Nonsmooth dynamic modeling and simulation of spatial mechanisms with frictional translational clearance joints

This paper presents a nonsmooth dynamic approach of modeling and computation for spatial mechanism including spatial frictional translational joints (SFTJs) with small clearances. In this research, the dynamic formulation is derived in the Lie group setting, which leads to a coordinate-free and compact formulation. In the following, the normal contact interaction between the slider and guide is described by the complementarity relations between the constraint force in the normal direction. The tangent contact interaction in the SFTJ is characterized by the Coulomb’s friction law in the type of a set-valued map. Based on the horizontal linear complementarity problem (HLCP), the non-smooth dynamics can be established and calculated by combining the Lemke’s algorithm and the RK-MK time integration algorithm. Finally, A spatial crank-slider mechanism with SFTJs is shown as a numerical case. The simulation results demonstrate the correctness and effectiveness of the proposed method which can capture the nonsmooth dynamical behavior of the system.


Introduction
In practical engineering applications, the clearance existed in joint of mechanisms is non-ignorable.The modelling and calculation of nonsmooth multibody dynamics is a complex and significant issue due to the influence of non-ideal joints, i.e., the frictional joints with clearance, which can modify the dynamic response of the system and result in unforeseen deviations between the theoretical and real outcome.
The non-smooth dynamics approach can effectively deal with the dynamics of the spatial mechanisms with non-ideal joints.It is presumed that two interacting objects are rigid and incapable of penetrating each other in this approach.As a result, the unilateral constraint conditions between two interacting objects are described by a linear complementarity problem (LCP) and solved by using differential variational inequalities [1][2][3][4][5].The linear complementarity problems were first studied by Signorini to describe the non-penetration conditions of rigid bodies [6].Subsequently, Moreau and colleagues employed the idea of complementarity to investigate nonsmooth multibody dynamics, proposing a time-stepping algorithm [7].Glocker and Pfeiffer analysed dynamics under conditions of dry friction contact, formulating unilateral contacts as linear complementarity problems and working out the transition conditions between stick and slip [8].Flores et al. based on the non-smooth dynamics approach, proposed a dynamic modelling method for planar multibody systems with joint clearances, describing clearances within the joints using normal gap functions [9][10][11].Wang et al. proposed a modeling and simulation approach for planar multibody dynamics considering frictional translational joints with small clearances which is described by bilateral constraint equations, they ignored impact effects and directly established linear complementarity conditions for normal contact forces without relying on normal gap functions [12,13].Swamy and Chand studied the joint parameter of a 3DOF spatial manipulator based on rigid body dynamic simulation [14].
Complementarity problems prove to be an effective approach in dealing with non-smooth dynamics, allowing for the accurate simulation of static friction forces without the need for a modified Coulomb friction model.This method helps avoid the rigidity issues present in the regularized approach, where high local stiffness will lead to highly-frequency oscillations and the contact parameters are difficult to determine.
However, handling complex constraints as linear complementarity problems for spatial multibody systems remains challenging.In our study, a simple method for modelling and analysis the nonsmooth dynamics of spatial multibody systems with SFTJs based on the horizontal linear complementarity problem is presented in the Lie group setting.
This paper is structured as follows.The geometric description and reduced model of SFTJs with small clearance, of which normal contact and tangential contact laws are presented in section 2. In section 3, the non-smooth dynamic equations of spatial multibody systems on SE(3) are formulated.In section 4, a numerical example is processed to demonstrate the correctness and effectiveness of the method proposed in this paper.The conclusions of this paper are given in the last section.

Geometric description of SFTJs with small clearances
The schematic of a spatial frictional translational joints with small clearance is shown in figure 1.For the sake of illustration, the geometric dimensions of clearance are intentionally exaggerated.The joint consists of the guide and the slider in the shape of straight quadrilateral prism.In this study, the clearance size is small enough to neglect the attitude deviation of the slider with respect to the guide.A coordinate frame is established by assigning the x-axis along the motion direction of spatial translational joint.For the sake of simplification, the clearance is evenly distributed in the x direction.Thus, the reduced model of the spatial translational joint with small clearance can be presented in yoz plane.Nine possible contact configurations are enumerated in figure 2 (a-i).
Reduced model of SFTJs with small clearances

Normal contact law of SFTJs
The normal constraint forces in nine possible contact configurations can be compactly described as an LCP.An LCP is a collection of linear equations subjected to a certain inequality complementarity condition, which can be expressed in matrix form as follow [15]  Any relative configurations of the SFTJs between the guide and the slider can be determined by the distribution of normal constraint forces.The complementarity conditions of the normal force magnitudes y F  , y F  , z F  and z F  can be given by 0, 0, 0 0, 0, 0 With regard to n spatial translational joints, the complementarity conditions can be expressed in the vector form as , ,..., ,

Tangential contact law of SFTJs
The tangential contact interaction between the slider and the guide of spatial translational joints is characterized by the Coulomb friction force law, which can be defined by where T F denotes the friction force, N F denotes the magnitude of the normal constraint force, T v is the relative velocity of the frictional contact,  and   are the kinetic friction coefficient and the static friction coefficient, respectively.

 
sgn x is the classical sign function, the multifunction

 
Sgn x is defined by Eq. 5, which is depicted in figure 3 In consideration of the static friction, an active set which describes the contact status of the SFTJs with small clearance is defined by The tangential contact force in x direction of the spatial translational joint is equal to the resultant frictional force between the guide and the slider diag ,..., In view of the active set, the tangential relative acceleration of the spatial translational joints can be split into positive and negative parts The friction saturation can be defined as Thus, the complementarity conditions of frictional translational joints can be written as where T 1 ,...,

Nonsmooth dynamics formulation of spatial MBS with SFTJs
The motion equations of spatial multibody system with SFTJs can be formulated as an index 1 system which can be written as the differential-algebraic equations (DAEs) on SE(3). where is the inertia matrix of system, is the ambient space of system, namely, the configuration of n rigid bodies, G Φ denotes the constraint Jacobian matrix of the system, T W is the Jacobian matrix of the relative velocity in tangential direction with respect to the generalized velocity, 6n  ξ  is the generalized velocity of system, λ is the Lagrange multiplier vector, H λ is tangential friction force of SFTJs, Q denotes the generalized external force vector and gyroscopic force vector, ,  and  are the feedback control parameters [16].
In order to establish the non-smooth dynamics of spatial MBS with SFTJs, the Lagrange multiplier vector is split into the positive parts and negative parts (11) According to equation ( 6), the tangential friction force can be given by introducing the positive parts and the negative parts of λ Substituting equation ( 11) and ( 12) into equation ( 10) yields Together with equation ( 12), the friction saturation can be written as 2 2 where Thus, equation ( 13) and ( 14) can be written in the following compact form where

Numerical examples
A spatial crank-slider mechanism consisting of a crank, a connecting rod, and a slider is depicted in figure 4, of which geometric and inertia characteristics are shown in Table 1.The global coordinate system is established on the bearing of the revolute joint, with z-axis oriented vertically upward, x-axis parallel to the rotational direction of the revolute joint, and y-axis satisfying the right-hand rule.When the mechanism is in the initial configuration, z-axis of the crank coincides with z-axis of the global coordinate system.The initial angular velocity of the crank is along the positive y-axis and has a magnitude of 0 2 rad s    .The initial positions and velocities of the other components can be calculated by the constraint equations.The gravitational acceleration 2 9.81m s g  is directed along the negative direction of z-axis.The crank is connected to the ground through the revolute joint, the crank is connected to the connecting rod through the spherical, the connecting rod is connected to the slider through a Hooke joint, and the slider is connected to the ground through a prismatic joint, which is nonideal (with small clearance and friction), while the other joints are ideal.The stability parameter is 5  In the framework of Lie group, the non-smooth dynamics of the system can be solved by using the RK-MK time integration algorithm and the Lemke's algorithm to calculate the equation (15).
Figure 5(a) and (b) depict the angular velocity of the crank and the slider's velocity along the x-axis, respectively.(c) and (d) show the angular acceleration of the crank and the slider's acceleration along the x-axis, respectively.The normal constraint force and tangential friction on the slider are presented in (e) and (f).It can be observed that when both the angular velocity of the crank and the velocity of the slider are zero, the system is in a sticking state, and abrupt changes occur in acceleration, normal constraint force, and tangential friction force.When the slider velocity is zero but the crank velocity is not, the mechanism is in a singular configuration.Although the slider is in a sliding state, the acceleration changes drastically.Figure 5(g) and (h) present the constraint forces in y-axis on the slider under ideal and non-ideal conditions.It can be seen that under ideal conditions, the dynamic response is periodic.However, in non-ideal conditions, under the influence of friction, non-smooth phenomena occur, and the amplitude decays over time.

Conclusions
In this paper, the modeling method and analysis for spatial MBS with SFTJs with small clearances is investigated in Lie group setting.The geometric constraints of SFTJs are processed as two sets of bilateral constraints by neglecting the impact effect between the slider and the guide under the small clearance hypothesis.The frictional contact in SFTJs is processed as a HLCP solved by Lemke's algorithm embedded in the RK-MK time integration algorithm.The Baumgarte stabilization method is employed to reduce the constraint drift of constraint equations.Although the attitude deviation of the slider with respect to the guide is ignored in this study, the approach presented can be extend to the nonsmooth dynamics of spatial MBS with SFTJs taking the attitude variation caused by clearance into account.

Figure 1 .
Figure 1.The schematic of SFTJs with small clearances

Figure 3 .
Figure 3.The map and the dynamic friction coefficient and static friction coefficient are 0

Figure 4 .
Figure 4. Spatial crank-slider mechanism with STCJsIn the framework of Lie group, the non-smooth dynamics of the system can be solved by using the RK-MK time integration algorithm and the Lemke's algorithm to calculate the equation(15).Figure5(a) and (b) depict the angular velocity of the crank and the slider's velocity along the x-axis, respectively.(c) and (d) show the angular acceleration of the crank and the slider's acceleration along the x-axis, respectively.The normal constraint force and tangential friction on the slider are presented in (e) and (f).It can be observed that when both the angular velocity of the crank and the velocity of the slider are zero, the system is in a sticking state, and abrupt changes occur in acceleration, normal constraint force, and tangential friction force.When the slider velocity is zero but the crank velocity is not, the mechanism is in a singular configuration.Although the slider is in a sliding state, the acceleration changes drastically.Figure5(g) and (h) present the constraint forces in y-axis on the slider under ideal and non-ideal conditions.It can be seen that under ideal conditions, the dynamic response is periodic.However, in non-ideal conditions, under the influence of friction, non-smooth phenomena occur, and the amplitude decays over time.

Figure 5 .
Figure 5. Dynamic response of the spatial crank-slider mechanism with SFTJs