An iterative predictor–corrector approach for modeling static and kinetic friction in interactive simulations

In this paper we propose a novel iterative predictor–corrector (IPC) approach to model static and kinetic friction during interactions with deformable objects. The proposed IPC method works within the purview of the implicit mixed linear complementarity problem (MLCP) formulation of collision response. In IPC, first the potential directions of frictional force are determined at each contact point by leveraging the monotonic convergence of an iterative MLCP solver. All the contacts are then categorized into either static or kinetic frictional states. Linear projection constraints (LPCs) are used to enforce ‘stiction’ for contacts in static friction. We propose a modified iterative constraint anticipation (MICA) approach that can resolve the LPCs while simultaneously solving the MLCP. Our method can handle arbitrary models including asymmetric and anisotropic friction models. IPC requires low memory and is highly tunable. Multiple example problems are solved to demonstrate the method.     

Frictional contact analysis of spatial prismatic joints in multibody systems

The contact analysis of spatial prismatic joints remains a hard problem due to its complex nature. In this paper, a methodology for the frictional contact analysis of rigid multibody systems with spatial prismatic joints is presented, which is free of calculating the relative motion between the slider and guide, and is particularly suitable to the case of clearances being tiny. Under the assumption of the slider and guide being rigid, we prove that all types of contacts in the joint can be converted to point-to-point contacts. At each of the candidate points, two gap functions are introduced. However, in the proposed method, not the values of these gap functions but the relations between them are essential. In view of the non-colliding contacts being predominant when clearances of joints are tiny, we formulate the contact forces in terms of resultant frictional forces in the joint, resulting in a linear complementarity problem. By the proposed method, details about the contacts including the impact instants can be obtained, although impacts are not taken into consideration explicitly, as indicated by the numerical examples in this paper.     

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.     

Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints

The main purpose of this paper is to present a modeling and simulation method for the rigid multibody system with frictional translational joints. The small clearance between a slider and guide is considered. The geometric constraints of the translational joints are treated as bilateral constraints and the impacts between sliders and guides are neglected when the clearance sizes of the translational joints are very small. The contact situations of the normal forces acting on the sliders are described by inequalities and complementarity conditions, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The dynamic equations of the multibody systems with normal and tangential contact forces are written on the acceleration-force level using the Lagrange multiplier technique. The problem of the transitions of the contact situation of the normal forces acting on sliders and the transitions of the stick-slip of the sliders in the system is formulated as a horizontal linear complementarity problem (HLCP), which is solved by event-driven method. Baumgarte’s stabilization method is used to decrease the constraint drift. Finally, two typical mechanisms are considered as demonstrative application examples. The numerical results obtained show some dynamical behaviors of the systems with frictional translational joints and constraint stabilization effect.     

Simulation of dynamics of interacting rigid bodies including friction I: General problem and contact model

This is the first of two papers that deal with the problem of modeling contact (impact, sliding, rolling) between unconstrained rigid bodies for computer simulation of their motion. Modeling impacts and sustained contact is difficult because of the lack of a concise theory for three-dimensional (3-D) impact with friction, the complex non-holonomic constraints involved, general complexity of friction, etc. The traditional rigid body hard contact approach leads to a quadratic program formulation for determining the contact forces, which branches into cases and does not model multiple simultaneous contacts adequately. We have formulated a computational theory based on the soft contact approach that models contact through localized non-permanent deformation in the vicinity of contact. The model of mechanical contact between polyhedral objects that we propose, strikes a balance between realism and computability. It is simple enough to permit numerical integration through periods of contact in reasonable time, yet rich enough to represent well a variety of contact behaviors. The main cost associated with our model is the need for small time steps during contact, which slows down the simulation. Starting with the motivation for our work, this paper describes the governing equations and constitutive laws for the interacting bodies. Issues pertaining to the solution approaches are then discussed. The final sections explain the mathematical details of our contact model. A companion paper [1] describes details of our computational theory and its integration into a software system.Authors are listed in alphabetical order.     

Implicit Contact Handling for Deformable Objects

We present an algorithm for robust and efficient contact handling of deformable objects. By being aware of the internal dynamics of the colliding objects, our algorithm provides smooth rolling and sliding, stable stacking, robust impact handling, and seamless coupling of heterogeneous objects, all in a unified manner. We achieve dynamicsawareness through a constrained dynamics formulation with implicit complementarity constraints, and we present two major contributions that enable an efficient solution of the constrained dynamics problem: a time stepping algorithm that robustly ensures non-penetration and progressively refines the formulation of constrained dynamics, and a new solver for large mixed linear complementarity problems, based on iterative constraint anticipation. We show the application of our algorithm in challenging scenarios such as multi-layered cloth moving at high velocities, or colliding deformable solids simulated with large time steps.     

Simulation of dynamics of interacting rigid bodies including friction II: Software system design and implementation

This is the second of two papers that deal with the problem of modeling contact (impact, sliding, rolling) between unconstrained rigid bodies, including friction. In a companion paper [1] we showed that the main underlying problem concerns the ability to do efficient contact mechanics when bodies interact through impact and/or sustained contact. Contact mechanics involves two aspects: detection of contact between bodies and estimation of contact forces. These forces are complicated in character and difficult to estimate because they depend on the material response of the contacting objects, on the duration of contact (very short duration impact, or more sustained contact), frictional interaction at the surfaces, geometry of contact, etc. In [1] we proposed a conceptual model in which linear elastic (springs) and viscous (dampers) elements acting at points of contact between objects generate all contact forces. In this paper we describe how the contact model has been implemented in the software of a working computer simulation system. The major aspects of this process are: formulation of a discrete version of the contact model; calculation of model parameters to reflect material properties; geometric representation of objects (in our system, objects are modeled as convex polyhedra); algorithms to detect and evaluate contacts among objects (a process called contact analysis); and estimation and control of model response for stable numerical integration of equations of motion. A graphical user interface displays a three-dimensional (3-D) perspective animation of the solution using full color shaded surface images. While the simulation may not be accomplished in real time, solutions can be saved in files for real-time visualization.Authors are listed in alphabetical order     

An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm

A review of existing augmented Lagrangian methods (ALM) for contact analysis problems reveals that they have not been implemented with automatic penalty updates as intended in their original development. Therefore, although the methods are an improvement over the penalty methods, solution with them still depends on the user-specified penalty values for the contact constraints. To overcome this drawback, an ALM is developed and discussed for contact analysis problems that automatically update the user-specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two-phase formulation is proposed. Solution of the Phase 1 problem removes penetration of the contacting nodes and brings them exactly to their initial contact points. In addition, a new contact constraint is introduced which allows determination of the precise friction force at the contacting nodes. Phase 2 of the formulation checks the friction conditions and solves the friction problem to bring the structure to an equilibrium state. Phases 1 and 2 are then combined to provide a general algorithm for multi-node frictional contact problems. The two-phase procedure also removes dependence of the contact solution on the number of load steps for the elastostatic problem. Numerical evaluation of the formulation and the algorithm is presented in Part 2 of the paper.     

An improved finite-element contact model for anatomical simulations

This work focuses on the simulation of mechanical contact between nonlinearly elastic objects, such as the components of the human body. In traditional methods, contact forces are often defined as discontinuous functions of deformations, which leads to poor convergence characteristics and high-frequency noises. We introduce a novel penalty method for finite-element simulation based on the concept of material depth, which is the distance between a particle inside an object and the objects boundary. By linearly interpolating precomputed material depths at node points, contact forces can be analytically integrated over contact surfaces without raising the computational cost. The continuity achieved by this formulation reduces oscillation and artificial acceleration, resulting in a more reliable simulation algorithm.     

A contact friction algorithm including nonlinear viscoelasticity and a singular yield surface provision

A finite element solution method for two-dimensional boundary value problems involving nonlinear viscoelasticity and contact friction is presented. The simulation of ceramic composites at elevated temperatures is the motivation of coupling viscoelasticity and contact friction. Three major topics are discussed; (i) the time-integration scheme developed for coupling the interface contact friction with nonlinear viscoelasticity in the surrounding continuum, (ii) the spatial discretization of a generalized two-dimensional deformation field using finite elements, and (iii) two methodologies for treating the nonlinearities introduced by the contact friction. The implementation of the contact friction utilizes a penalty method coupled with an incremental plasticity formulation. This formulation results in a highly nonlinear problem, and many solution techniques have difficulty with convergence due to a directional sensitivity arising from the unknown slip direction in the case of three-dimensional contact friction (or the unknown slip direction for hardening or dilatant two-dimensional friction problems). This directional sensitivity is illustrated along with an algorithm which alleviates this difficulty. Also, an algorithm based upon proportional stressing is developed to eliminate problems created by a singular yield surface for idealized Coulomb friction.     

Contact problems with friction: models and simulations

This article deals with the modeling and the numerical simulation of contact and friction problems. Some friction formulations are proposed and numerical methods of resolution by finite elements are presented. Two methods are more particularly developed. The first one is based on a formulation using displacements and the second one is based on a mixed formulation using displacements and contact forces. Then, some improvements of these two methods as well as a comparison of their respective performances are presented.     

Realistic haptic rendering of interacting deformable objects in virtual environments

A new computer haptics algorithm to be used in general interactive manipulations of deformable virtual objects is presented. In multimodal interactive simulations, haptic feedback computation often comes from contact forces. Subsequently, the fidelity of haptic rendering depends significantly on contact space modeling. Contact and friction laws between deformable models are often simplified in up to date methods. They do not allow a "realistic" rendering of the subtleties of contact space physical phenomena (such as slip and stick effects due to friction or mechanical coupling between contacts). In this paper, we use Signorini's contact law and Coulomb's friction law as a computer haptics basis. Real-time performance is made possible thanks to a linearization of the behavior in the contact space, formulated as the so-called Delassus operator, and iteratively solved by a Gauss-Seidel type algorithm. Dynamic deformation uses corotational global formulation to obtain the Delassus operator in which the mass and stiffness ratio are dissociated from the simulation time step. This last point is crucial to keep stable haptic feedback. This global approach has been packaged, implemented, and tested. Stable and realistic 6D haptic feedback is demonstrated through a clipping task experiment.     

An augmented Lagrangian optimization method for contact analysis problems, 2: numerical evaluation

The ALM2 solution procedure is evaluated by solving two simple contact analysis problems for different friction conditions. These example problems are devised to have closed form solutions. This way there is no uncertainty about the target solution for evaluation of the proposed algorithm as well as existing algorithms. The numerical results with ALM2 are compared with the analytical solutions as well as with the penalty, Lagrange multiplier and existing augmented Lagrangian methods. All the algorithms are analysed for stick and slip friction conditions. The example problems are used to show clearly the dependence of the existing solution methods on the number of load steps and penalty values. It is concluded that convergence of incremental solution schemes employed in these methods does not guarantee accuracy of the contact solution even with the use of solution enhancement schemes such as automatic load stepping and contact load prediction. The example problems are also used to demonstrate solution independence of the proposed ALM2 procedure from penalty values, and from the number of load steps. The proposed formulation for calculation of frictional forces and the ALM solution algorithm have worked quite well for the example problems. However, the algorithm needs to be developed and evaluated for more complex contact analysis problems.     

Multibody dynamics simulation of planar linkages with Dahl friction

It is well-known that classical Coulomb dry friction model does not portrays important physical phenomena occurring in the contact between mating surfaces. Moreover, the discontinuity of force at zero velocity has many drawbacks during numerical simulations. In the attempt of exploring alternatives to Coulomb friction model, this paper presents the application of the Dahl friction model in a multibody dynamics formulation. The analysis herein presented includes also the modeling of friction forces in lower pairs and some hints on the efficient computation of Lagrange parameters during the fixed-point iteration process. Two numerical examples are offered.     

Efficient formulation of the force distribution equations forgeneral tree-structured robotic mechanisms with a mobile base

In this paper, an efficient and systematic formulation of the force distribution equations for general tree-structured robotic mechanisms is presented. The applicable platforms include not only systems with star topologies, such as walking machines that have multiple legs with a single body but also general tree-structured mechanisms, such as variably configured wheeled vehicles having multiple modules. The force balance equations that govern the relationship between the contact forces and the resultant inertial forces/moments of the vehicle will be derived through a recursive and computationally efficient algorithm. Also, the joint torque constraints that specify the joint actuator limits, and contact friction constraints that may be used to avoid slippage and maintain contact, are efficiently incorporated in the formulation. Based on this formulation, several standard optimization techniques, such as linear programming or quadratic programming, can be applied to obtain the solution. An algorithm summarizing the results developed, and suitable for computer implementation, is included. The algorithm has been applied to an n-module actively articulated wheeled vehicle, and the computational cost evaluated. The efficiency of the algorithm is demonstrated with results showing real-time execution on a Pentium PC.     

A methodology for design and analysis of cooperative behaviors with mobile robots

New methodologies are needed for modeling of physically cooperating mobile robots to be able to systematically design and analyze such systems. In this context, we present a method called the ‘P-robot Method’ under which we introduce entities called the p-robots at the environmental contact points and treat the linked mobile robots as a multiple degree-of-freedom object, comprising an articulated open kinematic chain, which is manipulated by the p-robots. The method is suitable to address three critical aspects of physical cooperation: a) analysis of environmental contacts, b) utilization of redundancy, and c) exploitation of system dynamics. Dynamics of the open chain are computed independent of the constraints, thus allowing the same set of equations to be used as the constraint conditions change, and simplifying the addition of multiple robots to the chain. The decoupling achieved through constraining the p-robots facilitates the analysis of kinematic as well as force constraints. We introduce the idea of a ‘tipping cone’, similar to a standard friction cone, to test whether forces on the robots cause undesired tipping. We have employed the P-robot Method for the static as well as dynamic analysis for a cooperative behavior involving two robots. The method is generalizable to analyze cooperative behaviors with any number of robots. We demonstrate that redundant actuation achieved by an adding a third robot to cooperation can help in satisfying the contact constraints. The P-robot Method can be useful to analyze other interesting multi-body robotic systems as well.     

Study of variational inequality and equality formulations for elastostatic frictional contact problems

Summary An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.     

Touch-down induced contact and friction forces at the dimple/gimbal interface

A dynamic simulation is performed to investigate contact and friction forces at the dimple/gimbal interface. The time history of slider motion, the resultant force as well as the pitch and roll moments acting on the slider are determined. The time-dependent contact and friction forces at the dimple/gimbal interface are obtained. The effect of material properties on contact and friction forces at the dimple/gimbal interface is investigated. Experimental results for touch-down and take-off characteristics are presented.     

Energetically consistent simulation of simultaneous impacts and contacts in multibody systems with friction

This paper presents a methodology for treating energy consistency when considering simultaneous impacts and contacts with friction in the simulation of systems of interconnected bodies. Hard impact and contact is considered where deformation of the impacting surfaces is negligible. The proposed approach uses a discrete algebraic model of impact in conjunction with moment and tangential coefficients of restitution (CORs) to develop a general impact law for determining post-impact velocities. This process depends on impulse–momentum theory, the complementarity conditions, a principle of maximum dissipation, and the determination of contact forces and post-impact accelerations. The proposed methodology also uses an energy-modifying COR to directly control the system’s energy profile over time. The key result is that different energy profiles yield different results and thus energy consistency should be considered carefully in the development of dynamic simulations. The approach is illustrated on a double pendulum, considered to be a benchmark case, and a bicycle structure.