Jeffrey C. Trinkle

AN IMPLICIT TIME-STEPPING SCHEME FOR RIGID BODY DYNAMICS WITH INELASTIC COLLISIONS AND COULOMB FRICTION

In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small.

On the stability and instantaneous velocity of grasped frictionless objects

An efficient quantitative test for form closure valid for any number of contact points is formulated as a linear program, the optimal objective value of which provides a measure of how far a grasp is from losing form closure. When the grasp does not have form closure, manipulation planning requires a means for predicting the objects stability and instantaneous velocity, given the joint velocities of the hand. The classical approach to computing these quantities is to solve the systems of kinematic inequalities corresponding to all possible combinations of separating or sliding at the contacts. All combinations resulting in the interpenetration of bodies or the infeasibility of the equilibrium equations are rejected. The remaining combination is consistent with all the constraints and is used to compute the velocity of the manipulated object and the contact forces, which indicate whether or not the object is stable. A linear program whose solution yields the same information as the classical approach, usually without explicit testing of all possible combinations of contact interactions, is formulated. >

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulombs law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established.

An implicit time-stepping scheme for rigid body dynamics with Coulomb friction

In this paper a new time-stepping method for simulating systems of rigid bodies is given. Unlike methods which take an instantaneous point of view, our method is based on impulse-momentum equations, and so does not need to explicitly resolve impulsive forces. On the other hand, our method is distinct from previous impulsive methods in that it does not require explicit collision checking and it can handle simultaneous impacts. Numerical results are given for one planar and one three dimensional example, which demonstrate the practicality of the method, and its convergence as the step size becomes small.

An investigation of frictionless enveloping grasping in the plane

Grasping by a two-dimensional hand composed of a palm and two hinged fingers is studied. The mathematics of fric tionless grasping is presented and used in the development of a planner/simulator. The simulator computes the motion of the object using an active constraint set method and assuming exact knowledge of the physical properties of the polygonal object, hand, and support. Grasping is divided into three phases. During the first phase, the initial grasping configura tion is found. In the second phase, the object is manipulated away from the support, bringing it into contact with the palm. In the last phase, the grip is adjusted to minimize the contact forces acting on the object.

Interactive Simulation of Rigid Body Dynamics in Computer Graphics

Interactive rigid body simulation is an important part of many modern computer tools, which no authoring tool nor game engine can do without. Such high‐performance computer tools open up new possibilities for changing how designers, engineers, modelers and animators work with their design problems. This paper is a self contained state‐of‐the‐art report on the physics, the models, the numerical methods and the algorithms used in interactive rigid body simulation all of which have evolved and matured over the past 20 years. Furthermore, the paper communicates the mathematical and theoretical details in a pedagogical manner. This paper is not only a stake in the sand on what has been done, it also seeks to give the reader deeper insights to help guide their future research.

Dextrous manipulation by rolling and finger gaiting

Many practical dextrous manipulation tasks involve large-scale motion of the grasped object while maintaining a stable grasp. To plan such task, one must control both the motion of the object and the contact locations, while also adhering to the workspace constraints typical of multi-fingered hands. We integrate the relevant theories of contact kinematics, nonholonomic motion planning, coordinated object manipulation, grasp stability and finger gaits to develop a general framework for dextrous manipulation planning. To illustrate our approach, the framework is applied to the problem of manipulating a sphere with three hemi-spherical fingertips. The simulation results are presented.

A framework for planning dexterous manipulation

The authors present a general methodology based on R.S. Desais (1988) concept of contact formations and combine it with a model of contact mechanics to solve the dexterous manipulation planning problem. The model of contact mechanics supports the analysis of contact situations with multiple sliding contacts, allowing it to solve problems not solvable if only rolling contacts are allowed. Based on the proposed methodology, a planner would effectively solve two-point boundary value problems by using contact formation transitions to discretize the configuration space of, for example, a hand/object system. Within each discrete cell, or contact formation a model of contact mechanics is used to generate trajectories joining the cells and building a contact formation tree. If a solution exists, the tree grows until it contains a path from the initial grasp to the goal grasp. Then the individual input trajectories (assigned to the arcs of the tree) are combined to generate the complete manipulation trajectories.<<ETX>>

Complete Path Planning for Closed Kinematic Chains with Spherical Joints

We study the path planning problem, without obstacles, for closed kinematic chains with n links connected by spherical joints in space or revolute joints in the plane. The configuration space of such systems is a real algebraic variety whose structure is fully determined using techniques from algebraic geometry and differential topology. This structure is then exploited to design a complete path planning algorithm that produces a sequence of compliant moves, each of which monotonically increases the number of links in their goal configurations. The average running time of this algorithm is proportional to n3 . While less efficient than the O(n) algorithm of Lenhart and Whitesides, our algorithm produces paths that are considerably smoother. More importantly, our analysis serves as a demonstration of how to apply advanced mathematical techniques to path planning problems. Theoretically, our results can be extended to produce collision-free paths, paths avoiding both link—obstacle and link—link collisions. An approach to such an extension is sketched in Section 4.5, but the details are beyond the scope of this paper. Practically, link— obstacle collision avoidance will impact the complexity of our algorithm, forcing us to allow only small numbers of obstacles with “nice” geometry, such as spheres. Link—link collision avoidance appears to be considerably more complex. Despite these concerns, the global structural information obtained in this paper is fundamental to closed kinematic chains with spherical joints and can easily be incorporated into probabilistic planning algorithms that plan collision-free motions. This is also described in Section 4.5.

Dextrous manipulation with rolling contacts

Dextrous manipulation is a problem of paramount importance in the study of multifingered robotic hands. Given a grasped object, the main objectives are: (a) generate trajectories for the finger joints so that through the effects of contact constraints, the object can be transferred to a goal grasp configuration; and (b) derive control algorithms to realize planned trajectories. In this paper, we integrate the relevant theories of contact kinematics, nonholonomic motion planning and grasp stability to develop a general technique for dextrous manipulation planning with multifingered hands. Experimental results are discussed.