Stability analysis and control for bipedal locomotion using energy methods

In this thesis, we investigate the stability of limit cycles of passive dynamic walking. The formation process of the limit cycles is approached from the view of energy interaction. We introduce for the first time the notion of the energy portrait analysis originated from the phase portrait. The energy plane is spanned by the total energy of the system and its derivative, and different energy trajectories represent the energy portrait in the plane. One of the advantages of this method is that the stability of the limit cycles can be easily shown in a 2D plane regardless of the dimension of the system. The energy portrait of passive dynamic walking reveals that the limit cycles are formed by the interaction between energy loss and energy gain during each cycle, and they are equal at equilibria in the energy plane. In addition, the energy portrait is exploited to examine the existence of semi-passive limit cycles generated using the energy supply only at the take-off phase. It is shown that the energy interaction at the ground contact compensates for the energy supply, which makes the total energy invariant yielding limit cycles. This result means that new limit cycles can be generated according to the energy supply without changing the ground slope, and level ground walking, whose energy gain at the contact phase is always zero, can be achieved without actuation during the swing phase. We design multiple switching controllers by virtue of this property to increase the basin of attraction. Multiple limit cycles are linearized using the Poincare map method, and the feedback gains are computed taking into account the robustness and actuator saturation. Once a trajectory diverges from a basin of attraction, we switch the current controller to one that includes the trajectory in its basin of attraction. Numerical simulations confirm that a set of limit cycles can be used to increase the basin of attraction further by switching the controllers one after another. To enhance our knowledge of the limit cycles, we performed sophisticated simulations and found all stable and unstable limit cycles from the various ground slopes not only for the symmetric legs but also for the unequal legs that cause gait asymmetries. As a result, we present a novel classification of the passive limit cycles showing six distinct groups that are consecutive and cyclical

Impulsive torque control of biped gait with power packets

Many strategies for an actuated biped gait generation have been proposed based on the passive dynamic gait. Among them, this study focuses on an impulsive excitation at the toe-off instance. The strategy offers advantages in its experimental implementation; for example, it is not required to measure and control the trajectory of the legs all the time. However, there has been no study on a realistic design of the impulsive torque itself. In this paper, we propose an impulsive actuation method based on a power packet dispatching system. Power packet is a unit of electric power transfer in a pulse shape with information tags attached in voltage waveforms. According to the tag, power packets are transferred from sources to loads. On the basis of the power packetization, the torque input is configured as a result of a power packet supply to electric motors in a realistic setup. The proposed scheme controls the supply in a digitized way, that is, by changing the number of power packets supplied in a gait step. We confirm the successful gait generation with the power packets through numerical simulations

FEASIBILTY ANALYSIS OF WALKING OF PASSIVE DYNAMIC BIPED ROBOT

  Passive dynamic walking is an essential development for the biped robots. So the focus of our work is a systematic analysis of the passive walk of a planar biped robot on an inclined slope. The dynamics of passive biped robot is only caused of gravity. The biped robot with two point masses at kneeless legs and a third point mass at the hip-joint is kinematically equivalent to a double pendulum. In this paper, we represent a general method for developing the equations of motion and impact equations for the study of multi-body systems, as in bipedal models. The solution of this system depends on the initial conditions. But it is difficult to find the proper initial conditions for which the system has solutions, in other words, the initial conditions for which the robot can walk. In this paper, we describe the cell mapping method which able to compute the feasible initial conditions for which the biped robot can move forward on the inclined ramp. The results of this method described the region of feasible initial conditions is small and bounded. Moreover, the results of cell mapping method give the fixed of Poincare map which explains the symmetric gait cycle of the robot and describe the orientation of legs of robot

Fractal mechanism of basin of attraction in passive dynamic walking

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal

Effect of the Dynamics of a Horizontally Wobbling Mass on Biped Walking Performance

We have developed biped robots with a passive dynamic walking mechanism. This study proposes a compass model with a wobbling mass connected to the upper body and oscillating in the horizontal direction to clarify the influence of the horizontal dynamics of the upper body on bipedal walking. The limit cycles of the model were numerically searched, and their stability and energy efficiency was investigated. Several qualitatively different limit cycles were obtained depending mainly on the spring constant that supports the wobbling mass. Specific types of solutions decreased the stability while reducing the risk of accidental falling and improving the energy efficiency. The obtained results were attributed to the wobbling mass moving in the opposite direction to the upper body, thereby preventing large changes in acceleration and deceleration while walking. The relationship between the locomotion of the proposed model and the actual biped robot and human gaits was investigated.Comment: 6 pages, 8 figures, accepted to IEEE International Conference on Robotics and Automation (ICRA 2023

The validation of new phase-dependent gait stability measures:A modelling approach

Identification of individuals at risk of falling is important when designing fall prevention methods. Current measures that estimate gait stability and robustness appear limited in predicting falls in older adults. Inspired by recent findings on changes in phase-dependent local stability within a gait cycle, we devised several phase-dependent stability measures and tested for their usefulness to predict gait robustness in compass walker models. These measures are closely related to the often-employed maximum finite-time Lyapunov exponent and maximum Floquet multiplier that both assess a system's response to infinitesimal perturbations. As such, they entail linearizing the system, but this is realized in a rotating hypersurface orthogonal to the period-one solution followed by estimating the trajectory-normal divergence rate of the swing phases and the foot strikes. We correlated the measures with gait robustness, i.e. the largest perturbation a walker can handle, in two compass walker models with either point or circular feet to estimate their prediction accuracy. To also test for the dependence of the measures under state space transform, we represented the point feet walker in both Euler-Lagrange and Hamiltonian canonical form. Our simulations revealed that for most of the measures their correlation with gait robustness differs between models and between different state space forms. In particular, the latter may jeopardize many stability measures' predictive capacity for gait robustness. The only exception that consistently displayed strong correlations is the divergence of foot strike. Our results admit challenges of using phase-dependent stability measures as objective means to estimate the risk of falling

Influence of the controller design on the accuracy of a forward dynamic simulation of human gait

The analysis of a captured motion can be addressed by means of forward or inverse dynamics approaches. For this purpose, a 12 segment 2D model with 14 degrees of freedom is developed and both methods are implemented using multibody dynamics techniques. The inverse dynamic analysis uses the experimentally captured motion to calculate the joint torques produced by the musculoskeletal system during the movement. This information is then used as input data for a forward dynamic analysis without any control design. This approach is able to reach the desired pattern within half cycle. In order to achieve the simulation of the complete gait cycle two different control strategies are implemented to stabilize all degrees of freedom: a proportional derivative (PD) control and a computed torque control (CTC). The selection of the control parameters is presented in this work: a kinematic perturbation is used for tuning PD gains, and pole placement techniques are used in order to determine the CTC parameters. A performance evaluation of the two controllers is done in order to quantify the accuracy of the simulated motion and the control torques needed when using one or the other control approach to track a known human walking pattern.Postprint (author's final draft