A recursive algorithm for dynamics of multiple frictionless impact-contacts in open-loop robotic mechanisms

Highlights

• Dynamic modeling of multiple impact-contacts in open kinematic chains has been studied.

• Recursive Gibbs–Appell formulation has been used to derive the motion equations.

• In order to model the impact phenomenon, the regularized method has been utilized.

• Detecting the impact moments and solving the stiff differential equations are investigated.

• Nine of the famous contact force models have been compared to select a more suitable one.

Abstract

In this paper, the phenomenon of multiple impact-contacts has been dynamically modeled for an open kinematic chain with rigid links and revolute joints. The dynamic equations of the mentioned system have been extracted based on the recursive Gibbs–Appell formulation. The impact-contact phenomenon has been formulated by the regularized method in which the force of impact is a continuous function of the relative penetration and relative velocity of two colliding surfaces with respect to each other. The geometrical specifications and the mechanical properties of colliding surfaces are the two main parameters used in the modeling of viscoelastic contact force models. In this work, a recursive algorithm, which has been developed based on 3 × 3 rotation matrices to reduce the computational load, symbolically derives the motion equations of a multibody system that collides with surrounding surfaces at several points. The system under study includes the non-impact (flight) phase and the impact phase. Going from the flight phase to the impact phase and back, detecting the exact moment of impact, and also solving the differential equations of motion during a very short impact time have their particular challenges and complexities, which are dealt with in this work. In the next step, nine famous contact force models have been compared in order to select the most suitable model for simulation work. Finally, to show the accuracy and the capability of the presented algorithm, the dynamic behavior of an open-chain robotic mechanism consisting of 4 rigid links connected by revolute joints has been simulated and analyzed.

Introduction

In mechanics, ‘contact’ characterizes a continuous process of contact between two bodies, which occurs at a finite time; while ‘impact’ is a phenomenon in which a very large force is exerted on two colliding bodies in a very short time, resulting in a quick dissipation of energy and in the large changes that occur in the velocities of the two bodies [1]. The dynamic modeling and analysis of contact-impact phenomenon, which occurs extensively in multibody systems, is an interesting, complex and challenging subject in engineering. It has attracted the attention of many researchers in the past decades and is still considered a viable theme for research and development [2]. It is important to study the phenomenon of contact-impact in robots as multibody systems. The contact-impact phenomenon vastly occurs in humanoid robots, human-assisting robots and passive bipedal walking robots, in the simulation of accidents and the investigation of clearance in joints, in walking machines and space explorations, in surgeon robots when using surgical incision tools, in industrial robots when lifting objects, in search robots when hitting the obstacles on the way, and in agricultural robots involved in product harvesting, etc. Hence, in this paper, we study the contact-impact phenomenon in multibody robotic systems.

There are different methods for the dynamic modeling of contact in multibody systems. These methods can be divided into two main branches of contact force method (regularized, penalty or compliant method) and non-smooth method [1–3]. Each of these approaches has its own advantages and disadvantages. In the non-smooth method, which is based on the principle of impact-momentum, the colliding bodies are assumed to be rigid, and no deformation occurs at the impact location. However, in the contact force method, the impact-induced force is a function of the relative penetration of two bodies into each other. In this model, the impact phenomenon is analyzed by inserting the impact force into the equations of motion and considering a very short impact time. It is worth mentioning that the model prepared by this method is closer to reality [1]. The drawback of this method is the difficulty of selecting the parameters that are used to obtain the contact force; such as the generalized stiffness coefficient, surface damping coefficient, and the degree of indentation nonlinearity [2]. Nevertheless, since the modeling in this approach better reflects the reality, the contact force method has been employed in the present paper to model the contact-impact phenomenon.

The advent of the contact force models that are used today in the analysis of impact phenomenon can be traced back to the works of Hertz. He developed a purely elastic contact force model for the impact of two spheres [4]. The major flaw of this model, however, was that it did not consider the dissipation of energy in the process of impact. Nevertheless, this work provided a background for other researchers to develop more realistic models that do take energy loss into consideration [5]. The contact of cylindrical objects, friction in contact, surface adhesion, and the impact of isentropic bodies are some of the subjects that have been investigated by many researchers [6]. In the last several decades, effective analytical and experimental models have been introduced for modeling the viscoelastic contact forces. For example, the contact force model presented by Hunt and Crossley not only considers the elastic and damping properties but also the continuity at the beginning and the end of the impact process [7]. Most subsequent researchers then used the contact model of Hunt and Crossley to develop newer models. The difference of these models is in the way they express the hysteresis factor or, in fact, in the way this factor is expressed in terms of the coefficient of restitution. Herbert and McWhannell [8], Lee and Wang [9], Lankarani and Nikravesh [10], Gonthier et al. [11], Zhang and Sharf [12], Zhiying and Qishao [13], Ye et al. [14], Flores et al. [15], Gharib and Hurmuzlu [16] and Hu and Guo [17] are some of the researchers who have presented this type of contact force model.

The study of impact in multibody systems goes back to the last part of the 20th century. In 1977, Wittenburg investigated the subject of impact in multibody systems. The assumptions considered in studying the behavior of bodies subjected to impact forces are the same as those related to impact between two bodies plus some extra assumptions which have been added because of the existence of joint constraints [18]. The contact force model presented by Lankarani and Nikravesh is based on the loss of energy that occurs during the impact between two particles. After validating this model, it was generalized for multibody systems in which two of the bodies impact each other [10]. Pfeiffer and Glocker explored the behavior of multilink systems with an arbitrary number of rigid or elastic links undergoing multiple contacts [19]. In this work, they used the contact law which had been modified for multiple contacts. Hurmuzlu and Marghitu have studied the occurrence of multiple contacts in a planar kinematic chain in the presence of friction [20]. Ambrósio has modeled the phenomenon of impact in multibody systems with rigid and elastic members [21]. In this work, he has employed the plastic hinge, finite difference and finite element methods to obtain the deformations of elastic bodies. Due to the limited time of impact, it is essential to determine the moment of impact with an acceptable accuracy. To get the exact moment of impact, Flores and Ambrósio [22] used an algorithm with variable time step for numerical integration purposes. The phenomenon of impact-contact also occurs in multilink systems because of the clearance in their joints. Flores and Lankarani [23] investigated the internal impact that occurs in real joints. Their model is based on the elastic model of Hertz to which a damping term has been added. Also in this work, they have used the classic Coulomb model for the modeling of contact friction. For multibody systems, the modeling of continuous contact force in the presence of a constant external force has been presented by Shen et al. on the basis of Hertz contact law [24]. In this research, they have considered the damping force in computing the energy lost during impact. Also, they have used the coefficient of restitution and the external force effect in order to determine the hysteresis damping factor which is used in obtaining the damping force. Askari et al. [25] examined the effects of friction-induced vibration and contact mechanics on the maximum pressure and moment applied on human's artificial hip joints. They showed that their multibody dynamics model can provide fairly accurate and quick predictions of the distribution of contact pressure and of the moment applied on hip joint. Based on Johnson's contact model and using complementary finite element analysis, Pereira et al. [26] presented a new contact force model for cylindrical surfaces. Their proposed model was an explicit function of the amount of deformation. Based on experimental data, Jin at al. [27] proposed a new nonlinear force model. In this model, the radial force and radial deformations are measured first in static tests, and then the obtained data are fitted by using the Hertz contact model and a 3rd order polynomial function. In a research survey, Corral et al. [28] compared the effects of several different contact force models on the phenomena of impact and friction. However, in most of the abovementioned works, the researchers have developed and improved various contact models, but have paid less attention to the phenomenon of contact-impact in n-link systems subjected to multiple impacts.

The manual derivation of dynamic motion equations for multibody systems like robotic chains with more than two links is very complicated and time-consuming and requires a high degree of accuracy. In such cases, recursive algorithms have to be applied to extract a system's motion equations. With these algorithms, by knowing the kinematic and dynamic behavior of a member, the behavior of its adjacent members can be determined. Vereshchagin [29], Armstrong [30] and Featherstone [31] were the first researchers who employed a recursive algorithm in their works. Ploen and Park used the recursive method of Newton–Euler to obtain the inverse and forward dynamics equations of a multibody system [32]. Shabana [33] and Hwang [34] explored the use of generalized Newton–Euler equations in the development of recursive dynamics formulation for an open chain of elastic links with revolute, prismatic and cylindrical joints. Zhang and Song [35] showed that the recursive method based on virtual work principle is better than the recursive Newton–Euler approach in formulating the dynamic equations of parallel manipulators or closed-loop series manipulators. Yamane and Nakamura [36] presented an algorithm based on virtual work principle for deriving the forward dynamics equations of an n-link open kinematic chain. In this algorithm, for determining the constrained forces in each joint, the computations are performed from the last link to the first link; and then to obtain the accelerations, the computations are carried out from the first link to the last link. Zhang [37] used the recursive Lagrange method to derive the dynamic equations of a multi-flexible-link system with revolute joints. He also developed a software program for the dynamic simulation of system based on the mentioned method. The Gibbs–Appell method is another approach for expressing the dynamic equations of a system. Because of requiring fewer partial derivatives relative to the Lagrange approach, the G–A methodology is more suitable for multibody systems [38]. Using this method, Korayem and Shafei have recently modeled a variety of robotic systems. Some of these works include the modeling of rigid and flexible robotic arms [39], [40], [41], [42], [53], moving-base robots with nonholonomic constraints [43,44] and robots with prismatic-revolute joints [45,46]. In modeling the impact phenomenon in robotic chains, Shafei and Shafei have applied the recursive Gibbs–Appell algorithm for rigid-link or flexible-link open-chain mechanisms confined in a closed space [47], [48], [49]. They have also developed the mentioned algorithm for multi-branch open-loop robots [50,51] and closed-loop robots [52]. The modeling of impact in these works is based on the Newton's impact law in which the time of impact is assumed to be zero. Although the force of impact can be computed in this approach, its main drawback is that the deformations at the impact location and the force arising from collision cannot be calculated during such a short impact time.

In this paper, the recursive Gibbs–Appell method is used to model the phenomenon of impact-contact in an open kinematic chain system with rigid links and revolute joints, which is confined in a rectangular box. The impact model used in this work is a continuous contact force model. By incorporating this model into system equations, a systematic method is established for dynamically analyzing an n-link system's multiple collisions. By employing the proposed model, the resulting deformations, velocities and accelerations during impact time and the forces applied by surrounding walls on colliding joints can be determined. This paper has been organized as follows: Section 2 deals with the kinematics of the studied robot, and the dynamics of the problem are described in Section 3. The well-known contact force models used in the analysis of contact-impact phenomenon are introduced in Section 4. In Section 5, a relatively complex robotic system is simulated and the simulation results obtained by different models are compared with each other. Finally, the conclusion of the work is presented and the advantages of the proposed method are highlighted.