## Robot control part 7: OSC of a 3-link arm

So we’ve done control for the 2-link arm, and control of the one link arm is trivial (where we control joint angle, or x or y coordinate of the pendulum), so here I’ll just show an implementation of operation space control for a more interesting arm model, the 3-link arm model. The code can all be found up on my Github.

In theory there’s nothing different or more difficult about controlling a 3-link arm versus a 2-link arm. For the inertia matrix, what I ended up doing here is just jacking up all the values of the matrix to about 100, which causes the controller to way over control the arm, and you can see the torques are much larger than they would need to be if we had an accurate inertia matrix. But the result is the same super straight trajectories that we’ve come to expect from operational space control:

It’s a little choppy because I cut out a bunch of frames to keep the gif size down. But you get the point, it works. And quite well!

Because this is also a 3-link arm now and our task space force signal is 2D, we have redundant space of solutions, meaning that the task space control signal can be carried out in a number of ways. In other words, a null space exists for this controller. This means that we can throw another controller in our system to operate inside the null space of the first controller. We’ve already worked through all the math for this, so it’s straightforward to implement.

What kind of null space controller should we put in? Well, you may have noticed in the above animation the arm goes through itself, here’s another example:

Often it’s desirable to avoid this (because of physics or whatever), so what we can do is add a secondary controller that works to keep the arm’s elbow and wrist near some comfortable default angles. When we do this we get the following:

And there you have it! Operational space control of a three link arm with a secondary controller in the null space to try to keep the angles near their default / resting positions.

I also added mouse based control to the arm so it will try to follow your mouse when you move over the figure, which makes it pretty fun to explore the power of the controller. It’s interesting to see where the singularities become an issue, and how having a null space controller that’s operating in joint space can actually come to help the system move through those problem points more smoothly. Check it out! It’s all up on my Github.

## Robot control part 6: Handling singularities

We’re back! Another exciting post about robotic control theory, but don’t worry, it’s short and ends with simulation code. The subject of today’s post is handling singularities.

What is a singularity

This came up recently when I had build this beautiful controller for a simple two link arm that would occasionally go nuts. After looking at it for a while it became obvious this was happening whenever the elbow angle reached or got close to 0 or $\pi$. Here’s an animation:

What’s going on here? Here’s what. The Jacobian has dropped rank and become singular (i.e. non-invertible), and when we try to calculate our mass matrix for operational space

$\textbf{M}_\textbf{x}(\textbf{q}) = (\textbf{J} (\textbf{q}) \; \textbf{M}^{-1} (\textbf{q}) \; \textbf{J}^T(\textbf{q}))^{-1},$

the values explode in the inverse calculation. Dropping rank means that the rows of the Jacobian are no longer linearly independent, which means that the matrix can be rotated such that it gives a matrix with a row of zeros. This row of zeros is the degenerate direction, and the problems come from trying to send forces in that direction.

To determine when the Jacobian becomes singular its determinant can be examined; if the determinant of the matrix is zero, then it is singular. Looking the Jacobian for the end-effector:

$\textbf{J}(\textbf{q}) = \left[ \begin{array}{cc} -L_0 sin(q_0) - L_1 sin(q_0 + q_1) & -L_1 sin(q_0 + q_1) \\ L_0 cos(q_0) + L_1 cos(q_0 + q_1) & L_1 cos(q_0 + q_1) \end{array} \right].$

When $q_1 = 0$ it can be that $sin(q_0 + 0) = sin(q_0)$, so the Jacobian becomes

$\textbf{J}(\textbf{q}) = \left[ \begin{array}{cc} -(L_0 - L_1) sin(q_0) & -L_1 sin(q_0) \\ (L_0 + L_1) cos(q_0) & L_1 cos(q_0) \end{array} \right],$

which gives a determinant of

$(L_0 - L_1)(-sin(q_0))(L_1)(cos(q_0)) - (L_1)(-sin(q_0))(L_0 + L_1)(cos(q_0)) = 0.$

Similarly, when $q_1 = \pi$, where $sin(q_0 + \pi) = -sin(q_0)$ and $cos(q_0 + \pi) = -cos(q_0)$, the determinant of the Jacobian is

$\textbf{J}(\textbf{q}) = \left[ \begin{array}{cc} -(L_0 - L_1) sin(q_0) & L_1 sin(q_0) \\ (L_0 + L_1) cos(q_0) & - L_1 cos(q_0) \end{array} \right].$

Calculating the determinant of this we get

$(L_0 + L_1)(-sin(q_0))(L_1)(-cos(q_0)) - (L_1)(sin(q_0))(L_0 + L_1)(-cos(q_0)) = 0.$

Note that while in these cases the Jacobian is a square matrix in the event that it is not a square matrix, the determinant of $\textbf{J}(\textbf{q})\;\textbf{J}^T(\textbf{q})$ can be found instead.

Fixing the problem

When a singularity is occurring it can be detected, but now it must be handled such that the controller behaves appropriately. This can be done by identifying the degenerate dimensions and setting the force in those directions to zero.

First the SVD decomposition of $\textbf{M}_\textbf{x}^{-1}(\textbf{q}) = \textbf{V}\textbf{S}\textbf{U}^T$ is found. To get the inverse of this matrix (i.e. to find $\textbf{M}_\textbf{x}(\textbf{q})$) from the returned $\textbf{V}, \textbf{S}$ and $\textbf{U}$ matrices is a matter of inverting the matrix $\textbf{S}$:

$\textbf{M}_\textbf{x}(\textbf{q}) = \textbf{V} \textbf{S}^{-1} \textbf{U}^T,$

where $\textbf{S}$ is a diagonal matrix of singular values.

Because $\textbf{S}$ is diagonal it is very easy to find its inverse, which is calculated by taking the reciprocal of each of the diagonal elements.

Whenever the system approaches a singularity some of the values of $\textbf{S}$ will start to get very small, and when we take the reciprocal of them we start getting huge numbers, which is where the value explosion comes from. Instead of allowing this to happen, a check for approaching the singularity can be implemented, which then sets the singular values entries smaller than the threshold equal to zero, canceling out any forces that would be sent in that direction.

Here’s the code:

Mx_inv = np.dot(JEE, np.dot(np.linalg.inv(Mq), JEE.T))
if abs(np.linalg.det(np.dot(JEE,JEE.T))) > .005**2:
# if we're not near a singularity
Mx = np.linalg.inv(Mx_inv)
else:
# in the case that the robot is entering near singularity
u,s,v = np.linalg.svd(Mx_inv)
for i in range(len(s)):
if s[i] < .005: s[i] = 0
else: s[i] = 1.0/float(s[i])
Mx = np.dot(v, np.dot(np.diag(s), u.T))


And here’s an animation of the controlled arm now that we’ve accounted for movement when near singular configurations:

As always, the code for this can be found up on my Github. The default is to run using a two link arm simulator written in Python. To run, simply download everything and run the run_this.py file.

Everything is also included required to run the MapleSim arm simulator. To do this, go into the TwoLinkArm folder, and run python setup.py build_ext -i. This should compile the arm simulation to a shared object library that Python can now access on your system. To use it, edit the run_this.py file to import from TwoLinkArm/arm_python to TwoLinkArm/arm and you should be good to go!
More details on getting the MapleSim arm to run can be found in this post.