**7. References**


**Experiment Sim-A Sim-B** 

1-EXP(z) 4.4337 5.0121 2-EXP(x,y,z,α,β,γ) 4.5389 4.5877 3-EXP(x,y,z,α,β,γ) 3.6470 3.3697

In this paper, closed-form dynamic equations of the SP manipulator with the actuator dynamics were derived using Lagrangian method. A computational highly efficient method was developed for the explicit dynamic equations. Besides, two simple methods for the calculation of the Jacobian matrix of SP were proposed. Two dynamic models of the SP were obtained using these Jacobian matrices. Two SP models were simulated in a MATLAB-Simulink. In order to verify the simulation results, three experiments were conducted. Considering all of the results, there is very good agreement between the experiments and the simulations. Modeling errors for each experiment were computed. Based on the modeling error, modeling accuracy of the developed models is very high. Thus, the verified model of the SP can be used for control and design purposes. Especially, a model based

This work is supported by The Scientific and Technological Research Council of Turkey

Abdellatif, H. & Heimann, B. (2009). Computational efficient inverse dynamics of 6-DOF

Beji, L. & Pascal, M. (1999). The Kinematics and the Full Minimal Dynamic Model of a 6- DOF Parallel Robot Manipulator. *Nonlinear Dynamics*, 18, pp. (339-356) Bonev, I. A., & Ryu, J. A. (2000). New method for solving the direct kinematics of general 6-6

Carvalho, J. & Ceccarelli, M. (2001). A Closed-Form Formulation for the Inverse Dynamics

Dasgupta, B. & Mruthyunjaya, T. S. (1998). Closed-Form Dynamic Equations Of The General

Do, W. & Yang, D. (1988). Inverse Dynamic Analysis and Simulation of a Platform Type of

Robot*.Journal of Robotic Systems*, Vol. 5, pp. (209-227)

fully parallel manipulators by using the Lagrangian formalism. *Mechanism and* 

Stewart platforms using three linear extra sensors. *Mechanism and Machine Theory*,

of a Cassino Parallel Manipulator. *Multibody System Dynamics*, Vol. 5, pp. (185–

Stewart Platform Through The Newton-Euler. *Mechanism and Machine Theory*, 33

Table 5. The cost function values for the two different dynamic models

**5. Conclusion** 

controller needs the verified model.

35, pp. (423–436)

(7), pp. (993-1012)

210)

(TUBITAK) under the Grant No. 107M148.

*Machine Theory*, 44, pp. (192-207)

**6. Acknowledgment** 

**7. References** 


**Exploiting Higher Kinematic** 

*1Door To Door Company, Sharif University of Technology* 

*2Nanyang Technological University 3Sharif University of Technology, The Academy of Sciences of IR Iran* 

> *1,3Iran 2Singapore*

**Performance – Using a 4-Legged Redundant** 

Mohammad H. Abedinnasab1, Yong-Jin Yoon2 and Hassan Zohoor3

It is believed that Gough and Whitehall (1962) first introduced parallel robots with an application in tire-testing equipments, followed by Stewart (1965) , who designed a parallel mechanism to be used in a flight simulator. With ever-increasing demand on the robot's rigidity, redundant mechanisms, which are stiffer than their non-redundant counterparts,

Actuation redundancy eliminates singularity, and greatly improves dexterity and manipulability. Redundant actuation increases the dynamical capability of a PM by increasing the load-carrying capacity and acceleration of motion, optimizing the load distribution among the actuators and reducing the energy consumption of the drivers. Moreover, it enhances the transmission characteristics by increasing the homogeneity of the force transmission and the manipulator stiffness (Yi et al., 1989). From a kinematic viewpoint, redundant actuation eliminates singularities (Ropponen & Nakamura, 1990) which enlarge the usable workspace, as well. The kinematic analysis on general redundantly

A number of redundantly full-actuated mechanisms have been proposed over the years and some of them which are more significant are listed in this section. The spatial octopod, which is a hexapod with 2 additional struts, is one of them (Tsai, 1999). A 5-DOF 3-legged mechanism was proposed by Lu et al. (2008), who studied its kinematics, statics, and workspace. Staicu (2009) introduced a new 3-DOF symmetric spherical 3-UPS/S parallel mechanism having three prismatic actuators. As another work of Lu et al. (2009), they introduced and used 2(SP+SPR+SPU) serial–parallel manipulators. Wang and Gosselin (2004) addressed an issue of singularity and designed three new types of kinematically redundant parallel mechanisms, including a new redundant 7-DOF Stewart platform. They concluded that such manipulators can be used to avoid singularities inside the workspace of non-redundant manipulators. Choi et al. (2010) developed a new 4-DOF parallel mechanism with three translational and one rotational movements and found this mechanism to be ideal for high-speed machining.

actuated parallel mechanisms was investigated by Müller (2005).

**1. Introduction** 

are attracting more attention.

**PM Rather than Gough-Stewart Platforms** 

Wang , Y. (2007). A direct numerical solution to forward kinematics of general Stewart– Gough platforms, *Robotica*, 25(1), pp. ( 121–128) **3** 
