3.2 Inverse Kinematics

$q$ describes a unique joint configuration. Forward kinematics uses $q$ to find the end effector pose. The goal of Inverse Kinematics (IK) is to find a joint configuration that results in a desired end effector pose. i.e.

$q=f^{-1}(r)$ For a mobile robot, we also need to consider local velocities.

3.2.1 Solvability

If $n$ - or the number of DoF of joint space is less than $m$ - the number of DoF of task space then there may be poses in task space that are not in joint space, creating an infeasible situation.

3.2.2 IK of a Simple Manipulator arm

Consider this robot arm: We need to solve for $q=\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]$

Consider only the first link, $l_1$. Solving for $\alpha$ yields two solutions:

$\alpha =+/-co{s}^{-1}\frac{x_1}{l_1}$ one above the table and one below.

Solving even just for $\alpha ,\beta$ becomes cumbersome. Is there a scalable solution?

If we specify not only the position but also the orientation, then the desired pose will have the form $\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& x\\ sin\theta & cos\theta & 0& y\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ we can simplify this, yet this still does not scale well.

Let’s use optimization. You can define the euclidian distance between the current pose and the goal pose as a function of $\alpha ,\beta$, then minimize. This will find a local minima.