# How to derive the inverse kinematic equations for this simple 6-DOF robot?

For the simple 6-DOF shown, we have to solve for the inverse kinematic equations and its resulting figures.

What should I do to solve this? P.s. That's all the information our teacher gave us.

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– Ben
Mar 8, 2021 at 18:19

As mentioned above, you first need to develop the kinematic model of the robot. Try formulating the forward kinematics of the robot. For the inverse kinematics, you can either use geometric or algebraic method. A very good source to learn would be this book: "Introduction to Robotics - Mechanics and Control". In fact, there is a complete solution for forward and inverse kinematics of a 6dof robotics arm. Specifically, the inverse kinematics is solved in the 4th chapter. Here's the Link to the book

The first step is to create the model of the robot. I recommend you to model the robot using modified denavit hartenberg (There are plenty of book and youtube provide the information you need). Then you can simply use this robotic toolbox and use function like SerialLink.ikine to get the mathematical function of inverse kinematic.

Hope that help :)

Since you are solving an Inverse Kinematics (IK) problem, you should already have the pose (position and orientation) of the end effector (EE) relative to the fixed base of the robot (or the pose of both the base and the EE in one common frame). You should also already know how to generate the Forward Kinematics (FK), which can come in handy.

There are two ways that I generally recommend approaching an IK problem for a robot that you haven't seen before: one where you turn it into a simpler IK problem and one where you look carefully at the FK equations.

For the first one, see if you can find a reference point on the robot that you can locate using only the EE pose which will not move when you rotate the last 1-3 (or more) joints. Once you've found that reference point, you can solve a simpler IK problem to figure out how to use the remaining joints to place the reference point at the desired location.

For the second one, the FK equations will tell you which elements of the EE pose are affected by which of the joints: if the joint variable appears in the equation for the height of the EE, for example, then that joint affects the height. If you can find one element that only has 1 joint variable in it then you can solve that element using the reduced set of equations. This also works if there are 2 elements with only 2 joint variables, as long as they are the same 2 variables in both equations, and so on for more elements.

Hope that helps! As Ben said in his comment, without more details on what you've tried so far we won't be able to provide more than the first steps of a homework problem.

Cheers, Brandon

You would first solve for the center of the wrist. There are examples on this website and textbooks how to solve for the center of the wrist. The wrist 3d pose ignores the last joint value because you are traveling along the Z rotation axis of the EE. https://ieeexplore.ieee.org/ielx7/6287639/6514899/08630999.pdf

You can then solve for the joint value of theta 1 since it points the arm to the wrist center. (you should be able to figure this out).

Then, once you have t1 and the wrist center, you run into an issue.

It should be observed that you have 3 joints all on the same plane. (2,3,4) This means that there is a kinematic redundancy in this robot arm. I suggest you create your own solver to handle the planar kinematic redundancy. (The stretching out of the wrist). There are many ways you can do this, so you will just have to experiment. I suggest starting with the simplest design possible.

This solver would have to take into account the magnitude distance from the 3d location of the center of joint 2 all the way out to the 3d position of joint 5 (wrist center). This will allow you to solve joints 2,3,4. This can be done by assuming the last link points directly to the wrist center (from joint 2). You can then solve for the other 2 joints (both solutions) through the inverse of the cosine law.

Then, once you have the joints 1,2,3,4 solved, you need to solve for t5 and t6. If you do the FK given t1-t4, you know the 3d position of joint 4. Since you know the 3d position of joint 4,5,6, you can then solve for t5 using the 3d triangle shape. (sss form) Then, once you have t1-5, you can solve for t6 by subbing "0" into t6 with t1-5 and then doing an atan2() call.

This may not make a lot of sense, but you will have to do some research anyways.