Yan-Bin Jia

Pose and Motion from Contact

In the absence of vision, grasping an object often relies on tactile feedback from the fingertips. As the finger pushes the object, the fingertip can feel the contact point move. If the object is known in advance, from this motion the finger may infer the location of the contact point on the object, and thereby, the object pose. This paper primarily investigates the problem of determining the pose (orientation and position) and motion (velocity and angular velocity) of a planar object with known geometry from such contact motion generated by pushing. A dynamic analysis of pushing yields a nonlinear system that relates through contact the object pose and motion to the finger motion. The contact motion on the fingertip thus encodes certain information about the object pose. Nonlinear observability theory is employed to show that such information is sufficient for the finger to “observe” not only the pose, but also the motion of the object. Therefore, a sensing strategy can be realized as an “observer” of the nonlinear dynamical system. Two observers are subsequently introduced. The first observer, based on the work of Gauthier, Hammouri, and Othman (1992), has its “gain” determined by the solution of a Lyapunov-like equation; it can be activated at any time instant during a push. The second observer, based on Newton’s method, solves for the initial (motionless) object pose from three intermediate contact points during a push. Under the Coulomb-friction model, the paper deals with support friction in the plane and/or contact friction between the finger and the object. Extensive simulations have been done to demonstrate the feasibility of the two observers. Preliminary experiments (with an Adept robot) have also been conducted. A contact sensor has been implemented using strain gauges.

Multiple impacts: A state transition diagram approach

Impact happens when two or more bodies collide, generating very large impulsive forces in a very short period of time during which kinetic energy is first absorbed and then released after some loss. This paper introduces a state transition diagram to model a frictionless multibody collision. Each state describes a different topology of the collision characterized by the set of instantaneously active contacts. A change of state happens when a contact disappears at the end of restitution, or when a disappeared contact reappears as the relative motion of two bodies goes from separation into penetration. Within a state, (normal) impulses are coupled differentially subject to relative stiffnesses at the active contact points and the strain energies stored there. Such coupling may cause restart of compression from restitution during a single impact. Impulses grow along a bounded curve with first-order continuity, and converge during the state transitions. To solve a multibody collision problem with friction and tangential compliance, the above impact model is integrated with a compliant impact model. The paper compares model predictions to a physical experiment for the massé shot, which is a difficult trick in billiards, with a good result.

Geometric sensing of known planar shapes

Industrial assembly involves sensing the pose (orientation and position) of a part. Efficient and reliable sensing strategies can be developed for an assembly task if the shape of the part is known in advance. In this article we investigate two problems of determining the pose of a polygonal part of known shape for the cases of a continuum and a finite number of possible poses, respectively. The first problem, named sensing by inscription, involves determining the pose of a convex n-gon from a set of m sup porting cones. An algorithm with running time O(nm) that almost always reduces to O(n + m log n) is presented to solve for all possible poses of the polygon. We prove that the number of possible poses cannot exceed 6n, given m ≥ 2 supporting cones with distinct vertices. Simulation experiments demonstrate that two supporting cones are sufficient to determine the real pose of the n-gon in most cases. Our results imply that sensing in practice can be carried out by obtaining viewing angles of a planar part at multiple exterior sites in the plane. On many occasions a parts feeder will have reduced the number of possible poses of a part to a small finite set. Our second problem, named sensing by point sampling, is con cerned with a more general version: finding the minimum number of sensing points required to distinguish between n polygonal shapes with a total of m edges. In practice this can be implemented by embedding a series of point light detectors in a feeder tray or by using a set of mechanical probes that touch the feeder at a finite number of predeter mined points. We show that this problem is equivalent to an NP-complete set-theoretic problem introduced as Discriminat ing Set, and present an O(n2m2 ) approximation algorithm to solve it with a ratio of 2 In n. Furthermore, we prove that one can use an algorithm for Discriminating Set with ratio c log n to construct an algorithm for Set Covering with ratio c log n + O(log log n). Thus the ratio 2 In n is asymptotically optimal unless NP ⊂ DTIME(n poly log n ), a consequence of known results on approximating Set Covering. The complexity of subproblems of Discriminating Set is also analyzed, based on their relationship to a generalization of Independent Set called 3-Independent Set. Finally, simulation results suggest

Grasping deformable planar objects: Squeeze, stick/slip analysis, and energy-based optimalities

Robotic grasping of deformable objects is difficult and under-researched, not simply due to the high computational cost of modeling. More fundamentally, several issues arise with the deformation of an object being grasped: a changing wrench space, growing finger contact areas, and pointwise varying contact modes inside these areas. Consequently, contact constraints needed for deformable modeling are hardly established at the beginning of the grasping operation. This paper presents a grasping strategy that squeezes the object with two fingers under specified displacements rather than forces. A ‘stable’ squeeze minimizes the potential energy for the same amount of squeezing, while a ‘pure’ squeeze ensures that the object undergoes no rigid body motion as it deforms. Assuming linear elasticity, a finite element analysis guarantees equilibrium and the uniqueness of deformation during a squeeze action. An event-driven algorithm tracks the contact regions as well as the modes of contact in their interiors under Coulomb friction, which in turn serve as the needed constraints for deformation update. Grasp quality is characterized as the amount of work performed by the grasping fingers in resisting a known push by some adversary finger. Simulation and multiple experiments have been conducted to validate the results over solid and ring-like 2D objects.

Computation on Parametric Curves with an Application in Grasping

Curved shapes are frequent subjects of maneuvers by the human hand. In robotics, it is well known that antipodal grasps exist on curved objects and guarantee force closure under proper finger contact conditions. This paper presents an efficient algorithm that computes, up to numerical resolution, all pairs of antipodal points on a simple, closed, and twice continuously differentiable plane curve. Dissecting the curve into segments everywhere convex or everywhere concave, the algorithm marches simultaneously on a pair of such segments with provable convergence and interleaves marching with numerical bisection recursively. It makes use of new insights into the differential geometry at two antipodal points. We have avoided resorting to traditional nonlinear programming, which would neither be quite as efficient nor guarantee to find all antipodal points. A byproduct of our result is a procedure that constructs all common tangent lines of two curves, achieving quadratic convergence rate. Dissection and the coupling of marching with bisection constitute an algorithm design scheme potentially applicable to computational problems involving curves and curved shapes.

Three-dimensional impact: energy-based modeling of tangential compliance

Impact is indispensable in robotic manipulation tasks in which objects and/or manipulators move at high speeds. Applied research using impact has been hindered by underdeveloped computational foundations for rigid-body collision. This paper studies the computation of tangential impulse as two rigid bodies in the space collide at a point with both tangential compliance and friction. It extends Stronge’s spring-based planar contact structure to three dimensions by modeling the contact point as a massless particle able to move tangentially on one body while connected to an infinitesimal region on the other body via three orthogonal springs. Slip or stick is indicated by whether the particle is still or moving. Impact analysis is carried out using normal impulse rather than time as the only independent variable, unlike in previous work on tangential compliance. This is due to the ability to update the energies stored in the three springs. Collision is governed by a system of differential equations that are solvable numerically. Modularity of the impact model makes it easy to be integrated into a multibody system, with one copy at each contact, in combination with a model for multiple impacts that governs normal impulses at different contacts.

Grasping curved objects through rolling

Grasping a curved object free in the plane may be done through rolling a pair of fingers on the objects boundary. Each finger is equipped with a tactile sensor able to record any instantaneous point contact with the object. Contact kinematics reveal a relationship between the amount of finger rotations and the total curvatures of the boundary segments of the fingers and the object respectively traversed by the two contact points during the same period of rolling. Such relationship makes it possible to localize both fingers relative to the object from a few pairs of simultaneously taken finger contacts at different time instants. A least squares formulation of this localization problem can then be solved by the Levenberg-Marquardt algorithm. Simulation results are presented. After localization, a simple open loop strategy is used to control the continual rolling of the fingers until they simultaneously reach two locations on the objects boundary where a grasp is finally performed.

Curvature-based computation of antipodal grasps

It is well known that antipodal grasps can be achieved on curved objects in the presence of friction. This paper presents an efficient algorithm that finds, up to numerical resolution, all pairs of antipodal points on a closed, simple, and twice continuously differentiable plane curve. Dissecting the curve into segments everywhere convex or everywhere concave, the algorithm marches simultaneously on a pair of such segments with provable convergence and interleaves marching with numerical bisection. It makes use of new insights into the differential geometry at two antipodal points. We have avoided resorting to traditional nonlinear programming which would neither be quite as efficient nor guarantee to find all antipodal points. Dissection and the coupling of marching with bisection introduced in this paper are potentially applicable to many optimization problems involving curves and curved shapes.

Modeling Deformations of General Parametric Shells Grasped by a Robot Hand

The robot hand applying force on a deformable object will result in a changing wrench space due to the varying shape and normal of the contact area. Design and analysis of a manipulation strategy thus depend on reliable modeling of the objects deformations as actions are performed. In this paper, shell-like objects are modeled. The classical shell theory [P. L. Gould, Analysis of Plates and Shells. Englewood Cliffs, NJ: Prentice-Hall, 1999; V. V. Novozhilov, The Theory of Thin Shells . Gronigen, The Netherlands: Noordhoff, 1959; A. S. Saada, Elasticity: Theory and Applications. Melbourne, FL: Krieger, 1993; S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. New York: McGraw-Hill, 1959] assumes a parametrization along the two lines of curvature on the middle surface of a shell. Such a parametrization, while always existing locally, is very difficult, if not impossible, to derive for most surfaces. Generalization of the theory to an arbitrary parametric shell is therefore not immediate. This paper first extends the linear and nonlinear shell theories to describe extensional, shearing, and bending strains in terms of geometric invariants, including the principal curvatures and vectors, and their related directional and covariant derivatives. To our knowledge, this is the first nonparametric formulation of thin-shell strains. A computational procedure for the strain energy is then offered for general parametric shells. In practice, a shell deformation is conveniently represented by a subdivision surface [F. Cirak, M. Ortiz, and P. Schröder, “Subdivision surfaces: A new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng., vol. 47, pp. 2039-2072, 2000]. We compare the results via potential-energy minimization over a couple of benchmark problems with their analytical solutions and numerical ones generated by two commercial software packages: ABAQUS and ANSYS. Our method achieves a convergence rate that is one order of magnitude higher. Experimental validation involves regular and free-form shell-like objects of various materials that were grasped by a robot hand, with the results compared against scanned 3-D data with accuracy of 0.127 mm. Grasped objects often undergo sizable shape changes, for which a much higher modeling accuracy can be achieved using the nonlinear elasticity theory than its linear counterpart.

Semidifferential Invariants for Tactile Recognition of Algebraic Curves

In this paper we study the recognition of low-degree polynomial curves based on minimal tactile data. Euclidean differential and semidifferential invariants have been derived for quadratic curves and special cubic curves that are found in applications. These invariants, independent of translation and rotation, are evaluated over the differential geometry at up to three points on a curve. Their values are independent of the evaluation points. Recognition of the curve reduces to invariant verification with its canonical parametric form determined along the way. In addition, the contact locations are found on the curve, thereby localizing it relative to the touch sensor. Simulation results support the method despite numerical errors. Preliminary experiments have also been carried out with the introduction of a method for reliable curvature estimation. The presented work distinguishes itself from traditional model-based recognition in its ability to simultaneously recognize and localize a shape from one of several classes, each consisting of a continuum of shapes, by the use of local data.