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I am a student, and I am studying robotic arm manipulators. I am focusing on the concept of inverse kinematics, I have understood is the process of finding, from the direct kinematics, the joint varibles.

For easy problems, such as a 2R manipulatoer, I am able to solve algebraically the inverse kinematics problem, but for more complex problems I really get in troubles, and almost every time I can't solve the preoblem.

For example, consider the following arm manipulator:

enter image description here

if I compute the direct (or forward) kinematics of it, I get:

enter image description here

I have tried computing analitically the inverse kinematics for it, but without success.

My problem is that I don't understand how to tackle the problem of the inverse kinematics and so I don't know how to move.

What is the reasoning for solving inverse kinematics problems (both easy and hard) ?

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Deriving analytic IK for higher DOFs is tricky. Numerical IK solvers are a good alternative usually, unless you absolutely must use analytic IK.

In various robotics textbooks, there are examples on how to derive analytical IK for 6R robots. The way it's done is by considering position IK and rotation IK independently (note that this is possible only if the wrist is spherical).

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The reason for solving this problem becomes very obvious when you think of how to grab things. For example, consider a camera that is able to locate an object in the working range of the robot. Suppose you know the relation between camera and robot, but this is another problem...

The visual system tells you the coordinates where your target is located. Now the robot has to move its joints to reach the object. Arises the question, what are the correct joint parameters? You get the answer from inverse kinematics. And yes, this can be hard to answer and mostly (for real robots) you need to use iterative numerical approaches.

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