Inverse Kinematics - How to only find a unique joint angle solution in 4 dof robot?

Question

I have to develop an algorithm to determine the necessary joint angles to achieve a desired TCP position and orientation in a 4 joint manipulator. I have come across a concept called "degeneracies", and I have to think of a scheme that will handle degenerate points so that only one joint angle solution exists for any TCP position and orientation. For example: keep the shoulder tilted up at all times. How can I go about this? I understand the equations, it's mainly the "only one joint angle solution" I'm having a hard time with.

Answer 1

If you have a 4 degrees of freedom system you will most probably solve the inverse kinematics equations for $X, Y$ and $Z$ position and, aditionally 1 orientation, let's call this $\alpha$. IFF the chosen orientation and the robot structure allows an analytical solution, you can proced as follows:

As you mentioned you will get more than 1 solution for one set of inputs, so you must have a strategy to select one of the solutions.

You can derive the equations to have the following form:

$q_{11} = f_{11}^{-1}(x, y, z, \alpha)$

$q_{12} = f_{12}^{-1}(x, y, z, \alpha)$

here you decide which value of $q_1$ to use, e.g. based on its sign.

$ q_1 = \left\{ \begin{array}{c} q_{11} \text{ if } q_{11}>0;\\ q_{12} \text{ otherwise } \end{array} \right. $

usually, for an analytical solution the next angle which gets calculated is $q_{3}$. You can formulate the equations so that they are dependent on $q_1$ also. This way the $q_1$ choise made earlier will be pluged in the calculations for the remaining joint angles.

$q_{31} = f_{31}^{-1}(x, y, z, \alpha, q_1)$

$q_{32} = f_{32}^{-1}(x, y, z, \alpha, q_1)$

Also in this case you can make a choise which solution to use:

$ q_3 = \left\{ \begin{array}{c} q_{31} \text{ if } q_{31}>0; \\ q_{32} \text{ otherwise } \end{array} \right. $

You can continue solving the inverse kinematics problem by calculating $q_2$. Here there is are no multiple solutions expected if $q_3$ has already been chosen:

$q_3 = f_{2}^{-1}(x, y, z, \alpha, q_1, q_3)$

and you can continue this way until you have all angles.

Answer 2

You have a 4-Dof and want to find a set of joint angle values which fullfil the desired orientation and position. I doubt that you have multiple solution for this problem. Because you actually need 6-Dof for this task. Otherwise you can build a nullspace to specify the shoulder movement while the TCP track the desired movement ( but with 4-Dof not worthy probably to use the nullspace...)