# What should be the ideal ratio for link lengths ( L1/ L2) of a 2 link manipuator robot with revolute joints?

I am working on making a CAD model of a two link robotic manipulator, with link lengths around 15-20 cm. I am unable to figure how to set their mutual ratio and I want to know whether there is some fixed, determined ratio on how to take these values. I thought taking the human arm as an inspiration, but if so, is it right to take so?

This is a good question, and one you should definitely consider when designing a new robot arm. But unfortunately, it depends a lot on the configuration of your arm (i.e. how the joints will be bent), and how you determine "best" (i.e. are you optimizing for speed, strength, dexterity, etc). And even if you know these things, the metrics you use to determine "best" can differ.

You should read up on "manipulabilty". This is a feature of the arm's configuration. Roughly, it is a measure of how well the arm can move in a particular direction. Usually, this is represented as an ellipse.

The interesting thing about manupulabilty is that there are actually 2, orthogonal ellipoids. One for velocity and one for force. Intuitively, if you think about mechanics, where the arm can move fast, it is not very strong. But where it is strong, it is not very fast. And if you want a very general purpose arm, trying to achieve a more circular ellipse might be desirable.

You may have different link lengths and arm configurations depending on your use case.

There are other measures that you might want to consider too. Such as distance from joint limits, and condition number.

Your question can't really be answered without understanding what you want the arm to do. You will also need to account in turn for the masses of the payload and elbow actuator, as well as the maximum torque and angular velocity the actuator could support along with the gear ratio of any transmission. You could then work out the dynamics of the arm for a given actuator as a function of the link length ratio, maximizing, for example, something like the speed of the arm tip or even the expected lifetime of the arm.

It's all quite complicated. A good place to start, though, might be this old MIT AI Lab paper. You could build a dynamic and kinematic model in R or some other tool and experiment with different combinations. (Note that the linked paper doesn't deal with the elbow actuator mass, but it's easy to extend the model).