Limit Cycle Walking Research Articles

Controlling flat-foot limit cycle walkers with compliant joints based on local stability variation.

This study investigates local stability of a four-link limit cycle walking biped with flat feet and compliant ankle joints. Local stability represents the behavior along the solution trajectory between Poincare sections, which can provide detailed information about the evolution of disturbances. The effects of ankle stiffness and foot structure on local stability are studied. In addition, we apply a control strategy based on local stability analysis to the limit cycle walker. Control is applied only in the phases with poor local stability. Simulation results show that the energy consumption is reduced without sacrificing disturbance rejection ability. This study may be helpful in motion control of limit cycle bipedal walking robots with flat feet and ankle stiffness and understanding of human walking principles.

Energy-efficient limit cycle walking in disturbance based on nonlinear model predictive control

AbstractIn this study, we present a novel approach to generate limit cycle walking using nonlinear model predictive control (NMPC). Output-zeroing control is now widely used as a control method to limit cycle walking. This control offers strong feedback to the desired trajectory and the generation of energy-efficient and robust limit cycle walking. However, we observed that this method disables the natural dynamics of the robot, leading to problems regarding energy efficiency during walking. This study demonstrates that the energy consumption of walking using the output-zeroing control increases significantly in a disturbed environment. To overcome this limitation, the proposed approach leverages the robot’s dynamics using NMPC to achieve energy-efficient walking even in a disturbed environment. We demonstrate the practicality of the proposed method using two different simulation environments.

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

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

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.

Development of Vibratory Conveyor System Using Frequency Entrainment for Underactuated Walking of Rimless Wheel

In this paper, we propose a novel vibratory conveyor system based on frequency entrainment of limit cycle walking. The conveyor system transports a passive rimless wheel utilizing a vibratory plate which moves up and down by rolling and impact of an active rimless wheel. First, we introduce the mathematical model of the proposed conveyor system. Next, we investigate the influence for the passive rimless wheel gait by vibration of the conveyance plate via the numerical simulation using the mathematical model, and confirm the entrainment phenomenon between the passive rimless wheel and the vibratory plate. Futermore, we conduct experiments using the prototype vibratory conveyor system, and confirm that experimental results show similar trends to the numerical simulation.

Analysis of Fast Bipedal Walking Using Mechanism of Actively Controlled Wobbling Mass

In this study, a novel approach was developed to achieve fast bipedal walking by using an actively controlled wobbling mass. Bipedal robots capable of achieving energy efficient limit cycle walking have been developed, and researchers have studied methods to increase their walking speed. When humans walk, their arm swinging is coordinated with the walking phases, generating a regular symmetrical motion about the torso. The bipedal robots with a wobbling mass in the torso mimicked the arm swinging by the proposed control method. We demonstrated that the proposed method is capable of increasing the bipedal walking speed.

Input-to-State Stability of Periodic Orbits of Systems With Impulse Effects via Poincaré Analysis

In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincare maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding zero-input fixed point of the associated forced Poincare map. This result extends the classical Poincare analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincare map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the zero-input (unforced) Poincare map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincare analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.

A Multibody Dynamics Approach to Limit Cycle Walking

SummaryThough significant efforts are made to develop mathematical models of the limit cycle walking (LCW), there is still a lack of a general and efficient framework to study the periodic solution and robustness of a complex model like human with knees, ankles and flat feet. In this study, a numerical framework of the LCW based on general multibody system dynamics is proposed, especially the impacts between the feet and the ground are modeled by Hunt–Crossley normal contact force and Coulomb friction force, and the modeling of the knee locking is presented as well. Moreover, event-based operating strategies are presented to deal with controls for the ground clearance and the knee locking. Importantly, a fast and efficient two-step algorithm is proposed to search for stable periodic gaits. Finally, maximum allowable disturbance is adopted as the index for stability analysis. All these features could be readily implemented in the framework. The presented solution is verified on a compass-like passive dynamic walking (PDW) walker with results in the literature. Based on this framework, a fairly complicated level-walking walkers with ankles and knees under control are analyzed and their periodic gaits are obtained, and surprisingly, double stable periodic gaits with, respectively, low speed and high speed are found.

Lyapunov based estimation of the basin of attraction of Poincare maps with applications to limit cycle walking

Limit cycles occur in a wide range of nonlinear and hybrid systems including dynamic walking. While methods for determining local stability of limit cycles are available, they are relevant only for small perturbations. Characterizing the robustness of limit cycles to larger perturbations is an important, yet mostly open, challenge. Robustness can be characterized by the basin of attraction (BA), but standard techniques for computing BAs are computationally intensive. Here we present an algorithm to estimate inner bounds of BAs of fixed points of polynomial discrete maps. The algorithm is based on a convex optimization problem known as Sum of Squares programming, and results in BA estimates which are expressed as sublevel sets of polynomial Lyapunov functions. The method can be applied to estimate BAs of fixed points of discrete Poincaré Maps and thus the BAs of limit cycles on selected Poincaré sections, as demonstrated on a simple biped walking model under different modes of actuation. For each mode of actuation, the Poincaré map was fitted using a small number of simulations and an inner bound for the BA was estimated. Comparing the BAs achieved by the different modes of actuation demonstrates that the BA of a passive biped model can be enlarged using minimalistic event-driven actuation, and that this relationship is also apparent from the estimated inner bounds of the BAs.

Nonlinear analysis of an indirectly controlled limit cycle walker

Towards controlling the frequency of limit cycle locomotion as well as adapting to rough terrain and performing specific tasks, a novel and indirect method has been recently introduced using an active wobbling mass attached to limit cycle walkers. One of the strongest advantages of the method is the easiness of its implementation, prompting its applicability to a wide variety of locomotion systems. To deeply understand the nonlinear dynamics for further enhancement of the methodology, we use a combined rimless wheel with an active wobbling mass as an example to perform nonlinear analysis in this paper. First, we introduce the simplified equation of motion and the gait frequency control method. Second, we obtain Arnold tongue, which represents region of entrained locomotion. In regions where the locomotion is not entrained, we explore chaotic and quasi-periodic gaits. To characterize bistability of two different locomotions that underlie hysteresis phenomena, basins of attraction for the two locomotions were computed. The present nonlinear analysis may help understanding the detailed mechanism of indirectly controlled limit cycle walkers.

Asymmetric Swing-Leg Motions for Speed-Up of Biped Walking

[abstFig src='/00290003/04.jpg' width='120' text='Stick diagram of limit cycle walking with asymmetric swing-leg motion' ] This study presents a novel swing-leg control strategy for speed-up of biped robot walking. The trajectory of tip of the swing-leg is asymmetric at the center line of the torso in the sagittal plane for this process. A methodology is proposed that enables robots to achieve the synchronized asymmetric swing-leg motions with the stance-leg angle to accelerate their walking speed. The effectiveness of the proposed method was simulated using numerical methods.

Generating 1-DOF limit cycle walking at target walking speed by feed-forward and feedback limit cycle control

We propose the feed-forward and feedback (FaF) control systems for generating the limit cycle walking at a target walking speed by the combined rimless wheel (CRW) model. The proposed FaF control systems can calculate a control input constantly based on the mathematical analysis of the current and target walking states. As a result, first, the limit cycle walking at the target walking speed is generated by numerical simulations when the walker is driven by the constant FaF control system. Second, for controlling the convergence speed, we extend the FaF control to the two-period stepwise control systems. The limit cycle walking at the target walking speed is still generated, and the convergence speed is controlled by the settling time parameter. Finally, the real-time walking state updating is considered in the FaF control systems to handle the disturbances. In this case, we find that for generating a target walking speed, a precise mathematical model is not necessary for generating the target state, but can make the control input stable and efficient.

Efficient mechanical design and limit cycle stability for a humanoid robot: An application of genetic algorithms

In this paper the application of Genetic Algorithms is presented to the task of designing a humanoid robot able to exhibit an efficient walking. This task is presented as an optimization problem. The objective function is the so-called Specific Cost of Transportation and the restrictions of the problem are based on the Limit Cycle Walking stability criterion. The mechanical design of the prototype and its walking trajectories are inspired on passive dynamic walkers. A basic genetic algorithm was used to find: the optimized mechanical parameters for design, the walking trajectories and the feedback gains used in the control of the current prototype.

劣駆動リムレスホイールの陥穽踏破能力の解析を目的としたモデリングと歩行制御

This paper investigates modeling and control of an underactuated limit-cycle walker for escaping from the quagmire. First, we introduce the model of a eight-legged underactuated rimless wheel with a torso as a reaction wheel and develop the equations of motion and collision according to the phase sequence. Second, we design a method for generating a few steps walking motion based on collisionless limit cycle walking. The robot is then controlled from the initial static state to the terminal one including the impulsive effects and the influence of the reaction force from the quagmire. The fundamental properties of the generated motion are discussed through numerical simulations.

Reinforcement Learning of Potential Fields to achieve Limit-Cycle Walking

Reinforcement learning is a powerful tool to derive controllers for systems where no models are available. Particularly policy search algorithms are suitable for complex systems, to keep learning time manageable and account for continuous state and action spaces. However, these algorithms demand more insight into the system to choose a suitable controller parameterization. This paper investigates a type of policy parameterization for impedance control that allows energy input to be implicitly bounded: Potential fields. In this work, a methodology for generating a potential field-constrained impedance controller via approximation of example trajectories, and subsequently improving the control policy using Reinforcement Learning, is presented. The potential field-const rained approximation is used as a policy parameterization for policy search reinforcement learning and is compared to its unconstrained counterpart. Simulations on a simple biped walking model show the learned controllers are able to surpass the potential field of gravity by generating a stable limit-cycle gait on flat ground for both parameterizations. The potential field-constrained controller provides safety with a known energy bound while performing equally well as the unconstrained policy.

System overview and walking dynamics of a passive dynamic walking robot with flat feet

The concept of “passive dynamic walking robot” refers to the robot that can walk down a shallow slope stably without any actuation and control which shows a limit cycle during walking. By adding actuation at some joints, the passive dynamic walking robot can walk stably on level ground and exhibit more versatile gaits than fully passive robot, namely, the “limit cycle walker.” In this article, we present the mechanical structures and control system design for a passive dynamic walking robot with series elastic actuators at hip joint and ankle joints. We built a walking model that consisted of an upper body, knee joints, and flat feet and derived its walking dynamics that involve double stance phases in a walking cycle based on virtual power principle. The instant just before impact was chosen as the start of one step to reduce the number of independent state variables. A numerical simulation was implemented by using MATLAB, in which the proposed passive dynamic walking model could walk stably down a shallow slope, which proves that the derived walking dynamics are correct. A physical passive robot prototype was built finally, and the experiment results show that by only simple control scheme the passive dynamic robot could walk stably on level ground.

Analysis of steady and target walking speeds in limit cycle walking

This paper presents the mathematical analysis of steady walking speed and target walking speed generation in 1-DOF limit cycle walking driven by time-settling control inputs. An actuated combined rimless wheel (CRW) model is introduced to analyse the steady walking state when the CRW is walking on level ground. First the initial and terminal boundary conditions driven by discrete stepwise control systems are analysed. The steady step period can be calculated and the target walking period can be generated by the formulas. Second we extend the mathematical analysis to \((n+1)\)-period stepwise control system and derive the general formula of the steady step period. Finally, the continuous piecewise control systems are analysed mathematically by discretizing the control input, and thus the boundary conditions can be analysed by the formula of \((n+1)\)-period stepwise control system. As a result, if the generated walking gait is single-step-cycle, the steady step period can be calculated and target walking steady speeds can be generated by our general formulas in the time-settling control systems.

A novel gait generation method independent of target settling-time adjustment for underactuated limit cycle walking

This paper proposes a novel gait generation method for surely achieving constraint on impact posture in limit cycle walking. First, we introduce an underactuated rimless wheel model without ankle-joint actuation and formulate a state-space realization of the control output using the stance-leg angle as a time parameter through an input–output linearization. Second, we determine a control input that moves the control output to a terminal value at a target stance-leg angle during the single-support phase. Third, we conduct numerical simulations to observe the fundamental gait properties and discuss the relationship between the gait symmetry and mechanical energy restoration. Furthermore, we mathematically prove the asymptotic stability of the generated walking gait by analytically deriving the restored mechanical energy.

A different switching surface stabilizing an existing unstable periodic gait: an analysis based on perturbation theory

Limit cycle walkers are known as a class of walking robots capable of presenting periodic repetitive gaits without having local controllability at all times during motion. A well-known subclass of these robots is McGeer’s passive dynamic walkers solely activated by the gravity field. The mathematics governing this style of walking is hybrid and described by a set of nonlinear differential equations along with impulses. In this paper, by applying perturbation method to a simple model of these machines, we analytically prove that for this type of nonlinear impulsive system, there exist different switching surfaces, leading to the same equilibrium points (periodic solutions) with different stabilities. Furthermore, it has been shown that the number of existing periodic solutions depends on the characteristics of the switching surface.

Coupling mechanical design and control design for energy-efficient and stable walking of a compass-like biped

This paper aims at developing a highly efficient walking scheme for compass-like biped robots with feet. Towards this goal, a latch attached to the hip is introduced to lock the hip joint when the hip joint retracts to the desired value and spring devices are installed in the ankle joints to reduce the inputs of the ankles. Moreover, the stance foot allowed to rotate about the toe is considered. With these setups, a highly efficient limit-cycle walking gait is obtained by using the method of Poincaré shooting with the objective of minimizing the dimensionless specific mechanical cost of transport. However, this limit-cycle walking gait is unstable. To stabilize it, a new control strategy is proposed based on transverse coordinate linearization on moving Poincaré sections. A series of numerical simulations show that such a robot model under the proposed control law can walk stably and efficiently.