Walking of biped with passive exoskeleton: evaluation of energy consumption

The paper aims to theoretically show the feasibility and efficiency of a passive exoskeleton for a human walking and carrying a load. The human is modeled using a planar bipedal anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh, shin, and foot (massless). The exoskeleton is considered also as an anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to the biped (to human)—the topology of the exoskeleton is the same as of the biped (human). Each model of the human and exoskeleton has seven links and six joints. The hip-joint connects the trunk and two thighs of the two legs. If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip, knee, and ankle joints coincide. We compare the walking gaits of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; for both cases the same load is transported. The problem is studied in the framework of a ballistic walking model. During ballistic walking of the biped with exoskeleton, the knee of the support leg is locked, but the knee of the swing leg is unlocked. The locking and unlocking can be realized in the knees of the exoskeleton by any mechanical brake devices without energy consumption. There are no actuators in the exoskeleton. Therefore, we call it a passive exoskeleton. The walking of the biped consists of alternating single- and double-support phases. In our study, the double-support phase is assumed instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous support leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone. For transportation of a load with mass 40 kg, the economy of the energy is approximately 28%, if the length of the step and its duration are equal to 0.5 m and 0.5 s, respectively.     

Asymmetric trajectory generation and impedance control for running of biped robots

An online asymmetric trajectory generation method for biped robots is proposed to maintain dynamical postural stability and increase energy autonomy, based on the running stability criterion defined in phases. In a support phase, an asymmetric trajectories for the hip and swing leg of the biped robots is obtained from an approximated running model with two springless legs and a spring-loaded inverted pendulum model so that the zero moment point should exist inside the safety boundary of a supporting foot, and the supporting leg should absorb large reaction forces, take off and fly through the air. The biped robot is under-actuated with six degrees of under-actuation during flight. The trajectory generation strategies for the hip and both legs in a flight phase use the approximated running model and non-holonomic constraints based on the linear and angular momenta at the mass center. Next, we present an impedance control with a force modulation strategy to guarantee a stable landing on the ground and simultaneously track the desired trajectories where the desired impedance at the hip link and both legs is specified. A series of computer simulations for two different types of biped robots show that the proposed running trajectory and impedance control method satisfy the two conditions for running stability and make the biped robot more robust to variations in the desired running speed, gait transitions between walking and running, and parametric modeling errors. We have examined the feasibility of this method with running experiments on a 12-DOF biped robot without arms. The biped robot could run successfully with average forward speed of about 0.3359 [m/s]. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.

Passive joint stiffness in the hip and knee increases the energy efficiency of leg swinging

In the field of minimally-actuated robots, energy efficiency and stability are two of the fundamental criteria that can increase autonomy and improve task-performance capabilities. In this paper, we demonstrate that the energetic cost of leg swinging in dynamic robots can be reduced without significantly affecting stability by emulating the physiological use of passive joint stiffness, and we suggest that similar efficiency improvements could be realized in dynamic walking robots. Our experimental model consists of a two-segment dynamically swinging robotic leg with hip and knee joints. Closed-loop control is provided to the hip using neurally inspired, nonlinear oscillators that do not override the leg’s natural dynamics. We examined both linear and nonlinear, physiologically based stiffness profiles at the hip and knee and a hyperextension-preventing hard stop at the knee. Our results indicate that passive joint stiffness applied at one or both joints can improve the energy efficiency of leg swinging by reducing the actuator work required to counter gravitational torque and by promoting kinetic energy transfer between the shank and thigh. Energetic cost reductions (relative to the no-stiffness case) of approximately 25% can be achieved using hip stiffness, provided that the hip actuation bias angle is not coincident with gravity, and cost reductions of approximately 66% can be achieved using knee stiffness. We also found that constant stiffness combined with a limit on knee hyperextension produces comparable results to the physiological stiffness model without requiring complex implementation techniques. Although this study focused on the task of leg swinging, our results suggest that passive-stiffness properties could also increase the energy efficiency of walking by reducing the cost of forward leg swing by up to 66%. We also expect that the energetic cost of walking could be further reduced by adding stiffness to the ankle to assist in the propulsive portion of stance phase.     

The Relationships Between Length of Stride,Step Frequency,Time of Swing and Speed of Walking for Children and Adults

Fifty males and females between 1 and 35 years of age were studied during locomotion. During the first few months of walking the step frequency bears no apparent relationship to the speed of walking. A log-log regression equation describes the adult relationship better than a linear equation. A few adolescents were better described by a linear equation and either log-log or linear equations can be used for children.

The product of maximum step frequency and the square root of the stature is approximately constant after 5 years of age.

The time of swing initially shows a positive regression with the time for a complete cycle of one leg. The child abandons this pattern in favour of an approximately constant time of swing and by 4-5 years of age the negative linear regression of the adult appears. The time of swing is usually much less than half the natural period of either the whole leg about the hip or of the lower leg and foot about the knee. The effects of wearing shoes upon step frequency and time of swing were investigated.     

一种新的双足机器人模型设计与相关研究

针对双足机器人最简模型在行走过程中出现摆动腿足部擦地的问题,提出了一种通过摆动腿膝关节弯曲达到摆动腿缩短的新模型。当摆动腿开始摆动时,摆动腿膝关节弯曲锁定,摆动腿缩短;当摆动腿摆动到最大位置时,膝关节解锁,摆动腿伸直再锁定,此后摆动腿回摆,系统变为直腿模型。采用脚后跟冲击控制,在摆动腿落地前,拖后的支撑腿与地面接触处施加一指向髋关节的瞬时冲击力,冲击力可以减小摆动腿着地时能量的损耗,同时驱动被动机器人向前行走。设计了迭代学习控制算法,找到极限环与不动点,实现不同给定期望步长跟踪的冲击力的计算。仿真结果表明,迭代学习控制可以有效的实现不同期望步长的跟踪,可以很快的找到机器人系统的不动点,通过收敛的相平面,得到稳定的极限环,保证了机器人行走过程稳定。     

Construction of a Gait Adaptation Model in Human Split-Belt Treadmill Walking Using a Two-Dimensional Biped Robot

A number of studies have measured kinematics, dynamics and oxygen uptake while a person walks on a treadmill. In particular, during walking on a split-belt treadmill, in which the left and right belts have different speeds, remarkable differences in kinematics are observed between normal subjects and subjects with cerebellar disease. In order to construct a gait adaptation model of such human split-belt treadmill walking, we proposed a simple control model and developed a new two-dimensional biped robot walk on a split-belt treadmill. We combined the conventional limit-cycle-based control consisting of joint PD control, cyclic motion trajectory planning and a stepping reflex with a newly proposed adjustment of P-gain at the hip joint of the stance leg. The data obtained in experiments on the robot (normal subject model and cerebellum disease subject model) have highly similar ratios and patterns to data obtained in experiments on normal subjects and subjects with cerebellar disease carried out by Bastian et al. We also showed that the P-gain at the hip joint of the stance leg was the control parameter of adaptation for symmetric gaits in split-belt walking and that P-gain adjustment corresponded to muscle stiffness adjustment by the cerebellum. Consequently, we successfully proposed a gait adaptation model for human split-belt treadmill walking, and confirmed the validity of our hypotheses and the proposed model using the biped robot.     

Hopping on Even Ground and Up Stairs with a Single Articulated Leg

This paper presents a method for generating gaits for a one-legged articulated hopping robot. A static optimization procedure produces the initial joint velocities for the flight phase, using the principle of conservation of angular momentum and assuming (nearly) passive flight. Two novel objective functions for this static optimization enable one to choose different gaits by simply changing a few parameters. A dynamic optimization procedure yields a solution for the flight trajectory that minimizes control effort. The stance phase (when the foot is touching the ground) becomes a standard two point boundary value problem, also solved with a dynamic optimization procedure. During the stance phase, the physical joint limitations, ground reaction forces, and the trajectory of the zero-moment point all constrain the solution. After these single-phase optimizations, a complete-cycle optimization procedure, incorporating both flight and stance phases, further reduces the control effort and balances the motion phases. In simulation, the leg hops on even ground and up stairs, exhibiting energy-efficient and intuitively satisfying gaits.     

Optimal Trajectories for a Quadruped Robot with Trot,Amble and Curvet Gaits for Two Energetic Criteria

In this paper, optimal cyclic reference trajectories aredesigned for three gaits of a quadruped robot, the curvet, theamble, and the trot, taking into account the actuatorscharacteristics. The gaits are composed of stance phases andinstantaneous double supports. The principle of virtual leg isused to obtain simpler dynamic model describing the motion ofthe quadruped. The impact phases are modeled by passive impactequations. For the curvet the step is composed of twodifferent half steps. For the amble and trot gaits twofollowing half steps are symmetrical.The optimization problem is solved with an algebraicoptimization technique. The actuated joint evolution is chosenas a polynomial function of time. The coefficients of thepolynomial functions are optimization parameters. Thequadruped studied has non-actuated ankles. The kineticmomentum theorem permits to define the evolution of this non-actuated variable in function of the actuated variables. Twoenergetic criteria are defined: a torque cost and an energetic cost. The first is represented by the integral of the torquenorm and the second by the absolute value integral of theexternal forces work. The two criteria are calculated for adisplacement of one meter. During the optimization process,the constraints on the ground reactions, on the validity ofimpact, on the torques, on the joints velocities and on themotion velocity of the robot prototype are taken into account.Simulation results are presented for the three gaits. Allmotions are realistic. Curvet is the less efficient gait withrespect to the criteria studied. For slow motion, trot is themore efficient gait. But amble permits the fastest motion withthe same actuators.     

Synchronization of a class of cyclic discrete-event systems describing legged locomotion

It has been shown that max-plus linear systems are well suited for applications in synchronization and scheduling, such as the generation of train timetables, manufacturing, or traffic. In this paper we show that the same is true for multi-legged locomotion. In this framework, the max-plus eigenvalue of the system matrix represents the total cycle time, whereas the max-plus eigenvector dictates the steady-state behavior. Uniqueness of the eigenstructure also indicates uniqueness of the resulting behavior. For the particular case of legged locomotion, the movement of each leg is abstracted to two-state circuits: swing and stance (leg in flight and on the ground, respectively). The generation of a gait (a manner of walking) for a multi-legged robot is then achieved by synchronizing the multiple discrete-event cycles via the max-plus framework. By construction, different gaits and gait parameters can be safely interleaved by using different system matrices. In this paper we address both the transient and steady-state behavior for a class of gaits by presenting closed-form expressions for the max-plus eigenvalue and max-plus eigenvector of the system matrix and the coupling time. The significance of this result is in showing guaranteed stable gaits and gait switching, and also a systematic methodology for synthesizing controllers that allow for legged robots to change rhythms fast.     

Energy-preserving control of a passive one-legged running robot

Fast and energy-efficient control is an increasingly important and attractive area of research in legged locomotion. In this paper, we present a new simple controller for a planar one-legged passive running robot having a springy leg and a compliant hip joint. The most distinctive advantage of the controller over previously proposed ones is it does not require any pre-planned trajectories nor target dynamics. Instead, it utilizes exact non-linear dynamics. Our results are summarized as follows. First, we propose an energy-preserving control strategy for energy-efficient and autonomous gait generation. This strategy is successfully implemented as a new touchdown controller at the flight phase. Simulation results show that the robot can hop from a wide set of initial conditions. Moreover, the running gaits generated are found to be quasi-periodic orbits, which can be seen in Hamiltonian systems. Since the controlled running gaits exist for every admissible energy level, they have some robustness against disturbances. Next, it is shown that an adaptive control of the touchdown angle, which is similar to a delayed feedback controller for a chaotic system, can asymptotically stabilize these quasi-periodic gaits to the periodic ones of the desired period, with some limitations. In particular, for one-periodic gait, by using some additional adaptive controllers, the robot eventually hops without any control inputs. Since our energy-preserving strategy is clear and implementation of the controller is straightforward, we believe it can be easily applied to a wide class of legged mechanisms.     

Bionic Quadruped Robot Dynamic Gait Control Strategy Based on Twenty Degrees of Freedom

Quadruped robot dynamic gaits have much more advantages than static gaits on speed and efficiency, however high speed and efficiency calls for more complex mechanical structure and complicated control algorithm. It becomes even more challenging when the robot has more degrees of freedom. As a result, most of the present researches focused on simple robot, while the researches on dynamic gaits for complex robot with more degrees of freedom are relatively limited. The paper is focusing on the dynamic gaits control for complex robot with twenty degrees of freedom for the first time. Firstly, we build a relatively complete 3D model for quadruped robot based on spring loaded inverted pendulum (SLIP) model, analyze the inverse kinematics of the model, plan the trajectory of the swing foot and analyze the hydraulic drive. Secondly, we promote the control algorithm of one-legged to the quadruped robot based on the virtual leg and plan the state variables of pace gait and bound gait. Lastly, we realize the above two kinds of dynamic gaits in ADAMS-MATLAB joint simulation platform which testify the validity of above method.      

Stable active running of a planar biped robot using Poincare map control

This work formulates the active limit cycles of bipedal running gaits for a compliant leg structure as the fixed point of an active Poincare map. Two types of proposed controllers stabilize the Poincare map around its active fixed point. The first one is a discrete linear state feedback controller designed with appropriate pole placement. The discrete-time control first uses purely constant torques during stance and flight phase, then discretizes each phase into smaller constant-torque intervals. The other controller is an invariant manifold based chaos controller: a generalized Ott, Grebogi and Yorke controller having a linear form and a nonlinear form. Both controllers can stabilize active running gaits on either even or sloped terrains. The efficiency of these controllers for bipedal running applications are compared and discussed.     

Online Balance Controllers for a Hopping and Running Humanoid Robot

This paper describes online balance controllers for running in a humanoid robot and verifies the validity of the proposed controllers via experiments. To realize running in the humanoid robot, the overall control structure is composed of an offline controller and an online controller. The main purpose of the online controller is to maintain dynamic stability while the humanoid robot hops or runs. The online controller is composed of the posture balance control in the sagittal plane, the transient balance control in the frontal plane and the swing ankle pitch compensator in the sagittal plane. The posture balance controller makes the robot maintain balance using an inertial measurement unit sensor in the sagittal plane. The transient balance controller makes the robot keep its balance in the frontal plane using gyros attached to each upper leg. The swing ankle pitch compensator prevents the swing foot from hitting the ground at unexpected times while the robot runs forward. HUBO2 was used for the running experiment. It was designed for the running experiment, and is lighter and more powerful than the previous walking robot platform, HUBO. With the proposed controllers, HUBO2 ran forward stably at a maximum speed of 3.24 km/h and this result verified the effectiveness of the proposed algorithm. In addition, in order to show the contribution of the stability, the running performance according to the existence of each controller was described by experiment.     

Gait and foot trajectory planning for versatile motions of a six-legged robot

This article deals with the problem of planning and controlling a radially symmetric six-legged walker on an uneven terrain when a smooth time-varying body motion is required. The main difficulties lie on the planning of gaits and foot trajectories. As for the gaits, this article discusses the forward wave gait of a variable duty factor and a variable wave direction. With the commanded body motion, the maximum possible duty factor is computed using the speed limit of the leg swing motion. Guaranteeing this maximum duty factor contributes to obtain higher stability. We prove the “continuity” of this forward wave gait planning algorithm adds the versatility to gaits planned. The foot trajectory planning algorithm dynamically generates a smooth foot trajectory as a function of the instantaneous body motions by modifying standard leg motion templates. The robot can negotiate an uneven terrain by modifying a vertical leg motion by a signal of tactile sensors on the foot. The experiments prove that the robot can successfully track smooth curves with body rotations on an uneven terrain, and thus prove the robustness of the algorithms. © 1997 John Wiley & Sons, Inc.     

Individual muscle contributions to the swing phase of gait: An EMG-based forward dynamics modelling approach

Muscle activation patterns and kinematic conditions at the beginning of the swing phase of gait were used as input to a forward dynamics simulation of the swing leg. A neuromusculoskeletal model was used to account for the non-linearity between muscle excitation and muscle force outputs. Following model tuning a close agreement between simulated and measured swing phase kinematics was obtained. Simulation results suggest that swing leg muscles play an important role in controlling the motion of the swing leg during walking, and that the effect of individual muscles is not necessarily restricted to the joints they span or their basic anatomical classifications.     

sEMG-based continuous estimation of joint angles of human legs by using BP neural network

In this paper, we propose an mth order nonlinear model to describe the relationship between the surface electromyography (sEMG) signals and the joint angles of human legs, in which a simple BP neural network is built for the model estimation. The inputs of the model are sEMG time series that have been processed, and the outputs of the model are the joint angles of hip, knee, and ankle. To validate the effectiveness of the BP neural network, six able-bodied people and four spinal cord injury (SCI) patients participated in the experiment. Two movement modes including the treadmill exercise and the leg extension exercise at different speeds and different loads were respectively conducted by the able-bodied individuals, and only the treadmill exercise was selected for the SCI patients. Seven channels of sEMG from seven human leg muscles were recorded and three joint angles including the hip joint, knee joint and the ankle joint were sampled simultaneously. The results present that this method has a good performance on joint angles estimation by using sEMG for both able-bodied subjects and SCI patients. The average angle estimation root-mean-square (rms) error for leg extension exercise is less than 9°, and the average rms error for treadmill exercise is less than 6° for all the able-bodied subjects. The average angle estimation rms error of the SCI patients is even smaller (less than 5°) than that of the able-bodied people because of a smaller movement range. This method would be used to rehabilitation robot or functional electrical stimulation (FES) for active rehabilitation of SCI patients or stroke patients based on sEMG signals.     

Quadrupedal running with a flexible torso: control and speed transitions with sums-of-squares verification

This paper studies quadrupedal bounding in the presence of flexible torso and compliant legs with non-trivial inertia, and it proposes a method for speed transitions by sequentially composing locally stable bounding motions corresponding to different speeds. First, periodic bounding motions are generated simply by positioning the legs during flight via suitable (virtual) holonomic constraints imposed on the leg angles; at this stage, no control effort is developed on the support legs, producing efficient, nearly passive, bounding gaits. The resulting motions are then stabilized by a hybrid control law which coordinates the movement of the torso and legs in continuous time, and updates the leg touchdown angles in an event-based fashion. Finally, through sums-of-squares programming, formally verified estimates of the domain of attraction of stable fixed points are employed to realize stable speed transitions by switching among different bounding gaits in a sequential fashion.     

Constraint and exploitation of redundant degrees of freedom during walking

What kind of leg trajectories are selected during human walking? To address this question, we have analyzed leg trajectories from two points of view: constraint and exploitation of redundant degrees of freedom. First, we computed the optimal leg swing trajectories for forward and backward walking that minimize energy cost for the condition of having some stretch of elastic components at the beginning of the leg swing and found that the optimal trajectories explain the characteristics of measured trajectories. Second, we analyzed how and when leg joints cooperate to adjust the toe position relative to the hip position during walking and found that joint coordination (i.e., joint synergy) is exploited at some control points during human walking, e.g., the toe height when it passes through its lowest position from the ground and the leg posture at the beginning of the double-support phase. These results suggest that the basic constraint in selecting a leg trajectory would be the minimization of energy cost; however, the joint trajectory is not strictly controlled over the entire trajectory and redundant degrees of freedom are exploited to adjust the foot position at some critical points that stabilizing walking.     

Stability analysis of wave-crab gaits of a quadruped

Crab walking is as important as forward walking as applied to walking machine control. Crab walking is especially important to a quadruped since a quadruped has a similar leg geometric layout in both longitudinal and lateral directions. In the studies of forward walking gaits, the wave gait was found to be the optimally stable. In this article, the wave gait is applied to the crab walking of a quadruped and it is modified into four types of wave-crab gaits according to the range of crab angle. The stability formulae of these wave-crab gaits are then derived analytically based on the following three stability measurements: stability margin (S_m), body-longitudinal (or lateral) stability margin (S_bl or S_bt) and crab longitudinal stability margin (S_cl). S_m is the true stability index under quasi-static walking condition. However, the equations are more complicated. S_bl (or S_bt) is simpler and can be used as a good approximation of S_m. S_cl was commonly adopted as the stability index in the previous gait studies. Nevertheless, S_cl is found to be misleading for a large portion of crab angle range. The analytical results derived in this paper are useful to the geometric design and to the real-time control of a quadruped.     

Adding an Upper Body to Passive Dynamic Walking Robots by Means of a Bisecting Hip Mechanism

Passive dynamic walking is a promising idea for the development of simple and efficient two-legged walking robots. One of the difficulties with this concept is the addition of a stable upper body; on the one hand, a passive swing leg motion must be possible, whereas on the other hand, the upper body (an inverted pendulum) must be stabilized via the stance leg. This paper presents a solution to the problem in the form of a bisecting hip mechanism. The mechanism is studied with a simulation model and a prototype based on the concept of passive dynamic walking. The successful walking results of the prototype show that the bisecting hip mechanism forms a powerful ingredient for stable, simple, and efficient bipeds