**1. Introduction**

18 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

Wu, J.; Wang L.; You, Z. (2010). A new method for optimum design of parallel

Zhang, C. D. & Song, S. M. (1993). An efcient method for inverse dynamics of manipulators based on the virtual work principle, J. Robot. Syst., Vol.10, no.5, pp. 605–627

727

manipulatorbased on kinematics and dynamics, Nonlinear Dyn, Vol. 61, pp. 717–

Since a parallel structure is a closed kinematics chain, all legs are connected from the origin of the tool point by a parallel connection. This connection allows a higher precision and a higher velocity. Parallel kinematic manipulators have better performance compared to serial kinematic manipulators in terms of a high degree of accuracy, high speeds or accelerations and high stiffness. Therefore, they seem perfectly suitable for industrial high-speed applications, such as pick-and-place or micro and high-speed machining. They are used in many fields such as flight simulation systems, manufacturing and medical applications.

One of the most popular parallel manipulators is the general purpose 6 degree of freedom (DOF) Stewart Platform (SP) proposed by Stewart in 1965 as a flight simulator (Stewart, 1965). It consists of a top plate (moving platform), a base plate (fixed base), and six extensible legs connecting the top plate to the bottom plate. SP employing the same architecture of the Gough mechanism (Merlet, 1999) is the most studied type of parallel manipulators. This is also known as Gough–Stewart platforms in literature.

Complex kinematics and dynamics often lead to model simplifications decreasing the accuracy. In order to overcome this problem, accurate kinematic and dynamic identification is needed. The kinematic and dynamic modeling of SP is extremely complicated in comparison with serial robots. Typically, the robot kinematics can be divided into forward kinematics and inverse kinematics. For a parallel manipulator, inverse kinematics is straight forward and there is no complexity deriving the equations. However, forward kinematics of SP is very complicated and difficult to solve since it requires the solution of many non-linear equations. Moreover, the forward kinematic problem generally has more than one solution. As a result, most research papers concentrated on the forward kinematics of the parallel manipulators (Bonev and Ryu, 2000; Merlet, 2004; Harib and Srinivasan, 2003; Wang, 2007).

For the design and the control of the SP manipulators, the accurate dynamic model is very essential. The dynamic modeling of parallel manipulators is quite complicated because of their closed-loop structure, coupled relationship between system parameters, high nonlinearity in system dynamics and kinematic constraints. Robot dynamic modeling can be also divided into two topics: inverse and forward dynamic model. The inverse dynamic model is important for system control while the forward model is used for system simulation. To obtain the dynamic model of parallel manipulators, there are many valuable studies published by many researches in the literature. The dynamic analysis of parallel manipulators has been traditionally performed through several different methods such as

Dynamic Modeling and Simulation of Stewart Platform 21

For dynamic modeling of parallel manipulators, many approaches have been developed such as the principle of virtual work (Tsai, 2000, Wang and Gosselin, 1998; Geike and McPhee, 2003), screw teory (Gallardo et al., 2003), Kane's method (Liu et al., 2000; Meng et al., 2010) and recursive matrix method (Staicu and Zhang, 2008). Although the derived equations for the dynamics of parallel manipulators present different levels of complexity and computational loads, the results of the actuated forces/torques computed by different approaches are equivalent. The main goal of recent proposed approaches is to minimize the number of operations involved in the computation of the manipulator dynamics. It can be concluded that the dynamic equations of parallel manipulators theoretically have no trouble. Moreover, in fact, the focus of attention should be on the accuracy and computation

The aim of this paper is to present the work on dynamic formulation of a 6 DOF SP manipulator. The dynamical equations of the manipulator have been formulated by means of the Lagrangian method. The dynamic model included the rigid body dynamics of the mechanism as well as the dynamics of the actuators. The Jacobian matrix was derived in two different ways. Obtaining the accurate Jacobian matrix is very essential for accurate simulation model. Finally, the dynamic equations including rigid body and actuator

This chapter is organized in the following manner. In Section 2, the kinematic analysis and Jacobian matrices are introduced. In Section 3, the dynamic equations of a 6 DOF SP manipulator are presented. In Section 4, dynamic simulations and the experimental results

The SP manipulator used in this study (Figure 1), is a six DOF parallel mechanism that consists of a rigid body moving plate, connected to a fixed base plate through six independent kinematics legs. These legs are identical kinematics chains, couple the moveable upper and the fixed lower platform by universal joints. Each leg contains a precision ball-screw assembly and a DC- motor. Thus, length of the legs is variable and they

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dynamics were simulated in MATLAB-Simulink and verified on physical system.

are given in detail. Finally, conclusions of this study are summarized.

can be controlled separately to perform the motion of the moving platform.

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**2. Structure description and kinematic analysis** 

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efficiency of the model.

**2.1 Structure description** 

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Fig. 1. Solid model of the SP manipulator

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the Newton-Euler method, the Lagrange formulation, the principle of virtual work and the screw theory.

The Newton–Euler approach requires computation of all constraint forces and moments between the links. One of the important studies was presented by Dasgupta and Mruthyunjaya (1998) on dynamic formulation of the SP manipulator. In their study, the closed-form dynamic equations of the 6-UPS SP in the task-space and joint-space were derived using the Newton-Euler approach. The derived dynamic equations were implemented for inverse and forward dynamics of the Stewart Platform manipulator, and the simulation results showed that this formulation provided a complete modeling of the dynamics of SP. Moreover, it demonstrated the strength of the Newton-Euler approach as applied to parallel manipulators and pointed out an efficient way of deriving the dynamic equations through this formulation. This method was also used by Khalil and Ibrahim (2007). They presented a simple and general closed form solution for the inverse and forward dynamic models of parallel robots. The proposed method was applied on two parallel robots with different structures. Harib and Srinivasan (2003) performed kinematic and dynamic analysis of SP based machine structures with inverse and forward kinematics, singularity, inverse and forward dynamics including joint friction and actuator dynamics. The Newton-Euler formulation was used to derive the rigid body dynamic equations. Do and Yang (1988) and Reboulet and Berthomieu, (1991) presented the dynamic modeling of SP using Newton–Euler approach. They introduced some simplifications on the legs models. In addition to these works, others (Guo and Li, 2006; Carvalho and Ceccarelli, 2001; Riebe and Ulbrich, 2003) also used the Newton-Euler approach.

Another method of deriving the dynamics of the SP manipulator is the Lagrange formulation. This method is used to describe the dynamics of a mechanical system from the concepts of work and energy. Abdellatif and Heimann (2009) derived the explicit and detailed six-dimensional set of differential equations describing the inverse dynamics of non-redundant parallel kinematic manipulators with the 6 DOF. They demonstrated that the derivation of the explicit model was possible by using the Lagrangian formalism in a computationally efficient manner and without simplifications. Lee and Shah (1988) derived the inverse dynamic model in joint space of a 3-DOF in parallel actuated manipulator using Lagrangian approach. Moreover, they gave a numerical example of tracing a helical path to demonstrate the influence of the link dynamics on the actuating force required. Guo and Li (2006) derived the explicit compact closed-form dynamic equations of six DOF SP manipulators with prismatic actuators on the basic of the combination of the Newton-Euler method with the Lagrange formulation. In order to validate the proposed formulation, they studied numerical examples used in other references. The simulation results showed that it could be derived explicit dynamic equations in the task space for Stewart Platform manipulators by applying the combination of the Newton-Euler with the Lagrange formulation. Lebret and co-authors (1993) studied the dynamic equations of the Stewart Platform manipulator. The dynamics was given in step by step algorithm. Lin and Chen presented an efficient procedure for computer generation of symbolic modeling equations for the Stewart Platform manipulator. They used the Lagrange formulation for derivation of dynamic equations (Lin and Chen, 2008). The objective of the study was to develop a MATLAB-based efficient algorithm for computation of parallel link robot manipulator dynamic equations. Also, they proposed computer-torque control in order to verify the effectiveness of the dynamic equations. Lagrange's method was also used by others (Gregório and Parenti-Castelli, 2004; Beji and Pascal 1999; Liu et al., 1993).

the Newton-Euler method, the Lagrange formulation, the principle of virtual work and the

The Newton–Euler approach requires computation of all constraint forces and moments between the links. One of the important studies was presented by Dasgupta and Mruthyunjaya (1998) on dynamic formulation of the SP manipulator. In their study, the closed-form dynamic equations of the 6-UPS SP in the task-space and joint-space were derived using the Newton-Euler approach. The derived dynamic equations were implemented for inverse and forward dynamics of the Stewart Platform manipulator, and the simulation results showed that this formulation provided a complete modeling of the dynamics of SP. Moreover, it demonstrated the strength of the Newton-Euler approach as applied to parallel manipulators and pointed out an efficient way of deriving the dynamic equations through this formulation. This method was also used by Khalil and Ibrahim (2007). They presented a simple and general closed form solution for the inverse and forward dynamic models of parallel robots. The proposed method was applied on two parallel robots with different structures. Harib and Srinivasan (2003) performed kinematic and dynamic analysis of SP based machine structures with inverse and forward kinematics, singularity, inverse and forward dynamics including joint friction and actuator dynamics. The Newton-Euler formulation was used to derive the rigid body dynamic equations. Do and Yang (1988) and Reboulet and Berthomieu, (1991) presented the dynamic modeling of SP using Newton–Euler approach. They introduced some simplifications on the legs models. In addition to these works, others (Guo and Li, 2006; Carvalho and Ceccarelli, 2001; Riebe

Another method of deriving the dynamics of the SP manipulator is the Lagrange formulation. This method is used to describe the dynamics of a mechanical system from the concepts of work and energy. Abdellatif and Heimann (2009) derived the explicit and detailed six-dimensional set of differential equations describing the inverse dynamics of non-redundant parallel kinematic manipulators with the 6 DOF. They demonstrated that the derivation of the explicit model was possible by using the Lagrangian formalism in a computationally efficient manner and without simplifications. Lee and Shah (1988) derived the inverse dynamic model in joint space of a 3-DOF in parallel actuated manipulator using Lagrangian approach. Moreover, they gave a numerical example of tracing a helical path to demonstrate the influence of the link dynamics on the actuating force required. Guo and Li (2006) derived the explicit compact closed-form dynamic equations of six DOF SP manipulators with prismatic actuators on the basic of the combination of the Newton-Euler method with the Lagrange formulation. In order to validate the proposed formulation, they studied numerical examples used in other references. The simulation results showed that it could be derived explicit dynamic equations in the task space for Stewart Platform manipulators by applying the combination of the Newton-Euler with the Lagrange formulation. Lebret and co-authors (1993) studied the dynamic equations of the Stewart Platform manipulator. The dynamics was given in step by step algorithm. Lin and Chen presented an efficient procedure for computer generation of symbolic modeling equations for the Stewart Platform manipulator. They used the Lagrange formulation for derivation of dynamic equations (Lin and Chen, 2008). The objective of the study was to develop a MATLAB-based efficient algorithm for computation of parallel link robot manipulator dynamic equations. Also, they proposed computer-torque control in order to verify the effectiveness of the dynamic equations. Lagrange's method was also used by others

and Ulbrich, 2003) also used the Newton-Euler approach.

(Gregório and Parenti-Castelli, 2004; Beji and Pascal 1999; Liu et al., 1993).

screw theory.

For dynamic modeling of parallel manipulators, many approaches have been developed such as the principle of virtual work (Tsai, 2000, Wang and Gosselin, 1998; Geike and McPhee, 2003), screw teory (Gallardo et al., 2003), Kane's method (Liu et al., 2000; Meng et al., 2010) and recursive matrix method (Staicu and Zhang, 2008). Although the derived equations for the dynamics of parallel manipulators present different levels of complexity and computational loads, the results of the actuated forces/torques computed by different approaches are equivalent. The main goal of recent proposed approaches is to minimize the number of operations involved in the computation of the manipulator dynamics. It can be concluded that the dynamic equations of parallel manipulators theoretically have no trouble. Moreover, in fact, the focus of attention should be on the accuracy and computation efficiency of the model.

The aim of this paper is to present the work on dynamic formulation of a 6 DOF SP manipulator. The dynamical equations of the manipulator have been formulated by means of the Lagrangian method. The dynamic model included the rigid body dynamics of the mechanism as well as the dynamics of the actuators. The Jacobian matrix was derived in two different ways. Obtaining the accurate Jacobian matrix is very essential for accurate simulation model. Finally, the dynamic equations including rigid body and actuator dynamics were simulated in MATLAB-Simulink and verified on physical system.

This chapter is organized in the following manner. In Section 2, the kinematic analysis and Jacobian matrices are introduced. In Section 3, the dynamic equations of a 6 DOF SP manipulator are presented. In Section 4, dynamic simulations and the experimental results are given in detail. Finally, conclusions of this study are summarized.
