# Significance of link length ratio in serial manipulator

I am formulating a problem to optimize link lengths of 3R serial manipulator shown below :

I do not assume any explicit relationship between the link lengths, but in one of the papers (http://www.sciencedirect.com/science/article/pii/S1474667017358172 , page 119, equation no.14) I am referring assumes following relationship :

$l_1 /l_2 \geq 1.1$

$l_1 /l_2 \leq 2$

$l_2 /l_3 \geq 1.1$

$l_2 /l_3 \leq 2$

According to another paper (page 569, below equation 14), the above relationship exists in a number of the industrial manipulators they studied. What's the reason behind this relationship?

• I think you can find papers from Sheridan in the 1970’s and 1980’s that addressed this ratio. Roth and Duffy may have analyzed it, too. – SteveO Nov 3 '17 at 12:59
• @siddhesh Where in the paper that claimed that the above relationship held in most manipulators? – Petch Puttichai Nov 4 '17 at 4:00
• @PetchPuttichai : It's mentioned in another paper, Page no. 569 , paragraph below equation no.14 – Siddhesh Nov 4 '17 at 5:41
• "Limits of these nonlinear inequality constraints were chosen based on arm lengths of robot manipulators used in industry such as the Yamaha-HXYx series (l1 = 1250 mm, l2 = 1050 mm, l3 = 550 mm), Denso-VM- 6083D series (l1 = 475 mm, l2 = 385 mm, l3 = 329 mm), Mitsubishi-RV-2AJ series (l1 = 360 mm, l2 = 250 mm, l3 = 160 mm) and Staubli-Tx40 series (l1 = 320 mm, l2 = 225 mm, l3 = 160 mm)." – Siddhesh Nov 4 '17 at 5:43
• "Optimize" in what way? For reachable space, manipulability, maximizing payload, etc. I can imagine many metrics for optimization. – Ben Nov 6 '17 at 18:17

Concerning the paper you refer to, you can see the cost function $\kappa$ as equivalent the static force manipulability index, so the criterion used is the maximization of that index.