Dynamic modeling of spatial parallel mechanism with multi-spherical joint clearances

Abstract

The parallel mechanism has advantages of the high speed, high precision, strong carrying capacity, and high structural rigidity. Most of the previous studies concerning the dynamic modeling focused on planar mechanisms with revolute clearance joints or spatial mechanisms with one spherical joint clearance, while few studies focused on spatial parallel mechanisms with multi-spherical joint clearances. In this article, a general dynamic modeling method for spatial parallel mechanism with multi-spherical joint clearances based on Lagrange multiplier method is proposed. Taking 4universal joint-prismatic joint-spherical joint/universal joint-prismatic joint- universal joint (4UPS-UPU) spatial parallel mechanism as an example, the constraint equations of common kinematic pairs in spatial parallel mechanism, such as universal joint, spherical joint, and prismatic joint, are derived in detail. The dynamic model of the parallel mechanism with two spherical joint clearances combining the Flores contact force model and the LuGre friction model is established. The correctness of model has also been verified by comparing the analysis results of MATLAB with those of ADAMS. It can be seen that dynamic model of spatial parallel mechanism with multi-spherical joint clearances could be easily established by this method, which provides a theoretical reference to establish the dynamic model of other parallel mechanism with multi-clearance in the future.

Instruction

Compared with series mechanism, parallel mechanism has advantages of the high speed, high precision, strong carrying capacity, and high structural rigidity. ^{1 –6} Due to the closed-loop structure characteristics of the parallel mechanism, the dynamic model of the mechanism is usually more complicated. ⁷ In practical working conditions, joint clearance is unavoidable in parallel mechanism due to various reasons such as design, assembly process, friction, and wear. ⁸ The existence of clearance will cause vibration and noise of the mechanism to a large extent, which reduces accuracy, intensifies wear, and shortens service life of parallel mechanism. Therefore, the research on parallel mechanism with joint clearances must be considered, and how to improve the efficiency of the dynamic modeling with clearances becomes a research hotspot.

In recent years, many scholars have done a great deal of studies on dynamics model of mechanism and adopted many modeling methods. ^9,10 The dynamic modeling methods mainly include Lagrange multiplier method, Lagrange equation of second kind, the Newton–Euler method, the virtual-work approach, the Kane method, and so on. Marques et al. ¹¹ proposed a comprehensive and general method to eliminate constraints violation at the position and velocity levels based on Lagrange multiplier method. Tan et al. ¹² researched dynamic behavior of slider-crank mechanism containing the revolute clearance joints. And the dynamic equations are developed by combining Lagrange multiplier with modified contact force model and improved Coulomb friction force model. Chen et al. ¹³ established nonlinear dynamic model of parallel mechanism by the Lagrange equations of the second kind and analyzed the nonlinear dynamic performance of a four-degree-of-freedom (DOF) parallel mechanism with single spherical joint. Pedrammehr et al. ¹⁴ studied dynamic modeling of the Hexarot parallel manipulators by Newton–Euler approach. The dynamics of the mechanism were also verified by MATLAB and ADAMS. Pedrammehr et al. ¹⁵ used Newton–Euler method to establish the improved dynamic equation of the Stewart platform with general configuration. Simulation results showed that new method significantly enhanced the dynamic equation of previous method. Kalani et al. ¹⁶ investigated the inverse and direct dynamics of 6-UPS Gough–Stewart parallel robot based on the virtual work method, which reduced the computational time and improved accuracy of dynamics equations. Cheng and Shan ¹⁷ deduced and analyzed the dynamic model of a 3SPS + 1PS spatial parallel manipulator using Kane equation. The results showed that the inertia force of the branching chain, the friction moment of the artificial hip joint, and weight of the mobile platform should be balanced through the driving force. At present, Lagrange multiplier method has been applied to establish dynamics model of parallel mechanism. In this method, from the viewpoint of constraints among components of mechanism system, the constraint equation of joints for mechanism system have been described through generalized coordinates, and the constraint reaction between the components of multi-body systems can be obtained by calculating Lagrange multiplier.

In recent decades, many scholars have done plenty of studies on dynamic modeling of mechanisms with clearances. However, most of the previous studies focused on planar mechanisms with revolute clearance joints or spatial mechanisms with one spherical joint clearance, while few studies focused on spatial parallel mechanisms considering multi-spherical joint clearances. Flores and Lankarani ¹⁸ derived dynamic equations of planar crank slider mechanism with revolute clearance joint. The dynamic responses and nonlinear characteristics were both analyzed in detail. Chen et al. ¹⁹ established a 2-DOF complex planar dynamic model of flexible multi-link mechanism containing revolute clearance, and the effects of clearance value and driving speed on the nonlinear dynamic behavior of the multi-link mechanism were both studied. Erkaya ²⁰ studied the dynamic responses of a welding robot containing clearance and the different clearance values on dynamic responses were also researched. Muvengei et al. ²¹ investigated the influences of frictionless revolute clearance joints at different positions on dynamic characteristics of crank slider mechanism. Erkaya and Dogan ²² studied the dynamic response of conventional and compliant mechanism containing revolute clearance joints. The effects of different clearance values and input driving velocity on dynamics behaviors of slider crank are researched by ADAMS. Tian et al. ²³ developed nonlinear dynamics model of a flexible multi-body system considering clearance and lubricated revolute joint clearance and used a planar absolute nodal coordinate formulation based upon locking free shear deformable beam element to discretize flexible bodies. Hou et al. ²⁴ investigated the chaos phenomenon of planar RU-RPR parallel mechanism considering revolute clearance joint. The nonlinear characteristics of the RU-RPR mechanism have been discussed through Poincaré maps and bifurcation diagrams. Chen et al. ²⁵ analyzed the influence of the clearance size, input driving velocity, flexible links on dynamics behavior for crank slider mechanism through experiment, and ADAMS. Marques et al. ²⁶ proposed an investigation on dynamic modeling and analysis of a spatial four-bar mechanism containing spherical clearance joints including friction. Liu et al. ²⁷ studied the dynamics and control of a rigid–flexible multi-body system containing multiple cylindrical clearance joints via the absolute coordinate-based method. Three examples have been given to validate effectiveness and correctness of the proposed formulations.

According to the above statement, nowadays existing references mainly concentrate on dynamic modeling for planar mechanisms with clearances. However, spatial mechanisms are the most widely used at present. There are many clearance joints in the spatial mechanism. In addition, the multi-clearance joints have a direct impact on the performance of the spatial mechanism. But the studies are mainly focused on single clearance mechanism. Under this background, a general modeling method of spatial parallel mechanism with multi-spherical joint clearances is proposed based on Lagrange multiplier method. Moreover, the dynamic model established by this method is universal, easy to understand and apply, and suitable for the dynamic study of other parallel mechanism. The arrangement of this article is as follows. In the second section, the model of spherical joint clearance has been given, and Flores continuous contact force model and LuGre friction model are used. In the third section, the dynamic modeling of 4UPS-UPU spatial parallel mechanism considering multi-spherical clearance joints is provided. In the fourth section, numerical results of dynamic model are obtained. The correctness of the numerical results is verified by comparing with the results of ADAMS. Finally, in the fifth section, main conclusions of this article are drawn.

Modeling of the spherical joint clearance

Establishment of kinematics model of spherical joint clearance

Schematic diagram of the clearance model of spherical joint is shown in Figure 1, the body i and body j connected through spherical joint. o_i − x_iy_iz_i and o_j − x_jy_jz_j , respectively, represent local coordinate systems. r _i and r _j represent position vectors of coordinate origins of local coordinate systems o_i − x_iy_iz_i and o_j − x_jy_jz_j in the global coordinate system, respectively. Where P_i and P_j represent the center point of the socket and ball, Q_i and Q_j represent the collision point of the socket and ball, respectively, r _Pi and r _Pj are position vector of center point of socket and ball in the global coordinate system, respectively, n represents normal vector of contact surface, and t represents tangential vector of contact surface.

Figure 1.

Schematic diagram of the clearance model of spherical joint.

The eccentricity vector could be expressed as

$\mathit{e}={\mathit{r}}_{\mathit{P}\mathit{j}}-{\mathit{r}}_{\mathit{P}\mathit{i}}$

1

Its unit vector can be given by

$\mathit{n}=\mathit{e}/\mathit{e}$

2

where e is the eccentric amplitude, $e=\sqrt{{\mathit{e}}^{\text{T}}\mathit{e}}$

The penetration depth of socket and ball in collision can be expressed as

$\delta =e-c$

3

where c denotes clearance size between socket and ball, c = R_i − R_j , R_i and R_j are the radius of the socket and ball, respectively. Based upon penetration depth, it can be judged whether the socket and ball collide, when δ < 0, there is no contact between socket and ball, when δ = 0, socket and ball just contact or separate, when δ > 0, collision contact occurs.

Collisions between socket and ball can be judged by the following relationships

$\delta (\mathit{q},t)\cdot \delta (\mathit{q},t+\Delta t)\le 0$

4

where q is the generalized coordinates of the mechanism. When equation (4) is satisfied, the contact impact just started at t + ▵t moment. ²⁸

Q_i and Q_j represent the collision points of the socket and ball when they collide, respectively, r _Qi and r _Qj are the position vectors of the collision point in the global coordinate system when the socket collides with the ball, it could be written as

$\{\begin{array}{c}{\mathit{r}}_{Qi}={\mathit{r}}_{Pi}-R_i\mathit{n}\\ {\mathit{r}}_{Qj}={\mathit{r}}_{Pj}-R_j\mathit{n}\end{array}$

5

Normal component and tangential component of relative speed of ball collision point relative to socket collision point can be obtained, and it might be expressed as

$\{\begin{array}{c}{\mathit{v}}_{\mathit{n}}=\left[{\left({\dot{\mathit{r}}}_{Qj}-{\dot{\mathit{r}}}_{Qi}\right)}^{\text{T}}\mathit{n}\right]\mathit{n}\\ {\mathit{v}}_t={\left({\dot{\mathit{r}}}_{Qj}-{\dot{\mathit{r}}}_{Qi}\right)}^{\text{T}}-{\mathit{v}}_{\mathit{n}}\end{array}$

6

where ${\dot{\mathit{r}}}_{Qi}$ and ${\dot{\mathit{r}}}_{Qj}$ represent the velocities of collision point relative to the global coordinate system.

The contact force model of spherical joint clearance

The contact force of a joint with clearance will inevitably occur in the collision process, therefore, the establishment of a reasonable contact force model is the key step to analyze the mechanical properties of the mechanism. The earliest Hertz contact force model was based on pure elasticity theory, which did not consider the energy loss of collision and could not describe the continuous contact collision process. ⁸ As velocity of collision component increases, the accuracy of calculation decreases. Subsequently, Lankarani and Nikravesh proposed L-N contact force model based on Hertz contact theory, and recovery coefficient is widely applied. It takes into account the influence of damping force and includes the elastic deformation in the process of collision. ^19,24,26,27 But this model is only suitable for the analysis of materials whose recovery coefficient is approximately 1. However, the application of Flores contact force model is not limited by the restitution coefficient. ²⁹ Therefore, Flores contact force model is used to model normal contact force. Its expression is as follows

$F_N=K{\delta }^n\left[1+\frac{8\left(1-c_{\mathit{r}}\right)}{5c_{\mathit{r}}}\frac{\dot{\delta }}{{\dot{\delta }}^{(-)}}\right]$

7

where K is the stiffness coefficient, which is related to geometry and material characteristics of contact surface. c_r represents restitution coefficient, and it is related to material properties. n is a given fixed constant, usually set to 1.5. δ represents penetration depth and can be solved by equation (3), $\dot{\delta }$ is the derivative of δ and represents relative collision velocity, ${\dot{\delta }}^{(-)}$ is the initial collision velocity of each collision process. The stiffness coefficient K could be expressed as

$K=\frac{4\left(E_i{E}_j\right)}{3\left(E_i\left(1-{\upsilon }_j^2\right)+E_j\left(1-{\upsilon }_i^2\right)\right)}{\left(\frac{R_i{R}_j}{R_i-R_j}\right)}^{1/2}$

8

where $E_k(k=i,j)$ is the elastic modulus of contact body and ${\upsilon }_k(k=i,j)$ is Poisson’s ratio of contact body, and it is related to materials.

It is well-known that tangential friction occurs when two objects are in contact. LuGre friction model is proposed by Canudas de Wit based on Dahl model, which adopts the idea of bristle model. Under microscopic conditions, contact surface of rigid body is irregular, so it can be imagined that two friction surfaces are elastic bristle contact surface. The stiffness of the material below is greater than that of the material above. When there is tangential force, the bristle will migrate like a spring, which will increase the friction of the contact surface. When tangential force continues to increase to critical value, the bristle is further deformed and the sliding occurs. ^{30 –32} In addition, LuGre model could be observed to capture the Stribeck effect, and it is phenomenon related to stick-slip friction. It can more reasonably simulate the real situation of friction and smooth transition between different friction states. ³³ Therefore, the friction model is widely utilized. The schematic diagram of physical interpretation of LuGre model is shown in Figure 2.

Figure 2.

Physical interpretation of LuGre model.

The average deformation of bristle is expressed in z. Model is described as follows

$F_T=({\sigma }_{0}z+{\sigma }_1\dot{z}+{\sigma }_2{v}_t)F_N$

9

where σ ₀ is the bristle stiffness, σ ₁ is the microscopic damping coefficient, and σ ₂ is the viscous friction coefficient. The average deformation of bristle could be given by

$\dot{z}=\frac{\text{d}z}{\text{d}t}=v_t-\frac{{\sigma }_0|v_t|}{{\mu }_k+\left({\mu }_s-{\mu }_k\right)e^{-{|v_t/v_s|}^2}}z$

10

where μ_k is the dynamic friction coefficient, μ_s is the static friction coefficient, and v_s is the Stribeck velocity.

Besides, when the system reaches steady state, the average deformation z of bristle is constant, ³⁰ so it can be given by

$\dot{z}=\frac{\text{d}z}{\text{d}t}=v_t-\frac{{\sigma }_0|v_t|}{{\mu }_k+({\mu }_s-{\mu }_k)e^{-{|v_t/v_s|}^2}}z=0$

11

$z=\frac{v_t}{|v_t|}\times \frac{{\mu }_k+({\mu }_s-{\mu }_k)e^{-{|v_t/v_s|}^2}}{{\sigma }_0}$

12

The contact force between socket and ball can be expressed as follows

${\mathit{F}}_i=F_N\mathit{n}+F_T\mathit{t}$

13

According to reaction force and action force, contact force between socket and ball is as follows

${\mathit{F}}_j=-{\mathit{F}}_i$

14

Torque generated by contact force at the center of mass of bodies i and j is

$\{\begin{array}{c}{\mathit{M}}_i=({\mathit{r}}_{Qi}-{\mathit{r}}_i)\times {\mathit{F}}_i\\ {\mathit{M}}_j=({\mathit{r}}_{Qj}-{\mathit{r}}_j)\times {\mathit{F}}_j\end{array}$

15

where ${\mathit{r}}_{Qi}$ and ${\mathit{r}}_{Qj}$ are position vectors of collision points of socket and ball in the global coordinate system.

Dynamic modeling of 4UPS-UPU spatial parallel mechanism with spherical joint clearances

Establishment of the coordinate system

4UPS-UPU spatial parallel mechanism diagram is shown in Figure 3. The parallel mechanism consists of a moving platform, a fixed platform, four identical UPS driving branches, and one UPU driving branch. Each driving branch consists of a swing rod and a telescopic rod. Fixed platform A is a fixed part, swing rods 6, 7, 8, 9, 10 and fixed platform A are connected via universal joint $U_i(i=1,2\dots 5),$ telescopic rods 1, 2, 3, 4, 5 and swing rods 6, 7, 8, 9, 10 are connected by prismatic joint $P_i(i=1,2\dots 5),$ respectively. Telescopic rod 1 and moving platform 11 are connected by universal joint U ₆, and telescopic rods 2, 3, 4, 5 and moving platform 11 are connected via spherical joint $S_i(i=2,3,4,5),$ respectively.

Figure 3.

Structure diagram of the 4UPS-UPU spatial parallel mechanism.

The global coordinate system $O_A-X_A{Y}_A{Z}_A$ is built on fixed platform. Four universal joints $U_2,U_3,U_4\text{, and }{U}_5$ are evenly distributed on the circle, which takes O_A as its center, and r_A as its radius, and $\pi \text{/2}$ as distribution angle. Y_A axis is parallel to U ₂ U ₃ connection, universal joint $U_1$ is positioned at $l_{U1}$ of Y_A axis, X_A axis is perpendicular to fixed platform and downward direction, and Z_A obeys the right-hand rule. Local coordinate systems $o_i-x_i{y}_i{z}_i(i=1,2\dots 10)$ are built on each rod, and its coordinate origin is located at the centroid of rod. Local coordinate system $o_{11}-x_{11}{y}_{11}{z}_{11}$ is established on centroid of moving platform. Four spherical joints $S_2,S_3,S_4,S_5$ and universal joint U ₆ are distributed on the circle, which takes o ₁₁ as the center, r ₁₁ as the radius, and $2\pi \text{/5}$ as distribution angle. Axis x ₁₁ is perpendicular to moving platform and the downward direction, axis y ₁₁ points to projection point of U ₆ on moving platform. z ₁₁ obeys the right-hand rule. Coordinates of universal joint U ₆ in local coordinate system $o_{11}-x_{11}{y}_{11}{z}_{11}$ are $\left(-l_{U6x},l_{U6y},0\right)$ .

The lengths of telescopic rods 1, 2, 3, 4, 5 and swing rods 6, 7, 8, 9, 10 of 4UPS-UPU spatial parallel mechanism are expressed in $l_i(i=1,2\dots 10)$ and mass in $m_i(i=1,2\dots 10)$ . The quality of moving platform is expressed in $m_{11}$ .

The motion of the moving platform of the mechanism is driven by changing the length of the five branches. And prismatic joint is used as the driving joint. The moving platform has five DOFs, which can move along X_A , Y_A , and Z_A axes in space and rotate around Y_A axis and Z_A axis.

Conversion of coordinate system

The transformation matrix R _i from local coordinate systems $o_i-x_i{y}_i{z}_i(i=1,2\dots 11)$ to the global coordinate system $O_A-X_A{Y}_A{Z}_A$ is obtained by X–Z–Y type of Euler angle.

Dynamic modeling of spatial parallel mechanism considering spherical joint clearances

The 4UPS-UPU mechanism contains 11 active components. The generalized coordinates of the mechanism are taken as follows

$\mathit{q}={[\begin{array}{ccccccccccc}{\mathit{q}}_1{ }^{\text{T}}& {\mathit{q}}_2{ }^{\text{T}}& {\mathit{q}}_3{ }^{\text{T}}& {\mathit{q}}_4{ }^{\text{T}}& {\mathit{q}}_5{ }^{\text{T}}& {\mathit{q}}_6{ }^{\text{T}}& {\mathit{q}}_7{ }^{\text{T}}& {\mathit{q}}_8{ }^{\text{T}}& {\mathit{q}}_9{ }^{\text{T}}& {\mathit{q}}_{10}{ }^{\text{T}}& {\mathit{q}}_{\text{11}}{ }^{\text{T}}\end{array}]}^{\text{T}}$

16

where ${\mathit{q}}_k={[\begin{array}{cc}{\mathit{r}}_k{ }^{\text{T}}& {\mathit{\varphi }}_k{ }^{\text{T}}\end{array}]}^{\text{T}}$ , ${\mathit{r}}_k={\left[\begin{array}{ccc}{x}_k& y_k& z_k\end{array}\right]}^{\text{T}}$ is the coordinate of origin in local coordinate systems for component k in the global coordinate system, ${\mathit{\varphi }}_k={\left[\begin{array}{ccc}{\alpha }_k& {\beta }_k& {\gamma }_k\end{array}\right]}^{\text{T}}$ is the attitude angle in the local coordinate systems of component $k(k=1,2\dots 11)$ relative to the global coordinate system.

The schematic diagram of universal joint is shown in Figure 4. The universal joint connects body i and body j, where center point is P. $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ represent the local coordinate systems on the center of mass of body i and body j, respectively. Vectors r _i and r _j represent position vectors from origins of local coordinates $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ to global coordinate, respectively. Vectors ${\mathit{o}}_i^p$ and ${\mathit{o}}_j^p$ represent the position vectors from central P of universal joint to local coordinates $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ . Unit vectors ${\mathit{d}}_{oi}$ and ${\mathit{d}}_{oj}$ are defined on bodies i and j, and ${\mathit{d}}_{oi}\perp {\mathit{d}}_{oj}$ . The universal joint allows two bodies to rotate relative to each other around two vertical axes, so the universal joint will restrict movement in three directions and the rotation in one direction. Therefore, the four constraint equations of the universal joint can be obtained as follows

${\mathit{\Phi }}_U=\left[\begin{array}{c}{\mathit{r}}_i+{\mathit{R}}_i{\mathit{o}}_i^p-{\mathit{r}}_j-{\mathit{R}}_j{\mathit{o}}_j^p\\ {({\mathit{R}}_i{\mathit{d}}_{\mathit{o}i})}^{\text{T}}\cdot {\mathit{R}}_j{\mathit{d}}_{\mathit{o}j}\end{array}\right]={\mathbf{0}}_{4\times 1}$

17

Figure 4.

Diagram of universal joint.

where R _i and R _j are transformation matrices of local coordinate systems $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ relative to the global coordinate system $O_A-X_A{Y}_A{Z}_A$ .

A schematic diagram of hinge point $U_1,U_2,U_3,U_4,U_5$ on fixed platform of 4UPS-UPU mechanism is shown in Figure 5. Where ${\mathit{d}}_{\text{B}i}(i=1,2\dots 5)$ is the unit vector defined on the swing bar 6, 7, 8, 9, and 10, respectively, and ${\mathit{d}}_{\mathrm{A}i}(i=1,2\dots 5)$ is the unit vector defined on fixed platform.

Figure 5.

Diagram of the fixed platform.

Position vectors of hinge point $U_1,U_2,U_3,U_4,U_5$ on fixed platform in the global coordinate system can be written as ${\mathit{U}}_i^A(i=1,2\dots 5)$ .

The position vectors of hinge point $U_1,U_2,U_3,U_4,U_5$ on the fixed platform under local coordinate systems $o_k-x_k{y}_k{z}_k(k=6,7\dots 10)$ are expressed as ${\mathit{U}}_i^{{\mathit{o}}_k}(i=1,2\dots 5;k=6,7\dots 10)$ .

According to the first formula of equation (17), constraint equation of universal joint could be obtained as

${\mathit{\Phi }}_{\text{(}{\mathbf{U}}_i\text{,}\mathbf{3}\text{)}}={\mathit{U}}_i^{\mathit{A}}-{\mathit{r}}_k-{\mathit{R}}_k{\mathit{U}}_i^{{\mathit{o}}_k}={\mathbf{0}}_{3\times 1} (i=1,2\dots 5;k=6,7\dots 10)$

18

As shown in Figure 5, unit vector ${\mathit{d}}_{Ai}(i=1,2\dots 5)$ is expressed in the global coordinate system $O_A-X_A{Y}_A{Z}_A$ as follows

$\{\begin{array}{l}{\mathit{d}}_{\text{A}1}\text{=}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{\text{A}2}\text{=}{\left[\begin{array}{ccc}0& -\sqrt{2}/2& \sqrt{2}/2\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{\text{A}3}\text{=}{\left[\begin{array}{ccc}0& -\sqrt{2}/2& -\sqrt{2}/2\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{\text{A}4}\text{=}{\left[\begin{array}{ccc}0& \sqrt{2}/2& -\sqrt{2}/2\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{\text{A}5}\text{=}{\left[\begin{array}{ccc}0& \sqrt{2}/2& \sqrt{2}/2\end{array}\right]}^{\text{T}}\end{array}$

19

And the unit vector ${\mathit{d}}_{\text{B}i}(i=1,2\dots 5)$ in local coordinate systems $o_k-x_k{y}_k{z}_k(k=6,7\dots 10)$ can be expressed as

${\mathit{d}}_{\text{B}i}\text{=}{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^{\text{T}}(i=1,2\dots 5)$

20

According to second formula of equation (17), the constraint equation of universal joint could be obtained as

${\mathit{\Phi }}_{\text{(}{\mathbf{U}}_i\text{,}\mathbf{1}\text{)}}={({\mathit{d}}_{\text{A}i})}^{\text{T}}\cdot ({\mathit{R}}_k{\mathit{d}}_{\text{B}i})=\mathbf{0}(i=1,2\dots 5;k=6,7\dots 10)$

21

According to equations (18) and (21), the constraint equation of hinge joint $U_1,U_2,U_3,U_4,U_5$ of 4UPS-UPU spatial parallel mechanism can be given by

${\mathit{\Phi }}_{{\mathbf{U}}_i}\text{=}\left(\begin{array}{c}{\mathit{\Phi }}_{\text{(}{\mathbf{U}}_i\text{,}\mathbf{3}\text{)}}\\ {\mathit{\Phi }}_{\text{(}{\mathbf{U}}_i\text{,}\mathbf{1}\text{)}}\end{array}\right)\text{=}{\mathbf{0}}_{4\times 1}(i=1,2\dots 5)$

22

Similarly, the constraint equation of hinge joint $U_6$ of 4UPS-UPU spatial parallel mechanism can be written as

${\mathit{\Phi }}_{{\mathbf{U}}_6}\text{=}\left(\begin{array}{c}{\mathit{\Phi }}_{\text{(}{\mathbf{U}}_6\text{,}\mathbf{3}\text{)}}\\ {\mathit{\Phi }}_{\text{(}{\mathbf{U}}_6\text{,}\mathbf{1}\text{)}}\end{array}\right)\text{=}{\mathbf{0}}_{4\times 1}$

23

Figure 6 shows spherical joint constraints that connects two bodies i and j. The center of spherical joint represents point P. $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ are, respectively, local coordinate systems fixed on component i and j. r _i and r _j represent, respectively, position vectors of coordinate origins of local coordinate systems $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ in the global coordinate system, ${\mathit{o}}_{\mathit{i}}^p$ and ${\mathit{o}}_{\mathit{j}}^p$ represent the position vectors of P points on local coordinates $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ , respectively. The spherical joint can only constrain the movement in three directions. In the course of motion, point P on body i always coincides with point P on body j. Therefore, three constraint equations of spherical joints could be expressed as

${\mathit{\Phi }}_{\mathit{s}}={\mathit{r}}_i+{\mathit{R}}_i{\mathit{o}}_i^p-{\mathit{r}}_j-{\mathit{R}}_j{\mathit{o}}_j^p={\mathbf{0}}_{3\times 1}$

24

Figure 6.

Diagram of spherical joint.

where R _i and R _j are transformation matrices of local coordinate systems $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ relative to the global coordinate system $O_A-X_A{Y}_A{Z}_A$ .

For the 4UPS-UPU mechanism, coordinates of spherical joints $\left(S_2,S_3,S_4,S_5\right)$ in local coordinate system $o_{11}-x_{11}{y}_{11}{z}_{11}$ could be expressed by ${\mathit{S}}_i^{{\mathit{o}}_{11}}(i=2,3,4,5)$ .

Respectively, coordinates of spherical joints $\left(S_2,S_3,S_4,S_5\right)$ in local coordinate systems $o_i-x_i{y}_i{z}_i(i=2,3,4,5)$ are given by ${\mathit{S}}_i^{{\mathit{o}}_i}(i=2,3,4,5)$ .

According to equation (24), the spherical joint constraint equation of 4UPS-UPU spatial parallel mechanism can be given as

${\mathit{\Phi }}_{{\mathit{S}}_i}={\mathit{r}}_{11}+{\mathit{R}}_{11}{\mathit{S}}_i^{{\mathit{o}}_{11}}-{\mathit{r}}_i-{\mathit{R}}_i{\mathit{S}}_i^{{\mathit{o}}_i}={\mathbf{0}}_{3\times 1}\text{ (}i\text{=2,3,4,5)}$

25

Figure 7 shows prismatic joint that connects body i and body j. $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ represent local coordinate systems fixed on centroid of body i and body j, respectively. ${\mathit{d}}_{xi},{\mathit{d}}_{yi},{\mathit{d}}_{zi}$ is defined as three mutually perpendicular unit vectors fixed on body i. The direction of ${\mathit{d}}_{xi}$ is parallel to moving direction of prismatic joint. And ${\mathit{d}}_{xj},{\mathit{d}}_{yj},{\mathit{d}}_{zj}$ is defined as three mutually perpendicular unit vectors fixed on body j. The direction of ${\mathit{d}}_{xj}$ is parallel to the moving direction of the prismatic joint. Point Q and point P are fixed on body i and body j.

Figure 7.

Diagram of prismatic joint.

According to the characteristics of the prismatic joint: (1) the connection $\overrightarrow{\mathit{Q}\mathit{P}}$ from point Q to point P is always parallel to ${\mathit{d}}_{xi}$ in the relative motion of two bodies, that is, $\overrightarrow{\mathit{Q}P}\perp {\mathit{d}}_{yi}$ and $\overrightarrow{\mathit{Q}\mathit{P}}\perp {\mathit{d}}_{zi}$ . (2) Vectors ${\mathit{d}}_{xi}$ and ${\mathit{d}}_{xj}$ , ${\mathit{d}}_{yi}$ and ${\mathit{d}}_{yj}$ , and ${\mathit{d}}_{zi}$ and ${\mathit{d}}_{zj}$ are always parallel. In other words, only satisfy ${\mathit{d}}_{yi}\perp {\mathit{d}}_{xj}$ , ${\mathit{d}}_{zi}\perp {\mathit{d}}_{xj}$ , and ${\mathit{d}}_{zi}\perp {\mathit{d}}_{yj}$ . The movements of body j relative to body i in the directions ${\mathit{d}}_{yi}$ and ${\mathit{d}}_{zi}$ are limited, relative rotation in all three directions are also limited. Therefore, five constraint equations of prismatic joint could be obtained as

${\mathit{\Phi }}_{\mathit{P}}=\left[\begin{array}{c}\begin{array}{l}{\left({\mathit{R}}_i{\mathit{d}}_{yi}\right)}^{\text{T}}\cdot \left({\mathit{r}}_j+{\mathit{R}}_j{\mathit{o}}_j^P-{\mathit{r}}_i-{\mathit{R}}_i{\mathit{o}}_i^Q\right)\\ {\left({\mathit{R}}_i{\mathit{d}}_{zi}\right)}^{\text{T}}\cdot \left({\mathit{r}}_j+{\mathit{R}}_j{\mathit{o}}_j^P-{\mathit{r}}_i-{\mathit{R}}_i{\mathit{o}}_i^{\mathit{Q}}\right)\end{array}\\ {\left({\mathit{R}}_i{\mathit{d}}_{yi}\right)}^{\text{T}}\cdot {\mathit{R}}_j{\mathit{d}}_{xj}\\ \begin{array}{c}{\left({\mathit{R}}_i{\mathit{d}}_{zi}\right)}^{\text{T}}\cdot {\mathit{R}}_j{\mathit{d}}_{xj}\\ {\left({\mathit{R}}_i{\mathit{d}}_{zi}\right)}^{\text{T}}\cdot {\mathit{R}}_j{\mathit{d}}_{yj}\end{array}\end{array}\right]={\mathbf{0}}_{5\times 1}$

26

where R _i and R _j are transformation matrices of local coordinate systems $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ relative to global coordinate systems $O_A-X_A{Y}_A{Z}_A$ . Vectors ${\mathit{o}}_i^{\mathit{Q}}$ and ${\mathit{o}}_j^P$ are position vectors of points Q and P in local coordinates $o_i-x_i{y}_i{z}_i$ and $o_j-x_j{y}_j{z}_j$ , respectively.

For the 4UPS-UPU mechanism, the distance from $Q_i(i=1,2\dots 5)$ to the centroid of the swing rod (include 6, 7, 8, 9, and 10) is unit distance, and the $Q_i(i=1,2\dots 5)$ passes the axis direction of the swing rod (include 6, 7, 8, 9, and 10). The position vector of $Q_i(i=1,2\dots 5)$ in the global coordinate system is

${\mathit{Q}}_i\text{=}{\mathit{r}}_k+{\mathit{R}}_k{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^{\text{T}}(i=1,2\dots 5;k=6,7\dots 10)$

27

The distance from $P_i(i=1,2\dots 5)$ to the centroid of the telescopic rod (include 1, 2, 3, 4, and 5) is unit distance, and the $P_i(i=1,2\dots 5)$ passes the axis direction of the telescopic rod (include 1, 2, 3, 4, and 5). The position vector of $P_i(i=1,2\dots 5)$ in the global coordinate system is

${\mathit{P}}_i\text{=}{\mathit{r}}_i+{\mathit{R}}_i{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^{\text{T}}(i=1,2\dots 5)$

28

Then the direction vectors from point Q to point P can be expressed as

${\overrightarrow{\mathit{Q}\mathit{P}}}_i={\mathit{P}}_i-{\mathit{Q}}_i(i=1,2\dots 5)$

29

The d _xk , d _yk , and d _zk are unit vectors along three directions of the local coordinate system of the swing rod (include 6, 7, 8, 9, and 10), respectively. And position vectors in the local coordinate systems $o_k-x_k{y}_k{z}_k(k=6,7\dots 10)$ are as follows

$\{\begin{array}{l}{\mathit{d}}_{xk}\text{=}{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{yk}\text{=}{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{zk}\text{=}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\text{T}}\end{array}(k=6,7\dots 10)$

30

The d _xi , d _yi , and d _zi are unit vectors along three directions of the local coordinate system of the telescopic rod (include 1, 2, 3, 4, and 5), respectively. And their position vectors in the local coordinate systems $o_i-x_i{y}_i{z}_i(i=1,2\dots 5)$ are as follows

$\{\begin{array}{l}{\mathit{d}}_{xi}\text{=}{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{yi}\text{=}{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^{\text{T}}\\ {\mathit{d}}_{zi}\text{=}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\text{T}}\end{array}(i=1,2\dots 5)$

31

According to equation (26), the constraint equation of the prismatic joint of 4UPS-UPU spatial parallel mechanism is obtained as follows

${\mathit{\Phi }}_{P_i}=\left[\begin{array}{c}\begin{array}{l}{({\mathit{R}}_k{\mathit{d}}_{yk})}^{\text{T}}\cdot {\overrightarrow{\mathit{Q}\mathit{P}}}_i\\ {({\mathit{R}}_k{\mathit{d}}_{zk})}^{\text{T}}\cdot {\overrightarrow{\mathit{Q}\mathit{P}}}_i\end{array}\\ {({\mathit{R}}_k{\mathit{d}}_{yk})}^{\text{T}}\cdot {\mathit{R}}_i{\mathit{d}}_{xi}\\ \begin{array}{l}{({\mathit{R}}_k{\mathit{d}}_{zk})}^{\text{T}}\cdot {\mathit{R}}_i{\mathit{d}}_{xi}\\ {({\mathit{R}}_k{\mathit{d}}_{zk})}^{\text{T}}\cdot {\mathit{R}}_i{\mathit{d}}_{yi}\end{array}\end{array}\right]={\mathbf{0}}_{5\times 1}(i=1,2\dots 5;k=6,7\dots 10)$

32

The 4UPS-UPU spatial parallel mechanism has five DOFs, so it needs to impose five driving restrictions to enable the mechanism to have a definite motion. Through inverse kinematics analysis of spatial parallel mechanism, the relative distance curve from the centroid of swing rod to the centroid of telescopic rod, namely $f_n(t)(n=1,\cdots ,5)$ , is obtained, and the driving constraint equation can be obtained as follows

${\mathit{\Phi }}_D=\left[\begin{array}{c}|{\mathit{r}}_1-{\mathit{r}}_6|-f_1(t)\\ |{\mathit{r}}_2-{\mathit{r}}_7|-f_2(t)\\ |{\mathit{r}}_3-{\mathit{r}}_8|-f_3(t)\\ |{\mathit{r}}_4-{\mathit{r}}_9|-f_4(t)\\ |{\mathit{r}}_5-{\mathit{r}}_{10}|-f_5(t)\end{array}\right]={\mathbf{0}}_{5\times 1}$

33

By integrating equations (22), (23), (25), (32), and (33), the kinematic constraint equations of 4UPS-UPU spatial parallel mechanism in the idea case can be written as

$\mathit{\Phi }\text{(}q\text{)}={\left(\begin{array}{l}{\mathit{\Phi }}_{U1},{\mathit{\Phi }}_{U2},{\mathit{\Phi }}_{U3},{\mathit{\Phi }}_{U4},{\mathit{\Phi }}_{U5},{\mathit{\Phi }}_{U6},{\mathit{\Phi }}_{S2},{\mathit{\Phi }}_{S3},\\ {\mathit{\Phi }}_{S4},{\mathit{\Phi }}_{S5},{\mathit{\Phi }}_{P1},{\mathit{\Phi }}_{P2},{\mathit{\Phi }}_{P3},{\mathit{\Phi }}_{P4},{\mathit{\Phi }}_{P5},{\mathit{\Phi }}_D\end{array}\right)}^{\text{T}}={\mathbf{0}}_{\text{66}\times 1}$

34

Considering clearance presented in spherical joints (S ₂ and S ₅), the difference between the clearance spherical joint and the ideal spherical joint is that the constraint at the clearance spherical joint is replaced by force constraints. Therefore, corresponding constraint equations ${\mathit{\Phi }}_{S2} \text{and} {\mathit{\Phi }}_{S5}$ in equation (34) will be removed. The updated constraint equations of 4UPS-UPU mechanism considering multi-spherical joint clearances are given by

${\mathit{\Phi }}_c\text{(}q\text{)}={\left(\begin{array}{l}{\mathit{\Phi }}_{U1},{\mathit{\Phi }}_{U2},{\mathit{\Phi }}_{U3},{\mathit{\Phi }}_{U4},{\mathit{\Phi }}_{U5},{\mathit{\Phi }}_{U6},{\mathit{\Phi }}_{S3},\\ {\mathit{\Phi }}_{S4},{\mathit{\Phi }}_{P1},{\mathit{\Phi }}_{P2},{\mathit{\Phi }}_{P3},{\mathit{\Phi }}_{P4},{\mathit{\Phi }}_{P5},{\mathit{\Phi }}_D\end{array}\right)}^{\text{T}}={\mathbf{0}}_{\text{60}\times 1}$

35

Velocity constraint equations could be obtained by differentiating equation (35) with respect to the time, it can be written as

${\mathit{\Phi }}_{cq}\dot{\mathit{q}}=-{\mathit{\Phi }}_t\equiv \mathit{v}$

36

where ${\mathit{\Phi }}_{cq}$ represents Jacobian matrix of ${\mathit{\Phi }}_c$ , $\dot{\mathit{q}}$ is the vector of system velocities, v represents right side of velocity constraint equation.

A second differentiation of equation (35) with respect to the time is acceleration constraint equations

${\mathit{\Phi }}_{cq}\ddot{\mathit{q}}=-{\left({\mathit{\Phi }}_{cq}\dot{\mathit{q}}\right)}_q\dot{\mathit{q}}-2{\mathit{\Phi }}_{cqt}\dot{\mathit{q}}-{\mathit{\Phi }}_{ctt}\equiv \mathit{\gamma }$

37

where $\ddot{\mathit{q}}$ represents vector of system acceleration. ${\mathit{\Phi }}_{cqt}$ indicates derivative of ${\mathit{\Phi }}_c$ with respect to both t and q. ${\mathit{\Phi }}_{ctt}$ represents the second derivate of ${\mathit{\Phi }}_c$ with respect to t. $\mathit{\gamma }$ is the right side of acceleration equations.

The mass matrix of 4UPS-UPU spatial parallel mechanism can be expressed as

$\mathit{M}\text{=diag}\left[\begin{array}{cccc}\begin{array}{cccc}{\mathit{N}}_1& {\mathit{J}}_1& {\mathit{N}}_2& {\mathit{J}}_2\end{array}& \dots & {\mathit{N}}_{11}& {\mathit{J}}_{11}\end{array}\right]$

38

where ${\mathit{N}}_k\text{=diag}\left[\begin{array}{ccc}{m}_k& m_k& m_k\end{array}\right] (k=1,2\dots 11)$ , ${\mathit{J}}_k=\text{diag}\left[\begin{array}{ccc}{I}_{xk}& I_{yk}& I_{zk}\end{array}\right] (k=1,2\dots 11)$ .

The dynamics equations of parallel mechanism can be given by

$\mathit{M}\ddot{\mathit{q}}+{\mathit{\Phi }}_{cq}^{\text{T}}\mathit{\lambda }=\mathit{Q}$

39

where M represents mass matrix of system, including moment of inertia and mass of each movable component. $\mathit{\lambda }$ denotes the Lagrange multipliers vector. Q is a vector including the generalized external forces.

Equation (37) could be added into the equations of motion (equation (39)), the dynamic model of 4UPS-UPU spatial parallel mechanism including multiple spherical clearance joints can be written as

$\left[\begin{array}{cc}\mathit{M}& {\mathit{\Phi }}_{cq}^{\text{T}}\\ {\mathit{\Phi }}_{cq}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\ddot{\mathit{q}}\\ \mathit{\lambda }\end{array}\right]=\left[\begin{array}{c}\mathit{Q}\\ \mathit{\gamma }\end{array}\right]$

40

Equation (40) does not directly apply displacement constraint and velocity constraint equations, and it might violate the original constraint equations. At present, the more commonly used and effective method proposed by Baumgarte is Baumgarte stabilization method, which is to damp out acceleration constraint violations by feeding back violations of position and velocity constraints. ^22,26,34 So, it can be given by

$\left[\begin{array}{cc}\mathit{M}& {\mathit{\Phi }}_{cq}^{\text{T}}\\ {\mathit{\Phi }}_{cq}& 0\end{array}\right]\left[\begin{array}{c}\ddot{\mathit{q}}\\ \mathit{\lambda }\end{array}\right]=\left[\begin{array}{c}\mathit{Q}\\ \mathit{\gamma }-2{\alpha }_b{\dot{\mathit{\Phi }}}_c-{\beta }_b{ }^2{\mathit{\Phi }}_c\end{array}\right]$

41

where ${\alpha }_b$ and ${\beta }_b$ represent the correction parameters greater than 0, and ${\dot{\mathit{\Phi }}}_c=\frac{\text{d}{\mathit{\Phi }}_c}{\text{d}t}$

Dynamic simulation example

The 4UPS-UPU spatial parallel mechanism with two spherical joint clearances is taken as an example to verify the correctness of the model established above. The design parameters of 4UPS-UPU spatial parallel mechanism are listed in Table 1. And dynamic simulation parameters are listed in Table 2.

Table 1.

Design parameters of 4UPS-UPU spatial parallel mechanism.

Parameter	Value
Length of telescopic rod $l_i(i=1,2\dots 5)$ (m)	0.840
Length of swing rod $l_i(i=6,7\dots 10)$ (m)	0.760
Radius of moving platform rA (m)	0.650
Radius of fixed platform $r_{\text{11}}$ (m)	0.202
Length of $l_{U6x}$ (m)	0.0594
Length of $l_{U6y}$ (m)	0.2285
Mass of moving platform $m_{\text{11}}$ (kg)	18.94
Mass of telescopic rod $m_i(i=1,2\dots 5)$ (kg)	8.57
Mass of swing rod $m_i(i=6,7\dots 10)$ (kg)	9.91
Rotational inertia of moving platform (kg m2)	$\left[\begin{array}{ccc}{I}_{xx}=0.458& & \\ & I_{yy}=0.237& \\ & & I_{zz}=0.237\end{array}\right]$
Rotational inertia of telescopic rod (kg m2)	$\left[\begin{array}{ccc}{I}_{xx}=0.002& & \\ & I_{yy}=0.549& \\ & & I_{zz}=0.549\end{array}\right]$
Rotational inertia of swing rod (kg m2)	$\left[\begin{array}{ccc}{I}_{xx}=0.006& & \\ & I_{xx}=0.550& \\ & & I_{xx}=0.550\end{array}\right]$

Table 2.

Dynamic simulation parameters of 4UPS-UPU spatial parallel mechanism.

Ball radius	0.0225 m	Viscous friction coefficient σ 2	0
Socket radius	0.0228 m	Kinetic coefficient of friction ${\mu }_k$	0.05
Young’s modulus	207 GPa	Static coefficient of friction ${\mu }_s$	0.15
Poisson’s ratio	0.3	Stribeck velocity $v_s$	1% of maximum $v_t$
Restitution coefficient	0.9	MATLAB integration step	0.001 s
Baumgarte ${\alpha }_b$	5	MATLAB integrator scheme	Ode45
Baumgarte ${\beta }_b$	5	ADAMS integration step	0.001 s
Bristle stiffness ${\sigma }_0$	105 N/m	ADAMS integrator scheme	Gstiff
Microscopic damping coefficient ${\sigma }_1$	400 N s/m	ADAMS integrator formulation	I3

The two spherical joint clearances are both set 0.3 mm, and the trajectory of moving platform is set as follows (unit: s, m)

$\{\begin{array}{l}\begin{array}{l}X=1.10\\ Y=0.05\text{sin}(\pi t)\end{array}\hfill \\ Z=0.05\text{cos}(\pi t)\hfill \end{array}$

42

The dynamic response of the mechanism with two spherical joint clearances was obtained by programming with the help of MATLAB2018b software. At the same time, the model of 4UPS-UPU mechanism with two spherical joint clearances is also modeled in ADAMS. And the results of dynamic response obtained by ADAMS are compared with those obtained by MATLAB, as shown in Figures 8 and 9.

Figure 8.

Dynamic responses of 4UPS-UPU spatial parallel mechanism: (a) Y-displacement, (b) Z-displacement, (c) total displacement, (d) Y-velocity, (e) Z-velocity, (f) total velocity, (g) Y-acceleration, (h) enlargement of (g), (i) Z-acceleration, (j) enlargement of (i), (k) total acceleration, and (l) enlargement of (k).

Figure 9.

Dynamic responses of 4UPS-UPU spatial parallel mechanism: (a) contact force in joint S ₂, (b) enlargement of (a), (c) contact force in joint S ₅, (d) enlargement of (c), (e) the center trajectory diagram in joint S ₂, and (f) the center trajectory diagram in joint S ₅.

The displacement curves of the moving platform are shown in Figure 8(a) to (c). It can be seen that curves of MATLAB with two spherical joint clearances is coincided basically with those of ADAMS with two spherical joint clearances. The movement trend of ideal displacement curve and displacement curve with two spherical joint clearances is basically consistent and coincidence, which shows that the clearances have little effect on displacement of mechanism moving platform. The velocity and acceleration curves of moving platform are illustrated in Figure 8(d) to (l). In the initial stage of motion for mechanism, the velocity and acceleration of the center of mass of moving platform change greatly. The reason is that the contact force caused by clearance increases the collision between components, which makes velocity and acceleration of mechanism fluctuate greatly. The following velocity and acceleration curves with clearances are coincided with those without clearances. And the curve with clearance by MATLAB and the curve with clearance by ADAMS have the same overall movement trend, which shows the correctness of the established dynamic model to a certain extent. Figure 8(f) to (l) shows the total velocity and acceleration, since the total velocity and acceleration take into account the vibration in the X direction, the vibration in the X direction is random, so after 0.5 s, the ideal curve is slightly different from the curve with two spherical joint clearances.

Figure 9 shows contact force and center trajectory at spherical clearance joints (S ₂ and S ₅). From Figure 9(a) to (d), it can be seen that contact force changes abruptly at initial stage of motion, and subsequent contact force tends to be stable. According to Figure 9(a) to (d), the overall movement trend of MATLAB curve and ADAMS curve is basically the same. Because of difference of solving method and modeling between MATLAB and ADAMS, the simulation with clearance has randomness, which result in different initial collision positions, thus the peak time of collision force is different. When mechanism is stable, two curves basically coincide with each other, which verify correctness of established dynamic model again. From Figure 9(e) and (f), it is shown that the center of the socket and ball coincide at the beginning of the movement, and the socket and ball are in separated mode. When driving force is applied, the socket and ball have a violent impact, which cause fluctuations in speed, acceleration, and collision force. And then the socket and ball are in a continuous contact state, and the mechanism gradually enters a stable state.

Conclusions

In this article, a general modeling method for parallel mechanism with multi-spherical joint clearances based on Lagrange multiplier method is proposed. Taking 4UPS-UPU spatial parallel mechanism as an example, the dynamic model of the mechanism with two spherical joint clearances combining Flores contact force model and LuGre friction model is established. According to the built model, the dynamic response of the mechanism is obtained using MATLAB, including displacement, velocity, and acceleration of moving platform, the contact force, and the central trajectory diagram at spherical clearance joints. The ADAMS model of 4UPS-UPU spatial parallel mechanism with two spherical joint clearances is also established and simulated. Correctness of model has been verified through comparing results of MATLAB with those of ADAMS. It can be seen from this example that the dynamic model of spatial parallel mechanism with multi-spherical joint clearances can be easily established by this method, which provides a theoretical reference for other scholars to establish the dynamic model of parallel mechanism with multi-clearance in the future.