Velocity-Based Stability Margins for Fast Bipedal Walking

Abstract

We present velocity-based stability margins for fast bipedal walking that are sufficient conditions for stability, allow comparison between different walking algorithms, are measurable and computable, and are meaningful. While not completely necessary conditions, they are tighter necessary conditions than several previously proposed stability margins. The stability margins we present take into consideration a biped's Center of Mass position and velocity, the reachable region of its swing leg, the time required to swing its swing leg, and the amount of internal angular momentum available for capturing balance. They predict the opportunity for the biped to place its swing leg in such a way that it can continue walking without falling down. We present methods for estimating these stability margins by using simple models of walking such as an inverted pendulum model and the Linear Inverted Pendulum model. We show that by considering the Center of Mass location with respect to the Center of Pressure on the foot, these estimates are easily computable. Finally, we show through simulation experiments on a 12 degree-of-freedom distributed-mass lower-body biped that these estimates are useful for analyzing and controlling bipedal walking. Introduction"How stable is your robot?" is a fundamental yet challenging question to answer, particularly with fast moving legged robots, such as dynamically balanced bipedal walkers. With many traditional control systems, questions of stability and robustness can be answered by eigenvalues, phase margins, loop gain margins, and other stability margins. However, legged robots are nonlinear, under-actuated, combine continuous and discrete dynamics, and do not necessarily have periodic motions. These features make applying traditional stability margins difficult.

Design of a Neural Controller for Walking of a 5-Link Planar Biped Robot via Optimization

“…Extracting the eigenvalues of Poincaré return map is commonly used for analyzing PDW robots. But using of eigenvalues of Poincaré return maps assumes periodicity and is valid only for small deviation from limit cycle [20]. The ZMP criterion has become a very powerful tool for trajectory generation in walking of biped robots.…”

“…The ZMP criterion has become a very powerful tool for trajectory generation in walking of biped robots. However, it needs a stiff joint control of the prerecorded trajectories and this leads to poor robustness in unknown rough terrain [20] while humans and animals show marvelous robustness in walking on irregular terrains. It is well known in biology that there are Central Pattern Generators (CPG) in spinal cord coupling with musculoskeletal system [21]- [23].…”

Inverse Kinematics for a Point-Foot Quadruped Robot with Dynamic Redundancy Resolution

“…The notion of true stability for locomotion is difficult to quantify precisely (see [1]), so heuristics are often utilized for this purpose. These often include: maximizing the static stability margin [2], maximizing the ZMP margin [3], Resolved Momentum Control (RMC) [4], and Zero Spin Center of Pressure (ZSCP) Control [5], etc (see [6] for review). Essentially all of these approaches try to control the center of mass of the entire robot, so we pay particular attention to this issue in this work.…”

Analysis, control and design of walking robots

“…Similarly to the static case, if this point is outside the support polygon, the foot will start to rotate. Contrary to the static case however, there is no direct link between the FRI being outside the support polygon and falling of the robot (Pratt and Tedrake, 2006). This has been (and probably will remain) a critical point of misunderstanding (Ito et al, 2008).…”

“…This is neither a sufficient nor a necessary condition for walking without falling (Pratt and Tedrake, 2006). It must be said however, that having the ZMP strictly within the support polygon makes continuous pose control of the robot a lot easier 3 , which is probably the reason that so many people think that it is the only solution.…”