**Part 1**

**Kinematics and Dynamics** 

**1** 

Serdar Küçük *Kocaeli University* 

*Turkey* 

**Inverse Dynamics of RRR Fully Planar Parallel Manipulator Using DH Method** 

Parallel manipulators are mechanisms where all the links are connected to the ground and the moving platform at the same time. They possess high rigidity, load capacity, precision, structural stiffness, velocity and acceleration since the end-effector is linked to the movable plate in several points (Kang et al., 2001; Kang & Mills, 2001; Merlet, J. P. 2000; Tsai, L., 1999; Uchiyama, M., 1994). Parallel manipulators can be classified into two fundamental categories, namely spatial and planar manipulators. The first category composes of the spatial parallel manipulators that can translate and rotate in the three dimensional space. Gough-Stewart platform, one of the most popular spatial manipulator, is extensively preferred in flight simulators. The planar parallel manipulators which composes of second category, translate along the x and y-axes, and rotate around the z-axis, only. Although planar parallel manipulators are increasingly being used in industry for micro-or nanopositioning applications, (Hubbard et al., 2001), the kinematics, especially dynamics analysis of planar parallel manipulators is more difficult than their serial counterparts. Therefore selection of an efficient kinematic modeling convention is very important for simplifying the complexity of the dynamics problems in planar parallel manipulators. In this chapter, the inverse dynamics problem of three-**D**egrees **O**f **F**reedom (**DOF**) RRR **F**ully **P**lanar **P**arallel **M**anipulator (**FPPM**) is derived using DH method (Denavit & Hartenberg, 1955) which is based on 4x4 homogenous transformation matrices. The easy physical interpretation of the rigid body structures of the robotic manipulators is the main benefit of DH method. The inverse dynamics of 3-DOF RRR FPPM is derived using the virtual work principle (Zhang, & Song, 1993; Wu et al., 2010; Wu et al., 2011). In the first step, the inverse kinematics model and Jacobian matrix of 3-DOF RRR FPPM are derived by using DH method. To obtain the inverse dynamics, the partial linear velocity and partial angular velocity matrices are computed in the second step. A pivotal point is selected in order to determine the partial linear velocity matrices. The inertial force and moment of each moving part are obtained in the next step. As a last step, the inverse dynamic equation of 3-DOF RRR FPPM in explicit form is derived. To demonstrate the active joints torques, a butterfly shape Cartesian

**1. Introduction** 

trajectory is used as a desired end-effector's trajectory.

obtained by applying DH method.

**2. Inverse kinematics and dynamics model of the 3-DOF RRR FPPM** 

In this section, geometric description, inverse kinematics, Jacobian matrix & Jacobian inversion and inverse dynamics model of the 3-DOF RRR FPPM in explicit form are
