Lagrangian Dynamics Analysis of a XY-Theta Parallel Robotic Machine Tool

Dynamics of a highly stiff parallel machine tool is the subject of this paper. High stiffness, good accuracy, relatively large workspace and free of singularities on the whole workspace makes the manipulator suitable for machining applications as an XY-Theta precision table. First, obtaining kinematics constraints, inverse kinematics analysis and velocity analysis are performed. Next, using six redundant generalized coordinates, we obtain Lagrangian of the manipulator. Also, a Lagrangian approach is proposed to obtain dynamics equations of the machine tool using three Lagrangian multipliers. This method allows elimination of constraint forces and moments at the joints from the motion equations. Dynamic equations of the manipulator are formed as inverse dynamics and direct dynamics problems. Finally, two examples are presented that confirms the obtained dynamics equations.

linear joints. The HexaM (Toyoda Company) is another example, with non-coplanar linear joints. A 3-axis translational version of the Hexaglide is the TRIGLIDE (Mikron company), which has three coplanar and parallel linear joints.
The development of a dynamical model is important in several different ways. First, a dynamical model can be used for computer simulation of a robotic system. Various manufacturing tasks can be examined without the need of a real system. Second, it can be used for the development of suitable control strategies. Third, the dynamic analysis reveals all the joint reaction forces and moments necessary for sizing the links, bearings and actuators. Dynamic analysis plays an important role in predicting the behavior of mechanical systems and achieving their best performances. There are two types of dynamical problems: (i) The direct dynamics problem aims to find the response of a robot arm corresponding to given applied moments and/or forces. That is, given the vector of joint moments or forces, it computes the resulting motion of the manipulator as a function of time. (ii) The inverse dynamics problem aims to find the actuator moments and/or forces required to generate a desired trajectory of the manipulator.
The inverse dynamics algorithm solves the following problem; given the desired trajectory of the end-effector as well as the mass distribution of each link, find actuator moments and/or forces required to generate this trajectory. The inverse dynamics analysis is needed for control: if one wants the robot to follow a specified trajectory, one has to convert the desired motion along that trajectory into joint forces that will generate this motion.
The direct dynamics algorithm solves the following problem; given the vector of initial actuated joint positions, vector of initial actuated joint velocities, applied actuated torques, applied external forces to end-effector, as well as the mass distribution of all links, find the resulting motion of the endeffector. The direct dynamics is used for simulation, and as feed-forward in the robot's motion controller, i.e., the direct dynamics calculates what the robot does when specific joint torques are applied, and under the assumption that all physical parameters in the dynamics model are accurate. Due to the closed-loop structure and kinematic constraints of PKMs, the derivation of dynamic equations is quite complicated. There are three main methods of formulation of the dynamical equations; namely, Newton-Euler laws, the Lagrangian formulation, the principle of virtual work [18][19][20][21][22][23][24][25] and Kane's method [26]. The Lagrangian formulation allows eliminating all of the reaction forces and moments at the beginning. Since a PKM has several closed-loop chains, it is a difficult task to derive the equations of motions in terms of a set of independent generalized coordinates. To simplify the problem, additional coordinates with a set of Lagrangian multipliers must be introduced. Also, Kane's method has two crucial considerations: (1) operational simplicity, meaning reduced labor in the derivation of the equations of motion either by hand or in terms of computer operations via symbol manipulation; and (2) simplicity of the final form of the equations, simplicity giving rise to reduction in computational time [26]. Both Lagrangian and Kane's methods can be used for holonomic and nonholonomic constraints, but Kane's method is better method for nonholonomic constraints.
Wu, Li, and Wang optimized an asymmetrical hybrid machine tool based on dynamic isotropy [27]. Sung and Lu performed modeling and analysis of a four-half axis machine tool via modified Denavit-Hartenberg notation [28]. Mi, Yin, Sun and Wang analyzed effect of the joints as a significant parameter in the overall dynamic analysis of machine tools [29]. Kiran et. all performed inverse dynamics of three degrees of freedom (DOF) U-shaped planar parallel manipulator having three legs consisting of prismatic-prismatic-revolute (PPR) joint arrangement in which each leg has one active prismatic joint [30]. Jiang, Li and Wang proposed a novel planar 2-DOF parallel kinematic machine with kinematic redundancy and presented a method for redundant force optimization is presented to improve the precision of the machine. Also, they obtained dynamic model of the manipulator by the Newton-Euler method [31]. Wu, Chen, Li and Wang proposed a planar 2-DOF parallel manipulator with actuation redundancy and incorporated it into a 4-DOF hybrid machine tool. They also derived structural dynamics model of the manipulator to obtain its natural frequency [32].

Structure description of the star-triangle planar parallel robot
The use of long and slender legs in all of the mentioned designs causes vibration in high speed machining. To overcome this problem, Star-Triangle (ST) manipulator with virtually zero leg lengths is being considered here (see Fig. 1). High stiffness, good accuracy, relatively large workspace free of singularities [33] makes the manipulator suitable for machining applications. The manipulator has three degrees of freedom and can move the spindle of the machine in a plane. The machine can be augmented with a vertical motion as depicted in Fig. 2. This manipulator consists of a triangular base and a moving star that are connected via three PRP legs, in which P and R refer to prismatic and revolute pairs, respectively. The general model of this manipulator is depicted in Fig. 3, in which P i is the actuator. Moreover, C and R i are the end-effector (EE) and prismatic joint, respectively. Furthermore, A i is the joint with two degrees of freedom. If all leg lengths vanish, r i = 0 , then the structural stiffness of the manipulator increases, which make it suitable for machine tool applications (Figs. 1, 2 and 4). In Fig. 3, the geometric model of this manipulator is shown.

Kinematics Analysis 3.1 Kinematics constraints
The Cartesian coordinates of point C of the EE and its orientation are given by C (x, y) and φ , respectively (Fig. 4). The L f is the side length of the equilateral fixed base and the angles between three lines of the star is 120°. Once, the robot link parameters are fully defined, we can formulate the kinematics constraints for each limb. For this purpose, we obtain the position and orientation of the end-effector from each limb as (see Where Λ 1 -Λ 9 are constraints equations and x, y, φ, q 1 , q 2 and q 3 are generalized coordinates. Using combination and simplification of the above equations, we can reduce the constraint equations from (9) to (3) and rewrite as follows

Inverse kinematics problem
The inverse kinematics problem of a manipulator is the search for the parameters of the motions of all actuated kinematic pairs in the sub-chains that is based on the desired position and orientation of the moving platform. In this paper, a solution of the inverse position problem is required for the dynamics analysis of the manipulator. Referring to Fig. 4, the closure equation for each limb can be written as Multiplying both sides of the above equation by a i T E , yields

c P Es
Where E 2×2 is an orthogonal matrix that rotates any vector in a plane through an angle of 90° counterclockwise. Now, we multiply both sides of the Eq. (3) by s E i T , to obtain q i as (3)

Velocity analysis
The differential kinematics relations pertaining to parallel manipulators take on the form where J and K are the two Jacobian matrices of the manipulator at hand. Moreover,  q is the vector of joint rates and  X is the twist array and is defined as in which, ω is the scalar angular velocity of the moving star and  c is the two dimensional velocity vector of the operation point C. According to Fig. 3, the velocity  c can be written for the i th leg as where V Ai and V Ri are the absolute velocity vectors of points A i and R i , respectively. Moreover, we have where a i is the unit vector directed from P i to A i , and  q i is the rate of i th actuator. Furthermore, we have Moreover, r i is the vector from A i to R i and is assumed to have zero length for the manipulator at hand. Therefore, Eq. (10) leads to where ρ i is the distance from R i to C and s i is the unit vector representing the direction of the third prismatic joint. Substituting the corresponding values from Eqs. (9) to (12) We use these forms for inverse and direct dynamics analyses.

Energy and Lagrangian of the ST robot
Due to the ST manipulator works in a horizontal plane, its the potential energy is zero. Therefore, the energy of the ST manipulator divided into kinetic energies of the three actuators, the three intermediate links and the traveling plate (Star). Now, we will compute the energy of each part individually.

Kinetic energy of the traveling Plate (star)
The kinetic energy of the traveling plate, T p , can be divided into two terms; translational and rotational kinetic energies. Translational kinetic energy, T p t , is the energy of traveling plate's mass center when one considers that the entire mass is concentrated there. The rotational kinetic energy, T p r , is due to rotation of the traveling plate around an axis that is perpendicular to the horizontal plane. These kinetic energies can be express as

T T T m x y I
Where m p is mass of the star platform and I p is inertia of the star around an axis that pass from its mass center and is perpendicular to the horizontal plane.   Where m s is mass of the each actuated link.

Lagrangian of the ST manipulator
Since the potential energy of the whole system is zero, Lagrangian of the whole system is a summation of Lagrangians of the actuated links, the intermediated links and the star. Therefore, we can write Lagrangian as follows

Dynamics Analysis
As stated before, both Lagrangian and Kane's methods can be used for holonomic and nonholonomic constraints. However, because of simplicity, the Kane's method is better method for nonholonomic constraints. Since the ST manipulator is planar and the holonomic constraints are applied to obtain equations of motion, we employ the Lagrangian method. To evaluate Lagrangian of the planar ST manipulator, the velocities of the actuated joints, intermediate links and traveling plate (Star) are computed. The Lagrangian of the planar ST manipulator is related to generalized velocities of actuated joints and the velocity of the traveling plate. There are two ways to compute the dynamical model of the system; the first way is to express Lagrangian of the ST manipulator only as a function of generalized coordinates on the traveling plate (φ , x, y) , or generalized coordinates on the actuated link (q 1 , q 2 , q 3 ), which is complicated because the relationship between the actuated joints and the traveling plate (position and orientation) is nonlinear (see Eq. (2)). In the second way, Lagrangian of the ST manipulator will be written as a function of both actuated, (q 1 , q 2 , q 3 ), and redundant generalized coordinates (φ , x, y) which are related to each other through Γ k constraint (for k=1,2,3) denoting the k th constraint function (Eq. (2)). Therefore, we can now write Lagrangian equation based on the second way as Where k is the number of constraint functions and λ k are the Lagrangian multiplier. As stated before, Lagrangian of the ST manipulator is a function of a vector which is a vector containing all the actuated joints and the traveling plate (Star). Using Lagrangian, we can drive dynamics equations of the ST manipulator as  In which, On the other hand, using Eqs. (16) and (17) Equation (33) is called direct dynamic equation form of the ST manipulator. In this form, input forces by the actuators, f q , and the applied external wrench to the end-effector, f ext. , are known. We will compute trajectory of the end-effector. Furthermore, Eq. (34) is as an inverse dynamic equation form. For the inverse dynamics problem, a desired path of the star is given and the problem is determining the input torques required to produce the motion.

Case studies
Based on the previous section, a computer program is developed using MATLAB software. Two separate paths for the ST planar robot are considered. In the first path, Point C moves along a vertical line and in the second path, point C moves along a circular path. The parametric equations for these paths are given as Moreover, the direction of the star is chosen to be φ = π / 2 . This value will ensure maximum workspace [33] and thus the robot can travel its entire triangular base. Therefore, we will keeping this value constant during our simulations. The results are verified in two ways. First, using inverse dynamics problem, a trajectory for the MSS is supplied and required motor torques as well as the linear position of the actuators as a function of time are calculated for each path (inverse dynamics). Then, the initial conditions (position and velocity) and the calculated torques from the inverse dynamics problem are applied to the actuators for obtaining the path of the star. If the output of the inverse dynamics problem, is supplied to the direct dynamics problem, then the same trajectory for the star must be obtained. In this case, the obtained Eqs. Simulation of the linear and circular paths for both inverse and direct dynamics problems is shown in Figs. 5 to 8. As shown, the results confirm each other. Therefore, the obtained formulations for a dynamical model of the ST manipulator are correct.

Conclusion
A planar parallel robotic machine tool with high stiffness was considered. good accuracy, relatively large workspace free of singularities made the manipulator suitable for machining applications. First, obtaining kinematics constraints, inverse kinematics analysis and velocity analysis were performed. Next, the six redundant generalized coordinates and three Lagrangian multipliers were introduced. Using a Lagrangian approach was obtained dynamics equations of the machine tool. The Lagrangian method was allowed elimination of constraint forces and moments at the joints from motion equations. Dynamic equations of the manipulator were formed for inverse and direct dynamics analyses. Finally, two examples were presented that verified the obtained dynamics equations.