In this paper, we present a principled method to model gen- eral planar sliding motion with distributed convex contact patch. The effect of contact patch with indeterminate pressure distribution can be equivalently modeled as the contact wrench at one point contact. We call this point equivalent contact point. Our dynamic model embeds ECP within the equations of slider's motion and friction model which approx- imates the distributed contact patch, and eventually brings us a system of quadratic equations. This discrete-time dynamic model allows us to solve for the two components of tangential friction impulses, the friction moment and the slip speed. The state of the slider as well as the ECP can be computed by solving a system of linear equations once the contact impulses are computed. In addition, we derive the closed form solutions for the state of slider for quasi-static motion. Furthermore, in pure trans- lation case, based on the discrete-time model, we present the closed form expressions for the friction impulses the slider suffers and the state of it at each time step. Simulation examples are shown to demonstrate the validity of our approach.
In general, without knowing the external forces and moments (including applied and friction forces), the ECP or center of friction is not computable. In addition, the acceleration of a body whose positions of external forces (contact point of pushing, center of gravity, etc.) are above or below the support plane will, in general, cause the shift in the pressure distribution and corresponding shift in the center of friction or ECP @cite_21 . Due to the reasons above,
@cite_21
2569324215 2029342956 2098030884 2133010631
In this paper, we present a convex optimization problem to generate Center of Mass (CoM) and momentum trajectories of a walking robot, such that the motion robustly satisfies the friction cone constraints on uneven terrain. We adopt the Contact Wrench Cone (CWC) criterion to measure a robot's dynamical stability, which generalizes the venerable Zero Moment Point (ZMP) criterion. Unlike the ZMP criterion, which is ideal for walking on flat ground with unbounded tangential friction forces, the CWC criterion incorporates non-coplanar contacts with friction cone constraints. We measure the robustness of the motion using the margin in the Contact Wrench Cone at each time instance, which quantifies the capability of the robot to instantaneously resist external force torque disturbance, without causing the foot to tip over or slide. For pre-specified footstep location and time, we formulate a convex optimization problem to search for robot linear and angular momenta that satisfy the CWC criterion. We aim to maximize the CWC margin to improve the robustness of the motion, and minimize the centroidal angular momentum (angular momentum about CoM) to make the motion natural. Instead of directly minimizing the non-convex centroidal angular momentum, we resort to minimizing a convex upper bound. We show that our CWC planner can generate motion similar to the result of the ZMP planner on flat ground with sufficient friction. Moreover, on an uneven terrain course with friction cone constraints, our CWC planner can still find feasible motion, while the outcome of the ZMP planner violates the friction limit. 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. Zhukovskii's moment function (N.E. Zhukovskii, Collected Works, Vol. 1, Gostekhizdat, Moscow, 1948, pp. 339–354) is the net frictional moment about the body's 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 Zhukovskii's 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. We would like to give robots the ability to position and orient parts in the plane by pushing, particularly when the parts are too large or heavy to be grasped and lifted. Unfortunately, the motion of a pushed object is generally unpredictable due to unknown support friction forces. With multiple pushing contact points, however, it is possible to find pushing directions that cause the object to remain fixed to the manipulator. These are called stable pushing directions. In this article we consider the problem of planning pushing paths using stable pushes. Pushing imposes a set of nonholonomic velocity constraints on the motion of the object, and we study the issues of local and global controllability during pushing with point contact or stable line contact. We describe a planner for finding stable pushing paths among obstacles, and the planner is demon strated on several manipulation tasks. We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.