Stephen Berard

daVinci Code: A Multi-Model Simulation and Analysis Tool for Multi-Body Systems

This paper discusses the design and current capabilities of a new software tool, dVC, capable of simulating planar systems of bodies experiencing unilateral contacts with friction. Since different problems require different levels of accuracy, dVC provides user-selectable body types (rigid or locally-compliant), motion models (first-order, quasi-static, dynamic), and several state-of-the-art time-stepping methods. One can also choose to include friction between each body and the plane of motion. To support optimal and robust part design, dVC also allows on-the-fly changes to parameters of the geometric and physical models. The results obtained for three representative planar problems are presented: the design of a passive part-orienting device, the planning of a mesoscale assembly operation, and the design of a grasp strategy.

A time-stepping scheme for quasistatic multibody systems

Two new instantaneous-time models for predicting the motion and contact forces of three-dimensional, quasistatic multi-rigid-body systems are developed; one linear and one nonlinear. The nonlinear characteristic is the result of retaining the usual quadratic friction cone in the model. Discrete-time versions of these models provide the first time-stepping methods for such systems. As a first step to understanding their usefulness in simulation and manipulation planning, a theorem for solution uniqueness is presented along with simulation results for a simple example

An Implicit Time-Stepping Method for Multibody Systems with Intermittent Contact

In this paper we present an implicit time-stepping scheme for multibody systems with intermittent contact by incorporating the contact constraints as a set of complementarity and algebraic equations within the dynamics model. Two primary sources of stability and accuracy problems in prior time stepping schemes for differential complementarity models of multibody systems are the use of polyhedral representations of smooth bodies and the approximation of the distance function (arising from the decoupling of collision detection from the solution of the dynamic time-stepping subproblem). Even the simple example of a disc rolling on a table without slip encounters these problems. We assume each object to be a convex object described by an intersection of convex inequalities. We write the contact constraints as complementarity constraints between the contact force and a distance function dependent on the closest points on the objects. The closest points satisfy a set of algebraic constraints obtained from the KKT conditions of the minimum distance problem. These algebraic equations and the complementarity constraints taken together ensure satisfaction of the contact constraints. This enables us to formulate a geometrically implicit time-stepping scheme (i.e., we do not need to approximate the distance function) as a nonlinear complementarity problem (NCP). The resulting time-stepper is therefore more accurate; further it is the first geometrically implicit time-stepper that does not rely on a closed form expression for the distance function. We demonstrate through example simulations the fidelity of this approach to analytical solutions and previously described simulation results.

An Implicit Time-Stepping Method for Quasi-Rigid Multibody Systems With Intermittent Contact

We recently developed a time-stepping method for simulating rigid multi-body systems with intermittent contact tha t is implicit in the geometric information [1]. In this paper, we ex tend this formulation to quasi-rigid or locally compliant objec ts, i.e., objects with a rigid core surrounded by a compliant layer, si milar to Song et al. [2]. The difference in our compliance mode l from existing quasi-rigid models is that, based on physicalmotivations, we assume the compliant layer has a maximum possibl e normal deflection beyond which it acts as a rigid body. Therefore, we use an extension of the Kelvin-Voigt (i.e. linear springdamper) model for obtaining the normal contact forces by incorporating the thickness of the compliant layer explicitl y in the contact model. We use the Kelvin-Voigt model for the tangent ial forces and assume that the contact forces and moment satisfy an ellipsoidal friction law. We model each object as an intersection of convex inequalities and write the contact constraint as a complementarityconstraint between the contact force and a distance function de pendent on the closest points and the local deformation of the bo dy. The closest points satisfy a system of nonlinear algebraic e quations and the resultant continuous model is a Differential C omplementarity Problem (DCP). This enables us to formulate a g eometrically implicit time-stepping scheme for solving theDCP which is more accurate than a geometrically explicit scheme . The discrete problem to be solved at each time-step is a mixed nonlinear complementarity problem.

Sources of Error in a Simulation of Rigid Parts on a Vibrating Rigid Plate

We present a simulation study of an important rigid-body contact problem. The system in question is composed of a rigid plate and a single rigid body (or particle). The plate follows a prescribed periodic motion of small amplitude and high frequency, such that the net force applied to the part appears to be from a time-independent, position-dependent velocity field in the plane of the plate. Theoretical results obtained by Vose et al. were found to be in good agreement with simulation results obtained with the Stewart―Trinkle time-stepping method. In addition, simulations were found to agree with the qualitative experimental results of Vose et al. After such verification of the simulation method, additional numerical studies were done that would have been impossible to carry out analytically. Specifically, we were able to demonstrate the convergence of the method with decreasing step size (as predicted theoretically by Stewart). Further analytical and numerical studies will be carried out in the future to develop and select robust simulation methods that best satisfy the speed and accuracy requirements of different applications. With the accuracy of our time-stepper verified for this system, we were able to study the inverse problem of designing new plate motions to generate a desired part motion. This is done through an optimization framework, where a simulation of the part interacting with the plate (including the full dynamics of the system) is performed, and based on the results of the simulation the motion of the plate is modified. The learned (by simulation) plate motion was experimentally run on the device, and without any tuning (of the simulation parameters or device parameters) our learned plate motion produced the desired part motion.

A geometrically implicit time-stepping method for multibody systems with intermittent contact

Accurate dynamic simulation with robust handling of intermittent contact is necessary for a wide range of robotics problems, including the design of parts feeding devices, manipulation and kinodynamic planning, and designing grasp strategies. In this paper we present an implicit time-stepping scheme for dynamic simulation of multibody systems with intermittent contact by incorporating the contact constraints as a set of complementarity and algebraic equations within the dynamics model. We model each body as an intersection of convex inequalities and write the contact constraints as complementarity constraints between the contact force and a distance function dependent on the closest points on the bodies. The closest points satisfy a set of algebraic constraints obtained from the Karush–Kuhn–Tucker (KKT) conditions of the minimum distance problem. We prove that these algebraic equations and the complementarity constraints taken together ensure satisfaction of the contact constraints. This enables us to formulate a geometrically implicit time-stepping scheme (i.e. we do not need to approximate the distance function) as a nonlinear complementarity problem. The resulting time-stepper is therefore more accurate and does not rely on a closed-form distance function. We demonstrate through example simulations the fidelity of this approach to analytical solutions and previously described simulation and experimental results.

Sources of error in a rigid body simulation of rigid parts on a vibrating rigid plate

We present a simulation study of an important rigid body contact problem. The system in question is composed of a rigid plate and a single rigid body (or particle). The plate follows a prescribed periodic motion of small amplitude and high frequency, such that the net force applied to the part appears to be from a time-independent, position-dependent velocity field in the plane of the plate. Theoretical results obtained by Vose et al. were found to be in good agreement with simulation results obtained with the Stewart-Trinkle time-stepping method. In addition, simulations were found to agree with the qualitative experimental results of Vose et al. After such verification of the simulation method, additional numerical studies were done that would have been impossible to carry out analytically. Specifically, we were able to demonstrate the convergence of the method with decreasing step size (as predicted theoretically by Stewart). Further analytical and numerical studies will be carried out in the future to develop and select robust simulation methods that best satisfy the speed and accuracy requirements of different applications.

Contact modes and complementary cones

In this paper, we use a linear complementarity problem (LCP) formulation of rigid body dynamics with unilateral contacts to obtain definitions for contact modes. We show how the complementary cones of the LCP correspond to each of the intuitive contact modes: slide right, slide left, roll and separate. These complementary cones allow us to make rigorous definitions for contact modes in three-dimensional systems, where our intuitive understanding fails.

A Time-Stepping Scheme for Quasistatic Multibody Systems

Two new instantaneous-time models for predicting the motion and contact forces of three-dimensional, quasistatic multi-rigid-body systems are developed; one linear and one nonlinear. The nonlinear characteristic is the result of retaining the usual quadratic friction cone in the model. Discrete-time versions of these models provide the first time-stepping methods for such systems. As a first step to understanding their usefulness in simulation and manipulation planning, a theorem defining the equivalence of solutions of a time-stepping method for the nonlinear model and a global optimal solution of a related convex optimization problem is given. In addition, a Proposition giving necessary and sufficient conditions for solution uniqueness of the nonlinear time-stepping method is given. Finally, a simple example is discussed to help develop intuition about quasistatic systems and to solidify the reader’s understanding of the theorem and proposition.Copyright