Off-line Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park

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1 NSF GRANT # NSF PROGRAM NAME: Engineering Design Off-line Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park Atul Thakur University of Maryland, College Park Abstract: Rigid body dynamics simulations require use of accurate computation of contacts among bodies. Often collision detection algorithms are used for determining the contact between moving bodies. Mechanical parts have a large number of features and hence collision detection with the detailed part models often slows down the rigid body dynamics simulations. In many simulation scenarios, all the parts participating in the simulation are known in advance. In such cases, the simulation context (i.e., a priori knowledge of parts) can be exploited to simplify the part geometries such that the contact points among parts do not change. For example, parts with significant concavities may have regions on their boundaries that will be inaccessible to other parts in the simulation and hence contact points cannot lie on such inaccessible regions. Removing such regions from the parts can simplify the model and hence speed up the simulation. In this paper, we describe an off-line simplification algorithm that performs part simplification based on accessibility considerations. Our algorithm ensures that the potential contact points among parts do not change as a result of the simplification process and hence it preserves the underlying physics. 1. Introduction: Rigid body dynamics simulations are nowadays used in a wide variety of virtual environment based applications ranging from entertainment (computer games, 3D-movies, etc.), defense (ground vehicle simulations, explosive ordnance disposal robot simulation, flight simulations, etc.), health-care (surgical simulation, drug disposal using needle steering, etc.), manufacturing (assembly simulation), etc. Many of the applications also need simulation of compliant parts in addition to rigid body simulations especially in health-care and manufacturing applications. Such compliance is often modeled using rigid bodies connected by springs (i.e. by pseudo-rigid bodies) which intrinsically use the rigid body simulations. Thus, the rigid body dynamics simulation could be applied to a variety of applications involving both rigid as well as compliant parts. In a rigid body simulator, the major computational effort is spent on computing the contact points at each time step. Often collision detection algorithms are used for computing the contact points. Since the mechanical parts have large number of features, the collision detection of the detailed part models slows down the rigid body dynamics simulation. Hence, simplification of geometric models can improve the performance of dynamics simulation. To simplify a geometric model, many techniques involving vertex, edge and face decimation have been reported [1, 2]. A recent survey on model simplification techniques for physics based simulations can be found in [3]. One of the main limitations from the point of view of dynamics simulation of these techniques is that the shape of the simplified model is different than the original model. This is undesirable in case of rigid body dynamics simulation as its fidelity depends upon the accuracy of the contact points returned by collision detection engine. Thus, one of the main constraints for model simplification in case of rigid body dynamics simulation is that the potential contact points of colliding models should be preserved (remain unaltered) after simplification. The potential contact points depend upon the collision context (i.e., which parts are colliding). In many problems of dynamics simulations the collision context is known in advance or could be easily determined as the parts and their initial configurations are known beforehand and the underlying physics could be used to predict part configuration at each time step. This opens up a possibility of storing and retrieving multiple representations of parts based on collision contexts, where each representation can be simplified for the given collision context. This scheme is promising as memory is relatively inexpensive compared to the real time computation of contact points for fully featured part models. We plan to utilize the collision context to generate physics preserving simplified models. Our idea is to simplify models with respect to each other in an off-line manner, i.e. before the simulation is performed in such a way that possible contact points are preserved using part accessibility considerations. We developed an algorithm to implement and test this idea

2 and the details of the algorithm are presented in the following section. 2. Problem Statement and Solution Approach: Given a pair of boundary representation models A and B, generate simplified boundary representation models A s and B s such that the set of contact points obtained under any arbitrary rigid body collision between A and B is same as the set of contact points obtained by collision between and A s and B s. Our approach to solve this problem is mainly based on the fact that in a pair of parts colliding with each other there are some regions that are never accessed and should be removed from the models representing the parts for collision computations. Our algorithm for simplification of given model A with respect to B consists of the following three main steps: Step 1: Construction of the Connected Convex Edge Set (CCES) graph for A. We extract the convex edges of A and determine the sets of connected convex edges (CCES) in this step. The CCES thus determined represents the artifacts in the model representing cavity openings and protrusions. After this we find out the connectivity in terms of accessibility between the CCESs of the model and construct a graph whose nodes store the CCES and the edges store the connectivity information between the CCESs. The process of CCES construction is shown in Fig. 1. Step 2: Construction of approximate Part Section Signature (PSS) for B. We represent the part B with a 2D curve called PSS, which gives the minimum circular hole diameter (D) in which the part B can enter till a given distance (δ). The distance (δ) is represented along the X axis and minimum hole diameter (D) is plotted along the Y-axis of the PSS. Part B and the PSS construction procedure are shown in Fig. 2. The ideal PSS can be constructed by scanning infinite directions of approach which is not practical and thus we scan finite number of directions δ ( 1 α ) and introduce an estimation factor equal to D, where α is the discritization angle. We multiply this factor with the approximate part section signature obtained by discrete directions to estimate the lower bound on the PSS as shown in Fig. 2 (d) by the broken line. Step 3: Model simplification of A with respect to B. In this step, the CCES graph of A is traversed in depthfirst manner to determine which nodes (CCES) are inaccessible. To determine whether the part B enters a given opening of A (represented by a node of CCES graph of A), the bounding circle of the opening represented by the node is compared with the minimum bounding cylinder of B for the depth of the node. If the minimum bounding cylinder radius of B is found to be greater than radius of node's bounding circle, the children nodes of the node are considered to be inaccessible. The faces related to inaccessible CCES nodes are then removed from A and the simplified model A s is generated. Above steps are repeated by swapping A and B to generate B s. Fig. 3. shows the simplified models of A and B with respect to each other. Fig. 1. Construction of CCES graph of Part Model A 3. Implementation and Testing: We developed a prototype software implementation of the contact preserving model simplification algorithm discussed in this paper. The programming platform was chosen as VC++ (version 8) using ACIS modeling kernel on Windows XP operating system. We used CGAL library (version 3.3.1) for bounding circles and convex hull computations. To test the results of model simplification, we built a virtual environment consisting of an articulated mobile (wheeled) robot that picks and places parts. We selected Open Dynamics (version 0.10) physics engine for doing physics based modeling and simulation of the virtual environment [4]. We devised the test wherein the robot is used to pick up a geometrically complex part in an environment containing other geometrically complex parts and to move it to another place. We tested the simulation performance with the simplified parts obtained by using the techniques discussed in this paper. Our algorithm takes B-Rep model (ACIS SAT) file as input and after simplification generates output in the same format.

3 a. Part Model A b. Part Section variation along d1 c. Part Section variation along d2 d. Part Section signature of B Fig. 2. Construction of Part Section Signature of Part B

4 Fig. 3. Simplified models A s and B s with respect to each other Since most of the simulation engines take triangulated data as input, we chose ACIS relative precision to be for triangulating both the unsimplified input and the simplified output models for the purpose of comparison. We obtained the test models from an online repository [5]. We obtained the simplification results as shown in Fig. 4. The figure shows the reduction in file size and triangle count of simplified and unsimplified models. For the reported test part models we found that the reduction in number of triangle count ranges from 6% to 55% depending on part complexity and collision context. We then tested the simplified models in OpenDynamics Physics engine. We created a wheeled robot with two serially connected, articulated arms in the simulation world. At the end of the articulator, there is a gripper to lift parts. There are several objects spread in the virtual world. The task of the robot is to pick an object and place it at another location moving through the space. We performed the pick-and-place experiment with unsimplified and simplified models and monitored the frame rates using Fraps [6]. The screen shots of the simulation using simplified and unsimplified parts are shown in Fig. 5. The improvement in the frame rate was found to be about 35%. Fig. 5. Rigid body dynamics simulation with simplified and unsimplified models Fig. 4. Results obtained by simplification of various parts with respect to each other 4. Conclusion: We introduced the concept of contact preserving model simplification by using part accessibility considerations for accelerating the collision queries in rigid body dynamics simulations. The accuracy of rigid body simulation depends upon the accuracy of the contact points generated by the underlying collision detection engine. Our off-line approach gets rid of faces not playing any role in collision while retaining all the faces which might possibly affect the results of collision query. Our main contributions are as follows:

5 i. We introduced the concept of accessibility based on the part section signature and openings. ii. We developed and implemented a new off-line algorithm for contact preserving model simplification for rigid body dynamics simulation. iii. We demonstrated performance gains in a typical rigid body dynamics simulation scenario when our offline model simplification approach is used. A limitation of this work is the conservative approximation of approaching parts with bounding cylinders that sets an upper bound on the part crosssection, which can simplify the other parts relative to the approaching part but might leave some faces which should be removed. This way, we guarantee that the part is never over-simplified. Also, using this approach the number of models needed to be stored for the given n models in a scene is n 2 for the worst case. However, in practical situations, number of models needed to be stored can be further reduced based on part similarity and the collision context. 10. References: [1] P. Cignoni, C. Montani, and R. Scopigno. A comparison of mesh simplification algorithms, Computers & Graphics, 22(1):37 54, [2] D. P. Luebke. A developer s survey of polygonal simplification algorithms, IEEE Computer Graphics Applications, 21(3):24 35, [3] A. Thakur, A. G. Banerjee, and S. K. Gupta. A survey of CAD model simplification techniques for physics-based simulation applications, Computer- Aided Design, 41(2):65-80, [4] Opendynamics rigid body dynamics engine, November [5] 3d part repository, September [6] Frame rate monitoring, September

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