Analysis of Dynamic Characteristics of 6-PSS Parallel Mechanism Considering Spherical Hinge Clearance

This paper is based on a new type of 6-PSS parallel mechanism. Firstly, considering the clearance between the kinematic pair connecting the upper platform and the link, establish a kinematic model considering the spherical hinge clearance. Then, based on the Lankarani-Nikravesh(L-N) contact model and the modified Coulomb friction model calculate the contact force when the spherical elements contact, which is equivalent to the center of the corresponding member as the generalized external force. Apply for Newton Euler method with Lagrange Multiplier to establish the dynamics model of the parallel mechanism with clearance. Finally, the R-K method is used to solve the dynamic equation and analyze the influence of different spherical hinge gap sizes on the dynamic characteristics of the mechanism.

 is the coordinate system fixed to the center of the lower platform. The Z axis is perpendicular to the plane of the lower platform and the X axis is connected to one of the horizontal guide rails. The Y axis is determined by the right-hand spiral theorem. D XYZ  is a coordinate system fixedly connected to the center of the upper platform, and the direction of the coordinate system is the same as the direction of the fixed coordinate system when the entire mechanism is in the neutral position. Six guide rails are arranged on the lower platform to form two inner and outer equilateral triangles. The circumscribed circle radius is 1 R and 2 R respectively. The six spherical hinge points on the upper platform are distributed on a circle with a semi-diameter of r .The lengths of the inner and outer links connecting the upper and lower spherical hinges are 1 l and 2 l respectively.

Inverse solution of mechanism kinematics
According to the coordinate rotation formula, the rotation matrix of the upper platform coordinate system relative to the fixed platform coordinate system is: (1) Then the coordinates of the spherical hinge point on the upper platform under fixed coordinates can be expressed as: A is the coordinate of the upper spherical hinge point i A in the coordinate system of upper platform. The angle between the directions of the six guide rails and the X axis of the fixed coordinate system is i  , and set the coordinates of the guide rail vertex ,then the coordinates of the lower spherical hinge: ( The distance between the upper and lower spherical hinges is equal to the length of the link, which can gain the inverse solution of the mechanism.

3.Kinematics model considering spherical hinge clearance
For an ideal sphere hinge, the geometric centers of the motion pair connecting the sphere and the sphere socket are completely coincident, and the sphere socket rotates in three directions in the middle of the sphere. In fact, there is a clearance between the elements that make up the sphere hinge, which causes the sphere to move in three directions in the sphere socket. The motion constraint is transformed into a  O and 2 O . The eccentric vector is e , of which direction is from the center of the sphere socket to the center of the sphere.When the sphere socket and the sphere are in contact with each other ,suppose the contact points are 1 P and 2 P respectively, and contact deformation is  . The normal and tangential unit vectors of the contact surface e n and t n respectively.
(2) Then the eccentric unit vector between the sphere and the sphere socket is e n e e  , e is the modulus of vector e. Assuming that the radii of the sphere socket and the sphere are 1 r and 2 r , the sphere hinge clearance is 1 2 gap r r   , and the contact deformation is | | e gap    . This paper uses the "contactseparation" state model, assuming that there are only two states of contact and separation between the sphere socket and the sphere. Therefore, it can be judged whether the sphere socket and the sphere are in contact according to the contact deformation at two adjacent moments: When contact occurs, suppose the contact points of the sphere socket and the sphere are 1 P and 2 P , respectively, then: Take the derivative of (6) to get the contact velocity between the sphere socket and the sphere: The normal and tangential velocities can be obtained by projecting formula (5) to the contact surface and the normal plane of the contact surface respectively: (6) Then the tangential unit vector of the contact surface is:

Normal contact force
The classic Hertz contact model regards the contact problem as a completely elastic contact, ignoring the damping, and does not consider the energy loss during the contact process. The Lankarani-Nikravesh contact model (L-N model) proposes a nonlinear damping model that considers the coefficient of restitution, introduces the initial contact velocity and material properties [7] and takes into account the energy change during the contact process, which is closer to the real situation. Therefore, this paper uses the L-N contact model to describe the contact force of the spherical secondary elements of the parallel mechanism: Where n is the index; r c is the recovery coefficient of the spherical hinge;  and 0  are the contact deformation velocity and the initial contact deformation velocity, respectively; K is the stiffness coefficient: Where c E and b E Young's modulus of the sphere socket and sphere; c  and b  Poisson's ratio of the sphere socket and sphere respectively.

Tangential contact force
Since the surface of the sphere socket and the sphere is not smooth, friction is unavoidable. The most classic friction model at present is the Coulomb friction model, which expresses the friction as the product of the positive pressure and the friction factor. In order to prevent the frictional force from becoming discontinuous due to the direction change when the tangential velocity is near zero, the Coulomb friction model with correction coefficient is used to describe the frictional force of the spherical hinge in the contact process: t Where 0  and 1  Specific limit velocity within the error range

Equivalence of contact force
According to the normal and tangential contact force models established above, the contact force is obtained: c n e t t F F n F n   Force and torque can be used to equate the contact force to the center of the upper platform and the link: When the link rotates around its own axis, it does not affect the movement of the entire mechanism. Therefore, in order to simplify the kinematics constraint equation. Assuming that the angle of the link turning around its own axis is zero, the generalized coordinate of the link becomes kinematic pair and the components connected to the kinematic pair, which helps to quickly and accurately establish the kinematic constraint equation of the mechanism. The topology diagram of the 6-PSS parallel mechanism is Fig 5, the numbers 0-13 respectively represent the lower platform, 6 sliders, and 6 links, and the numbers H1-H18 represent various kinematic pairs that connect these components. The points in parentheses indicate the center of the coordinate system. The center points of the two components can be regarded as a group of associative arrays of the kinematic pairs, and the associative array is in turn related to the topology of the 6-PSS parallel mechanism There is a one-to-one correspondence between the graphs. The associative array of the 6-PSS parallel mechanism can be shown in Tab1.Since this paper does not need to establish a coordinate system at the center of the slider, the center of the slider is represented by "-".
Tab1. Associative array of 6-PSS parallel mechanism j The constraint equation of the spherical hinge connecting the slider and the link of the 6-PSS parallel mechanism: 6 3 1 The constraint equation of the spherical hinge connecting the upper platform and the link of the 6-PSS parallel mechanism is: In order to make the mechanism have a definite movement, a set of driving constraints must be imposed on the mechanism. The driving constraint equation is the slider displacement obtained by the inverse solution of the platform pose on the known mechanism, which can be expressed as: Since there is a clearance at the sixth spherical hinge connecting the upper platform and the link considered in this paper, the motion constraint is transformed into a force constraint, which needs to be removed from the ideal kinematic constraint equation. The removed kinematic constraint equation:

Establish the clearance dynamic equation
According to Newton's Euler formula with Lagrange multipliers, a dynamic equation including the kinematic constraints of the mechanism can be established without considering the clearance: where  is Lagrange multiplier; M and F are the generalized mass matrix and generalized external force, respectively.
When considering the sphere hinge clearance, remove the kinematic constraint equation of the spherical hinge connecting the upper platform and the link, and the contact force at the spherical hinge is equivalent to the center of the corresponding member, then the generalized force received of the upper platform and the sixth link: The position and velocity equations can only meet the requirements at a certain discrete moment, but the acceleration equation has a default phenomenon, which makes it impossible to obtain stable numerical solutions through numerical method. Baumgart algorithm [9] is used the most commonly solution. In the Baumgart algorithm, the velocity and position constraints are introduced into the acceleration term. Then the kinetic equation (15) becomes: Where a and b are default correction coefficients. When a and b are positive values, the system can generally reach stability, and when a=b, the system can quickly reach a stable state.

6.Numerical simulation
This paper uses R-K method to analyze the dynamic characteristics of the mechanism. The entire solution flow chart is shown in Fig 7.   Fig 7. Flow chart of clearance calculation In order to quantitatively evaluate the influence of different gap sizes on the dynamic characteristics of the mechanism, this paper selects the mean square root error of acceleration of the upper platform as the quantitative evaluation index of the simulation results. Where  is the mean square root error of acceleration; n is the sample size; and represent the acceleration with the clearance and the ideal acceleration, respectively. This chapter mainly analyzes the effect of the gap size on the dynamic performance of the mechanism through numerical simulation. The clearance parameters used are shown in Tab  The paper mainly analyzes the impact on the dynamic performance of the mechanism when there is a clearance at the upper spherical hinge H18, and the gap size gap=0.01mm、 0.05mm, 0.1mm and 0.2mm. The simulation parameters of the mechanism are shown in Tab 3. Because the mechanism is under the action of the single kinematic pair clearance, the upper platform's Y-direction and Z-direction movement laws are similar, so this paper uses the upper platform Z-direction movement parameters to illustrate the influence of the kinematic pair clearance on the dynamic characteristics of the entire mechanism. Given the center movement curve of the upper platform and kinetic simulation parameters Tab 3:  From Fig 8(a) of displacement with different gap sizes, it can be seen that the Z-direction displacement curve with clearance is relatively smooth and basically coincides with the ideal displacement curve, indicating that the clearance has little effect on the displacement accuracy of the upper platform. Further from the displacement error Fig 8(b), it can be seen that the absolute value of the maximum displacement error in the Z direction increases with the increase of the gap size. When the gap size changes from 0.01mm to 0.2mm, the Z-direction displacement error is that about 0.005 mm rises to 0.09mm, and the maximum displacement error is less than the gap size.  From Fig 9，it can be seen that the velocity with clearances becomes unsmooth, with many burrs appearing, and fluctuates up and down the ideal curve, and it is most obvious near the velocity extreme. It can be seen from the partial enlarged view of the velocity that the amplitude of the fluctuation increases with the increase of the gap size.
From Fig 10, it can be seen that there are many spikes in the acceleration curve under different gap sizes. The value of the maximum spike increases with the increase of the gap size. When the clearance ranges from 0.01mm to 0.2mm, The Z direction ranges from 27.6 m/s^2 to 60.2 m/s^2, which is due to the increase in the maximum contact force as the gap size increases and leads to a corresponding increase Finally, increase some gap sizes and use the mean square root error of acceleration proposed quantify the degree of influence of different gaps on the dynamic characteristics of the upper platform. It can be seen from Tab 4 that mean square root error of acceleration increases with the increase of the gap size, which decreases dynamic performance and stability ; when the gap is 0.01mm and 0.02mm, mean square root error of acceleration smally changes, which means that it is of a little significance to improve the dynamic performance of the mechanism in reducing the gap size which increases the cost when the gap size is 0.02mm. Tab

7.Conclusion
Based on a new type of 6-PSS space parallel mechanism, this paper uses Newton's Euler formula with Lagrange Multiplier to establish a dynamic model with spherical hinge clearance, and analyzes the influence of different gap sizes on the dynamic characteristics of the entire mechanism. Draw the following conclusion, different gap sizes have a little effect on displacement and velocity, but have greater influence on acceleration and contact force. The increase of the gap size will deteriorate the dynamic characteristics of the mechanism and decrease the stability.