Prediction of stable walking for a toy that cannot stand

From a 3D Passive Biped Walker to a 3D Passivity-Based Controlled Robot

Asymptotically stable control of biped robots, especially based on reproducing passive periodic motions, have become of interest nowadays. In this paper, firstly, a three-dimensional (3D) stable passive biped walker which is a compass gait one with flat feet, compliant ankles and particular arrangement of moments of inertia has been presented. Then, a passivity-based control of the related biped robot based on elaborating 3D form of potential energy shaping method has been applied. In other words, by adding minimal actuations to the aforementioned passive walker, its passive periodic gait that belongs to a particular slope has been reproduced on any arbitrary surface such as the level ground. Simulation results support the effectiveness of the proposed approach.

A Three-Dimensional Passive-Dynamic Walking Robot with Two Legs and Knees

The authors have built the first three-dimensional, kneed, two-legged, passive-dynamic walking machine. Since the work of Tad McGeer in the late 1980s, the concept of passive dynamics has added insight into animal locomotion and the design of anthropomorphic robots. Various analyses and machines that demonstrate efficient human-like walking have been developed using this strategy. Human-like passive machines, however, have only operated in two dimensions (i.e., within the fore-aft or sagittal plane). Three-dimensional passive walking devices, mostly toys, have not had human-like motions but instead a stiff legged waddle. In the present three-dimensional device, the authors preserve features of McGeer’s two-dimensional models, including mechanical simplicity, human-like knee flexure, and passive gravitational power from descending a shallow slope. They then add specially curved feet, a compliant heel, and mechanically constrained arms to achieve a harmonious and stable gait. The device stands 85 cm tall. It weighs 4.8 kg, walks at about 0.51 m/s down a 3.1-degree slope, and consumes 1.3 W. This robot further implicates passive dynamics in human walking and may help point the way toward simple and efficient robots with human-like motions.

Skateboards, Bicycles, and Three-dimensional Biped Walking Machines: Velocity-dependent Stability by Means of Lean-to-yaw Coupling

One of the great challenges in the development of passive dynamic walking robots (useful for an understanding of human gait and for future applications in entertainment and the like) is the stabilization of three-dimensional motions. This is a difficult problem due to the inherent interaction between fore-aft motions and sideways motions. In this paper we propose a simple solution. Conceptually, one can avert a sideways fall by steering in that direction, similar to skateboards and bicycles. We propose to implement this concept for walking robots by the introduction of an ankle joint that kinematically couples lean to yaw. The ankle joint has an unusual orientation; its axis points forward and downward, without any left-right component. The effect of the ankle joint is investigated in a simple three-dimensional model with three internal degrees of freedom: one at the hip and two at the ankles. It has cylindric feet and an actuator at the hip joint, which quickly moves the swing leg to a preset forward position. The simulations show that it is easy to find a stable configuration, and that the resultant walking motion is highly robust to disturbances. Similar to skateboards and bicycles, there exists a critical velocity (as a function of the parameters) above which stable walking motions occur. The critical velocity can be lower for a more vertical ankle axis orientation. As an additional benefit, the ankle joint allows a straightforward implementation for steering; a simple sideways offset of the mass distribution will cause the model to gently steer in that direction. The results show great potential for the construction of a real-world prototype with the proposed ankle joint.

Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge

No legged walking robot yet approaches the high reliability and the low power usage of a walking person, even on flat ground. Here we describe a simple robot which makes small progress towards that goal. Ranger is a knee-less four-legged ‘bipedal’ robot which is energetically and computationally autonomous, except for radio controlled steering. Ranger walked 65.2 km in 186,076 steps in about 31 h without being touched by a human with a total cost of transport [TCOT ≡ P/mgv ] of 0.28, similar to human’s TCOT of ≈ 0.3. The high reliability and low energy use were achieved by: (a) development of an accurate bench-test-based simulation; (b) development of an intuitively tuned nominal trajectory based on simple locomotion models; and (c) offline design of a simple reflex-based (that is, event-driven discrete feed-forward) stabilizing controller. Further, once we replaced the intuitively tuned nominal trajectory with a trajectory found from numerical optimization, but still using event-based control, we could further reduce the TCOT to 0.19. At TCOT = 0.19, the robot’s total power of 11.5 W is used by sensors, processors and communications (45%), motor dissipation (≈34%) and positive mechanical work (≈21%). Ranger’s reliability and low energy use suggests that simplified implementation of offline trajectory optimization, stabilized by a low-bandwidth reflex-based controller, might lead to the energy-effective reliable walking of more complex robots.

Capturability-based analysis and control of legged locomotion, Part 1: Theory and application to three simple gait models

This two-part paper discusses the analysis and control of legged locomotion in terms of N-step capturability: the ability of a legged system to come to a stop without falling by taking N or fewer steps. We consider this ability to be crucial to legged locomotion and a useful, yet not overly restrictive criterion for stability. In this part (Part 1), we introduce a theoretical framework for assessing N-step capturability. This framework is used to analyze three simple models of legged locomotion. All three models are based on the 3D Linear Inverted Pendulum Model. The first model relies solely on a point foot step location to maintain balance, the second model adds a finite-sized foot, and the third model enables the use of centroidal angular momentum by adding a reaction mass. We analyze how these mechanisms influence N-step capturability, for any N > 0. Part 2 will show that these results can be used to control a humanoid robot.

Intelligence by mechanics

Research on the biomechanics of animal and human locomotion provides insight into basic principles of locomotion and respective implications for construction and control. Nearly elastic operation of the leg is necessary to reproduce the basic dynamics in walking and running. Elastic leg operation can be modelled with a spring-mass model. This model can be used as a template with respect to both gaits in the construction and control of legged machines. With respect to the segmented leg, the humanoid arrangement saves energy and ensures structural stability. With the quasi-elastic operation the leg inherits the property of self-stability, i.e. the ability to stabilize a system in the presence of disturbances without sensing the disturbance or its direct effects. Self-stability can be conserved in the presence of musculature with its crucial damping property. To ensure secure foothold visco-elastic suspended muscles serve as shock absorbers. Experiments with technically implemented leg models, which explore some of these principles, are promising.

A direct comparison of local dynamic stability during unperturbed standing and walking

Standing and walking are very different tasks. It might be reasonable, therefore, to assume that the mechanisms used to maintain the stability of standing and walking should be quite different. However, many studies have shown that postural stability measures can generally predict risk of falls, even though most falls occur during locomotor tasks and not during postural tasks. This suggests that there is at least some commonality among the mechanisms governing the control of both standing and walking. The present study was conducted to determine whether the postural stability either is or is not directly related to locomotor stability. Twenty healthy adults, age 18-73 years, walked on a motorized treadmill at their preferred walking speed for three trials of 5 min. They also stood on a force plate for three trials of 5 min. Both tasks were performed without imposing any additional external perturbations. The motion of each subject's trunk segment was recorded and described using a multi-dimensional state space defined in the same manner for both tasks. Local dynamic stability was quantified from the mean divergence over time of locally perturbed trajectories in state space, which was parameterized as a double exponential process. Divergence parameters were compared to determine the relationship between local dynamic stability during standing and walking. Standing and walking exhibited local dynamic stability properties that were significantly different (P<0.001) and not correlated (P>0.1). Divergence parameters were also compared to traditional center of pressure (COP) measures obtained from standing trials. COP measures were significantly correlated to local divergence parameters for standing, but not to those for walking. This study provides direct evidence that the mechanisms governing standing and walking stability are significantly different.

Human leg design: optimal axial alignment under constraints

Alignment of joints with respect to the leg axis reduces the moment arm of external forces and therefore joint torques. Moreover, it affects the gearing of muscle forces and displacements. Thus, it influences tissue stress, cost of support and locomotion, and stability. Assuming that alignment is of general advantage we propose a mathematical criterion quantifying the axial alignment using the static torque equilibrium of a three-segment leg. Using this criterion derived from joint torque minimisation we asked for optimal leg designs (segment lengths and joint angles) at varied leg lengths. The trivial "straight is best" solution is excluded and the configuration space is restricted by geometrical constraints such as the ground contact. For different total leg lengths we could identify different optimal segment length combinations and appropriately adjusted joint angles. The extended human leg configuration characterised by a short foot and a combination of unequal ankle and knee angles emerges as a global optimum from our analysis. For crouched configurations allowing for larger leg extensions an angle symmetrical 1:1:1 segment length combination is best. The plantigrade optimum is enforced by the requirement of the distal segment (foot) being shorter than the opposite outer segment (thigh), as well as by the ground contact constraint. Different (e.g. digitigrade) geometries might be of advantage in different biological contexts with different constraints. The fact that small mammals use a crouched equal segment design implies that other locomotor requirements such as stability, strain rates, and acceleration distance per step might dominate.