We introduce Pressure Field Contact (PFC), an approximate model for predicting the contact surface, pressure distribution, and net contact wrench between nominally rigid objects. PFC combines and generalizes two ideas: a bed of springs (an `elastic foundation') and hydrostatic pressure. Continuous pressure fields are computed offline for the interior of each nominally rigid object. Unlike hydrostatics or elastic foundations, the pressure fields need not satisfy mechanical equilibrium conditions. When two objects nominally overlap, a contact surface is defined where the two pressure fields are equal. This static pressure is supplemented with a dissipative rate-dependent pressure and friction to determine tractions on the contact surface. The contact wrench between pairs of objects is an integral of traction contributions over this surface. PFC evaluates much faster than elasticity-theory models, while showing the essential trends of force, moment, and stiffness increase with contact load. It yields continuous wrenches even for non-convex objects and coarse meshes. The method shows promise as sufficiently fast, accurate, physical, and robust for robotics applications including motion and tactile sensor simulation, controller learning and synthesis, state estimation, and design-in-simulation.

From which states and with what controls can a biped avoid falling or reach a given target state? What is the most robust way to do these? So as to help with the design of walking robot controllers, and perhaps give insights into human walking, we address these questions using two simple 2-D models: the inverted pendulum (IP) and linear inverted pendulum (LIP). Each has one state variable at mid-stance, i.e., hip velocity, and two state-dependent controls at each step, i.e., push-off magnitude and step length (IP) and step time and length (LIP). Using practical targets and constraints, we compute all combinations of initial states and control actions for the next step, such that the robot can, with the best possible future controls, avoid falling for n steps or reach a target within n steps. All such combinations constitute regions in the combined space of states and controls. Farther from the boundaries of these regions, the robot tolerates larger errors and disturbances. Furthermore, for these models, and thus possibly real bipeds, usually if it is possible to avoid falling, it is possible to reach the target, and if it is possible to reach the target, it is possible to do so in two steps.

We present a method for designing a walking controller for the walking robot Cornell Ranger. Our goal is a controller that can be designed using model-based optimization, and then transferred directly to the robot without the need for after-the-fact hand-tuning. The structure of the controller is hierarchical, with a high-level balance controller that plans step-to-step motions, and a lower-level joint controller that coordinates the individual joint motors to achieve the desired limb motions. The balance controller is designed through optimization, with the explicit goals of a) achieving a desired walking speed while b) minimizing energy use and c) avoiding falls due to disturbances. We demonstrate this walking controller on the Cornell Ranger, and find that the resulting gait is comparable to that of a previous (hand-tuned) controller, with regard to energy use, speed regulation, and fall prevention.

We present an approach to high-level control for bipedal walking exemplified with a 2D point-mass inextensible-legs inverted-pendulum model. Balance control authority here is only from step position and trailing-leg push-off, both of which are bounded to reflect actuator limits. The controller is defined implicitly as the solution of an optimization problem. The optimization robustly avoids falling for given bounded disturbances and errors and, given that, minimizes the number of steps to reach a given target speed. The optimization can be computed in advance and stored for interpolated real-time use online. The general form of the resulting optimized controller suggests a few simple principles for regulating walking speed: 1) The robot should take bigger steps when speeding up and should also take bigger steps when slowing down 2) push-off is useful for regulating small changes in speed, but it is fully saturated or inactive for larger changes in speed. While the numerically optimized model is simple, the approach should be applicable to, and we plan to use it for, control of bipedal robots in 3D with many degrees of freedom.

Before designing a controller in detail, we ask this question: for a given robot state, and every possible control action, what is the minimum number of steps needed to get to a given target state (if it is possible to do so)? Our biped model is a 2D inverted pendulum with massless legs. We have two controls: (i) the magnitude of an impulsive push-off just before heel-strike and (ii) the step length (location of the next heel-strike). The maximum impulse and minimum step time are bounded to reflect limited motor strength. We compute the set of initial mid-stance velocities from which the biped can reach a given target mid-stance velocity in n or fewer steps. The result: for most target speeds and initial velocities, and with realistically strong actuators, it is possible to reach the target in two steps, if it is possible to reach it at all. This `two steps is enough' result expands on Koolen et al.'s [1] results for capturability of the linear inverted pendulum and is consistent with some human balance and visual guidance experiments.

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.

This paper addresses some performance limits of the kneed and non-kneed passive-dynamic walking machines discovered by McGeer (1990). Energetic inefficiency is measured by the slope /spl gamma/ needed to sustain gait, with /spl gamma/=0 being perfectly efficient. We show some necessary conditions on the walker mass distribution to achieve perfectly efficient walking. From our experience and study of a simpler model, only two gaits exist; the longer-step gait is stable at small enough slopes. Speed is regulated by energy dissipation. Dissipation can be dominated by a term proportional to speed or a term proportional to speed from normal foot collisions, depending on the gait, slope, and walker design. For special mass distributions of kneeless walkers, the long-step gaits are especially fast at small slopes. A period doubling route to chaos is numerically demonstrated for the kneed walker.

The authors present two geometric descriptions of the frictional properties of a rigid body sliding on a planar surface. The limit surface from classical plasticity theory, is the boundary of the set of all possible frictional forces and moments that can be sustained by the frictional interface. Zhukovskii's moment function is the frictional moment as a function of the instantaneous center of rotation's location. Both of these descriptions implicitly contain the full relationship between slip motion and frictional load for an object that makes contact governed by a useful class of friction laws that includes Coulomb friction. These surfaces can be used to deduce results concerning the overall frictional motion behavior of rigid bodies, such as the existence of characteristic final slip motion directions.>