Analysis of a generalized kinematic impact law for multibody-multicontact systems,with application to the planar rocking block and chains of balls

In this paper, we analyze the capabilities of a generalized kinematic (Newton’s like) restitution law for the modeling of a planar rigid block that impacts a rigid ground. This kinematic restitution law is based on a specific state transformation of the Lagrangian dynamics, using the kinetic metric on the configuration space. It allows one to easily derive a restitution rule for multiple impacts. The relationships with the classical angular velocity restitution coefficient r for rocking motion are examined in detail. In particular, it is shown that r has the interpretation of a tangential restitution coefficient. The case when Coulomb’s friction is introduced at the contact impulse level together with an angular velocity restitution is analyzed. A simple chain of aligned balls is also examined, illustrating that the impact law applies to various types of multibody systems.     

Collision with friction; Part A: Newton’s hypothesis

This paper deals with collision with friction. In Part A, equations governing a one-point collision of planar, simple nonholonomic systems are generated. Expressions for the normal and tangential impulses, the normal and tangential velocities of separation of the colliding points, and the change of the system mechanical energy are written for three types of collision (i.e., forward sliding, sticking, etc.). These together with Routh’s semigraphical method and Coulomb’s coefficient of friction are used to show that the algebraic signs of four, newly-defined, configuration-related parameters, not all independent, span five cases of system configuration. For each, the ratio between the tangential and normal components of the velocity of approach, called α, determine the type of collision, which once found, allows the evaluation of the associated normal and tangential impulses and ultimately the changes in the motion variables. The analysis of these cases indicates that the calculated mechanical energy may increase if sticking or reverse sliding occur. In Part B, theories based on Poisson’s and Stronge’s hypotheses are presented with more encouraging results.     

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.     

Stronge’s hypothesis-based solution to the planar collision-with-friction problem

This paper deals with collision with friction. Differential equations governing a one-point collision of planar, simple non-holonomic systems are generated. Expressions for quantities of interest (e.g., normal and tangential impulses, normal and tangential relative velocities of the colliding points, and the change of the system mechanical energy), are written for five types of collision (i.e., sticking in compression, forward sliding, etc.) associated with Stronge’s collision hypothesis and Coulomb’s coefficient of friction, in conjunction with a two-integration procedure. These expressions, together with Routh’s semi-graphical method are used to show that the algebraic signs of four configuration-related parameters span five cases of system configuration. For each, the ratio between the tangential and normal components of the velocity of approach, called α, determine the type of collision which, once found, allows the evaluation of the changes in the motion variables. The analysis of these cases indicates that an algebraic, Stronge’s hypothesis-based solution to the planar collision-with-friction problem always exists, and is unique, coherent and energy consistent. Finally, substitutions are found which transform the Stronge’s hypothesis-based solution to a Poisson’s hypothesis-based and to a Newton’s hypothesis-based solutions appearing in the literature.     

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.     

Intelligent compliant force/motion control of nonholonomic mobile manipulator working on the nonrigid surface

The task under consideration is to control a mobile manipulator for the class of nonrigid constrained motion. The working surface is deformable. The geometric and physical model of the surface is unknown and all contact force is nonlinear and difficult to model. To accomplish a task of this kind, we propose a force/motion fuzzy controller based on the philosophy of the parallel approach in two decoupled subspaces. In one subspace, we control the constant contact force normal to the surface and estimate the end-effector tool’s deformable depth of the surface; in the other, we keep the end-effector’s constant velocity parallel to the tangential plane of the surface and suppress the tangential force of the surface deformation. The nonholonomic mobile base is utilized to avoid the singularity. Stability is established and conditions for the control parameters are derived. Performance of the proposed controller is verified through computer simulations compared with the model-based control.     

Dynamics of a three-module vibration-driven system with non-symmetric Coulomb’s dry friction

In the present paper, a three-module vibration-driven system moving on a rough horizontal plane is modeled to investigate the relation among the system’s steady-state motion, external Coulomb’s dry friction force and internal excitations. Each module of the system represents a vibration-driven system composed of a rigid body and a movable internal mass. Major attention is focused on the primary resonance situation that the excitation frequency is close to the first-order natural frequency of the system. In the case that the external friction is low, the internal excitation is weak and the stick–slip motion is negligible, both methods of averaging and modal superposition are employed to study the steady-state motion of the system. Through a set of algebraic equations, an approximate value of the system’s average steady-state velocity is obtained. Several numerical examples are calculated to verify the validity of the analytical results both qualitatively and quantitatively. It is seen that big quantitative errors will appear if stick–slip motions occur. Then, two mechanisms for the possible stick–slip motions are put forward, which explain the errors on the average steady-state velocity. Numerical simulations verify our analysis on the stick–slip effects and their mechanisms. Finally, to maximize the average steady-state velocity of the system, optimal control problem is studied. It is shown that, in addition to modifying the friction coefficients, the improvement of the system’s efficiency can be provided by changing the initial phase shifts among the three internal excitations.     

Collision with friction; Part B: Poisson’s and Stornge’s hypotheses

In Part B of this paper, planar collision theories, counterparts of the theory associated with Newton’s hypotheses described in Part A, are developed in connection with Poisson’s and Stronge’s hypotheses. First, expressions for the normal and tangential impulses, the normal and tangential velocities of separation, and the change of the system mechanical energy are written for five types of collision. These together with Routh’s semigraphical method and Coulomb’s coefficient of friction are used to show that the algebraic signs of the four parameters introduced in Part A span the same five cases of system configuration of Part A. For each, α determines the type of collision which once found allows the evaluation of the normal and tangential impulses and ultimately the changes in the motion variables. The analysis of the indicated cases shows that for Poisson’s hypothesis, a solution always exists which is unique, coherent and energy-consistent. The same applies to Stronge’s hypothesis, however, for a narrower range of application. It is thus concluded that Poisson’s hypothesis is superior as compared with Newton’s and Stronge’s hypotheses.     

Dynamic analysis of planar multi-body systems with LuGre friction at differently located revolute clearance joints

In this paper, the dynamic response of a planar rigid multi-body system with stick?Cslip friction in revolute clearance joints is studied. LuGre friction law is proposed to model the stick?Cslip friction at the revolute clearance joints. This is because using this law, one can capture the variation of the friction force with slip velocity, thus making it suitable for studies involving stick?Cslip motions. The effective coefficient of friction is represented as a function of the relative tangential velocity of the contacting bodies, that is, the journal and the bearing, and an internal state. In LuGre friction model, the internal state is considered to be the average bristle deflection of the contacting bodies. By applying the LuGre friction law on a typical slider?Ccrank mechanism, the friction force in the revolute joint having clearance is seen not to have a discontinuity at zero slip velocity throughout the simulation unlike in static friction models. In addition, LuGre model was observed to capture the Stribeck effect which is a phenomenon associated directly with stick?Cslip friction. The friction forces are seen to increase with increase in input speed. The effect of stick?Cslip friction on the overall dynamic behavior of a mechanical system at different speeds was seen to vary from one clearance joint to another.     

The Painlevé paradox studied at a 3D slender rod

This paper aims at investigating the dynamical behaviors of a 3D rod moving on a rough surface with so-called Painlevé paradox. The condition for the occurrence of the Painlevé paradox in the rod is studied according to the theoretical results obtained from LCP’s method for spatial multibody systems. Numerical results obtained by inserting a compliant contact model into the rigid body model present a support for the assumption that a tangential impact is related to the spatial paradoxical situations. Furthermore, the tangential impact is analyzed by using the Darboux–Keller’s shock dynamics and are found with the same properties as the one in the planar rod: A tangential stick appears at the contact point during the impulsive process. With the help of the Stronge’s coefficient, an impact rule is developed to describe the dynamical behaviors of the 3D rod with paradoxical situations. Comparisons between numerical results obtained from Darboux’s model and the ones obtained from the compliant contact model are carried out and show well agreements.     

Three-dimensional, one-point collision with friction

This paper deals with one-point collision with friction in three-dimensional, simple non-holonomic multibody systems. With Keller’s idea regarding the normal impulse as an independent variable during collision, and with Coulomb’s friction law, the system equations of motion reduce to five, coupled, nonlinear, first order differential equations. These equations have a singular point if sticking is reached, and their solution is ‘navigated’ through this singularity in a way leading to either sticking or sliding renewal in a uniquely defined direction. Here, two solutions are presented in connection with Newton’s, Poisson’s and Stronge’s classical collision hypotheses. One is based on numerical integration of the five equations. The other, significantly faster, replaces the integration by a recursive summation. In connection with a two-sled collision problem, close agreement between the two solutions is obtained with a few summation steps.     

Implementation of the Contensou–Erismann tangent forces model in the Hertz contact problem

In frame of the Hertz contact problem, an approximate model to compute resulting wrench of the dry friction tangent forces is built up. The wrench consists of the total friction force and the drilling friction torque. An approach under consideration develops in a natural way the contact model constructed earlier. The dry friction force and torque are integrated over the contact elliptic area. Generally, an analytic computation of the integrals mentioned leads to the cumbersome calculation, tens of terms, including rational functions depending in turn on complete elliptic integrals. To implement the elastic bodies contact interaction computer model fast enough, one builds up an approximate model in the way initially proposed by Contensou.     

Capillary filling with the effect of pneumatic pressure of trapped air

This article presents an investigation into the effects of pneumatic pressure of trapped air on the dynamics of capillary filling. Controlled experiments were carried out in horizontal closed-end capillaries with diameters of 200–700 μm. Glycerol–DI water mixture solutions having viscosities ranging from 8 to 80 mPa s were used as the filling liquids. The pneumatic air backpressure is built up as a result of the air compressed at the closed end of the capillary. A model is presented based on the conventional theory of capillary filling (i.e., Washburn’s equation) with consideration of the effect of air backpressure force on the advancing meniscus. The molecular kinetics theory of Blake and De Coninck’s model (Adv Colloid Interface Sci 96:21–36, 2002) is also incorporated in the model to account for the dependence of dynamic contact angle on wetting velocity. The model predictions agree reasonably well with the experimental data. It is observed that due to the presence of air backpressure, the smaller the capillary diameter, the longer the length that the liquid fills the capillary, regardless of the liquid viscosity. It is also shown that the increased pneumatic air backpressure reduces the equilibrium contact angle (θ ⁰). A relation is then proposed among liquid penetration, capillary length and radius, and contact angle. In addition, a dimensionless analysis is performed on experimental data, and the power law dependence of dimensionless meniscus position on dimensionless time is obtained.     

A study of capillary flow from a pendant droplet

Surface tension driven capillary flow from a pendant droplet into a horizontal glass capillary is investigated in this paper. Effect of the droplet surface on dynamic behavior of such capillary flow is examined and compared with surface tension driven capillary flow from an infinite reservoir. In the experiment, capillaries of 300–700 μm in diameter were used with glycerol–DI water mixture solutions having viscosities ranging from 80 to 934 mPa s. It is observed that compared to the capillary flow from an infinite reservoir, the capillary flow from a droplet exhibits higher rates of meniscus displacement. This is due to an additional driving force resulted from change in droplet surface area (or curvature). The two main parameters influencing the flow are the dimensionless droplet geometry parameter (k) and the dynamic contact angle (θ _D). The molecular kinetics theory of Blake and De Coninck’s model [Adv Colloid Interface Sci 96(1–3):21–36, 2002] is used to interpret the dynamic contact angle. This theory considers a molecular friction coefficient (ζ) at the liquid front flowing over a solid surface. Moreover, three models are proposed to describe the shape of the pendant droplet during capillary action. It is found that the egg-shaped model provides a more realistic model to compute the shape of the pendant droplet deformed during the capillary action. Thus the predictions by the egg-shaped model are in good agreement with the experimental data.     

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.     

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.     

Rolling Contact Motion Generation and Control of Robotic Fingers

Dexterity in human hand is connected with the fingertip rolling ability. In this work we consider rolling motion of spherical robotic fingertips as one of the control objectives together with the set point position control and force trajectory tracking. The generation of a rolling motion trajectory is proposed and a control solution is designed which achieves prescribed transient and steady state tracking behavior. The proposed control law is structurally and computationally simple and does not utilize the dynamics of the robot model or its approximation. A simulation of a five degrees of freedom robot show excellent contact rolling performance even at cases of adverse friction conditions while alternative controllers lead to contact sliding. Experiments with a KUKA LWR4 + are performed to validate the proposed method.     

Contact Friction Compensation for Robots Using Genetic Learning Algorithms

In this paper, the issues of contact friction compensation for constrained robots are presented. The proposed design consists of two loops. The inner loop is for the inverse dynamics control which linearizes the system by canceling nonlinear dynamics, while the outer loop is for friction compensation. Although various models of friction have been proposed in many engineering applications, frictional force can be modeled by the Coulomb friction plus the viscous force. Based on such a model, an on-line genetic algorithm is proposed to learn the friction coefficients for friction model. The friction compensation control input is also implemented in terms of the friction coefficients to cancel the effect of unknown friction. By the guidance of the fitness function, the genetic learning algorithm searches for the best-fit value in a way like the natural surviving laws. Simulation results demonstrate that the proposed on-line genetic algorithm can achieve good friction compensation even under the conditions of measurement noise and system uncertainty. Moreover, the proposed control scheme is also found to be feasible for friction compensation of friction model with Stribeck effect and position-dependent friction model.     

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.     

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.