Exploiting natural dynamics for gait generation in undulatory locomotion

ABSTRACT

Robotic vehicles inspired by animal locomotion operate via periodic body movements. The pattern of body oscillation (gait) can be mimicked from animals, but understanding the principles underlying gait generation would allow for broad, flexible applications beyond nature's design. We hypothesise that travelling-wave oscillations observed in undulatory locomotion can be characterised as a natural oscillation of the locomotion dynamics and propose a formal definition of the natural gait for locomotion systems. We identify the essential dynamics and define the mode shape of natural oscillation by the free response of an idealised system. We then use body-environment resonance to define the amplitude and frequency. Explicit formulas for the natural gait are derived to provide insight into the mechanisms underlying undulatory locomotion. Examples of a swimming leech and a fictitious swimmer illustrate how undulatory gaits similar to those observed can be produced as the natural gait and modulated to achieve different swim speeds.     

A method for determination of optimal gaits with application to a snake-like serial-link structure

In this paper, we present a method of determining optimal gaits for shape actuated locomotion systems. This method is the synthesis of techniques for computing reduced equations for robotic locomotion systems and a numerical optimal control strategy. Symmetry reduction processes induce a form of locomotion system dynamics that reveals a cyclic-like coupling between group, shape, and momenta coordinates. This form allows one to focus on designing gaits, abandoning concern over shape dynamics. Using this vantage point we indicate how a numerical optimal control method based on Gaussian quadrature may be acclimatized to periodicity, thus providing optimal gaits. The method is demonstrated by means of its application to a snake-like serial-link structure or snake robot. This application provides scientific merit to hypotheses concerning observed locomotion phenomena amongst animals employing undulatory propulsive mechanisms.     

Gait transition and modulation in a quadruped robot: A brainstem-like modulation approach

In this article, we propose a bio-inspired architecture for a quadruped robot that is able to initiate/stop locomotion; generate different gaits, and to easily select and switch between the different gaits according to the speed and/or the behavioral context. This improves the robot stability and smoothness while locomoting.We apply nonlinear oscillators to model Central Pattern Generators (CPGs). These generate the rhythmic locomotor movements for a quadruped robot. The generated trajectories are modulated by a tonic signal, that encodes the required activity and/or modulation. This drive signal strength is mapped onto sets of CPG parameters. By increasing the drive signal, locomotion can be elicited and velocity increased while switching to the appropriate gaits. This drive signal can be specified according to sensory information or set a priori.The system is implemented in a simulated and real AIBO robot. Results demonstrate the adequacy of the architecture to generate and modulate the required coordinated trajectories according to a velocity increase; and to smoothly and easily switch among the different motor behaviors.     

An adaptive locomotion controller for a hexapod robot: CPG,kinematics and force feedback

Insects can perform versatile locomotion behaviors such as multiple gaits, adapting to different terrains, fast escaping, etc. However, most of the existing bio-inspired legged robots do not possess such walking ability, especially when they walk on irregular terrains. To tackle this challenge, a central pattern generator （CPG）-based locomotion control methodology is proposed, integrated with a contact force feedback function. In this approach, multiple gaits are produced by the CFG module. After passing through a post-processing circuit and a delay-line, the control signal is fed into six trajectory generators to generate predefined feet trajectories for the six legs. Then, force feedback is employed to adjust these trajectories so as to adapt the robot to rough terrains. Finally the regulated trajectories are sent to inverse kinematics modules such that the position control instructions are generated to control the actuators. In both simulations and real robot experiments, we consistently show that the robot can perform sophisticated walking patterns. What is more, the robot can use the force feedback mechanism to deal with the irregularity in rough terrain. With this mechanism, the stability and adaptability of the robot are enhanced. In conclusion, the CPG-base control is an effective approach for legged robots and the force feedback approach is able to improve walking ability of the robots, especially when they walk on irregular terrains.     

Evolution of a 3D Gallop in a Quadrupedal Model with Biological Characteristics

The gallop is the preferred high-speed gait for dynamic locomotion in most cursorial mammals. Due to the lack of good analytical models and proven control strategies, however, the gallop remains an elusive goal in the field of legged robotics. While there have been several attempts at creating a gallop, none have captured all of the important dynamic characteristics of the gait. In this work, we present a practical approach for producing a stable 3D gallop in a quadrupedal model which includes these characteristics. The dynamic model utilizes biologically-based assumptions including articulated legs with nonzero mass, compliance at the knee joints, and a body with an asymmetric mass distribution. Furthermore, the resulting 3D gallop contains the prominent features found in the biological gait: early leg retraction, phase-locked leg motion creating an asymmetric footfall pattern, a significant gathered flight phase, unconstrained spatial dynamics, and a smooth gait. To obtain these results, we employ a multiobjective genetic algorithm with a carefully designed vector fitness function to search for various control parameters. Furthermore, we partition the search space in roughly orthogonal subspaces to find parameters for each sub-controller. A critical component of the controller is an energy control law that ensures a fixed amount of energy in the knee springs during each stride. A characterization of the resulting gait is presented, which highlights biological properties and the visual realism of the solution.     

Optimal fault tolerant gait sequence of the hexapod robot withoverlapping reachable areas and crab walking

This paper extends the authors' previous results on fault tolerant locomotion of the hexapod robot on even terrain by relaxing nonoverlap of redefined reachable cells of legs and considering crab walking. It is shown that in fault tolerant locomotion two adjacent legs of the hexapod robot can have overlapping redefined reachable cells with each other and consequently the stride length of the gaits is increased. Also, the optimal fault tolerant periodic gaits for hexapod robots to have the maximum stride length in one cycle in crab walking on even terrain are derived with distinct reachable cells. The derived sequence for crab walking has different orders of leg swing according to the relative values of the crab angle and some design parameters of the robot     

A lattice-type reconfigurable robotic system for achieving gaits of chain-type robots

Traditional lattice-type reconfigurable robots can only achieve the flow-style locomotion with low efficiency. Since gaits of chain-type robots are proved to be efficient and practical, this paper presents a novel lattice distortion approach for lattice-type reconfigurable robots to achieve locomotion gaits of chain-type robots. Using this approach, the robotic system can be actuated by local lattice distortion to move as an ensemble. In this paper, a rule that makes the lattice distortion equivalent to joint rotation is presented firstly. Then, a kind of module structure is designed according to requirements of the lattice distortion. Finally, a motion planning for achieving locomotion is developed, which works well in physics-based simulations of completing a serpentine locomotion gait of a snake-like robot and a tripod gait of a hexapod robot.     

A versatile locomotion mechanism for amphibious robots: eccentric paddle mechanism

Most of recently developed rescue robots can only be deployed to limited attacked regions after tsunami and the floods, due to their limited mobility on complex amphibious terrains. To access such amphibious environments with improved mobility, we propose a novel eccentric paddle mechanism (ePaddle) which has a set of paddles eccentrically placed in a wheel to perform multiple terrestrial, aquatic, and amphibious gaits. One of the advantages of our proposed ePaddle mechanism is its unique locomotion versatility introduced by the eccentric distance between the paddle shaft and the wheel center. We demonstrate this versatility by proposing five typical gaits for traveling on different terrains. For instance, wheeled rolling gait is used to achieve high-speed locomotion on even terrain. Legged gait is applied to travel on the rough terrains. To access the soft terrains where wheels slip and legs sink, a wheel-leg-integrated gait is performed by digging the paddle into the ground. To swim in the water, rotational paddling and oscillating paddling gaits are proposed. For each of these gaits, standard gait sequence is defined and joint parameters are calculated based on kinematics. An ePaddle prototype is then built and tested with the proposed gait sequences. Experimental results verify the design of the ePaddle mechanism as well as its versatile gaits.     

Evolving gaits for hexapod robots using cyclic genetic algorithms

A major facet of multi-legged robot control is locomotion. Each leg must move in such a manner that it efficiently produces thrust and provides maximum support. The motion of all the legs must be coordinated so that they are working together to provide constant stability while propelling the robot forward. In this paper, we discuss the use of a cyclic genetic algorithm (CGA) to evolve control programs that produce gaits for actual hexapod robots. Tests done in simulation and verified on the actual robot show that the CGA successfully produces gaits for both fully capable and disabled robots.     

Dynamic Effects in Statically Stable Walking Machines

Discontinuous gaits for walking machines present some advantages over wave gaits such as better stability margins and greater speed for small duty factors, for instance. The problem is that a machine using discontinuous gaits starts and stops its body motion several times per locomotion cycle. This means that high accelerations appear, therefore the theoretical static stability margin can be inadequate for measuring stability. This paper addresses how dynamic effects modify the measurement of the static stability of a discontinuous gait and determines the acceleration under which the criterion of using the static stability margin for measuring the stability is valid. For this study, a dynamic planar model of a four-legged walking machine was derived. Then, both the longitudinal and dynamic stability margins were computed and compared. Final results show that the static stability margin is an adequate measurement for studying stability in massless leg machines with the constraint that the acceleration of the body be smaller than the inversion acceleration. When the mass of the legs is significant, stability is determined by the dynamics of the legs and the distribution of the mass of the legs as well.     

Autonomous evolution of dynamic gaits with two quadruped robots

A challenging task that must be accomplished for every legged robot is creating the walking and running behaviors needed for it to move. In this paper we describe our system for autonomously evolving dynamic gaits on two of Sony's quadruped robots. Our evolutionary algorithm runs on board the robot and uses the robot's sensors to compute the quality of a gait without assistance from the experimenter. First, we show the evolution of a pace and trot gait on the OPEN-R prototype robot. With the fastest gait, the robot moves at over 10 m/min, which is more than forty body-lengths/min. While these first gaits are somewhat sensitive to the robot and environment in which they are evolved, we then show the evolution of robust dynamic gaits, one of which is used on the ERS-110, the first consumer version of AIBO.     

Online Optimization of Swimming and Crawling in an Amphibious Snake Robot

An important problem in the control of locomotion of robots with multiple degrees of freedom (e.g., biomimetic robots) is to adapt the locomotor patterns to the properties of the environment. This article addresses this problem for the locomotion of an amphibious snake robot, and aims at identifying fast swimming and crawling gaits for a variety of environments. Our approach uses a locomotion controller based on the biological concept of central pattern generators (CPGs) together with a gradient-free optimization method, Powell's method. A key aspect of our approach is that the gaits are optimized online, i.e., while moving, rather than as an off-line optimization process. We present various experiments with the real robot and in simulation: swimming, crawling on horizontal ground, and crawling on slopes. For each of these different situations, the optimized gaits are compared with the results of systematic explorations of the parameter space. The main outcomes of the experiments are: 1) optimal gaits are significantly different from one medium to the other; 2) the optimums are usually peaked, i.e., speed rapidly becomes suboptimal when the parameters are moved away from the optimal values; 3) our approach finds optimal gaits in much fewer iterations than the systematic search; and 4) the CPG has no problem dealing with the abrupt parameter changes during the optimization process. The relevance for robotic locomotion control is discussed.     

Control Approach for Legged Robots with Fast Gaits

A constructive control approach is proposed for legged robots with fast dynamic gaits. These systems interact intermittently with the environment. Our approach is based on Controlled Limit Cycles (CLC) and stabilizes periodic system trajectories. The designed control law generates (on-line) the desired trajectories and control input.     

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.     

Minimalistic control of biped walking in rough terrain

Toward our comprehensive understanding of legged locomotion in animals and machines, the compass gait model has been intensively studied for a systematic investigation of complex biped locomotion dynamics. While most of the previous studies focused only on the locomotion on flat surfaces, in this article, we tackle with the problem of bipedal locomotion in rough terrains by using a minimalistic control architecture for the compass gait walking model. This controller utilizes an open-loop sinusoidal oscillation of hip motor, which induces basic walking stability without sensory feedback. A set of simulation analyses show that the underlying mechanism lies in the “phase locking” mechanism that compensates phase delays between mechanical dynamics and the open-loop motor oscillation resulting in a relatively large basin of attraction in dynamic bipedal walking. By exploiting this mechanism, we also explain how the basin of attraction can be controlled by manipulating the parameters of oscillator not only on a flat terrain but also in various inclined slopes. Based on the simulation analysis, the proposed controller is implemented in a real-world robotic platform to confirm the plausibility of the approach. In addition, by using these basic principles of self-stability and gait variability, we demonstrate how the proposed controller can be extended with a simple sensory feedback such that the robot is able to control gait patterns autonomously for traversing a rough terrain.     

Planar legged walking of a passive-spine hexapod robot

This paper proposes a new legged walking method for a novel passive-spine hexapod robot. This robot consists of several body segments connected by passive body joints. Each of the body segments carries two 1-DoF (degree of freedom) actuated legs. The robot is capable of achieving planar legged walking by rapidly abducting and adducting its legs. To model the mobility of a robot based on this simple design, the candidate configurations from all possible configurations are first selected in a mobility analysis of the robot based on the screw theory. All the feasible sequences of these candidate configurations are then searched to form planar locomotion gaits. Next, locomotive performance of the gaits is analyzed. Finally, the proposed locomotion design and gait planning methods are verified through simulations and experiments.     

基于离散化的六足机器人自由步态规划

为精细模仿生物步态,充分发挥六足机器人运动潜能,本文在离散化机器人足端轨迹的基础上,融合中枢模式发生器(central pattern generator,CPG)模型与反射模型的核心思想,建立了离散化步态模型,结合稳定性分析,构建了机器人稳定的位置状态空间,将复杂的步态规划问题转化为稳定的位置状态空间中位置状态间的排序问题,在此基础上,提出了一种新的自由步态生成算法,并基于平均稳定裕量对该算法进行了优化.样机步态实验结果表明,自由步态生成算法与自由步态优化算法均可生成在一定程度上符合生物运动特点的稳定步态,实现机器人运动过程中速度的动态调整,跨越宽度为步距的障碍,且基于平均稳定裕量的自由步态优化算法生成步态的稳定性要远大于自由步态生成算法.     

From human to humanoid locomotion—an inverse optimal control approach

The purpose of this paper is to present inverse optimal control as a promising approach to transfer biological motions to robots. Inverse optimal control helps (a) to understand and identify the underlying optimality criteria of biological motions based on measurements, and (b) to establish optimal control models that can be used to control robot motion. The aim of inverse optimal control problems is to determine—for a given dynamic process and an observed solution—the optimization criterion that has produced the solution. Inverse optimal control problems are difficult from a mathematical point of view, since they require to solve a parameter identification problem inside an optimal control problem. We propose a pragmatic new bilevel approach to solve inverse optimal control problems which rests on two pillars: an efficient direct multiple shooting technique to handle optimal control problems, and a state-of-the art derivative free trust region optimization technique to guarantee a match between optimal control problem solution and measurements. In this paper, we apply inverse optimal control to establish a model of human overall locomotion path generation to given target positions and orientations, based on newly collected motion capture data. It is shown how the optimal control model can be implemented on the humanoid robot HRP-2 and thus enable it to autonomously generate natural locomotion paths.     

Spine as an engine: effect of spine morphology on spine-driven quadruped locomotion

In quadruped animals, spinal movements contribute to locomotion in terms of controlling body posture and integrating limb and trunk actions. In this paper, we develop quadruped models with different numbers of spinal joints to demonstrate the spine-driven locomotion. Actuated spinal joint(s) are exclusively employed to these models with a minimalistic control strategy. We choose some typical individuals from two models and analyze them on gait properties. Results show that employing the spine morphology with two joints can greatly enhance the stability and speed of locomotion by readjusting the center of mass, increasing the stride length, and generating double flight phases similar to running cheetahs’ gait, which makes significant difference in the speed and the gait. Furthermore, we explore and compare models with more spinal joints. Results show that all gaits emerged from them can be categorized into three types (bounding, bounding with double flight phase, and stotting gaits). Overall, bounding gait with double flight phases is a more biologically inspired locomotive behavior; model with two spinal joints could be sufficient to emulate biological spine-driven locomotive behaviors.     

Polychaete-Like Undulatory Robotic Locomotion in Unstructured Substrates

A biological paradigm of versatile locomotion and effective motion control is provided by the polychaete annelid worms, whose motion adapts to a large variety of unstructured environmental conditions (sand, mud, sediment, water, etc.), and could thus be of interest to replicate by robotic analogs. Their locomotion is characterized by the combination of a unique form of tail-to-head body undulations (opposite to snakes and eels), with the rowing-like action of numerous lateral appendages distributed along their long segmented body. Focusing on the former aspect of polychaete locomotion, computational models of crawling and swimming by such tail-to-head body undulations have been developed in this paper. These are based on the Lagrangian dynamics of the system and on resistive models of its interaction with the environment, and are used for simulation studies demonstrating the generation of undulatory gaits. Several biomimetic robotic prototypes have been developed, whose undulatory actuation achieves propulsion on sand and other granular unstructured environments. Extensive experimental studies demonstrate the feasibility of robot propulsion by tail-to-head body undulations in such environments, as well as the agreement of its qualitative and quantitative characteristics to the predictions of the corresponding computational models.