Evaluating the Performance of Constraint Formulations for Multibody Dynamics Simulation

Contemporary software systems used in the dynamic simulation of rigid bodies suffer from problems in accuracy, performance, and robustness. Significant allowances for parameter tuning, coupled with the careful implementation of a broad phase collision detection scheme is required to make dynamic simulation useful for practical applications. A geometrically accurate constraint formulation, the Polyhedral Exact Geometry method, is presented. The Polyhedral Exact Geometry formulation is similar to the well-known Stewart-Trinkle formulation, but extended to produce unilateral constraints that are geometrically correct in cases where polyhedral bodies have a locally non-convex free space. The PEG method is less dependent on broad-phase collision detection or system tuning than similar methods, demonstrated by several examples. Uncomplicated benchmark examples are presented to analyze and compare the new Polyhedral Exact Geometry formulation with the well-known Stewart-Trinkle and Anitescu-Potra methods. The behavior and performance for the methods are discussed. This includes specific cases where contemporary methods fail to match theorized and observed system states in simulation, and how they are ameliorated by PEG. ∗Address all correspondence to this author. INTRODUCTION The simulation of the dynamics of rigid bodies has many applications in a variety of scientific and engineering fields. The demand for fast and accurate simulation is high, as dynamic problems grow larger and larger under limited computation power. The performance of an improvement to the popular StewartTrinkle [1] multibody dynamic system formulation is evaluated in this work. This recent multibody dynamics constraint formulation, Polyhedral Exact Geometry [2, 3] (PEG) is compared to widely accepted constraint generation methods. The purpose of this new formulation is to increase simulation performance and accuracy. The accuracy of Polyhedral Exact Geometry is evaluated compared to the Stewart-Trinkle and Anitescu-Potra methods, specifically the position error present in a small planar particle simulation. Multibody dynamic simulation is used in a broad range of engineering, research, and entertainment fields. Accurate simulation of the motion of machinery is crucial in mechanical design [4–6]. It is particularly useful in designing complex machinery that is expensive to prototype, such as internal combustion engines [7]. Accurate simulation is required for robotics applications, particularly where contact is expected, such as in grasp planning [8]. And virtual reality is more useful and convincing when objects undergo more realistic motion in simulation [9]. Entertainment products such as motion pictures and video games benefit from accurate simulation [10], where realism is enhanced. 1 Copyright c © 2013 by ASME There are many challenges involved in providing a reasonable approximation of physical reality through simulation of rigid bodies [11, 12]. This is particularly a difficult issue when considering contact and friction. This consideration makes a system mathematically nonsmooth, and difficult to simulate accurately. A simulation must use constraints that closely represent the geometry of objects interacting with each other. First, accuracy is lost when representing real world objects as topologically connected vertices, faces, and edges in computer simulation [13–15]. In addition, constraints do not always exactly coincide with this approximated geometric representation, causing further error. These problems can be mitigated through increased accuracy by taking small time steps, or by more accurately mapping the constraints to the geometry. The formulations presented in this paper are of the latter solution. Time-stepping simulations calculate estimates of system state at discrete times. The simulation starts at an initial state, and then calculates the future system state one time step ahead. The systems states at the next time step are dependent on the solution of the dynamics and constraints using the states of the previous time step. And the simulation proceeds until criteria are met, e.g., reaching a predetermined final time. Within a time step, two phases are common. First, the unilateral and bilateral constraints, and the time-stepping subproblem are formulated. The constraints are determined by the geometry and kinematics of the simulated system. Second, the dynamics of the system are formulated from the equations of motion, and calculated to produce the derivative states. These derivative states are used to calculate all system states for the following time step. Popular physics engines such as ODE [16], and Bullet [10] use variations of the Stewart-Trinkle [1, 17] and AnitescuPotra [18, 19] formulations. Inaccuracies resulting from simplifications of Coulomb friction and geometric modeling of constraints result in undesirable behaviors in simulation that make the use of these popular engines for scientific work challenging [2]. The research presented herein continues the work in [3] and [20]. The mean error is used as a metric in evaluating the performance three formulations. The performance of the Polyhedral Exact Geometry formulation is shown to be more accurate than Stewart-Trinkle and Anitescu-Potra . The higher accuracy allows larger step sizes to be utilized while producing similarly accurate results as Stewart-Trinkle and Anitescu-Potra at smaller step sizes. METHODS The formulation of the Polyhedral Exact Geometry method is similar to the Stewart-Trinkle [1,17] formulation, but extended to produce unilateral constraints that are geometrically correct in cases where polyhedral bodies have a locally non-convex free space. Constraints for faces in a polygon model are defined as halfspaces in the Stewart-Trinkle and Anitescu-Potra methods. These halfspaces extend beyond the limits of the faces, producing constraints which are not geometrically accurate. The Polyhedral Exact Geometry method uses a heuristic to choose which constraints are active based on local geometry. This increases the accuracy of the simulation, and eliminates the need for specialized methods of broadphase collision detection required to avoid undesired behaviors in simulation caused by active constraints that should be ignored.