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.     

Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach

The main purpose of this paper is to present and discuss a methodology for a dynamic modeling and analysis of rigid multibody systems with translational clearance joints. The methodology is based on the non-smooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the non-smooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The results obtained show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.     

Docking dynamics between two spacecrafts with APDSes

Contact/impact models are established mainly for the capturing phase of a docking system. All the interactions between them are supposed as point contacts/impacts. For contacts, instead of using compliant models, constraint equations are formulated on the thorough investigation of the contour shapes of the docking system. All the contact constraints are originated from the sufficient and necessary conditions of point contacts. The constraints and Coulomb’s frictional law are incorporated into the contact dynamic equations of the system to obtain the contact forces by Lagrangian multipliers. For impacts, multiple impacts are considered in the term of the complex configuration of the docking dynamics. Impulsive equations are deduced, and the distributing law which is also called the LZB method is applied for multiple impacts. Finally, simulation is realized and the numerical results for one working condition are presented.     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

Qualitative analysis and quantitative evaluation of fixturing

Form-closure is considered as a purely geometric property of a set of unilateral contact constraints such as those applied on a workpiece by a mechanical fixture. This paper provided qualitative analysis of form-closure fixturing. The necessary and sufficient condition for form-closure fixturing is derived. Some fundamental problems related to form-closure are solved such as minimum number of frictionless contact points and the way to arrange them to achieve form-closure. On the basis of qualitative analysis, the quantitative evaluation of form-closure is investigated. To assess quantitatively the form-closure fixturing, two quantitative indices, one to minimize the sum of all normal contact forces and the other to minimize the maximum normal contact force, are presented. Finally, the given example verifies the analysis method and evaluating indices.     

一种基于遗传算法的多机器人内力优化计算方法

本文在对多机器人静力分析的基础上,考虑最大作用力约束和点接触静摩擦约束,提 出了一种采用混合遗传 模拟退火算法的内力优化计算方法,并通过2个平面机器人夹持一 负载的仿真实验,给出了内力和负载分配的优化计算结果．     

An automatic incrementation technique for contact problems with friction

A numerical method for analysis of elastostatic contact problems with friction has been developed. This class of problems are load history dependent because of the irreversible nature of frictional forces. An automatic incrementation technique of the applied load has been developed and implemented in the algorithm. The method is a direct method based on an iterative procedure applied to a set of linear equations established with the finite element method. The size of the applied load increments, automatically chosen by the algorithm, is in general influenced both by the nature of the problem and of the discretization of the bodies involved. The frictional forces occurring in the slip zone of the contact area are treated as known tangential forces calculated from the normal forces in the previous iteration. This piecewise linear treatment of the frictional contact problem requires an innermost iteration loop over the applied tangential force.The tangential force must coincide with the coefficient of friction times the normal force obtained in the last iteration, otherwise a new tangential force has to be calculated and the system of equations must be solved for a new right hand side vector. The automatic incrementation technique is based on the fact that each iteration is a linear problem. A tentative load increment is used in the solution of a certain iteration. A linear scaling of this solution is performed afterwards. A load scale factor is calculated in each contact node pair where a change of contact condition will occur. The change in contact status corresponding to the node pair with the smallest load scale factor is the only change which is accomplished in a certain iteration. The uniqueness of this kind of contact problem with friction has not been mathematically proven for a general case.The method has been applied to a case of loading and unloading of an elastic halfspace by a rigid cylindrical stamp and compared to solutions by Spence and Turner.     

On Clamping Planning in Workpiece-Fixture Systems

Deformations of contacts between the workpiece and locators/clamps resulting from large contact forces cause overall workpiece displacement, and affect the localization accuracy of the workpiece. An important characteristic of a workpiece-fixture system is that locators are passive elements and can only react to clamping forces and external loads, whereas clamps are active elements and apply a predetermined normal load to the surface of workpiece to prevent it from losing contact with the locators. Clamping forces play an important role in determining the final workpiece quality. This paper presents a general method for determining the optimal clamping forces including their magnitudes and positions. First, we derive a set of “compatibility” equations that describe the relationship between the displacement of the workpiece and the deformations at contacts. Further, we develop a locally elastic contact model to characterize the nonlinear coupling between the contact force and elastic deformation at the individual contact. We define the minimum norm of the elastic deformations at contacts as the objective function, then formulate the problem of determining the optimal clamping forces as a constrained nonlinear programming problem which guarantees that the fixturing of the workpiece is force closure. Using the exterior penalty function method, we transform the constrained nonlinear programming into an unconstrained nonlinear programming which is, in fact, the nonlinear least square. Consequently, the optimal magnitudes and positions of clamping forces are obtained by using the Levenberg–Marquardt method which is globally convergent. The proposed planning method of optimal clamping forces, which may also have an application to other passive, indeterminate problems such as power grasps in robotics, is illustrated with numerical example.      

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.     

Improved implicit integrators for transient impact problems--dynamic frictional dissipation within an admissible conserving framework

This work presents a frictional extension of an improved conservative integration framework for dynamic contact, extending the details of the frictionless scheme to encompass Coulomb stick/slip friction and the associated conservation/dissipation. It addresses algorithmic details that do not appear in the frictionless context, establishing an appropriate and objective treatment of relative tangential motion that is necessary to rigorously ensure conservation of angular momentum in the fully discrete setting. It also further extends the functionality of a discrete contact velocity term proposed in the frictionless implementation, using it to preserve conservation or dissipation locally as warranted by the frictional model and effectively enabling an enforcement of the contact constraints independent of energy considerations. The result is a robust implicit algorithmic framework for dynamic frictional impact, viable for large deformation analysis, appropriately conservative or dissipative for both stick and slip phenomena on a local scale, and readily extensible to more complex frictional models.     

Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics

In this paper, we provide a full instruction on how to formulate and evaluate planar frictional contact problems in the spirit of non-smooth dynamics. By stating the equations of motion as an equality of measures, frictional contact reactions are taken into account by Lagrangian multipliers. Contact kinematics is formulated in terms of gap functions, and normal and tangential relative velocities. Associated frictional contact laws are stated as inclusions, incorporating impact behavior in form of Newtonian kinematic impacts. Based on this inequality formulation, a linear complementarity problem in standard form is presented, combined with Moreau’s time stepping method for numerical integration. This approach has been applied to the woodpecker toy, of which a complete parameter list and numerical results are given in the paper.     

A contact force solution for non-colliding contact dynamics simulation

Rigid-body impact modeling remains an intensive area of research spurred on by new applications in robotics, biomechanics, and more generally multibody systems. By contrast, the modeling of non-colliding contact dynamics has attracted significantly less attention. The existing approaches to solve non-colliding contact problems include compliant approaches in which the contact force between objects is defined explicitly as a function of local deformation, and complementarity formulations in which unilateral constraints are employed to compute contact interactions (impulses or forces) to enforce the impenetrability of the contacting objects. In this article, the authors develop an alternative approach to solve the non-colliding contact problem for objects of arbitrary geometry in contact at multiple points. Similarly to the complementarity formulation, the solution is based on rigid-body dynamics and enforces contact kinematics constraints at the acceleration level. Differently, it leads to an explicit closed-form solution for the normal forces at the contact points. Integral to the proposed formulation is the treatment of tangential contact forces, in particular the static friction. These friction forces must be calculated as a function of microslip velocity or displacement at the contact point. Numerical results are presented for four test cases: (1) a thin rod sliding down a stationary wedge; (2) a cube pushed off a wedge by an applied force; (3) a cube rotating off the wedge under application of an external moment; and (4) the cube and the wedge both moving under application of a moment. To ascertain validity and correctness, the solutions to frictionless and frictional scenarios obtained with the new formulation are compared to those generated by using a commercial simulation tool MSC ADAMS.     

Dealing with multiple contacts in a human-in-the-loop application

This paper deals with continuous contact force models applied to the human-in-the-loop simulation of multibody systems, while the results are valid in general to all the real-time applications with contacts. The contact model proposed in this work is suited to collisions between massive solids for which the assumption of quasi-static contact holds, and it can be supposed that the deformation is limited to a small region of the colliding bodies while the remainder of them are assumed to be rigid. The model consists of two components: normal compliance with nonlinear viscoelastic model based on the Hertz law, and tangential friction force based on Coulomb’s law including sticktion and a viscous friction component. Furthermore, the model takes into account the geometry and the material of the colliding bodies. The tangential model is a novel contribution while the normal model is completely taken from previous works. For this work, the formulation of the equations of motion is an augmented Lagrangian with projections of velocities and accelerations onto their constraints manifolds and implicit integrator. The whole solution proposed is tested in three applications: the first one is the simulation of a spring–mass system with Coulomb’s friction, which is an academic problem with known analytical solution; the second one is the Bowden and Leben stick–slip experiment; the third one is a simulator of a hydraulic excavator Liebherr A924, which is a realistic application that gives an idea of the capabilities of the method proposed.     

Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation

Direct differentiation method of sensitivity analysis is developed for frictional contact problems. As a result of the augmented Lagrangian treatment of contact constraints, the direct problem is solved simultaneously for the displacements and Lagrange multipliers using the Newton method. The main purpose of the paper is to show that this formulation of the augmented Lagrangian method is particularly suitable for sensitivity analysis because the direct differentiation method leads to a non-iterative exact sensitivity problem to be solved at each time increment. The approach is applied to a general class of three-dimensional frictional contact problems, and numerical examples are provided involving large deformations, multibody contact interactions, and contact smoothing techniques.     

Development of Realistic Pressure Distribution and Friction Limit Surface for Soft-Finger Contact Interface of Robotic Hands

Various models have been presented for pressures distribution in the contact interface of a soft finger and object in the literature. These models have been proposed without considering the effect of the tangential forces which are usually exerted in the contact interface of a soft finger and object during grasping and manipulation. Having an accurate pressures distribution model across the contact interface is important for designing tactile sensors and improving the modeling of the friction limit surface (LS). In this paper, a new and more accurate model is proposed to describe the asymmetry of the pressure distribution in the contact interface of a hemispherical soft finger under both normal and tangential forces. This model is derived based upon observations in the previous literature stating that the contact interface would move and skew toward the direction of the tangential force. According to the proposed pressure distribution model in this study, an improved and more accurate LS is presented. The LS profile obtained by this model is compared with the corresponding results based on the previous models. The new results show that the consideration of the skewness or asymmetry in the pressure distribution (due to the tangential force) causes the LS profile to shrink compared with that constructed with symmetric pressure distribution assumption. This shrinkage, as a result of the skewness and asymmetry of the pressure distribution, makes the contact interface more vulnerable. Furthermore, this new model can also provide a more accurate tool for the analysis of grasping and manipulation involving soft contact interface.     

Intelligent Fixture Design through a Hybrid System of Artificial Neural Network and Genetic Algorithm

In designing fixtures for machining operations, clamping scheme is a complex and highly nonlinear problem that entails the frictional contact between the workpiece and the clamps. Such parameters as contact area, state of contact, clamping force, wear and damage in the contact area and deformation of the component are of special interest. A viable fixture plan must include the optimum values of clamping forces. Along research efforts carried out in this area, this comprehensive problem in fixture design needs further investigation. In this study, a hybrid learning system that uses nonlinear finite element analysis (FEA) with a supportive combination of artificial neural network (ANN) and genetic algorithm (GA) is discussed. A frictional model of workpart–fixture system under cutting and clamping forces is solved through FEA. Training and querying an ANN takes advantage of the results of FEA. The ANN is required to recognize a pattern between the clamping forces and state of contact in the workpiece–fixture system and the workpiece maximum elastic deformation. Using the identified pattern, a GA-based program determines the optimum values for clamping forces that do not cause excessive deformation/stress in the component. The advantage of this work against similar studies is manifestation of exact state of contact between clamp elements and workpart. The results contribute to automation of fixture design task and computer aided process planning (CAPP).     

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.     

Modeling of cutting geometry and forces for 5-axis sculptured surface machining

5-Axis sculptured surface machining is simulated using discrete geometric models of the tool and workpiece to determine the tool contact area, and a discrete mechanistic model to estimate the cutting forces. An extended Z-buffer model represents the workpiece, while a discrete axial slice model represents the cutting tool. Determination of the contact area for a given tool move requires a swept envelope (SWE) of the tool path. The SWE is used to find the intersections of the tool envelope with Z-buffer elements (ZDVs) representing the workpiece. A 3-axis approximation of the 5-axis tool movement is used to simplify the calculations while maintaining a desired level of accuracy. The intersection of the SWE with each ZDV yields segments which are used to find the contact area between the cutter and the workpiece for a given tool path. The contact area is subsequently used with the discrete force model to calculate the vector cutting force acting on the tool.     

Compliant grasping with passive forces

Because friction is central to robotic grasp, developing an accurate and tractable model of contact compliance, particularly in the tangential direction, and predicting the passive force closure are crucial to robotic grasping and contact analysis. This paper analyzes the existence of the uncontrollable grasping forces (i.e., passive contact forces) in enveloping grasp or fixturing, and formulates a physical model of compliant enveloping grasp. First, we develop a locally elastic contact model to describe the nonlinear coupling between the contact force with friction and elastic deformation at the individual contact. Further, a set of “compatibility” equations is given so that the elastic deformations among all contacts in the grasping system result in a consistent set of displacements of the object. Then, combining the force equilibrium, the locally elastic contact model, and the “compatibility” conditions, we formulate the natural compliant model of the enveloping grasp system where the passive compliance in joints of fingers is considered, and investigate the stability of the compliant grasp system. The crux of judging passive force closure is to predict the passive contact forces in the grasping system, which is formulated into a nonlinear least square in this paper. Using the globally convergent Levenberg‐Marquardt method, we predict contact forces and estimate the passive force closure in the enveloping grasps. Finally, a numerical example is given to verify the proposed compliant enveloping grasp model and the prediction method of passive force closure. © 2005 Wiley Periodicals, Inc.