid nonlinear dynamics. Such complexity inhibits investigating the biped locomotive mechanism by means of the Poincaré map. Nevertheless, it is very difficult (and even impossible) to establish the exact closed form of the Poincaré mapping. This paper is concerned with the development of an improved closed-form analytical expression of the impact-to-impact Poincaré map for analyzing the complex walking behavior of the passive-dynamics biped walker and its stability. Our methodology consists in linearizing and then approximating the hybrid dynamics of the compass-gait biped robot around a predefined hybrid limit cycle. Based on the second-order Taylor series expansion, we design first an enhanced closed-form expression of the Poincaré map as well as an analytical expression for the computation of the step period of the bipedal locomotion. Furthermore, a simplification of these developed expressions is presented by decreasing the dimension. We provide also an expression of the Jacobian matrix for investigating the stability of the period-1 fixed point of the designed simplified Poincaré map. At the end of this work, we illustrate some simulation results in order to evaluate the validity of the new developed expression of the impact-to-impact Poincaré map in analyzing the stability and the complex walking behavior of passive-dynamics compass biped robot.

Occasional stabilisation of limit cycle walking and control of chaos in the passive dynamics of the compass-gait biped model

It is known that the compass-gait biped model exhibits cyclic walking patterns and attractive complex phenomena, namely chaos and bifurcations. This work is concerned with the stabilisation of limit cycles and then the control of chaos. A new stabilisation process of the limit cycle walking is developed based on self-detection of the fixed point of the (un)stable limit cycle and on energy-shaping-based trajectory-tracking controller. The control process is applied in the beginning of the swing stage during a desired short time interval making hence the compass-gait biped robot to be completely passive on the remaining swing phase. We demonstrate that such occasional stabilisation of the limit cycles considerably increases the energetic efficiency of the bipedal locomotion. We show also that the proposed control method allows the compass-gait biped to walk efficiently and with a periodic gait down sloped surfaces of different angles.

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.

Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark–Sacker bifurcation

This paper continues our research work on analysis of passive dynamic walking of the two-degree-of-freedom planar compass-gait biped robot under the OGY-based state-feedback control. We showed, in our previous works, that the walking dynamics of the compass-gait model under control exhibits chaos and periodic-doubling and cyclic-fold bifurcations. In a first part, our analysis of the walking behavior was achieved using the impulsive hybrid continuous-time nonlinear dynamics of the compass-gait model under the OGY-based control. In a second part, our study of the controlled gait and the displayed local bifurcations was realized via the controlled hybrid Poincaré map. In the present work, we demonstrate, for the first time, the appearance of the Neimark–Sacker bifurcation in the controlled dynamic walking of the compass-gait model. Our investigation is achieved via the controlled hybrid Poincaré map instead of the impulsive continuous dynamics model of the bipedal walking. For such study, we mainly use bifurcation diagrams and 2D phase portraits of the discrete Poincaré map. We show that such Neimark–Sacker bifurcation is generated from a period-1 gait and is localized in a small range of the bifurcation parameter, the slope angle. We introduce a two-parameter bifurcation diagram to study occurrence of the Neimark–Sacker bifurcation.

Modeling and Control of Torso Compass Gait Biped Robot with AI Controller

Modeling and Control of Torso Compass Gait Biped Robot with AI Controller

Modeling and Control of Torso Compass Gait Biped Robot with AI Controller

This work presents the mathematical model for a torso compass gait biped robot with three degrees of freedom (DOF) which is comprised of two legs and torso. Euler Lagrange method's is used to drive the dynamic equation of robot with computed control is used as a controller. The relative angles are used to simplify the robot equation and get the symmetry of the matrix. Convention controller uses critical sampling to find the value of KP and Kv in computed controller, in this paper the Genetic optimization method is used to find the optimal value of KP and Kv with suitable objective function which employ the error and overshoot to make the biped motion smooth as possible. To investigate the work of robot a Matlab 2013b is used and the result show success of modeling.

Switched Control for the Walking of a Compass Gait Biped Robot

This study presents a new approach for modeling and controlling of a compass gait biped robot based on the use of the switched systems. The linearization of the equations stemming from the formalism of Lagrange allows the construction of a set of local models used to describe the behavior of this non linear system. The selection of each model depends on its activation function depending on the system states. The synthesis of the stability of the walking robot is based on the use of second method of Lyapunov. The synthesis approach leads to a set of bilinear matrices inequalities non resolvable by actual numerical solvers. To come over these difficulties, some relaxations are brought to get useful and exploitable numerical solutions.

On existence of oscillations in hybrid systems

The paper extends the notion of oscillations in the sense of Yakubovich to hybrid dynamics. Several sufficient stability and instability conditions for a forward invariant set are presented. The consideration is illustrated by the analysis of a model of two-link compass-gait biped robot.

Conditions of existence of oscillations for hybrid systems

The paper extends the notion of oscillations in the sense of Yakubovich to hybrid dynamics. Several sufficient stability and instability conditions for a forward invariant set are presented. The consideration is motivated by analysis of a model of two-link compass-gait biped robot.

CYCLIC-FOLD BIFURCATION AND BOUNDARY CRISIS IN DYNAMIC WALKING OF BIPED ROBOTS

The aim of this paper is to show that the onset/destruction of bipedal chaos in the dynamic walking of a passive compass-gait biped robot and a semi-passive torso-driven biped robot walking down a slope can occur via a transition mechanism known as boundary crisis. It is known that such biped robots exhibit a scenario of period-doubling bifurcations route to chaos as one of their geometrical or inertial parameters changes. In this paper, we show that a cyclic-fold bifurcation is the key of the occurrence of a double boundary crisis. We demonstrate through bifurcation diagrams how the same period-three unstable periodic orbit generated from the cyclic-fold bifurcation causes the sudden birth/death of the bipedal chaos in the dynamic walking of the two biped robots. We stress that a double boundary crisis is responsible for the fall of each biped robot while walking down an inclined surface and as some bifurcation parameter varies. Stability of the cyclic-fold bifurcation under small perturbations is also discussed.

Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.

INTERMITTENCY AND INTERIOR CRISIS AS ROUTE TO CHAOS IN DYNAMIC WALKING OF TWO BIPED ROBOTS

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position. In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.