Andy Ruina

The simplest walking model : Stability, complexity, and scaling

We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope gamma. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < gamma < 0.015 rad. With increasing gamma, stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to gamma 1/3, and that the gravitational power used is proportional to v4 where v is the velocity along the slope.

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.

Energetic consequences of walking like an inverted pendulum: step-to-step transitions

Walking like an inverted pendulum reduces muscle-force and work demands during single support, but it also unavoidably requires mechanical work to redirect the body’s center of mass in the transition between steps, when one pendular motion is substituted by the next. Production of this work exacts a proportional metabolic cost that is a major determinant of the overall cost of walking.

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

Abstract We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V 0 We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V . In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value − B In V , where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables. Linearized stability analysis predicts constant slip rate motion at V 0 to change from stable to unstable with a decrease in the spring stiffness k below a critical value k cr . At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k , if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased. Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V 0 is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k :, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.

Computer optimization of a minimal biped model discovers walking and running

Although peoples legs are capable of a broad range of muscle-use and gait patterns, they generally prefer just two. They walk, swinging their body over a relatively straight leg with each step, or run, bouncing up off a bent leg between aerial phases. Walking feels easiest when going slowly, and running feels easiest when going faster. More unusual gaits seem more tiring. Perhaps this is because walking and running use the least energy. Addressing this classic conjecture with experiments requires comparing walking and running with many other strange and unpractised gaits. As an alternative, a basic understanding of gait choice might be obtained by calculating energy cost by using mechanics-based models. Here we use a minimal model that can describe walking and running as well as an infinite variety of other gaits. We use computer optimization to find which gaits are indeed energetically optimal for this model. At low speeds the optimization discovers the classic inverted-pendulum walk, at high speeds it discovers a bouncing run, even without springs, and at intermediate speeds it finds a new pendular-running gait that includes walking and running as extreme cases.

A Bipedal Walking Robot with Efficient and Human-Like Gait

Here we present the design of a passive-dynamics based, fully autonomous, 3-D, bipedal walking robot that uses simple control, consumes little energy, and has human-like morphology and gait. Design aspects covered here include the freely rotating hip joint with angle bisecting mechanism; freely rotating knee joints with latches; direct actuation of the ankles with a spring, release mechanism, and reset motor; wide feet that are shaped to aid lateral stability; and the simple control algorithm. The biomechanics context of this robot is discussed in more detail in [1], and movies of the robot walking are available at Science Online and http://www.tam.cornell.edu/~ruina/powerwalk.html. This robot adds evidence to the idea that passive-dynamic approaches might help design walking robots that are simpler, more efficient and easier to control.

Planar sliding with dry friction Part 1. Limit surface and moment function

Abstract We present two geometric descriptions of the net frictional force and moment between a rigid body and a planar surface on which it slides. The limit surface, from classical plasticity theory, is the surface in load space which bounds the set of all possible frictional forces and moments that can be sustained by the frictional interface. Zhukovskiis moment function (N.E. Zhukovskii, Collected Works, Vol. 1, Gostekhizdat, Moscow, 1948, pp. 339–354) is the net frictional moment about the bodys instantaneous center of rotation as a function of its location. Both of these descriptions implicitly contain the full relation between slip motion and frictional load. While Zhukovskiis moment function applies only to ordinary isotropic Coulomb friction, the limit surface applies to a wider class of friction laws that includes, for example, contact mediated by massless rigid wheels. Both the limit surface and the moment function can be used to deduce results concerning the motion of sliding rigid bodies.

A New Algebraic Rigid-Body Collision Law Based on Impulse Space Considerations

We present a geometric representation of the set of three-dimensional rigid-body collisional impulses that are reasonably permissible by the combination of non-negative post-collision separation rate, non-negative collisional compression impulse, non-negative energy dissipation and the Coulomb friction inequality. The construction is presented for a variety of special collisional situations involving special symmetry or extremes in the mass distribution, the friction coefficient, or the initial conditions. We review a variety of known friction laws and show how they do and do not fit in the permissible region in impulse space as well as comment on other attributes of these laws. We present a few parameterizations of the full permissible region of impulse space. We present a simple generalization to arbitrary three-dimensional point contact collisions of a simple law previously only applicable to objects with contact-inertia eigenvectors aligned with the surface normal and initial relative tangential velocity component (e.g., spheres and disks). This new algebraic collision law has two restitution parameters for general three-dimensional frictional single-point rigid-body collisions. The new law generates a collisional impulse that is a weighted sum of the impulses from a frictionless but non rebounding collision and from a perfectly sticking, nonrebounding collision. We describe useful properties of our law; show geometrically the set of impulses it can predict for several collisional situations ; and compare it with existing laws, For simultaneous collisions we propose that the new algebraic law be used by recursively breaking these collisions into a sequence ordered by the normal approach velocities of potential contact pairs.

Efficiency, speed, and scaling of two-dimensional passive-dynamic walking

We address performance limits and dynamic behaviours of the two dimensional passive-dynamic bipedal walking mechanisms of Tad McGeer. The results highlight the role of heelstrike in determining the mechanical efficiency of gait, and point to ways of improving efficiency. We analyse several kneed and straight-legged walker designs, with round feet and and point-feet. We present some necessary conditions on the walker mass distribution to achieve perfectly efficient (zero-slope-capable) walking for both kneed and straight-legged models. Our numerical investigations indicate, consistent with a previous study of a simpler model, that such walkers have two distinct gaits at arbitrarily small ground-slopes, of which the longer-step gait is stable at small slopes. Energy dissipation can be dominated by a term proportional to (speed) 2 from tangential foot velocity at heelstrike and from kneestrike, or a term proportional to (speed) 4 from normal foot collisions at heelstrike, depending on the gait, ground-slope, and walker design. For all zeroslope capable straight-legged walkers, the long-step gaits have negligible tangential foot velocity at heelstrike and are hence especially fast at low power. Some apparently chaotic walking motions are numerically demonstrated for a kneed walker.

Motions of a rimless spoked wheel: a simple three-dimensional system with impacts

This paper discusses the mechanics of a rigid rimless spoked wheel, or regular polygon, ‘rolling’ downhill. By ‘rolling’, we mean motions in which the wheel pivots on one ‘support’ spoke until another spoke collides with the ground, followed by transfer of support to that spoke, and so on. We carry out three-dimensional (3D) numerical and analytical stability studies of steady motions of this system. At any fixed, large enough slope, the system has a one-parameter family of stable steady rolling motions. We find analytic approximations for the minimum required slope at a given heading for stable rolling in three dimensions, for the case of many spokes and small slope. The rimless wheel shares some qualitative features with passive–dynamic walking machines; it is a passive 3D system with intermittent impacts and periodic motions. In terms of complexity, it lies between one-dimensional impact oscillators and 3D walking machines. In contrast to a rolling disk on a flat surface which has steady rolling motion...