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

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?

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    $\begingroup$ 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. $\endgroup$
    – SteveO
    Commented Nov 3, 2017 at 12:59
  • $\begingroup$ @siddhesh Where in the paper that claimed that the above relationship held in most manipulators? $\endgroup$ Commented Nov 4, 2017 at 4:00
  • $\begingroup$ @PetchPuttichai : It's mentioned in another paper, Page no. 569 , paragraph below equation no.14 $\endgroup$
    – Siddhesh
    Commented Nov 4, 2017 at 5:41
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    $\begingroup$ "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)." $\endgroup$
    – Siddhesh
    Commented Nov 4, 2017 at 5:43
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    $\begingroup$ "Optimize" in what way? For reachable space, manipulability, maximizing payload, etc. I can imagine many metrics for optimization. $\endgroup$
    – Ben
    Commented Nov 6, 2017 at 18:17

1 Answer 1


I am not sere about the conclusion of the paper in general as it can be considered a bit outdated with respect to the "famous" manipulators configuration nowadays. You can easily find the dimensions on the manufacturer websites and check if that relation holds.

In general when you optimize you should state clearly your criteria, what do you want to optimize? Workspace ? Payload for given motors? maximal velocity at the end-effector? Possibilities are endless ...

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.

  • $\begingroup$ I am using Global Conditioning(GCI) Index as my optimization criteria, but I am unsure whether any constraints should be put on link lengths ? and what was the rationale behind selecting certain link ratios? $\endgroup$
    – Siddhesh
    Commented Nov 3, 2017 at 15:57
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    $\begingroup$ The GCI is a property of matrix, you should also provide the matrix you are considering if you want to be complete in your description. Obviously you should put constraints on link lengths because you have not infinite actuation power so if you keep putting mass further away you will exceed your motors capacity... The link ratios comes from the criteria detailed in the paper you cited, but in general it is faster to use those ratio to design a new arm than to redo a complete study of the kinematic/dynamic of a potential arm. $\endgroup$
    – N. Staub
    Commented Nov 6, 2017 at 8:22
  • $\begingroup$ I am using length constraints as well as some other constraints on my link lengths, for the sake of brevity I didn’t add them in question. $\endgroup$
    – Siddhesh
    Commented Nov 7, 2017 at 14:03

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