This paper investigates the mathematical modeling and basic motion properties of planar seven-link robots that forms passive closed and active open chains. The passive closed model is formed by connecting seven rigid frames via seven viscoelastic joints, and the active open model is formed by connecting them via actuated joints. The former is a convex heptagonal model and can exhibit passive-dynamic rolling on a gentle downhill, whereas the latter virtually forms a forward-leaning octagonal shape by controlling the six relative joint angles. In the first half of this paper, we describe the model assumptions and develop the mathematical equations of motion and collision of the passive closed model, and numerically analyze the motion characteristics by changing the slope angle while checking the conditions necessary for stable motion generation. In the second half, we outline the active open model, develop the PD control system, and numerically analyze the motion characteristics by changing the target angle parameter that controls the degree of forward lean of the virtual octagon.

Quadrupeds exhibit remarkable locomotion performance through the coordination between their limbs and torso. From past biological knowledge, it is understood that during walking, the forelimbs primarily contribute to braking, while the hindlimb are responsible for propulsion. However, in the field of quadruped robot dynamics, effectively leveraging this coordination remains a challenge. To investigate the torso-limb coordination, this study explores the walking performance of a quadruped walker with a compliant torso, driven by the forelimb or the hindlimb. Through numerical simulations, we analyze the walking behavior under different control drive methods. The findings provide insights into the design of compliant-bodied robots and the optimal distribution of propulsion forces between the forelimbs and hindlimbs.

The authors have proposed a passive rotating locomotion robot that forms a convex heptagonal body by connecting seven identical linear rigid frames via viscoelastic rotational joints. In our previous study, it was confirmed through both numerical simulations and actual experiments that stable and passive rotating motion on a downhill could be generated. This paper proposes two new models in which the seven rigid frames are used as robust exoskeletons as they are, but the elastic elements attached to the rotating joints are removed and repositioned on the diagonals of the convex heptagon to reproduce the flexibility of the internal tissue. The elastic elements form a star-shaped polygon called a heptagram, which is formed by connecting seven vertices with a single stroke. The seven vertices can be connected in two different ways to form two different heptagram shapes. We report the basic numerical results of the change in the motion characteristics of the two models with respect to the slope angle and elastic modulus. An overview of the prototypes developed and the results of basic experiments are also reported.

This paper proposes a novel passive dynamic walker with a body shape similar to an eight-legged rimless wheel that performs a natural swinging motion of the swing leg through storage and release of elastic energy. The generated motion is period-1 and asymptotically stable, but the inherent limit cycle stability is nontrivial because it does not achieve constraint on impact posture. Since it has almost linear dynamics, however, its walkability can be instantaneously determined using a linearized model without numerical integration. With the equations of linearized motion and exact collision, the step period and the state at the next collision can be obtained numerically and instantaneously using a bisection method based on the geometric constraint condition at impact. Then, by updating the state for each collision and repeating the same calculation, it is possible to instantaneously determine whether or not the walking motion continues stably for a long period of time. By comparing the results of this calculation with those of the numerical integration of the nonlinear and linearized models, the effectiveness of the proposed method is confirmed. Furthermore, using the proposed method, we analyze the period-doubling bifurcation phenomenon and the change in the singular values of the Poincaré map that occurs with the change in the elastic modulus.

This study introduces a novel burrowing robot that achieves effective locomotion inside granular media through the synergistic integration of high-frequency vibration-induced environmental-phase-transition (EPT) and adaptive morphing. The robotic system employs three key innovations: 1) an asymmetric arm trajectory mechanism generating directional propulsion, 2) a vibration-mediated granular fluidization system reducing environmental resistance, and 3) passively adaptive claws demonstrating phase-dependent configuration changes. Experimental results demonstrate that the synchronization of morphologically adaptive claws and high-frequency vibration significantly improves locomotion performance. Additionally, numerical simulations based on Adams-EDEM coupling provide deeper insights into the interaction mechanisms between the robot and granular media. This work advances fundamental understanding of terradynamic locomotion by demonstrating environmental modification as a viable strategy for resistance reduction, while providing a bio-inspired framework for developing versatile robotic systems capable of navigating complex particulate environments.

Quadrupedal locomotion involves coordinated interaction between limbs and torso, enabling them to achieve remarkable movement performance and adapt effectively to various environments. In previous studies, mathematical dynamic models of quadrupeds have been established to investigate the mechanisms of limb-torso interaction during walking. However, due to the strong nonlinearity within the model, analyzing how the torso’s motion, especially vibrations, affects walking remains a significant challenge. In this study, the linearization and frequency analysis methods are applied to the quadruped walker to analyze its vibration characteristics, including natural frequency and vibration amplitude. Subsequently, numerical simulations are conducted to examine the relationship between torso vibration and walking performance. Furthermore, a comparison between the vibration characteristics and the simulation results reveals a potential resonance phenomenon. This finding not only validates the effectiveness of the linearization approach but also offers new insights into the interaction between the limbs and torso.

This paper addresses the problem of realizing a novel robotic bipedal locomotion called wheel gait, which is achieved by rotating the stance and swing legs in the same direction. First, a model of a planar 3-DOF X-shaped walker with a reaction wheel is introduced, and the mathematical equations are described. Second, the condition for stabilizing zero dynamics is formulated as the time integral value of control input to the reaction wheel for one step becomes zero, and the control system for achieving this is designed based on the method of continuous-time output deadbeat control. Third, a typical steady wheel gait of the linearized model is numerically generated, and its extension to the nonlinear model is discussed. Although the nonlinear model has only one nonlinear term in the gravity term, numerical simulations show that there is a big gap between this and the linearized model. Through analysis of the typical nonlinear wheel gaits, the difficulty of achieving the same walking speed as the linearized model is discussed.

In the natural world, benefited from the advantages of the spine, quadrupeds exhibiting extraordinary flexibility which allowing them to move efficiently on variable terrains. The previous researches have indicated the legged robots which efficiently utilizing their spine can achieve rapid and stable locomotion. However, within the field of legged robot dynamics, the design of the spine and understanding how it positively influences locomotion is unclear, which is significant for quadruped robot to achieve efficient and stable walking. In this study, we proposed a model formed by tensegrity spine and rimless wheel to represent quadrupeds, using passive dynamic walking as a method, which has been well-demonstrated for observing the inherent characteristics, exhibited the locomotion characteristic of the model proposed. By numerical simulation, we observed change trend of locomotion performance with the configurations of spine’s shape, and found direction of spine design that have a positive impact on walking. These findings contribute to the design of spine structures in quadruped robots.

This paper introduces a novel X-shaped walker with telescopic legs and investigates its control method with the aim of generating a stable wheel gait on a horizontal plane without including zero dynamics which is essentially unstable and difficult to stabilize. First, we outline a planar 6-DOF robot model with three control inputs, and describe the equations of motion and inelastic collision. Second, we design an output-following control system that smoothly controls the extension/contraction lengths of the legs and relative hip-joint angle to their target terminal values, and creates an asymmetric impact posture in the anteroposterior direction so that the robot can easily overcome the next potential barrier. The coefficients of the desired-time trajectory for each control output are updated with the position and velocity values immediately after each impact as the target initial values, so the generated leg motion and control inputs exhibit smooth time variation. The validity of the proposed gait generation method and the change trend of fundamental motion characteristics with respect to control parameters are investigated through numerical simulations.

Inspired by the fascinating underwater locomotion of cephalopods such as octopuses, this research explores the possibility of using legged robots in underwater environments. Using a rimless wheel model, we investigated their navigation and adaptation capabilities in a dynamic fluid environment. Through sophisticated numerical simulations, we reproduce legged robot behaviours such as walking underwater and jumping over uneven seabed terrain. Our research aims to provide valuable insights for the development of versatile underwater robotic systems suitable for various applications in ocean exploration and surveying.

The quadrupeds have wider active territory than humans. Their bodies can adapt various environments through evolution, enabling the efficient, elegant gait for their legged locomotion. Previous researches have indicated lots of examples of utilizing the advantages of body to achieve environment adaptive and stable gait, and for legged locomotion, especially with quadruped robot determining how to generate environment-adaptive mobile locomotion remains a significant challenge. In this study, we discussed the adaptability to environments of quadruped robots, specific walking stability and gait convergence in low-friction environments with compliant torso. The numerically simulations are proposed for observing the trend of walking performance with various friction coefficient. By analyzing the typical walking gait, the adaptability of quadruped walkers with compliant torso are found. These conclusions contribute to the design and development of compliant torso for quadruped walkers.

The movement of the legs involves the interaction between the feet and the ground. Consequently, most animals possess a wide variety of foot morphologies and multifunctional capabilities. The selection and switching of these foot functions are passive and environment-dependent, ensuring environmental compliance. Despite this, current research on compliant feet lacks mathematical models that simultaneously encompass locomotion and foot compliance. Therefore, conducting in-depth studies on locomotion properties under current conditions is challenging. In this study, we present novel passive compliant feet applicable to the passive walking of a rimless wheel. We first introduce a dynamic model, achieve passive walking through numerical simulations, and subsequently analyze the gait patterns for compliance and multi-period gait. This study bridges a gap in understanding the interaction between motion and compliance in foot design, providing insights into the dynamics of compliant motion.

This paper presents a water walking robot with 6 feet which consists of a rimless wheel and a flywheel, and has a foot attached to the tip of each leg. First, in order to make the robot walk on water, we propose a foot control method by imitating the legs of some animals that can walk on water. Second, in order to increase the robot's forward speed, we improved the control method. In addition we have analysed the effect of some different physical parameters on the motion of the robot. This research is aimed at building a versatile water walking model for such applications as marine exploration.

We have been investigating a crawling-like loco-motion robot to make it efficiently slide forward based on a simple system and control mechanisms on a slippery level surface, where the motion of the center of mass plays an important role. In this paper, we induce an effective motion of the center of mass considering a streamlined body shape of a locomotion robot in which a pendulum is installed. First, we derive the equation of motion and a control input to achieve a desired motion of the inner pendulum. Second, we formulate constraint conditions between the streamlined body and a slippery floor. Third, we demonstrate the numerical simulation, and the robot steadily slides forward by adopting the streamlined shape as the body frame. Fourth, we verify the numerical results through experiments, and the experimental results exhibit a similar tendency compared with the numerical results. Fifth, we find a local minimum value of locomotion efficiency based on Bayesian optimization which is a class of machine-learning-based optimization, and we achieve exceedingly efficient locomotion of the robot on the slippery floor at the local minimum in both the simulation and experiment.

This paper proposes novel passive-dynamic walk-ers formed by two cross-shaped frames and eight viscoelastic elements. Since it is a combination of two four-legged rimless wheels via viscoelastic elements, we call it viscoelastically-combined rimless wheel (VCRW). Two types of VCRWs consisting of different cross-shaped frames are introduced; one is formed by combining two Greek-cross-shaped frames (VCRW1), and the other is formed by combining two-link cross-shaped frames that can rotate freely around the central axis (VCRW2). First, we describe the model assumptions and equations of motion and collision. Second, we numerically analyze the basic gait properties of passive dynamic walking. Furthermore, we consider an activation of VCRW2 for gen-erating a stable level gait, and discuss the significance of the study as a novel walking support device.

Biological observations on tetrapods locomotion deduce that anti-phase synchronization (APS) between fore and rear parts is beneficial for achieving a high-speed walking. On the other hand, theoretical analysis and experimental studies on quadruped robots suggest that a flexible spine potentially improves the gait efficiency and adaptability via smoothing the ground collisions. However, these two mechanisms have never been placed together by a comprehensive investigation in terms of their synergetic effect. Namely, an advanced principle is still lacking in combining the APS and the spine flexibility for quadruped walking robots. To address this issue, we construct a mathematical model for a quadruped dynamic walker under different spine conditions. First, the APS effect is generated via entrainment-based control method under a rigid spine condition. Then, flexible spines realized by three kinds of springs are compared with the rigid one via theoretical analysis. The results suggest that the APS mechanism and the flexible spine can be synergized via an appropriate deformation control. The theoretical findings not only uncover locomotion control mechanisms for quadruped walking robots, but also provide additional understandings of tetrapods dynamic walking from a mechanical engineering point of view.

This paper introduces a planar tensegrity robot that walks passively and cyclically on a gentle downhill, where its gait versatility can be strengthened by applying actuation forces on the connection cables. The novelty of this work is that we design the structure of this passive robot inspired by the rimless wheel, which naturally generates cyclic locomotion. Consequently, its mathematical model is analytically derived based on passive dynamic walking. Besides, the limb support conditions and dynamics effects induced by the collisions can be precisely determined accordingly. Moreover, numerical simulation is performed to show the typical gait pattern, and resonance phenomenon is observed. Finally, a preliminary experimental study is conducted to prove the validity of the mathematical model. The robot we developed and the mathematical model we derived enable further extensions on the gait analysis and model-based control by conveniently adopting efficient passivemimic walking techniques.

After occasional perturbation, it is crucial to spontaneously control the limit cycle walking so that it quickly returns to its closed orbit in phase space. Otherwise, its stability can not be sufficiently guaranteed if the speed of recovery is slow while successive perturbation is applied. The accumulated deviation may eventually drive the phase outside the basin of attraction, leading to failure of the walking. In this sense, a control law that quickly recovers the disturbed phase before encountering the following perturbations is indispensable. With this consideration, here we analytically derive an optimal fast entrainment waveform that maximizes the speed of phase recovery based on phase reduction theory. Our theoretical method is numerically evaluated using a limit cycle walker, which is indirectly controlled by the oscillation of a wobbling mass via entrainment effect. The obtained waveform is used as the desired trajectory of the wobbling motion. The simulation results show that the waveform we derived achieves the best performance among all candidates. Our method helps to enhance the stability of limit cycle walking.

The authors have investigated underactuated locomotion robots with an inner wobbling mass, it is discovered that the wobbling mass controls the gait speed by entrainment. Supplying the wobbling from outside, outer wobbling entrains load objects and controls the transferring speed. In this paper, we propose a vibratory conveyor system based on the frequency entrainment of a limit cycle walker. The conveyance plate is vibrated by an active rimless wheel, and the system conveys a passive rimless wheel which is defined as a load object. The vibration entrains transferring of the passive rimless and controls the conveyance speed. First, we introduce the prototype experimental system and its mathematical model. Second, we report basic behavior of the passive rimless with regards to the outer vibration and results of frequency analysis through the numerical simulation. Third, we experimentally verify the results of the numerical simulation. The active rimless wheel entrains the walking frequency of the passive rimless wheel in both the simulations and the experiments.

The author has investigated the method of stealth walking for generating adaptive walking gaits of underactuated walkers without having the control torques at the feet. This approach is also effective for achieving careful walking on the frictionless road surface by applying angular momentum constraint control (AMCC); the generated gait completes in one step while maintaining the horizontal ground reaction force to zero. The result is mathematically thorough, but is not realistic because any uncertainties in the system cannot be permitted. This paper then discusses more realistic slidingresistant situation: stealth walking on the low-friction road surface. First, we introduce a model of a planar underactuated rimless wheel, and describe the equation of motion and the control input for AMCC. Second, we specify the linearized equation of motion with AMCC, and derive the analytical solution of the stance-leg motion which is used as a desired trajectory for the nonlinear model. Furthermore, we discuss the optimality of the upper-body control during the double-limb support phase from the sliding-resistant characteristics point of view through mathematical and numerical investigations.

Stealth walking is a way of walking carefully and noiselessly, and is an approach to stable legged locomotion of underactuated robotic walkers on irregular terrains. This paper proposes a method for generating a high-speed stealth walking gait without including double-limb support phase, and discusses the effect of upper-body control and semicircular feet on the gait properties. First, we introduce a model of a 3-link planar underactuated biped with an upper body and semicircular feet, and derive the approximate target initial state of the upper body by using the linearized equation of motion. Second, we conduct numerical simulations of the nonlinear model to observe the typical stealth walking gaits, and analyze the changing tendency of the upper body motion with respect to the foot radius. Furthermore, we discuss the advantage of semicircular feet through parametric studies of the gait efficiencies.

With the purpose of achieving stable and energy efficient locomotion on the slippery road surface, a sliding locomotion robot without joint torque but indirectly controlled by an active wobbling mass is recently proposed. In this paper, we deepen the analysis of the mechanism of the indirectly controlled sliding locomotion for further optimization and generalization. First, we derive the equations of dynamics and control. Second, we estimate the natural frequency of the robot, the moving speed and energy efficiency are also evaluated with respect to forcing amplitude and frequency of the wobbling mass. Third, the Arnol'd tongue is introduced to analyze the relationship between achieving efficient locomotion and being entrained. In addition, phase oscillation and synchronization phenomenon are analyzed via hysteresis plot to further interpret the unusual shapes of the Arnol'd tongues. Finally, we analyze the entrained, however, inefficient locomotion by reconfirming the rolling constraints from the mechanical energy dissipation point of view. Our results help better understanding of the indirectly controlling mechanism, and the methods can be applied to other indirectly controlled locomotion robots.

Precisely manipulating the center of mass (CoM) of the underactuated locomotion robot can't be easily achieved by common control mechanisms which apply only joint torques. A novel and indirect method has been recently introduced using an active wobbling mass attached to limit cycle walkers. The next important issue is to design an optimal control input to reduce the forcing energy. In this paper, we use combined rimless wheels as a simplified example to apply our method, which is based on the theory of phase oscillators. First, we introduce the typical modeling and control of this underactuated robot. Second, we obtain the phase response curve by numerically applying perturbations at different phases of the walker's gait interval and calculating the deviations from the unperturbed. Third, we analytically derive an optimal forcing waveform for the wobbling mass to entrain the combined rimless wheel based on the phase response curve. As an ecological extension, an ideal forcing waveform for m: 1 entrainment was further generated. Finally, the proposed method was evaluated by locking range of the Arnold tongues. The results show that the optimal forcing waveform we derived achieves the best performance for 1:1 entrainment among all the candidates. One of the strongest advantages of our method is the easiness of its implementation, prompting its applicability to a wide variety of locomotion systems.

This paper proposes an approximate-kinetic-model-based self-adaptive (AKS) control system to rapidly generate target walking speed by an underactuated compasslike bipedal walker. First, a model of the underactuated compass-like bipedal walker is built, and an open-loop system is introduced and analysed as the prototype of AKS system. Second, the control law of AKS is described in detail. The dynamic updating of trajectory is proposed and the calculation of control parameters by an approximate linearized model is analysed. Finally, simulations are conducted, and thus the capability of disturbance rejection and versatility is tested. As a conclusion, target walking speeds with 4∼6% steady-state error can be generated rapidly and steadily. Limit cycle walker can obviously improve the capability of handling disturbance and various tasks by AKS control system.

This paper discusses a method for generating an underactuated bipedal gait completing in one step in the presence of an upper body that behaves as zero dynamics. First, we introduce a model of a 3-link planar biped robot with an upper body, and develop the equation of motion and holonomic constraint conditions. We also describe a straightforward derivation of the relationship between the angular momentum and whole center of mass. Second, we design two control laws for achieving stable stealth walking. During the single-limb support phase, the stance- and swing-leg angles are strictly controlled to follow the desired trajectories so that the swing foot lands on the ground stealthily at the end of this phase. During the double-limb support phase, the angular position and velocity of the upper body are controlled to follow the desired trajectory for returning them to the initial state. Through numerical simulations, the fundamental gait properties and change tendencies are investigated.

This paper discusses the condition necessary for achieving stable sliding passive-dynamic walking of a compass-like biped robot with semicircular feet, and analyzes the fundamental gait properties through numerical simulations. First, we introduce the semicircular-footed compass model and develop the equation of motion, holonomic constraint condition, and equation of collision considering the effect of impulsive frictional force. The paradox in transformation of the impulsive frictional force effect is also discussed. Second, we numerically show that short- and long-period sliding passive compass gaits can be generated under the same condition except the initial condition, and the sliding directions of the fore foot immediately after impact in the two gaits are different from each other. Furthermore, we conduct parametric studies to understand the change tendency of the fundamental gait properties with respect to the system parameters such as the foot radius and frictional coefficient. The simulation results show that the effect of semicircular feet makes it possible for the passive biped to walk on slippery downhill stably in the absence of the hip-damper.

Toward stable gait generation on a steep slope, this paper discusses a method for generating a stable walking gait completing in one step for an underactuated rimless wheel. First, we introduce a model of a six-legged rimless wheel with a reaction wheel, and develop two controllers for the single-limb and double-limb support phases. For easily achieving precise control of the state variables, we apply a discrete-time output deadbeat control for each phase. We numerically analyze the fundamental properties of the generated gait, and show that the target period of the double-limb support phase must be chosen as a significantly long so that the rear and fore feet are grounded during control of the reaction wheel. The gait efficiency is also analyzed in terms of the walking speed and specific resistance through numerical simulations.

The authors demonstrated that a stable passive compass gait can be generated on a slippery downhill, and showed that the 2-DOF constraint conditions for the contact point are not always necessary for it. The generated walking gait while sliding on the slope becomes 3-DOF increased by addition of the positional coordinate of the contact point. This paper then investigates the effects of semicircular feet on the 3-DOF passive compass gait. First, we develop the equations of a 4-DOF passive compass-like biped robot with semicircular feet that slides on a slippery downhill. Second, we derive the term of the sliding friction force and discuss the complexity and relationship with holonomic constraint condition. Third, we perform numerical simulations to observe the typical 3-DOF passive walking motion and change in the gait properties according to the foot radius.

This paper proposes a novel method for generating an underactuated bipedal gait that completes in one step. First, we introduce an underactuated biped robot model that has a circular torso as a reaction wheel. Second, we consider an input-output linearization to formulate an output-following control law for achieving collisionless limit cycle walking. We then mathematically analyze the stability of the zero dynamics and investigate the fundamental gait properties through numerical simulations. Furthermore, we discuss an extension to generation of a few-steps walking motion aimed at safe and quick passage from a safety island to the next.

In this paper, we propose a novel speeding-up method for biped walking using a swinging-arms motion based on the principle of an up-and-down wobbling mass. We have shown that biped robots with a wobbling mass can achieve fast walking using an active up-and-down motion of the wobbling mass. We have also shown that the active up-and-down motion increases walking speed of biped robots. We apply this principle to a biped robot with two linked arms like humans for achieving high-speed limit cycle walking. We show that the proposed method achieves high-speed limit cycle walking of biped robots with arms.

This paper investigates the analytical solution of target steady walking speed in 1-DOF limit cycle walking. We introduce an active combined rimless wheel (CRW) model to analyze the target steady walking state when the CRW walks on level ground. The walking speed is determined by the step period because the step length is constant. First we propose a two-period stepwise control system and the target walking period can be generated by solving the equations of boundary conditions. Second we extend this method to (n + 1)-period stepwise control system and generate a general formula for the boundary equation. At last we generate the target steady step period in the continuous control systems by calculating the approximate solution based on discretization of control input. We verify all the results through numerical simulations. If the generated walking gait is single-step-cycle, we can generate the target steady step period by our general formula in most of control systems.

This paper proposes a novel underactuated rimless wheel (URW) that has high stiffness and reproduces walking motion sliding on a slippery road surface. First, we describe the overview of the experimental URW we developed and its mechanical features. Second, we develop the equations of motion and collision for the ideal URW with small passive rollers, and discuss the collision dynamics for stance-leg exchange with the prospect of transition to non-instantaneous double-support motion. Third, we perform numerical simulations to understand the fundamental properties of the gait rolling on a rigid road surface and its similarity to limit cycle walking on a slippery road surface.

In this paper, we discuss the possibility and fundamental properties of passive bipedal walking on a slippery downhill. First, we develop a mathematical model of the passive compass-like biped robot that contacts with the ground incorporating sliding friction which is specified as the Coulomb model. Second, we discuss the possibility of instantaneous stance-leg exchange through mathematical analysis of the collision dynamics. Third, we perform numerical simulations using the model developed, and show that a stable walking gait can be generated with suitable initial conditions and physical parameters. Furthermore, we numerically show that period-doubling bifurcation occurs with the decrease of the coefficient of sliding friction.

This paper proposes the method independent of numerical integration for analyzing the stability of a limit cycle walker that falls down as a 1-DOF rigid body in the same posture. We introduce the model of an underactuated rimless wheel with a torso for analysis and show that the transition function of the state error can be analytically derived from the recurrence formula of kinetic energy immediately before impact. We then investigate the accuracy of the derived solution through comparison with the numerical solutions and discuss how the convergence property changes with respect to the control parameter. Furthermore, we extend the method to an underactuated biped and show that the stability analysis can be conducted in a similar manner without performing numerical simulations.

This paper investigates the roles and effects of dynamic absorbers attached to the leg frames on the gait properties of passive dynamic walking. First, we model a passive compass-like biped robot that consists of two identical leg frames with passive dynamic absorbers that represent micromechanical vibration or human flesh dynamics. We then conduct gait analysis through numerical simulations to observe how small oscillation of the leg frames affects the gait properties, and show that speeding-up is achieved by utilizing the indirect softness produced by the dynamic absorbers. Second, we investigate the dominant effect of small oscillation using the same model. The simulation results show high nonlinearity in the generated walking gait.

This paper discusses the problem of how to achieve high speed, energy efficient, and fast convergent limit cycle walking. We newly introduce the convergence speed of the generated gait as a criterion for evaluation. Through mathematical and numerical investigations, we clarify the role and importance of deceleration effect in fast convergent gait generation. First, we generate ballistic gaits of the simplest 1-DOF active walker by applying a simple control torque, and show that the stance phase is destabilized in the case only of acceleration effect. Second, we propose another method for generating a fast convergent gait by adding deceleration effect and investigate the validity through numerical simulations. Furthermore, we extend our analysis to the cases taking joint viscosity and time delay into account and discuss the effects on the gait properties.

Through modeling and analysis of a rimless wheel with active knees and semicircular feet, we showed that measurable period of double-limb support (DLS) emerges immediately after landing of the fore leg as in the case of a telescopic-legged one. In this paper, we extend the analysis to an underactuated biped model only with revolving joints and investigate the potentiality of the emergence of DLS motion in level dynamic walking. We numerically show that non-instantaneous DLS motion emerges after the instant of landing of the swing leg if the knee-joints are free and semicircular feet have a sufficient radius. Furthermore, we numerically analyze the tendency of the emergence of DLS motion according to the robot's physical parameters and shapes.

In this paper, we derive the analytical solution to the transition function of the state error in 1-DOF semi-passive dynamic walking for understanding how the gait stability changes according to acceleration or deceleration. We introduce the model of an active rimless wheel (RW) as the simplest walker for analysis and linearize the equation of motion incorporating a simple control torque. Through mathematical investigations, we finally derive the analytical solution to the transition function of the state error for the stance phase as a function only of the control parameters. We discuss the accuracy of the solution obtained through comparison with the values numerically-integrated in the linearized and the nonlinear walking models.

This paper investigates modeling and control of a limit cycle walker that walks sliding on the ice. We introduce the model of an underactuated spoked walker for analysis and analyze the collision model on the assumption of sliding contact with the ground to identify the condition for achieving instantaneous exchange of the stance leg. We also develop the equation of motion incorporating dynamic friction in sliding contact. Numerical simulations show that the walker can generate stable walking gaits by applying a simple control of the torso.

It was clarified that limit cycle walkers can improve the gait efficiency by using the oscillatory effect of a wobbling mass moving in the body frame. In this research, we investigate the effects of a 2-DOF wobbling mass on the gait properties. As the simplest walker for analysis, we introduce the model of a planar eight-legged rimless wheel (RW) with a passive 2-DOF wobbling mass that is connected to the RW incorporating a spring and a damper. Through numerical simulations, we analyze changes in the gait properties with respect to the system parameters such as the slope angle and the elastic coefficient. Furthermore, entrainment to a wobbling mass motion actively controlled to rotate is also investigated.

In this paper, we propose a novel method for fast limit cycle walking using active control of a wobbling mass. Limit cycle walkers achieving energy-efficient walking have been developed in the last decade. Many researchers have recently studied methods for improving walking speed of limit cycle walkers. In human walking, humans swing their arms according to walking phases and the motion is a regularly symmetric motion about the torso. We consider that this motion is an active up-and-down motion for a mass and improves biped walking speed. We numerically and mathematically show that a biped robot achieves fast limit cycle walking by the proposed method.

It was clarified that limit cycle walkers with redundant free joints generate the measurable periods of double-limb support (DLS) through numerical simulations and experiments. This paper then conducts numerical analyses to examine the effects of non-instantaneous DLS motion on the gait properties such as stability and energy efficiency. First, we divide the gait cycle into the collision and the stance phases and numerically evaluate their stability in terms of the convergence rate. Second, we analyze the stability in more detail by dividing the stance phase into the periods of DLS and single-limb support. The simulation results show that the energy efficiency monotonically worsens with the increase of the ratio of the period of DLS to the gait cycle but the convergence rate improves. Furthermore, we discuss the similarities between robot walking and human walking based on the analysis results obtained.

Limit cycle walking including passive-dynamic walkers is mathematically modeled as a nonlinear hybrid dynamical system with state jumps in general. The generated motion is natural and energy efficient, but it is still pointed out that there are many differences between limit cycle walking and human walking. Non-existence of the period of double-limb support in the former comes from the assumption of instantaneous inelastic collision and is one of the biggest differences from the latter. In human walking, the period of double-limb support accounts for more than 10% of one cycle, and this must have significant effects on the gait stability and efficiency. Also in robot walking, utilizing the effects of double-limb support is essential to achieve more flexible, adaptive and human-like behavior. This paper then develops a novel mathematical model of a passive rimless wheel that emerges double-limb support by using the leg viscoelasticity, and numerically investigates the fundamental properties.

It was clarified that a rimless wheel with viscoelastic legs generates stable passive-dynamic gaits including measurable period of double-limb support motion. This paper then investigates the effect of the leg viscoelasticity on the adaptation ability to irregular terrain through numerical simulations and experiments. We introduce the model of an active viscoelastic-legged rimless wheel (VRW) that consists of eight identical viscoelastic legs and an upper body for analysis. We then develop the mathematical model and numerically examine the adaptation ability to irregular terrain. In this paper, we consider the two situations; one is overcoming steps and the other is sustaining stable walking on a flexible surface. The adaptability of the active VRW is compared with that of the rigid-legged model.

It was clarified that speeding-up of passive dynamic walking (PDW) of a combined rimless wheel (CRW) can be achieved by adjustment of the phase difference between the fore and rear legs. We discussed the mechanism from the viewpoint of overcoming potential barrier, and showed that the trajectory of the whole center of mass (CoM) is significantly flattened by choosing the phase difference. This paper investigates the potentiality of speeding-up the CRW without having the phase difference but by using a passive wobbling mass that vibrates up and down in the body in expectation of flattening the whole CoM trajectory. We show that the walking speed is increased by the effect of the wobbling mass and that a transition from anti-phase to in-phase oscillation arises with the increase of the elasticity. We also show nonlinear characteristics such a high sensitivity to initial conditions and hysteresis phenomenon. Furthermore, the validity of the simulation results is verified using an experimental CRW machine.

The inherent self-stabilization mechanism of passive gait is one of the most fundamental question in the area of studies on limit cycle walking. It is well known that the limit cycle stability can be tested by calculating the eigenvalues of the approximated Poincare return map, and they have been calculated numerically in previous studies. Hirata and Kokame, however, firstly succeeded to derive the Jacobian matrix of the Poincare return map using linearization of the dynamic equation. In this paper, we reconsider their method and propose an easier approach to derive the discrete-time linear error system. Our approach enables to derive the error system without integral terms. Through the theoretical investigations, we show that a passive compass-gait is formed by the combination of unstable stance phases and marginally stable collision phases. The validity is investigated by numerical simulations.

This paper investigates active dynamic running of a planar telescopic-legged rimless wheel. Unlike the most dynamic runners with compliant legs, our model achieves stable limit-cycle running only by a simple extension control of the stance leg. We first generate a stable running gait by adjusting the desired settling time for the leg-extension control. We then analyze the gait efficiency with respect to the change of the desired settling time. Furthermore, several extentions, comparison with walking gait and running up slopes, are discussed.

A passive compass gait consists of the stance and collision phases. The author clarified that the former is unstable and the latter is marginally stable, and that the state error norm tends to increase during the stance phases and decreases during the collision phases. The convergence property is, however, complicated and the overall self-stabilization mechanism is still unclear. This paper then investigates it from the mechanical energy point of view. First, we introduce the linearized mechanical energy that leads to the linearized dynamic equation of the compass-like biped robot, and numerically show that its error norm almost monotonically converges to zero. Second, we numerically show that the monotonic convergence comes from the fact that the error of the angular positions is one digit smaller than that of the angular velocities by using approximate difference functions that varies depending on the slope.

This paper investigates passive dynamic walking of a combined rimless wheel that consists of two identical 8-legged rimless wheels. By using the mathematical model, we numerically show that stable passive gaits can be generated on a gentle slope, and that the walking speed is dramatically improved by adjusting the phase difference between the fore and rear legs. We then discuss the inherent speeding-up mechanism from the potential barrier point of view. Furthermore, the validity of the derived results is evaluated by using an experimental machine.

This paper proposes a novel spoke-like passive-dynamic walker that consists of two identical crossed frames whose center of mass is positioned on the central axis. The purpose of this study is to develop an easy test-bed for investigating the roles of double-support phase and swing-leg retraction in limit cycle walking. The crossed frame as a swing leg does not have inherent rotational dynamics, and it rotates only in the presence of viscous friction at the central axis. We first analyze the properties of this walker through mathematical modeling. Next, we numerically investigate the gait efficiency, and discuss the effect of inertia moment and mass distribution of the crossed frame. Finally, we briefly report our experimental results.

The author has proposed a novel method for generating a dynamic gait based on anterior-posterior asymmetric impact posture tilting the robot's center of mass forward. The primary purpose of this method is to asymmetrize the impact posture by actuating the robot's telescopic-legs to make overcoming the potential barrier at mid-stance easy, and the mechanical energy is accordingly restored. We have already validated our method through numerical investigations of a planar rimless wheel model with telescopic legs. The results implied that the robot motion becomes remarkably high-speed and the need of ankle brake was indicated. This paper then extends the method to a planar telescopic-legged biped model incorporating the brake effect of ankle spring, and numerically investigate the gait properties. We also discuss the role of anterior-posterior asymmetric shape of human foot from the viewpoint of zero moment point.

Period-doubling bifurcation and chaotic behavior are interesting phenomena in limit cycle walking. Their mechanisms are very complicated and their roles in dynamic biped locomotion are still unclear. This paper then investigates how the gait efficiency changes with the asymmetrization through analysis of a simple rimless wheel model. We mathematically show that the symmetric configuration gives the optimal solution when the mean value of the inter-leg angles is constant in terms of kinetic energy. This is derived from two magnitude relations of the energy-loss coefficient and restored mechanical energy. We also show that the symmetric gait is not always optimal from another viewpoint.

This paper investigates the efficiency of a 2-period gait from the kinetic energy view-point. First, we formulate a steady 2-period gait for a compass-like bipedal robot by using a simple recurrence formula for the kinetic energy of an asymmetric rimless wheel. Second, we theoretically show that, in the case that the mean value of the hip angle is constant, the generated 2-period steady gait is less efficient than a 1-period symmetric one in terms of kinetic energy. We also show that the symmetric gait is not always optimal from another viewpoint. We then investigate the validity of the derived method through numerical simulations of virtual passive dynamic walking. Other approaches, delayed feedback control and a quasi-constraint on the impact posture, are also considered for stabilization to a 1-period gait and their effects are discussed.

Limit cycle walkers utilizing their natural dynamics can achieve energy-efficient dynamic walking. Their heel-strike collision with the ground is generally modeled as an inelastic collision, and the discrete walking dynamics can be specified in the same manner as a rimless wheel by using the energy-loss coefficient and restored mechanical energy. Energy-loss coefficient is especially significant because it controls the gait efficiency and stability, but the value varies significantly according to the swing-leg retraction just prior to impact and robot's mass-distribution. This paper then mathematically investigates how energy-loss coefficient changes with respect to them, and discusses the effect on the gait efficiency and stability.

In the passive dynamic walking proposed by McGeer, mechanical energy lost by heel strike is restored by transporting potential energy to kinetic energy as walking down a slope. When energy input is large such as an angle of slope is steep, bifurcation of walking period occurs. In parametric excitation walking, which is one method to realize passive dynamic-like walking on level ground, bifurcation has also been observed when walking speed is fast. Asano et al. have shown that bifurcation exerts an adverse influence upon walking performance by using rimless wheel model. In this paper, we apply delayed feedback control (DFC) originally used in chaos control to parametric excitation walking to suppress bifurcation. We show in numerical simulation that the proposed method makes two-period walking to one-period walking, and energy efficiency is improved. The analyses using Poincare¿ map reveal that the one-period walking with DFC is unstable periodic orbit and that the robot dealt in this paper satisfies the sufficient condition of applicability of DFC.

This paper reports some interesting results on our experimental study of parametrically excited dynamic bipedal walking. We describe the details of the walking machine that has telescopic legs, semicircular feet, free hip-joint and counterweights. The walker can sustain stable dynamic walking on level ground based on mechanical energy restoration in accordance with the principle of parametric excitation utilizing the effects of semicircular feet and counterweights. Results of numerical analysis of the effect of the counterweights on the gait efficiency are also described.

This paper studies the efficiency and asymptotic stability of a dynamic bipedal gait with a constraint on the impact posture. First, we generate a gait by using tracking control to achieve the desired trajectory of the hip-joint angle, and show that there is a trade-off between efficiency and robustness through a numerical simulation. Second, we investigate the asymptotic stability of the gait from the mechanical energy balance viewpoint, and discuss the importance of the control input properties. Furthermore, we point out that there is a feedback in mechanical energy in the discrete walking system, and it is difficult to detect a stable 2-period gait.

Achieving energy-efficient dynamic walking has become one of the main subjects of research on robotic bipedal locomotion. Approaches based on passive-dynamic walkers can accomplish bipedal locomotion. However, passive dynamic walking has only been studied with the legs, and the effect of an upper body has not been clarified. This paper investigates the effect of an upper body on the efficiency and stability of dynamic bipedal locomotion based on observations. We first investigated a suitable upper body, which was a simple 1-link torso with a bisecting hip mechanism that would not destroy natural dynamics of the biped model. Second, we analyzed the robot's driving mechanism and chose underactuated virtual passive dynamic walking as the method for generating an efficient dynamic gait. We confirmed that efficient dynamic walking was possible with a specific resistance of 0.01 and investigated the effect of the physical parameters of the upper body through numerical simulations.

This paper studies efficiency and symmetry of ballistic gait. We introduce a compass-like biped model and propose a simple control law for generating the dynamic gait. The efficiency is then numerically analyzed and the optimality of mechanical energy restoration is discussed. It is also shown that perfectly ballistic gait by kick action achieves the highest walking speed and exhibits symmetric gait automatically. We discuss how the symmetric gait is generated from the angular momentum point of view.

This paper investigates the effect of an upper body on efficient dynamic bipedal walking utilizing its natural dynamics. We introduce an upper body as a one-link torso and add it to a simple biped model by means of a bisecting hip mechanism (BHM). We first mathematically analyze the effect of the upper body with the BHM as a counterweight and discuss how it affects natural swinging motion of the swing leg. Second, we propose a simple method for generating efficient dynamic bipedal gait imitating the property of virtual passive dynamic walking, and numerically analyze the gait efficiency. Simulation results show that the walking system exhibits period-doubling bifurcation, and we discuss how the efficiency changes in the multiple-period gait.

The parametric excitation based gait generation method proposed by Asano et al. restores mechanical energy lost by heel-strike collisions. Harata et. al. applied this method to a kneed biped robot which is proper for the parametric excitation, and show that sustainable gait has been generated with only knee torque. A swing-leg of a kneed biped robot has similar mechanism to an acrobot, and many acrobots bends a joint in inverse direction like ornithoid walking. This suggests that inverse bending a knee restores more mechanical energy than forward bending like human walking, and hence, inverse bending may be more efficient. In this paper, we propose a parametric excitation based ornithoid gait generation method for a kneed biped robot, and show that it can walk sustainably by numerical simulation. We also show that parametric excitation based inverse bending walking is more efficient than parametric excitation based forward bending walking with respect to performance indices in our model.

In this paper, we consider the effects of feet in various convex shapes on the performance of underactuated passive dynamic walking. We first derive formula for calculating virtual ankle-joint torque and a performance index of dissipated mechanical energy for a convex foot shape model with general foot shape parameterization. Secondly, using the foot shape determined by a 2nd Bezier curve with which a form can be controlled easily, we calculate a value of the virtual ankle-joint torque and the performance index of dissipated mechanical energy to verify effects of change in foot shape. Furthermore, relationship between the foot shapes and the walking speeds are clarified by a numerical simulation. Finally, we propose a method to design a convex foot shape that attains a desired virtual ankle-joint torque and verify an improvement of the walking speed by a numerical simulation with the foot shape that is designed by the proposed method. We believe that the proposed method will be applied to design effective active ankle-joint torque and/or to design a foot shape for biped robots.

We clarified that the common necessary condition for generating a dynamic gait results from the requirement to restore mechanical energy through studies on passive dynamic walking mechanisms. This paper proposes a novel method of generating a dynamic gait that can be found in the mechanism of a swing inspired by the principle of parametric excitation using telescopic leg actuation. We first introduce a simple underactuated biped model with telescopic legs and semicircular feet and propose a law to control the telescopic leg motion. We found that a high-speed dynamic bipedal gait can easily be generated by only pumping the swing leg mass. We then conducted parametric studies by adjusting the control and physical parameters and determined how well the basic gait performed by introducing some performance indexes. Improvements in energy efficiency by using an elastic-element effect were also numerically investigated. Further, we theoretically proved that semicircular feet have a mechanism that decreases the energy dissipated by heel-strike collisions. We provide insights throughout this paper into how zero-moment-point-free robots can generate a novel biped gait.

Realization of an energy-efficient and high-speed dynamic walking has come to be one of the main subjects in the research area of robotic biped locomotion, and passive dynamic walking has been widely attracted as a clue to solve the problem. It has been empirically known that the effect of convex curve shape of foot, which characterizes passive-dynamic walkers, is important to increase walking speed. This paper then investigates the driving mechanism of compasslike biped robots and the rolling effect of semicircular feet are mainly investigated. We first analyze the mechanism of a planar fully-actuated compass-like biped model to clarify the importance of ankle-joint torque introducing generalized virtual gravity concept. In the second, a planar underactuated biped model with semicircular feet is introduced and we show that virtual passive dynamic walking by hip-joint torque only can be realized based on the rolling effect. We then compare with a flat feet model through linear approximation, and show that the rolling effect is equivalent to its virtual ankle-joint torque. Throughout this paper, we provide novel insights into how ZMP-free robots can generate a dynamic bipedal gait.

This paper investigates the effect of semicircular feet on dynamic bipedal walking. It has been clarified by Asano and Luo (2006) that underactuated virtual passive dynamic walking can be realized by using the rolling effect, which acts as the ankle-joint torque virtually. It has been also shown that, throughout parameter studies, the rolling effect dramatically increases the stable domain of limit cycles. Now that the effect of semicircular feet during stance phase has been discussed, this paper then focuses the effect on mechanical energy dissipation by heel-strike. It is theoretically clarified that, through modeling and analysis of an inelastic collision, increasing walking speed is achieved not by the rolling effect during stance phase but by the effect of reducing mechanical energy dissipation by heel-strike.

Traditionally, an inverted pendulum has been used as a reduced biped locomotion system, whereas this paper proposes a different approach. The essence of dynamic biped gait generation is mechanical energy restoration, and parametric excitation approach is a good idea for it. Our novel approach does not require any rotational actuation and thus enables to be free from the constraint of zero moment point (ZMP). This paper considers some basic methods of parametric excitation and shows that energy-efficient biped locomotion can achieved very easily without taking the ZMP condition into account. We then conduct parametric studies by adjusting the control and physical parameters, and determine how well the basic gait perform by introducing some performance indices.

This paper investigates dynamic bipedal gait and its stability from the mechanical energy balance point of view. The equilibrium points at impact in a dynamic gait are uniquely and systematically determined by two constraint conditions; one is the constraint on restored mechanical energy, the other is the constraint on impact posture. The dynamic gait then becomes always asymptotically stable around the equilibrium points, and this is shown by a simple recurrence formula of the pre-impact kinetic energy. The validity of the method is numerically confirmed via gait generation by virtual passive dynamic walking.

Restoring mechanical energy lost by heel-strike collisions is necessary for stable gait generation. One principle to realize this is parametric excitation. Recently, Asano et al. applied this principle to a biped robot with telescopic-legs, and succeeded in generating a sustainable biped gait by computer simulation. In this paper, we deal with a model of a biped robot that has not only semicircular feet but also actuated knees. Though this robot has no actuator at the hip, knee actuators can sustain gait by parametric excitation. We first verify that an actuated knee can cause parametric excitation, and then show by computer simulation that the proposed biped robot can walk continuously with actuated knees only.

This paper investigates effectiveness of semicircular feet on dynamic biped locomotion. We first introduce the simplest biped model with semicircular feet and show its level walking by hip-joint actuation. In the second, parameter study is performed by adjusting the physical and control parameters. Throughout numerical analysis, it is shown that the semicircular feet dramatically increases the walking speed and the stable domain. By the effect of the rolling, energy-efficient and high-speed dynamic biped locomotion on a level can be realized easily without ankle-joint actuation nor concerning the zero moment point condition

Our goal is to create advanced engineering systems such as a soft human interactive robot. The robot developed here is named RI-MAN. RI-MAN exhibits the skill and ability to realize human care and welfare tasks. RI-MAN can search out a specific person in real time by fuing audio and visual information, and understand human speech based on a sound recognition function. In addition, RI-MAN's body is coverd with soft touch sensors, and RI-MAN can react to the amplitude and location of external forces. Using all these sensor functions, RI-MAN can successfully follow human commands and hold up a dummy of the same size as an adult human. RI-MAN will become an invaluable partner robot.

It is already clarified throughout studies of passive dynamic walking mechanisms that the common nec essary condition for dynamic gait generation comes from the requirement on mechanical energy restoration. Until now we have treated only rotational joints of the robot, whereas in this paper we consider a novel dynamic gait generation method based on mechanical energy restoration by parametric excitation using telescopic leg actuation. We first introduce a simple walking model and a control law for the telescopic leg motion, and show the typical walking pattern by numerical simulations. We then analyze the gait performance by adjusting some control and physical parameters. In addition, some extensions of the mechanism and control applications are investigated.

Principal mechanisms of passive dynamic walking are studied from the mechanical energy point of view, and novel gait generation and control methods based on passive dynamic walking are proposed. First, a unified property of passive dynamic walking is derived, which shows that the walking system's mechanical energy increases proportionally with respect to the position of the system's center of mass. This yields an interesting indeterminate equation that determines the relation between the system's control torques and its center of mass. By solving this indeterminate equation for the control torque, active dynamic walking on a level can then be realized. In addition, the applications to the robust energy referenced control are discussed. The effectiveness and control performances of the proposed methods have been investigated through numerical simulations.

This paper studies the dynamic principles of passive dynamic walking and proposes novel gait generation and control methods based on it. The authors have clarified a unified property of passive dynamic walking which shows that the walking system's mechanical energy increases proportionally with respect to the position of system's center of mass. Following this, the gait generation problem yields solutions of an indeterminate equation. By solving it for the control torque, active dynamic walking on a level can then be realized. In this paper, we first discuss the solutions and consider the unification of the previous gait generation methods from a variable virtual gravity point of view. Second, its applications to a kneed biped system considering ZMP condition and robust energy referenced control are discussed. The effectiveness of the proposed methods have been investigated through numerical simulations.

This paper studies modeling and bio-mimetic control of a 3D 8-dof whole arm cooperative manipulation system using sensitive skin. A sphere is considered here as a manipulated object. The control law is designed based on integration of voluntary and reflex movements considering the system's redundancy. The voluntary task for holding the object is realized by impedance control at the four contact points with the object using the contact force information from the sensitive skin. The reflection on the other hand is introduced as a regulation problem of the direction between the points of end-effectors and elbows. The solution for the redundant control is formulated and derived from the optimization point of view. The validity of the proposed method is investigated by numerical simulations.

The authors have clarified the mechanism of passive dynamic walking from the mechanical energy point of view, and reported its basic results. This paper then considers the generalization and unification of dynamic gait generation methods by introducing variable virtual gravity concept and some solution formulas as well as essential mechanical energy orbits. As two leading methods, energy tracking control and virtual passive dynamic walking are considered, and we analyze the control performances of robust stability and energy-efficiency criterions by numerical simulations. Finally we discuss the application possibility of the methods to actual walking machines from the ZMP point of view.

This paper proposes novel energy-based gait generation and control methods for biped robots based on an analysis of passive dynamic walking. First, we discuss the essence of dynamic walking using a passive walker on a gentle slope from the mechanical energy point of view. Second, we propose a simple and effective gait-generation method, which imitates the energy behavior in every walking cycle considering the zero-moment point condition and other factors of the active walker. The control strategy is formed by taking into account the features of mechanical energy dissipation and restoration. Following the proposed method, the robot can exhibit a natural and reasonable walk on a level ground without any gait planning and design in advance. The effectiveness of the method is examined through numerical simulations and experiments.

Unlike the present manipulator control technologies that operate objects only by the robots' end-effectors, human beings can perform whole body manipulation flexibly and easily. Bio-mimetic research of such skillful human motor behavior is important not only for deeper understanding of human sensations and nervous control functions but also for developing of higher level robots. In this paper, we first formulate the basic model of multipoints whole body interaction between a robot manipulator and its object. We then study three control approaches by considering the limitations of force feedback, the complexity of control algorithms. As an example, we consider a 2-link planar manipulator that operates a circle object with dynamic arm friction. We show a computer simulation algorithm, and compare the results of each control methods, numerically.

This paper proposes a novel energy-based control law for biped robots based on an analysis of passive dynamic walking. Firstly we discuss the essence of dynamic walking using a passive walker on a gentle slope. In the second, we propose a simple and effective control law which imitates the energy behavior in every cycle considering the ZMP condition and other factors of the active walker. The control strategy is formed by the feature of mechanical energy dissipation and restoration. By the effect of the proposed method, the robot can exhibit natural and reasonable walk on a level ground without any gait design in advance. The validity of the proposed method is examined by numerical simulations and experiments.

In our previous works (2000), we have proposed "virtual passive dynamic walking" utilizing modified gravity condition with virtual gravity field. The virtual passive walking motion critically depends on the physical parameters and the steady walking pattern is not easily obtained without suitable physical parameters. In this paper we propose a more generalized method of virtual passive walk and a virtual passivity mimicking control law. With the effect of the control laws, we can generate the steady walking pattern even if the physical parameters are not suitable. We call the walking pattern generated by the control methods as "extended virtual passive dynamic walking". The validity of the proposed method is examined by numerical simulations and tested by a prototype experimental machine.

In our previous works (2000), we proposed some system augmentation methods for dynamic bipedal walking in order to realize variable walking patterns in real-time, however, the robust performance of the methods were discussed. Based on the observation we propose a new coupling control law considering the flywheel effect for robustness. In this paper the validity of the methods is analyzed and investigated by numerical simulations and experiments.

In our previous works, we have proposed "virtual passive dynamic walking" with virtual gravity for biped robots in order to realize active walking on level ground without any gait design in advance. In this paper, we discuss some control problems of a kneed biped robot and propose "modified compass-like virtual passive dynamic walking" with active knee-lock algorithms in order to avoid the "foot scuffing" problem during the single support phase. Furthermore, a virtual coupling control law is proposed which can realize a variable walking pattern with respect to the robot's energy levels. By the effect of the control law, the robot, which is a hybrid dynamical system, can be regarded as a passive system which does not include any collisions, and a variable walking pattern can be realized without loss of the properties of a virtual passive walk. The validity of the proposed methods has been examined by numerical simulations.

The study of bipedal walking in the framework of humanoid robot is a recent active research area. In this paper, we apply passive velocity field control to the control of a biped walking robot which walks on the level ground by actuators. Using this method, we can change the walking speed easily by modifying a virtual energy. The validity of the proposed method is demonstrated by numerical simulations.

It has been shown that a simplest walker with suitable parameter choice can walk down a gentle slope without any control forces and generate its steady walking pattern utilizing gravity effect automatically. On the floor, however, the robot cannot exhibit passive walk, so any application methods of passive walk to active walker on the horizontal floor has not been studied yet. In this paper we introduce "virtual passive dynamic walking" with virtual gravity field which acts as a driving force for the biped robot. The robot can walk on the floor without any control torque except virtual gravity effect. Since the modified gravity field seems to be very close to real condition, the generated walking pattern seems to be natural. Further, multi-pattern walking with respect to energy level is proposed. With the proposed method, safety and energy-effective control of biped walking robot can be realized.