1 Rigid Body Dynamic Simulation with Line and Surface Contact Jiayin Xie, Student Member, IEEE, Nilanjan Chakraborty , Member, IEEE, Abstract—In this paper, we develop a principled method to model line and surface contact with point contact (we call this point, equivalent contact point) that is consistent with physics- based models of surface (line) contact. Assuming that the set of contact points form a convex set, we solve the contact detection and dynamic simulation step simultaneously by formulating the problem as a mixed nonlinear complementarity problem. This allows us to simultaneously compute the equivalent contact point as well as the wrenches (forces and moments) at the equivalent contact point (consistent with the friction model) along with the Fig. 1: State-of-the-art dynamic simulation algorithms assume configuration and velocities of the rigid objects. Furthermore, that the contact between two objects is always a point contact we prove that the contact constraints of no inter-penetration (left). However, the contact may actually be a (middle) line between the objects is also satisfied. We present a geometri- cally implicit time-stepping scheme for dynamic simulation for or (right) surface contact. We develop a principled method to contacts between two bodies with convex contact area, which incorporate line and surface contact in rigid body dynamic includes line contact and surface contact. We prove that for simulation. surface and line contact, for any value of the velocity of center of mass of the object, there is a unique solution for contact point and contact wrench that satisfies the discrete-time equations of motion. Simulation examples are shown to demonstrate the dynamical system, an almost universal assumption is that the validity of our approach and show that with our approach we contact between the two objects is a point contact. can seamlessly transition between point, line, and surface contact. However, in practical manipulation scenarios the point con- Index Terms—Dynamic Simulation, Nonlinear Complementar- tact assumption may not be valid. ity Problems, Contact Modeling, Intermittent Contact. Consider the example of a cylinder being pushed on a flat surface. The contact between the flat surface and the cylinder is a line contact (Figure 1, middle) that changes with time as I. INTRODUCTION the cylinder rolls. Further, consider a box being pushed on a A fundamental characteristic of a wide range of robotics table. Here, the contact between the box and the table is a problems including robotic grasping, in-hand manipulation [1], surface contact (Figure 1, right). [2], non-prehensile manipulation (say, by pushing) [3], [4] is In general, during motion, the contact between two objects the presence of controlled intermittent contact between the may switch among point, line, and surface contact. gripper or the robotic end effector with other objects. The In such cases, multiple contact points (usually less than ability to predict motion of objects undergoing intermittent three) are usually chosen in an ad hoc manner. For example, contact can help in the design of robust grasp strategies, for the box on the table one can choose any three points, arXiv:2010.02291v2 [cs.RO] 7 Oct 2020 effective part feeder devices [5], and planning and control such that the projection of the center of mass of the box algorithms for manipulation. Thus, dynamic modeling and on the flat surface lies within the convex hull of the points. simulation for problems with intermittent unilateral contact is If more than three contact points are chosen (e.g., the four a key problem. vertices), the force distribution at the points cannot be uniquely Existing mathematical models for motion of rigid objects determined. However, such ad hoc a priori choices may not with intermittent contact [6]–[12] can be classified into two be valid because the actual contact patch may change during broad categories, namely, (a) Differential Algebraic Equation motion, e.g., when part of the box is over the edge of the table. (DAE) models [13] and (b) Differential Complementarity Also, the contact mode may change from surface to point or Problem (DCP) models [14]–[16]. In DAE models, it is line, e.g., when the box is being tilted. assumed that the contact mode (i.e., sliding contact, rolling Furthermore, such ad hoc choices (along with other as- contact, or no contact) is known, whereas DCP models solve sumptions like linearization of friction cones) can lead to for the contact mode along with the state of the system. inaccuracies in simulation (please see Figure 6c and 6d, which Irrespective of whether a DAE or DCP is used to model the demonstrates this for Open Dynamics Engine (ODE) [17] and BULLET [18], two popular dynamic simulation software Jiayin Xie and Nilanjan Chakraborty are with the Department of Me- codes). Hence, simulation for predicting motion for line or chanical Engineering, State University of New York at Stony Brook, Stony Brook, NY, 11790 USA. Email: [email protected]; nilan- surface contact may not be reliable enough for use in manip- [email protected]. ulation planning or control. 2 A key reason for the modeling difficulties with line or sur- planar (please see [21]). face contact is that most current dynamic simulation methods Contributions: Our key contributions are as follows: (a) decouple the contact detection from integrating the equations We show that the geometrically implicit time-stepping model of motion. A collision detection algorithm is used to compute proposed in [20] can be used to simulate contact problems the closest points or the contact points at a given configuration with line or surface contact. We also formally prove that obtained by integrating the equations of motion. When there is the geometric contact constraints will be always satisfied and line or surface contact, there are infinitely many possible pairs there will be no interpenetration between the objects. (b) We of closest points on the two objects and thus the collision formalize our claim that our approach of using a geometrically detection problem does not have a unique solution. In such implicit time-stepping model by combining collision detection cases, it is not clear which point should be used as a contact within the equations of motion leads to a well-posed problem. point to ensure that there is no non-physical penetration and More specifically, for 3D line and surface contact, we prove error in the simulation is not introduced by the choice of the that for any value of the state (position, orientation, linear, point. The goal of this paper is to develop a principled method and angular velocity) of the two objects, there is a unique to incorporate line and surface contact in dynamic simulation. solution for the closest points and the contact wrenches. The set of contact points will often be referred to as contact Thus, for resolving non-point contacts, when we solve for patch and we assume that the contact patch is a convex set. the contact wrenches and contact point simultaneously, the When there is a contact patch between two rigid objects, problem is well-posed, whereas it is ill-posed when we want there is a distribution of the normal force and the friction to resolve contacts by solving for the contact points separately force in the contact patch. There is a unique point in the (as is traditionally done), since there are infinite number of contact patch where the net moment due to the normal contact possible contact points for the given state. (c) We also present force is zero. The effect of the contact patch can be modeled numerical simulation results, depicting the correctness of our equivalently by the sum of the total distributed normal and method by comparing the numerical solution of the dynamic tangential force acting at this single point and the net moment simulation with analytical solutions that we have derived for about this point due to the tangential contact forces. We call pure translation with patch contact. For more general motion this point the equivalent contact point (ECP). with non-point contact, we show that our algorithm can track In statics, where the friction may or may not be relevant, the change of the equivalent contact point as the contact mode the point in the contact patch where the net moment of changes from point contact to line contact to surface contact. the normal force distribution is zero is called the centre of We use the PATH solver [22] to get the numerical solutions pressure. In the manipulation literature, where the friction is to the complementarity problem that we are formulating. also relevant, this point is called the center of friction [19]. This paper is organized as follows: In Section II, we discuss In this paper, we show that the ECP as well as the contact in detail the relationship of our work to the related work. force and moment (i.e., contact wrench) at the ECP can be In Section III, we present the mathematical model of the computed by incorporating the collision detection within the equations of motion of objects in intermittent contact with each dynamic simulation time step. We use a DCP formulation of other. The modeling of the contact constraints is presented the dynamics, since it does not make any assumptions about in detail in IV. In Section V, we formally prove that the the contact modes, which are usually not known a priori. discrete time-model that we present by combining the equa- In [20], the authors presented a method for incorporating the tions corresponding to the computation of the closest point collision detection within the dynamic simulation time step. within the equations of motion leads to a well-posed problem They showed that such a formulation improves accuracy of the for determining the ECP and contact wrench.