# Optimal Design of Three-Link Planar Manipulators Using Grashof’s Criterion

Sarosh R. Patel, Tarek Sobh
DOI: 10.4018/978-1-4666-0176-5.ch003
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## Nomenclature

• D: Dexterity index of the manipulator at a point.

• DMean: Mean dexterity index over a region or trajectory.

• N: Number of points along the trajectory.

• dx,dy,dz: Dexterity indices about the X, Y and Z axis.

• α, β, γ:Yaw, pitch and roll angels of the end-effector.

• a,b,c,d: Link lengths of the four-bar kinematic chain.

• θ1, θ2, θ3: Joint angles of the three-link planar manipulator.

• d: Distance between a task-point and base of the manipulator

• dmin: Minimum distance between a task-point and base of the manipulator.

• dmax: Maximum distance between a task-point and base of the manipulator.

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## 1. Introduction

The problem of designing an optimal manipulator configuration is very complex, as the equations governing the motion of the end-effector in the workspace are both non-linear and complex, often having no closed solutions. Prototyping methods such as kinematic synthesis and numerical optimization are complex and very time consuming. The inherent complexity of kinematic synthesis has helped to make a strong case for rapid prototyping methods in which manipulators are designed with very specific performance requirements or tasks point specifications. Rapid prototyping allows designers to spend more time on design, simulation and evaluation of different manipulator configurations instead of solving mathematical models describing kinematics chains.

The study of mobility of closed chain mechanisms has interested researchers for a very long time. Understanding the mobility of chain mechanisms in relation to their link lengths can help us to design better and highly dexterous manipulators. In 1833, Grashof first introduced a simple rule to understand the mobility of four-link mechanisms [6]. This rule, commonly known as the Grashof's theorem, helps analyze the rotatability of links in a closed four-bar mechanism. This was further extended by Paul (1979), who introduced an inequality into the Grashof's theorem and proved that Grashof's criterion is both a necessary and sufficient condition for the existence of a crank in the four-bar mechanism (Chang, Lin, & Wu, 2005).

Researchers have applied Grashof's criterion to understand and study the workspace mobility of both closed and open chain planar mechanisms. Barker (1985), using Grasshof's criterion, classified four-bar planar mechanisms based on their mobility. Grashof's criterion was applied to the study of three-link planar mechanism by Li and Dai (2009). Furthermore, they developed equations for the orientation angle and presented a simple program to analyze the orientation angle for a manipulator, given the link parameters. The mobility and orientation of open chain mechanisms can also be analyzed using Grashof's criterion. Dai and Shah (2002, 2003) studied the mobility of serial manipulators by introducing a virtual ground link between the end-effector and the base so as to form a virtual closed chain. In (Li, & Dai, 2009; Dai, & Shah, 2003), the authors proposed workspace decomposition based on the orientation capability of the manipulator.

Grashof's Theorem has been extended to include more than four-bar chain mechanisms. Grashof's criterion for five bar chain was proposed by Ting (1986). Ting and Liu (1991) extended this work to evaluate the mobility of N-bar chain mechanisms. Nokleby and Podhorodeski (2001) applied Grashof's criterion for the optimized synthesis for five-bar mechanisms.

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