Dynamic bipedal robot locomotion has achieved remarkable success due in part to recent advances in trajectory generation and nonlinear control for stabilization. A key assumption utilized in both theory and experiments is that the robot’s stance foot always makes no-slip contact with the ground, including at impacts. This assumption breaks down on slippery low-friction surfaces, as commonly encountered in outdoor terrains, leading to failure and loss of stability. In this work, we extend the theoretical analysis and trajectory optimization to account for stick-slip transitions at point foot contact using Coulomb’s friction law. Using AMBER-3M planar biped robot as an experimental platform, we demonstrate for the first time a slippery walking gait which can be stabilized successfully both on a lubricated surface and on a rough no-slip surface. We also study the influence of foot slippage on reducing the mechanical cost of transport, and compare energy efficiency in both numerical simulation and experimental measurement.

Hopping is a highly dynamic motion requiring precise input over brief moments of ground contact in order to achieve desired performance. While this problem has been approached from multiple perspectives, this work provides a comparative analysis of two robot models. The first model uses an actuator to store energy in a spring and release it during the ground phase, while the second uses an actuator to move an additional mass vertically to generate force on the spring. In the first model, analytic expressions are used to find the desired controllers, while trajectory optimization is used in the latter. Orbital stability of each model under the conditions of uncertain damping and poor estimation of the hop height is examined. To this end, Poincaré analysis is used to give a metric of stability in the presence of different initial conditions and parameter uncertainty. Simulations show that the first model converges quickly to a point near the desired height determined by the amount of uncertain damping present. The second model is less robust to uncertainty, but is be made to converge to a desired height with the addition of PD control around the optimal trajectory. This robustness is improved with different gains in the controller. In experiments performed on hardware for the second model, stability is observed through convergence to a periodic orbit within several hops.

Soft robotics is an emerging bioinspired concept of actuation, with promising applications for robotic locomotion and manipulation. Focusing on actuation by pressurized embedded fluidic networks, existing works examine quasi-static locomotion by inviscid fluids. This paper presents analytic formulation and closed-form solutions of an elastic actuator with pressurized fluidic networks, while accounting for the effects of solid inertia and elasticity, as well as fluid viscosity. This allows modeling the system's step response and frequency response as well as suggesting mode elimination and isolation techniques. The theoretical results describing the viscous-elastic-inertial dynamics of the actuator are illustrated by experiments. The approach presented in this paper may pave the way for the design and implementation of soft robotic legged locomotion that exploits dynamic effects.

Robotic locomotion typically involves using gaits-periodic changes of kinematic shape, which induce net motion of the body in a desired direction. An example is robotic microswimmers, which are inspired by motion of swimming microorganisms. One of the most famous theoretical models of a microswimmer is Purcell's planar three-link swimmer, whose structure possesses two axes of symmetry. Several works analyzed gaits for robotic three-link systems based on body-fixed velocity integrals. Using this approach, finite motion in desired directions can only be obtained approximately. In this study, we propose gaits which are based on analysis of the system's structural symmetries, and generate exact motion along principal directions without net rotation. Another gait that produces almost pure rotation is presented, and bounds on the small-amplitude residual translation are obtained by using perturbation expansion. Next, the theory is extended to more realistic swimmers which have only one symmetry axis. Gaits for such swimmers which generate net translation are proposed, and their small-amplitude motion is analyzed using perturbation expansion. The theoretical results are demonstrated by using numerical simulations and conducting controlled motion experiments with a robotic macroswimmer prototype in a highly viscous fluid.

This paper studies the hybrid dynamics of simple biped robot models under stick-slip transitions. Unlike most of existing works in the literature that assume sticking contact, we explore the case of insufficient friction which induces foot slippage. Numerical simulations of passive dynamic walking reveal the onset of stable periodic solutions involving stick-slip transitions. In the case of a biped with joint torques actuation, we demonstrate how one can induce and stabilize stick-slip gaits which significantly reduce the energetic cost of walking and enforce slippage even for large friction.

This paper is concerned with the stability of planar mechanisms supported by multiple frictional contacts against gravity. Stability of equilibrium postures is investigated under initial perturbations which may involve sliding or separation at the contacts. The frictional dynamics is formulated using the notion of contact modes, and the related problems of solution ambiguity and inconsistency are reviewed. The paper then uses the condition of strong equilibrium to eliminate ambiguities, and defines a new condition of kinematic-strong equilibrium which additionally eliminates frictional inconsistencies. It is then proven that strong equilibrium is necessary for stability of frictional equilibrium postures, and that kinematic-strong equilibrium guarantees finite-time recovery of an initially perturbed contact. The results are demonstrated on a reduced model of a rigid body having a variable center-of-mass and supported by two frictional contacts. A companion paper completes the analysis of this reduced problem by investigating the overall hybrid dynamics and deriving sufficient conditions for its stability.

This paper is concerned with the hybrid dynamics and stability of a planar rigid body supported by two frictional contacts in a gravitational field. The stability of equilibrium postures is investigated under initial perturbations which may involve sliding or separation of the contacts. The paper formulates the hybrid dynamics induced by collisions and impacts at the two contacts. The Zeno behavior of bouncing and clattering motion converging to limit points of one- or two-contact re-establishment are analyzed, and a condition guaranteeing convergence to a Zeno point is derived. Finally, this condition is combined with the results of the companion paper to derive sufficient conditions for stability of frictional two-contact equilibrium postures of a planar rigid body.

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 object's configuration space into strata corresponding to different contact arrangements. Stance stability becomes a geometric problem of determining whether the object's 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 ak-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.

The dynamics of a rigid body sliding on a frictional contact can have multiple solutions as well as sudden contact-mode transitions. This paper is concerned with dynamic jamming, an event where a sliding rigid body suddenly jamms and experiences an impact-like transition into free flying mode. Using a simple experiment that mimics a sliding rigid-body situation, dynamic jamming is recorded for the first time. The phenomenon occurs almost at the theoretical position-and-velocity prediction, indicating that this type of jamming is not a mere artifact of the rigid-body modeling paradigm. A new interpretation of dynamic jamming is offered in terms of the body's instantaneous acceleration center. Once this center reaches a graphically determinable jamming line, the body ceases its sliding mode and experiences a dynamic jamming event. Based on this insight, some ways to prevent dynamic jamming are discussed.

Quasistatic multi-legged locomotion consists of a sequence of equilibrium postures where the mechanism supports itself against gravity while moving free limbs to new positions. A posture maintains equilibrium if the contacts can passively support the mechanism against gravity. This paper is concerned with computation and graphical characterization of equilibrium postures for mechanisms supported by frictional contacts in a three-dimensional gravitational field. For a given set of contacts, this problem reduces to the computing feasible region R, defined as the center-of-mass positions that maintain equilibrium while satisfying the frictional constraints. This paper continues a previous work by the authors, and provides a new method for computing the boundary of R for 3-contact stances. The paper also gives physical and geometric interpretations for the boundary of R and discusses its relation with the classical support polygon principle. Finally, the paper reports experimental results that validate the theoretical computation.

Quasistatic multi-legged locomotion consists of a sequence of equilibrium postures where the mechanism supports itself against gravity while moving free limbs to new positions. A posture maintains equilibrium if the contacts can passively support the mechanism against gravity. This paper is concerned with identifying and computing equilibrium postures for mechanisms supported by frictional contacts in a three-dimensional gravitational field. The complex kinematic structure of the mechanism is lumped into a single rigid body B having the same contacts with the environment and a variable center of mass. The identification of equilibrium postures associated with a given set of contacts is reduced to the identification of center-of-mass locations of B that maintain equilibrium stances while all nonlinear frictional constraints are satisfied. Focusing on 3-contact stances, this paper provides an exact formulation for the boundary of the center-of-mass feasible region R as a solution of implicit polynomials. The paper also provides a conservative polyhedral approximation of R by using techniques for projection of convex polytopes. The exact and approximate computations of R are demonstrated with graphical examples

This paper is concerned with the problem of identifying robust equilibrium postures of a planar mechanism supported by fixed frictional contacts in a two-dimensional gravitational field. The complex kinematic structure of the mechanism is lumped into a single rigid body, B, with 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 identification of the robust equilibrium postures associated with a given set of contacts is reduced to the identification of center-of-mass locations that maintain equilibrium of B with respect to any wrench in the given neighborhood. The static response of B to an external wrench involves static indeterminacy and frictional constraints. The region of center-of-mass locations that generate equilibrium with respect to a particular external wrench is formulated as a linear programming problem, and a full graphical characterization is provided. The result is then generalized to robust equilibrium postures that resist a neighborhood of external wrenches. Finally, we present experimental results that validate the criteria for feasible equilibrium postures.

We describe a practical method for computing the statically stable postures of a planar mechanism in a two-dimensional gravitational field. Our method lumps the complex kinematic structure of the mechanism into a rigid body B having a variable center of mass, while inertial forces generated by moving parts of the mechanism are lumped into a neighborhood of disturbance wrenches centered at the nominal gravitational wrench. Given this reduction, the statically stable postures of the mechanism correspond to center-of-mass locations of B that guarantee static stability for the neighborhood of wrenches. However, the response of B to an applied wrench involves dynamic ambiguity associated with different reaction modes at the contacts. Hence we compute only those center-of-mass locations where B maintains a non-ambiguous equilibrium for the entire neighborhood of wrenches. These center-of-mass locations are called the robust stability region of the posture. Focusing on two-contact postures, we compute the robust stability region as an arrangement of planar cells. Finally, we sketch an application of the results to robust locomotion planning of a 3-legged mechanism.