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IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 02 Issue: 09 | Sep-2013, Available @ 79 Determinant of inverse jacobian matrix is ¸±C ´¶ S ´ ¸ S ´¶ C ´ ²/ L ´ L S . If the value of determinant is zero for given task space coordinates then the resulting configuration is singular. Desired pose can be achieved by iterative determination of the joint angles. Multi valued inverse mapping and redundancy are common problems with inverse kinematic. 4. PERFORMANCE MEASUREMENT Manipulability measures the effective robotic pose for object manipulation with in workspace. The term was proposed by Yoshikawa [10] and manipulability index ( μ² is defined as, μ ³ |det ±J²| ³ 9 σ - -;´ Where, J is jacobian of robot kinematics. For a two DOF RR- robotic manipulator, the manipulability index is in form of μ ³ L ´ L |sinq | . The manipulability index always varies within the workspace. The normalized mobility index is also an alternative way to represent a singular configuration and determining well conditioned workspace. Its value near to zero near boundaries shows boundary singularities. The condition number is another measure of presence of singularity as well as dexterity of robotic pose. Normally, the larger value of condition number above 30 indicates presence of singularity. The condition number is representation of an amplification factor of error during actuation. It is used for optimum trajectory planning and gross motion calculation of robotic manipulation within workspace. It ranges from 1 · κ · ∞ . Hence, the local conditioning index also known as reciprocal of condition number is used to determine control accuracy, dexterity and isotropy of manipulator. Figure 4(a) and 4(b) shows the variation of mobility index and condition number within workspace for l ´ ³ 0.2m and l ³ 0.15 m without time lag between servomotors with cosine input using MATLAB program respectively. The position curve is also plotted in same figure as shown on xy-plane for specified joint ranges. Fig -4 (a): Variation of condition number Fig -4 (b): Mobility Index μ Moreover, jacobian matrix loses its rank then the corresponding robot configurations are termed as singular configurations. It means for the smaller movements in operational space large amount of velocities are required at joints. Fig -5: Variation of condition number and mobility index with respect to time q ³ A ∗ cos C360 ∗ F G µ BI µ C is called a general cosine function, which is used as an input to the both servo motors simulation for kinematic and dynamic analysis. In the expression A, B, C and T affects amplitude, phase, offset (horizontal translation) and period (vertical translation) respectively. -0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 0.6 0 200 400 600 x (m) y (m) Condition number -0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 0.6 0 0.5 1 x (m) y (m) Norm alized m u 0.0 200.0 400.0 600.0 0 2 4 6 Condition Number Time 0 0.5 1 1.5 0 2 4 6 Mobility Index Time (sec)
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IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 02 Issue: 09 | Sep-2013, Available @ 80 Fig -6:
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  • International Journal of Research in Engineering and Technology

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