Yizhar Or

Undulatory Locomotion of Magnetic Multilink Nanoswimmers

Micro- and nanorobots operating in low Reynolds number fluid environments require specialized swimming strategies for efficient locomotion. Prior research has focused on designs mimicking the rotary corkscrew motion of bacterial flagella or the planar beating motion of eukaryotic flagella. These biologically inspired designs are typically of uniform construction along their flagellar axis. This work demonstrates for the first time planar undulations of composite multilink nanowire-based chains (diameter 200 nm) induced by a planar-oscillating magnetic field. Those chains comprise an elastic eukaryote-like polypyrrole tail and rigid magnetic nickel links connected by flexible polymer bilayer hinges. The multilink design exhibits a high swimming efficiency. Furthermore, the manufacturing process enables tuning the geometrical and material properties to specific applications.

Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems

This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior-an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint.

Dynamics of Purcell’s three-link microswimmer with a passive elastic tail

One of the few possible mechanisms for self-propulsion at low Reynolds number is undulations of a passive elastic tail, as proposed in the classical work of Purcell (1977). This effect is studied here by investigating a variant of Purcell’s three-link swimmer model where the front joint angle is periodically actuated while the rear joint is driven by a passive torsional spring. The dynamic equations of motion are formulated and explicit expressions for the leading-order solution are derived by using perturbation expansion. The dependence of the motion on the actuation amplitude and frequency is analyzed, and optimization with respect to the swimmer’s geometry is conducted.

Zeno Stability of the Set-Valued Bouncing Ball

Hybrid dynamical systems consist of both continuous-time and discrete-time dynamics. A fundamental phenomenon that is unique to hybrid systems is Zeno behavior, where the solution involves an infinite number of discrete transitions occurring in finite time, as best illustrated in the classical example of a bouncing ball. In this note, we study the hybrid system of the set-valued bouncing ball, for which the continuous-time dynamics has a set-valued right-hand side. This system is typically used for deriving bounds on the solution of nonlinear single-valued hybrid systems in a small neighborhood of a Zeno equilibrium point in order to establish its local stability. We utilize methods of Lyapunov analysis and optimal control to derive a necessary and sufficient condition for Zeno stability of the set-valued bouncing ball system and to obtain a tight bound on the Zeno time as a function of initial conditions.

Stability of Zeno equilibria in Lagrangian hybrid systems

This paper presents both necessary and sufficient conditions for the stability of Zeno equilibria in Lagrangian hybrid systems, i.e., hybrid systems that model mechanical systems undergoing impacts. These conditions for stability are motivated by the sufficient conditions for Zeno behavior in Lagrangian hybrid systems obtained in we show that the same conditions that imply the existence of Zeno behavior near Zeno equilibria imply the stability of the Zeno equilibria. This paper, therefore, not only presents conditions for the stability of Zeno equilibria, but directly relates the stability of Zeno equilibria to the existence of Zeno behavior.

A General Stance Stability Test Based on Stratified Morse Theory With Application to Quasi-Static Locomotion Planning

This paper considers the stability of an object supported by several frictionless contacts in a potential field such as gravity. The bodies supporting the object induce a partition of the objects configuration space into strata corresponding to different contact arrangements. Stance stability becomes a geometric problem of determining whether the objects configuration is a local minimum of its potential energy function on the stratified configuration space. We use stratified Morse theory to develop a generic stance stability test that has the following characteristics. For a small number of contacts - less than three in 2D and less than six in 3D - stance stability depends both on surface normals and surface curvature at the contacts. Moreover, lower curvature at the contacts leads to better stability. For a larger number of contacts, stance stability depends only on surface normals at the contacts. The stance stability test is applied to quasi-static locomotion planning in two dimensions. The region of stable center-of-mass positions associated with a k-contact stance is characterized. Then, a quasi-static locomotion scheme for a three-legged robot over a piecewise linear terrain is described. Finally, friction is shown to provide robustness and enhanced stability for the frictionless locomotion plan. A full maneuver simulation illustrates the locomotion scheme.

Computation and Graphical Characterization of Robust Multiple-Contact Postures in Two-Dimensional Gravitational Environments

This paper is concerned with computation and graphical characterization of robust equilibrium postures suited to quasistatic multi-legged locomotion. Quasistatic locomotion consists of postures in which the mechanism supports itself against gravity while moving its free limbs to new positions. A posture is robust if the contacts can passively support the mechanism against gravity as well as disturbance forces generated by its moving limbs. This paper is concerned with planar mechanisms supported by frictional contacts in two-dimensional gravitational environments. The kinematic structure of the mechanism is lumped into a rigid body B having the same contacts with the environment and a variable center of mass. Inertial forces generated by moving parts of the mechanism are lumped into a neighborhood of wrenches centered at the nominal gravitational wrench. The robust equilibrium postures associated with a given set of contacts become the center-of-mass locations of B that maintain equilibrium with respect to all wrenches in the given neighborhood. The paper formulates the computation of the robust center-of-mass locations as a linear programming problem. It provides graphical characterization of the robust center-of-mass locations, and gives a geometric algorithm for computing these center-of-mass locations. The paper reports experiments validating the equilibrium criterion on a two-legged prototype. Finally, it describes initial progress toward computation of robust equilibrium postures in three dimensions.

Dynamic Bipedal Walking under Stick-Slip Transitions

This paper studies the hybrid dynamics of bipedal robot walking under stick-slip transitions. We focus on two simple planar models with point feet: the rimless wheel and the compass biped. Unlike most of the existing works in the literature that assume sticking contact between the foot and the ground, we explore the case of insufficient friction, which may induce foot slippage. Numerical simulations of passive dynamic walking reveal the onset of stable periodic solutions involving stick-slip transitions. In the case of the compass biped with controlled joint torque actuation, we demonstrate how one can exploit kinematic trajectories of passive walking in order to induce and stabilize gaits with slipping impact.

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model.

This work studies the motion of Purcells three-link microswimmer in viscous flow, by using perturbation expansion of its dynamics under small-amplitude strokes. Explicit leading-order expressions and next-order correction terms for the displacement of the swimmer are obtained for the cases of a square or circular gait in the plane of joint angles. The correction terms demonstrate the reversal in movement direction for large stroke amplitudes, which has previously only been shown numerically. In addition, asymptotic expressions for Lighthills energetic efficiency are obtained for both gaits. These approximations enable calculating optimal stroke amplitudes and swimmers geometry (i.e. ratio of links’ lengths) for maximizing either net displacement or Lighthills efficiency.

Painlevé’s paradox and dynamic jamming in simple models of passive dynamic walking

Painlevé’s paradox occurs in the rigid-body dynamics of mechanical systems with frictional contacts at configurations where the instantaneous solution is either indeterminate or inconsistent. Dynamic jamming is a scenario where the solution starts with consistent slippage and then converges in finite time to a configuration of inconsistency, while the contact force grows unbounded. The goal of this paper is to demonstrate that these two phenomena are also relevant to the field of robotic walking, and can occur in two classical theoretical models of passive dynamic walking — the rimless wheel and the compass biped. These models typically assume sticking contact and ignore the possibility of foot slippage, an assumption which requires sufficiently large ground friction. Nevertheless, even for large friction, a perturbation that involves foot slippage can be kinematically enforced due to external forces, vibrations, or loose gravel on the surface. In this work, the rimless wheel and compass biped models are revisited, and it is shown that the periodic solutions under sticking contact can suffer from both Painlevé’s paradox and dynamic jamming when given a perturbation of foot slippage. Thus, avoidance of these phenomena and analysis of orbital stability with respect to perturbations that include slippage are of crucial importance for robotic legged locomotion.