The Inverse Dynamics of a 3-DOF Parallel Mechanism Based on Analytical Forward Kinematics

: A new type of 3-dof parallel mechanism(PM) with analytical forward displacement analysis is proposed. The reverse dynamic equation of the PM is solved. Different from the traditional dynamic analysis using inverse kinematics, the displacement, velocity and acceleration equations of the PM are established and solved by forward kinematics.The inverse dynamic equation of the PM is constructed and solved by analyzing the forces on each link and based on Newton-Euler method. The correctness of the dynamic model is verified by an example using MATLAB and ADAMS. The maximum driving force error of each actuated pair is 1.32%, 5.8% and 5.2% respectively.This paper provides a theoretical basis for the design, manufacture and application of the PM.


Introduction 1
Compared with the series mechanism, the parallel mechanism (PM) has the advantages of compact structure, high rigidity and motion accuracy, etc., The topic has attracted extensive attention from the academic and industrial community for many years [1][2][3]. At present, research on PMs mainly focuses on the topology, kinematics, dynamics and control [4][5][6].The dynamic analysis mainly studies the relationship between the input forces and the output forces. This is the determination of the maximum load carrying capacity and reasonable design of the driver during the development of parallel robots. According to this, the constraint reaction force of the motion pair solved during the analysis process plays an important role in the design, mechanical efficiency estimation, friction calculation, and Therefore, for a PM to be developed, it is necessary to establish a dynamic model that can accurately and meet the requirements of real-time control.
In terms of modeling methods, the commonly used dynamic modeling methods are Lagrangian method, universal equations of dynamics, Newton-Euler method, etc. [7][8][9].
Among them, the universal equations of dynamics and Lagrange method are based on the system's virtual displacement and kinetic and potential energy respectively to build a simple dynamic model. While the Newton-Euler method can obtain the force of each joint by analyzing each member separately, and then establishes a complete dynamic model by eliminating the interaction force of each member [10], it can also solve the support force and moment of force between members.
In terms of research objects, most of the kinetic analysis mainly focus on the 6-DOF Stewart PM [11][12][13]. For 5-DOF PM, Chen et.al [14] use the universal equations of dynamics to analyze the 4-UPS-UPU PM. Li et.al [10] used the Newton-Euler method to analyze the dynamics of the 5-PSS / UPU PM. For 4-DOF PM, Geng [15] used Newton-Euler to analyze the 4UPS-UPR PM. In terms of 3-DOF, Li et.al [16] used the Newton-Euler method to analyze the dynamics of a 3-RPS PM. Liu et.al [17] performed a Lagrangian method to analyze the dynamics of a 3-RRS PM.    axis of the coordinate system B2-x2y2z2 is parallel to the X axis of the base coordinate system, and the positive half of the z2 axis is pointed from B2 to C2, where the y axis of each coordinate system meets the right-hand screw rule; The positive half of the z3 axis of the coordinate system B3-x3y3z3 is pointed from B3 to C3, and the x3 axis lies in the XOZ plane and the angle with the X axis is 1  , its Euler transformation relative to the base coordinate system is shown in Figure 3, from which the coordinate transformation matrix from the coordinate system {B3} to the base coordinate system O-XYZ is: Euler transform

Forward Position Solution
The forward kinematics of PM is to solve the position and orientation of the moving platform when the structural parameters and input of the mechanism are given.
Based on the constrained length of the bars, we can get: Angle  of moving platform is: where,

Velocity and acceleration of the moving platform
Taking the time derivative of Eq. (2)~(3), the output velocity and acceleration of the moving platform can be obtained as

Velocity and acceleration of members
(1)Velocity and acceleration of member B1C1 Because the movements of the B11C11 and B12C12 rods are the same, the two rods are equivalent to the rod B1C1 for analysis.
The velocity of the point C1 is: Substituting Eq.(5) into Eq.(6) to obtain the velocity of the center of mass of the rod B1C1, (6) Taking the time derivative of Eq. (4) , the acceleration of the C1 can be obtained as: The angular acceleration of the rod B1C1 can be determined by taking the cross product of the two sides of Eq. (7) with 1 c ,which yields: Where, 1 c is the skew symmetric matrix associated with the vector 1 c .
Taking the time derivative of Eq. (4) , the centroid acceleration of rod B1C1 can be obtained as: (2)Velocity and acceleration of member B2C2 Similarly, using the same method as the velocity and acceleration of the rod B1C1, the angular velocity of the rod B2C2 can be obtained as: The centroid velocity of the rod B2C2: (11) Angular acceleration of rod B2C2: Centroid acceleration of rod B2C2: The angular velocity of the rod B3C3 can be determined by taking the cross product of the two sides of Eq. (14) with 3 c ,which yields: Substituting Eq.(15) into Eq. (16) gives the velocity at the centroid of rod B3C3: (16) Taking the time derivative of Eq. (14) , the acceleration of the C3 can be obtained as: The angular acceleration of the rod B3C3 can be determined by taking the cross product of the two sides of Eq. (17) with 3 c ,which yields:

Dynamics modeling of mechanism
When using the Newton-Euler method, the friction of each moving pair is not considered, then the Newton-Euler equation of each member is established.Then the dynamic model of the PM is obtained by eliminating the internal forces between the members.Finally, the relationship between the driving force and the external forces of the moving platform is obtained, which is illustrated as follows.

Dynamic equation of moving platform
As shown in Figure 4, the gravity of the moving platform is  The dynamic equation of the moving platform is:

Dynamic equation of connecting rod
The R-R-link is subject to the constraint reaction force of the sub-moving platform -Fci (i=1,2), its own gravity mcg, and the constraint force Fbi (i = 1,2), and its force analysis is shown in Figure 6.
Dynamic equation of single link (B2C2) is as follows.  Then, the dynamic equations of the -S-S-link are described as

Dynamic equation of driving sliders
The three driving sliders are subject to the constraint reaction forces of each link-F bi (i = 1,2,3), its own gravity, and the driving force of the driving motor m i g (i = 1,2, 3), and the force diagram of the slider is shown in Figure 8. The dynamic equation of slider 1 is The dynamic equations of slider 2 and slide 3 are as follows.

The integrated dynamic model of the PM
The establishment of the integrated dynamic model is to eliminate the internal forces of members and to obtain the dynamic relationship between the input force, torque and output force.
Taking the dot product of the both sides of Eq(29) with is the unit vector for driving force.

Dynamic Simulation
Firstly, the following motion laws of three driving pairs are given. Virtual prototype of 2T1R parallel mechanism A three-dimensional prototype of the PM is designed, as shown in Figure 9. The dimension parameters of the PM are shown in Table 1.   Table 2. As shown in Table 2, only when the step size ≤ 2000 005 . 0 10  ,the dynamic modeling analysis program is real-time .

Conclusions
For a new type of 3-DOF PM proposed in this paper, the analytical solution of forward kinematics is given, and it is used to analyze the velocity and acceleration. The driving force is obtained by calculating the dynamic equation. The dynamic simulation of the three-dimensional prototype is carried out by ADAMS, and the simulation value of driving force is obtained. By comparing the two cases, the correctness of the dynamic modeling is verified, which lays a foundation for the design, manufacture and application of PM.

Declaration Acknowledgement
The first authors sincerely thanks to Professor Shen of Changzhou University for his critical discussion and reading during manuscript preparation.

Funding
Supported by National Natural Science Foundation of China (Grant No.51975062) and National Science Foundation of Jiangsu Province (Grant No.BK20161192)

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.

Authors' contributions
The author' contributions are as follows: Huiping Shen was in charge of the whole work; Ke Wang wrote the manuscript.