Continuous Contact Force Model Research Articles

Multibody dynamics in robotics with focus on contact events

Abstract Multibody dynamics methodologies have been fundamental tools utilized to model and simulate robotic systems that experience contact conditions with the surrounding environment, such as in the case of feet and ground interactions. In addressing such problems, it is of paramount importance to accurately and efficiently handle the large body displacement associated with locomotion of robots, as well as the dynamic response related to contact-impact events. Thus, a generic computational approach, based on the Newton–Euler formulation, to represent the gross motion of robotic systems, is revisited in this work. The main kinematic and dynamic features, necessary to obtain the equations of motion, are discussed. A numerical procedure suitable to solve the equations of motion is also presented. The problem of modeling contacts in dynamical systems involves two main tasks, namely, the contact detection and the contact resolution, which take into account for the kinematics and dynamics of the contacting bodies, constituting the general framework for the process of modeling and simulating complex contact scenarios. In order to properly model the contact interactions, the contact kinematic properties are established based on the geometry of contacting bodies, which allow to perform the contact detection task. The contact dynamics is represented by continuous contact force models, both in terms of normal and tangential contact directions. Finally, the presented formulations are demonstrated by the application to several robotics systems that involve contact and impact events with surrounding environment. Special emphasis is put on the systems’ dynamic behavior, in terms of performance and stability.

A continuous contact-force model for the impact analysis of viscoelastic materials with elastic aftereffect

A continuous contact-force model for the impact analysis of viscoelastic materials with elastic aftereffect

Dynamic analysis for mechanical system with clearance joint considering dependent interval parameters

Uncertainties caused by measuring errors or incomplete knowledge affect the dynamic response and functional reliability of the mechanical system, particularly for the mechanisms with clearance joints. Considering that dependency between the interval parameters in the uncertain quantification of mechanical systems with clearance joints is prone to be ignored, in this article, a dynamic analysis procedure considering the correlation between interval parameters is presented. Firstly, the continuous contact force model and modified friction force model are embedded into the deterministic dynamic governing equations of the mechanical system with clearance joints derived by the Lagrangian method. To characterize the correlation between interval uncertainties, the convex ellipsoid model is employed, and a Chebyshev-based convex scanning method is developed to obtain the bounds of dynamic response. Finally, a robot manipulator with clearance joint considering dependent uncertainties is illustrated as an example to investigate the clearance effect and correlations between uncertainties on the dynamic behavior. In the numerical simulation, the geometrical parameters of the manipulator arm are regarded as dependent interval parameters. The results show that the dependence between interval uncertainties induces relatively conservative bounds and should be considered carefully in the case of a large clearance size existing in the mechanical system.

A continuous contact force model for the impact analysis of hard and soft materials

• A new continuous contact model is proposed with high accuracy. • The new model can simulate impacts with remaining surface deformation after impact. • The new model can simulate impacts with zero separation indentation. • The model is verified by two published experimental data and typical contact models. Impacts with remaining surface deformation or zero indentation at the separation time are common in mechanical systems with hard and soft materials, and corresponding contact force models have been established for these two types of collision phenomena, but there is a lack of models that can simulate both types of impacts. A new continuous contact model is proposed that can simulate these two types of collisions with high accuracy. By constructing an approximate equation, the new model is developed based on the typical equation of continuous contact models. Two published experiments related to two types of collisions are used for model validation; in addition, typical continuous models for the impact analysis of two types of collisions are utilized for the comparison. The validation and comparison show that the new model can simulate the impact with remaining surface deformation or zero separation indentation with high accuracy. The influences of the restitution coefficient and remaining surface deformation ratio on the system dynamic behaviors have also been investigated. The new model will be helpful for the dynamic simulation of mechanical systems with contact-impacts phenomena involving hard materials and soft materials.

Dynamics and control of spacecraft solar array deployment considering physical contacts between locking mechanisms

The numerical simulation of the deployment and locking process of flexible appendages is an important topic in the overall design of spacecraft. In the existing research, the connecting mechanism between the solar panels is usually regarded as an ideal revolute constraint, and the locking process is simulated by applying a virtual lock torque related to the deployment angle. However, this method cannot obtain the physical contact force history, and because the physical meaning of the stiffness term in the lock torque function is not clear, it is difficult to accurately illustrate the frequency characteristics of the flexural vibration of the panels in the post-lock phase. In fact, the locking process is a frequent contact-impact process between the lock pin and groove. In this study, a typical deployable spacecraft with torsion-spring-driven panels were investigated numerically. The locking mechanism is modeled as a physical entity, and the continuous contact force model between the lock pin and groove is established. The contact stiffness of cylindrical line-contact is given by preloading calculation using the finite element method. Based on the multibody dynamics, rigid body model and flexible body model of the solar panels are established respectively. Through the comparison between the numerical simulation results of the two models, it is found that the rigid body model will produce excessive contact force and false high-frequency vibration. Furthermore, the attitude PD controller of spacecraft is designed, and the difference of system dynamic response in the deployment process under the main-body free state and the main-body controlled state is studied. The vibration frequency under the main-body controlled state is close to the first-order natural frequency of the structural finite element analysis, which verifies the rationality of the calculation results of the model.

Numerical and experimental study on dynamic characteristics of planar mechanism with mixed clearances

Clearances in joints are unavoidable due to assemblage, manufacturing errors and wear, which will lead to poor dynamic performances and undesirable vibrations due to impact forces in clearance joints. In dynamics analysis of planar mechanical system, it is usually assumed that the revolute joint is planar joint with radial clearance. However, the axial clearance in the revolute joint and the clearance in the translational joint are ignored. In fact, even planar mechanical systems may exhibit out-of-plane motion due to misalignments. In this work, a planar slider-crank mechanism is used as the example to investigate the effects of mixed clearances on the dynamic responses of planar mechanical system considering radial and axial clearances in spatial revolute joint due to axis misalignment as well as clearance in planar translational joint. The dynamic responses of the planar slider-crank mechanism system with two spatial revolute clearance joints and a planar translation clearance joint are investigated using computational and experimental methodology. The mathematic model of the spatial revolute clearance joint and the planar translational clearance joint are presented. The clearance joints are considered as force constraints and the joint components of an imperfect joint with clearance are modeled as contact bodies. The normal contact force model of clearance joint is established using continuous contact force model. The friction effect is considered using modified Coulomb friction model. Different case studies are presented and discussed to investigate the effects of the mixed clearances on the dynamic responses of the planar slider-crank mechanism.

A Continuous Contact Force Model for the Impact Analysis of Hard and Soft Materials

A Continuous Contact Force Model for the Impact Analysis of Hard and Soft Materials

A continuous contact force model for impact analysis

Hunt and Crossley proposed a general expression of the contact force in 1975. The dissipative force term of the general expression has two exponents: the exponent of indentation depth and that of velocity. Because it is almost impossible to obtain an analytical solution based on the general expression, more than twenty continuous contact models have been developed based on the simplification of the general expression. In these studies, the exponent of the indentation depth was set as 0.25, 0.5, 0.65, 1.0 or 1.5, and the exponent of the velocity was set to 1.0. This paper proposes a new continuous contact force model with arbitrary values of the exponents of the indentation depth and velocity. The model is based on the general expression of the contact force. Considering the rule of energy equivalence, an approximatedynamic equation is developed by introducing the equivalent indentation and equivalent velocity. Subsequently, a new continuous contact force model is constructed based on the system dynamic equation and approximate dynamic equation. The influences of the two exponents on the performance of the continuous contact force models are investigated by analyzing the simulation results. Moreover, the validity of the new model is demonstrated by comparing the simulation results with two published experimental datasets. The comparison also indicates that the performance of the continuous contact model can be enhanced by selecting appropriate values of the two exponents.

Kinematic accuracy and nonlinear dynamics of a flexible slider-crank mechanism with multiple clearance joints

Clearance joint appears in the multibody system and it could cause the uncertainty of motion. A general method for dynamic modelling and analysis of slider-crank mechanism with multiple clearance joints is presented and discussed in this work. The contact-impact characteristics of clearance joint are described by the nonlinear continuous contact force model. With flexible link taken into account, the dynamic characteristics of slider-crank mechanism with multiple clearance joints are analyzed based on the presented model. Then, the dynamic response of a flexible slider-crank mechanism is measured to clarify how to clearance joints can cause the vibration phenomenon of multibody system. Moreover, the stability and chaos analysis are conducted with the purpose of performing a parametric study, which quantifies the effects of clearance size and rotational speed on dynamic behavior of multibody system with multiple clearance joints. The numerical results indicate that the dynamic behavior of a flexible slider-crank mechanism depends on the characteristics of clearance joints.

Continuous contact force model with an arbitrary damping term exponent: Model and discussion

More than twenty continuous contact force models have been presented for modeling contact-impact problems; however, there was no uniformity regarding the value of the damping term exponent in these models. A new type of continuous contact force model is proposed in this research, in which the value of the damping term exponent can be arbitrary. Then, the effect of the value of the damping term exponent on the model accuracy is investigated by comparing the simulation and experimental results. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, according to the rule of energy equivalence, an approximate dynamic equation is developed by introducing an equivalent indentation and equivalent velocity. Based on the system dynamic equation and the approximate dynamic equation, a primary formula for the hysteresis damping factor of the model can be obtained. Through nondimensional analysis, new models for different values of the damping term exponent are established by modifying the primary formula. The comparison between the simulation results and published experimental data demonstrates the validities of the new models and reveals that the influence of the value of the damping term exponent on the model accuracy can be considered to be negligible.

Influence of Projectile-Barrel Coupling on Gun Vibration of the Moving Tank Under the Influence of a Stabilizer

The initial disturbance of the projectile and the barrel vibration is the premiere factors influencing the firing accuracy of the moving tank. To better understand the launch progress of the moving tank, a novel mechanical–electrical-hydraulic co-simulation model of the moving tank under the influence of a stabilizer was established, and a contact model with clearance for projectile-barrel rigid-flexible coupling was presented. Therein, the projectile-barrel coupling relationship was described by a cylindrical joint with clearance, and an improved contact algorithm based on the L–N model (Lankarani–Nikravesh continuous contact force model) was introduced to calculate the projectile-barrel contact force. The established model was verified tentatively by comparing with previously published literature. The simulation results indicate that the projectile-barrel coupling affects both the cradle vibration characteristics and the barrel dynamic bending features of the moving tank. It causes the gun's pointing deviation towards the target and the pointing deviation of the projectile velocity at muzzle exit and ultimately affects the firing accuracy of the moving tank.

Analysis of ride comfort of a continuous tracked bogie system with variable configuration

In this study, the effect of change in height of a so-called variable configuration of a continuous tracked bogie system on ride comfort performance is investigated. To this end, constraint equations among different vehicle parts are derived recursively, and equations of motion are formulated according to the Newton–Euler equations using the embedding technique. Besides, to model the track as a closed-loop chain, the penalty method has been used instead of loop-closure equations. To mathematically model the contact between tracks, wheels, and the terrain, the continuous contact force model has been employed. To validate the mathematical model, the vehicle dynamic behavior is simulated by the Runge–Kutta method in MATLAB software and compared with those obtained from the identical model, built in RecurDyn software, and the results show good agreement. Moreover, the vibrational behavior of the tracked vehicle is investigated using two different vehicle configurations. This reveals that an increase in vehicle height does not change the vehicle vibrational frequencies. However, it increases the vehicle vibration amplitude and the energy absorbed by the vehicle components and decreases the ride comfort.

A continuous contact force model for impact analysis in multibody dynamics

A new continuous contact force model for contact problems with complex geometries and energy dissipations in multibody systems is presented and discussed in this work. The model is developed according to the Hertz law, and a hysteresis damping force is introduced for modeling the energy dissipation during the contact process. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, an approximate dynamic equation for the collision system is proposed, achieving a good approximation of the system dynamic equation. An approximate function relating the deformation velocity and indentation depth is determined based on the approximate dynamic equation and is utilized to calculate the energy loss due to the damping force. Then, a primary formula of the hysteresis damping parameter is derived by combining energy balance and the law of conservation of linear momentum. The new model is developed by modifying the primary formula through nondimensional analysis. The comparison with published experimental data and the analysis of the simulation data of eight different continuous contact models reveal the capability and high precision of the new model as well as the effect of the geometry of the contacting surfaces on the dynamic system response.

A general formulation for the contact between superellipsoid surfaces and nodal points

The superellipsoids, part of the family of superquadratics, have geometric and mathematical properties that not only make them suitable for the representation of solids, ranging from spheres to relatively thin sheets, but also allow describing a very broad number of complex geometries. Their mathematical description favors a very efficient numerical treatment, in particular when used in the context of contact mechanics. Here, the contact between superellipsoids and nodal meshes, such as those used in finite elements, is analyzed. Based on the analytical properties of the implicit form of the superellipsoid equations, defined in a local coordinate frame, all the quantities necessary to handle a contact problem are derived. Recognizing that one of the most expensive tasks in contact mechanics is the contact detection, a three-stage contact search strategy is envisaged and implemented. Only in the close proximity between a node and the superellipsoid is a detailed contact search performed. In case of contact, the contact forces and their application points are evaluated using a continuous contact force model based on a penalty formulation. The existence of friction forces is considered. The methods developed and implemented for the contact between superellipsoids and nodal meshes are demonstrated and discussed with two contact scenarios comprising the impact of a superellipsoid on a hanging ‘towel’, and the contact of several superellipsoids with a flexible elastic platform.

Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method

A mechanical system with clearance joints exhibits non-linear characteristics. Even a small variation of one parameter may lead to a drastic change of the overall system response. The coupling of clearance and uncertainty would therefore significantly influence the system performance. In this paper, an analysis method for dynamic response and parameter sensitivity of mechanical systems is proposed with the consideration of both clearance joints and uncertainties. In this method, elements of a revolute clearance joint are modeled as colliding bodies, where continuous contact force model and modified friction force model are employed to describe the impact-contact behavior. The clearance joint model is then coupled with the system motion equations to obtain the system dynamic response. Furthermore, by introducing the multi-dimensional Chebyshev polynomials approach, dependence of the system dynamic response on its parameters can be established. The bounds of the dynamic response could be obtained by using the interval operations, and the parameter sensitivity is interpreted as the rate of the change of the system dynamic response with respect to the parameter variation, revealing the contribution of each parameter on the dynamic performance. Finally, dynamics of a crank-slider mechanism with clearance and uncertainties is investigated. Results show that a larger clearance would cause severe oscillation in the response bounds and larger expansion of the interval length, and clearance effect restraint zones can be obtained through the sensitivity analysis, where the expansion of the response interval length could be reduced.

Analysis of boot rail collision resistance in rocket sled test system

The shoe-rail resistance to impact of rocket sled was analyzed. Shoe-rail friction resistance was backward calculated with speed, acceleration, weight and gap of shoe-rail. The influencing factor was gain by the continuous contact force model. The relation of shoe-rail resistance to impact with speed, weight and gap of shoe-rail was gained by means of data fitting. Based on the results, the collision resistance coefficient of a new type of sled is calculated. According to the resistance coefficient, the speed of sled is predicted. The predicted results are consistent with the dynamic experimental results.

Wear Analysis of Spatial Parallel Mechanisms With Multiple Three-Dimensional Spherical Clearance Joints

The goal of this work is to investigate the dynamic responses of the parallel mechanism with irregular clearances caused by wear and to further reveal the influences of multiple clearance interaction on wear. The motion model and the force model of spherical clearance joint based on a continuous contact force model and a static friction model are established. The dynamic equation of the spatial parallel mechanism considering two spherical clearance joints is derived. A general wear analysis strategy to establish spherical clearance joint with sustainable updation of the surface profile is presented, and the dynamic responses of parallel mechanism after wear are studied. The interaction between two wear joints with different initial clearance values is further investigated. The results show that it is necessary to consider the factor of irregular clearances caused by wear in the analysis of dynamics behavior for precision mechanisms. Proper distribution of clearance values can reduce wear of clearance joint and improve the useful life of mechanism to a certain extent. This work provides a foundation for life prediction and reliability analysis of parallel mechanisms.

Model smoothing method of contact-impact dynamics in flexible multibody systems

An original model smoothing method for contact-impact dynamics in flexible multibody system is proposed in this work. First, the model smoothing method is applied to improve the computational efficiency, in which the instant stresses of flexible bodies are replaced by the time-averaged stresses in a short time during the modeling stage, the resulting equations of motion do not contain high frequency components. In the following, the fundamental issues of continuous contact force models are discussed. A smoothing method of modeling linear complementarity problem (LCP) is proposed for the dynamic analysis of flexible multibody system. This approach takes into account the permanent contact and impact, which has the great merit that can be used straightforward without switching contact models. Numerical results show that the model smoothing method is a new effective approach for the numerical analysis of contact-impact problems in flexible multibody system.

Simulating the anchor lifting maneuver of ships using contact detection techniques and continuous contact force models

Designing the geometry of a ship’s hull to guarantee a correct anchor maneuver is not an easy task. The engineer responsible for the design has to make sure that the anchor does not jam up during the lifting process and the position adopted by the anchor on the hull is acceptable when it is completely stowed. Some years ago, the design process was based on wooden scale models of the hull, anchor and chain links, which are expensive, their building process is time consuming and they do not offer the required precision. As a result of this research, a computational tool to simulate the anchor maneuver of generic ships given by CAD models was developed, and it is proving to be very helpful for the naval engineers. In this work, all the theory developed to simulate anchor maneuvers is thoroughly described, taking into account both the behavior of the anchor and the chain. To consider the contact forces between all the elements involved in the maneuver, a general contact algorithm for rigid bodies of arbitrary shapes and a particular contact algorithm for the chain links have been developed. In addition to the contact problem, aspects like the dynamic formulation of the equations of motion or the static equilibrium position problem are also covered in this work. To test the theory, a simulation of the anchor lifting maneuver of a ship is included as a case study. In spite of the motivation to develop the theory, the algorithms derived in this work are general and usable in any other multibody simulation with contacts, under the scope of validity of the models proposed. To illustrate this, the simulation of a valve rocker arm and cam system is accomplished, too. The computational and contact detection algorithms derived have been implemented in MBSLIM, a library for the dynamic simulation of multibody systems, and MBS model, a library for the contact detection and 3D rendering of multibody systems.

A contact force model considering constant external forces for impact analysis in multibody dynamics

A continuous contact force model, which considers the influence of constant external forces, is presented for the dynamic analysis of a multibody system. In this model, the Hertz contact law is applied to represent the nonlinear nature of contact, and a damping force is derived for evaluating the energy loss during impact. Together with the restitution coefficient, the external force influence factor defined in this paper is required for calculating the hysteresis damping factor associated with damping force. Moreover, the expression of hysteresis damping factor is deduced based on the energy-based method, which is adopted frequently in literature, and then it is improved by a weighted combination method with an exponential function due to the fact that the energy-based method has great errors when the restitution coefficient is low. Meanwhile, the exponential function is obtained by fitting the parametric surface of hysteresis damping factor gained from a numerical approach. Finally, four contact force models, including the new model, are utilized to compare the dynamic response of a special bouncing ball. The results illustrate that the described model is more suitable for impact analysis in multibody dynamics. In addition, the external forces and the energy loss are the main reasons for the multibody system to enter a steady contact state from repeated impact state.