## Quick calculations with SymPy and Cython

Alrighty! Last time I posted about SymPy we weren’t worrying too much about computation speed. It was worth the time saved by automating the Jacobian, inertia matrix, etc calculations and it didn’t affect simulation speed really all that much. But! When working with a real arm it suddenly becomes critical to have highly efficient calculations.

Turns out that I still don’t want to code those equations by hand. There are probably very nice options for generating optimized code using Mathematica or MapleSim or whatever pay software, but SymPy is free and already Python compatible so my goal is to stick with that.

Options

I did some internet scouring, and looked at installing various packages to help speed things up, including

1. trying out the Sympy simplify function,
2. trying out SymEngine,
3. trying out the Sympy compile to Theano function,
4. trying out the Sympy autowrap function, and
5. various combinations of the above.

The SymEngine backend and Theano functions really didn’t give any improvements for the kind of low-dimensional vector calculations performed for control here. Disclaimer, it could be the case that I gave up on these implementations too quickly, but from some basic testing it didn’t seem like they were the way to go for this kind of problem.

So down to the simplify function and compiling to the Cython backend options. First thing you’ll notice when using the simplify is that the generation time for the function can take orders of magnitude longer (up to 12ish hours for inertia matrices for the Kinova Jaco 2 arm with simplify vs about 1-2 minutes without simplify, for example). And then, as a nice kick in the teeth after that, there’s almost no difference between straight Cython of the non-simplified functions and the simplified functions. Here are some graphs:

So you can see how the addition of the simplify drastically increases the compile time in the top half of the graphs there. In the lower half, you can see that the simplify does save some execution time relative to the baseline straight lambdify function, but once you start compiling to Cython with the autowrap the difference is on the order of hundredths to thousandths of milliseconds.

Results

So! My recommendation is

• Don’t use simplify,
• do use autowrap with the Cython backend.

To compile using the Cython backend:

from sympy.utilities.autowrap import autowrap
function = autowrap(sympy_expression, backend="cython",
args=parameters)


In the last Sympy post I looked at a hard-coded calculation against the Sympy generated function, using the timeit function, which runs a given chunk of code 1,000,000 times and tells you how long it took. To save tracking down the results from that post, here are the timeit results of a Sympy function generated using just the lambdify function vs the hard-coding the function in straight Numpy:

And now using the autowrap function to compile to Cython code, compared to hard-coding the function in Numpy:

This is a pretty crazy improvement, and means that we can have the best of both worlds. We can declare our transforms in SymPy and automate the calculation of our Jacobians and inertia matrices, and still have the code execute fast enough that we can use it to control real robots.

That said, this is all from my basic experimenting with some different optimisations, which is a far from thorough exploration of the space. If you know of any other ways to speed up execution, please comment below!

You can find the code I used to generate the above plots up on my GitHub.

## Full body obstacle collision avoidance

Previously I’ve discussed how to avoid obstacles using DMPs in the end-effector trajectory. This is good when you’re controlling a single disconnected point-mass, like a mobile robot navigating around an environment. But if you want to use this system to control a robotic manipulator, then pretty quickly you run into the problem that your end-effector is not a disconnected point-mass moving around the environment. It’s attached to the rest of the arm, and moving such that the arm segments and joints also avoid the obstacle is a whole other deal.

I was doing a quick lit scan on methods for control methods for avoiding joint collision with obstacles, and was kind of surprised that there wasn’t really much in the realm of recent discussions about it. There is, however, a 1986 paper from Dr. Oussama Khatib titled Real-time obstacle avoidance for manipulators and mobile robots that pretty much solves this problem short of getting into explicit path planning methods. Which could be why there aren’t papers since then about it. All the same, I couldn’t find any implementations of it online, and there are a few tricky parts, so in this post I’m going to work through the implementation.

Note: The implementation that I’ve worked out here uses spheres to represent the obstacles. This works pretty well in general by just making the sphere large enough to cover whatever obstacle you’re avoiding. But if you want a more precise control around other shapes, you can check out the appendix of Dr. Khatib’s paper, where he works provides the math for cubes and cones as well.

Note: You can find all of the code I use here and everything you need for the VREP implementation up on my GitHub. I’ve precalculated the functions that are required, because generating them is quite computationally intensive. Hopefully the file saving doesn’t get weird between different operating systems, but if it does, try deleting all of the files in the ur5_config folder and running the code again. This will regenerate those files (on my laptop this takes ~4 hours, though, so beware).

The general idea

Since it’s from Dr. Khatib, you might expect that this approach involves task space. And indeed, your possible suspicions are correct! The algorithm is going to work by identifying forces in Cartesian coordinates that will move any point of the arm that’s too close to an obstacle away from it. The algorithm follows the form:

Setup

• Specify obstacle location and size
• Specify a threshold distance to the obstacle

While running

• Find the closest point of each arm segment to obstacles
• If within threshold of obstacle, generate force to apply to that point
• Transform this force into joint torques
• Add directly to the outgoing control signal

Something that you might notice about this is that it’s similar to the addition of the null space controller that we’ve seen before in operational space control. There’s a distinct difference here though, in that the control signal for avoiding obstacles is added directly to the outgoing control signal, and that it’s not filtered (like the null space control signal) such that there’s no guarantee that it won’t affect the movement of the end-effector. In fact, it’s very likely to affect the movement of the end-effector, but that’s also desirable, as not ramming the arm through an obstacle is as important as getting to the target.

OK, let’s walk through these steps one at a time.

Setup

I mentioned that we’re going to treat all of our obstacles as spheres. It’s actually not much harder to do these calculations for cubes too, but this is already going to be long enough so I’m only addressing sphere’s here. This algorithm assumes we have a list of every obstacle’s centre-point and radius.

We want the avoidance response of the system to be a function of the distance to the obstacle, such that the closer the arm is to the obstacle the stronger the response. The function that Dr. Khatib provides is of the following form:

$\textbf{F}_{psp} = \left\{ \begin{array}{cc}\eta (\frac{1.0}{\rho} - \frac{1}{\rho_0}) \frac{1}{\rho^2} \frac{\partial \rho}{\partial \textbf{x}} & \rho \leq \rho_0 \\ \textbf{0} & \rho > \rho_0 \end{array} \right. ,$

where $\rho$ is the distance to the target, $\rho_0$ is the threshold distance to the target at which point the avoidance function activates, $\frac{\partial \rho}{\partial \textbf{x}}$ is the partial derivative of the distance to the target with respect to a given point on the arm, and $\eta$ is a gain term.

This function looks complicated, but it’s actually pretty intuitive. The partial derivative term in the function works simply to point in the opposite direction of the obstacle, in Cartesian coordinates, i.e. tells the system how to get away from the obstacle. The rest of the term is just a gain that starts out at zero when $\rho = \rho_0$, and gets huge as the obstacle nears the object (as $\rho \to 0 \Rightarrow \frac{1}{\rho} \to \infty$). Using $\eta = .2$ and $\rho_0 = .2$ gives us a function that looks like

So you can see that very quickly a very, very strong push away from this obstacle is going to be generated once we enter the threshold distance. But how do we know exactly when we’ve entered the threshold distance?

Find the closest point

We want to avoid the obstacle with our whole body, but it turns out we can reduce the problem to only worrying about the closest point of each arm segment to the obstacle, and move that one point away from the obstacle if threshold distance is hit.

To find the closest point on a given arm segment to the obstacle we’ll do some pretty neat trig. I’ll post the code for it and then discuss it below. In this snippet, p1 and p2 are the beginning and ending $(x,y,z)$ locations of arm segment (which we are assuming is a straight line), and v is the center of the obstacle.

# the vector of our line
vec_line = p2 - p1
# the vector from the obstacle to the first line point
vec_ob_line = v - p1
# calculate the projection normalized by length of arm segment
projection = (np.dot(vec_ob_line, vec_line) /
np.dot(vec_line, vec_line))
if projection < 0:
# then closest point is the start of the segment
closest = p1
elif projection > 1:
# then closest point is the end of the segment
closest = p2
else:
closest = p1 + projection * vec_line


The first thing we do is find the arm segment line, and then line from the obstacle center to the start point of the arm segment. Once we have these, we do:

$\frac{\textbf{v}_\textrm{ob\_line} \; \cdot \; \textbf{v}_\textrm{line}}{\textbf{v}_\textrm{line} \; \cdot \; \textbf{v}_\textrm{line}},$

using the geometric definition of the dot product two vectors, we can rewrite the above as

$\frac{||\textbf{v}_\textrm{ob\_line}|| \; || \textbf{v}_\textrm{line} || \; \textrm{cos}(\theta)}{||\textbf{v}_\textrm{line}||^2} = \frac{||\textbf{v}_\textrm{ob\_line}||} {||\textbf{v}_\textrm{line}||} \textrm{cos}(\theta)$

which reads as the magnitude of vec_ob_line divided by the magnitude of vec_line (I know, these are terrible names, sorry) multiplied by the angle between the two vectors. If the angle between the vectors is < 0 (projection will also be < 0), then right off the bat we know that the start of the arm segment, p1, is the closest point. If the projection value is > 1, then we know that 1) the length from the start of the arm segment to the obstacle is longer than the length of the arm, and 2) the angle is such that the end of the arm segment, p2, is the closest point to the obstacle.

Finally, in the last case we know that the closest point is somewhere along the arm segment. To find where exactly, we do the following

$\textbf{p}_1 + \textrm{projection} \; \textbf{v}_\textrm{line},$

which can be rewritten

$\textbf{p}_1 + \frac{||\textbf{v}_\textrm{ob\_line}||} {||\textbf{v}_\textrm{line}||} \textrm{cos}(\theta) \; \textbf{v}_\textrm{line},$

I find it more intuitive if we rearrange the second term to be

$\textbf{p}_1 + \frac{\textbf{v}_\textrm{line}} {||\textbf{v}_\textrm{line}||} \; ||\textbf{v}_\textrm{ob\_line} || \; \textrm{cos}(\theta).$

So then what this is all doing is starting us off at the beginning of the arm segment, p1, and to that we add this other fun term. The first part of this fun term provides direction normalized to magnitude 1. The second part of this term provides magnitude, specifically the exact distance along vec_line we need to traverse to form reach a right angle (giving us the shortest distance) pointing at the obstacle. This handy figure from the Wikipedia page helps illustrate exactly what’s going on with the second part, where B is be vec_line and A is vec_ob_line:

Armed with this information, we understand how to find the closest point of each arm segment to the obstacle, and we are now ready to generate a force to move the arm in the opposite direction!

Check distance, generate force

To calculate the distance, all we have to do is calculate the Euclidean distance from the closest point of the arm segment to the center of the sphere, and then subtract out the radius of the sphere:

# calculate distance from obstacle vertex to the closest point
dist = np.sqrt(np.sum((v - closest)**2))
# account for size of obstacle
rho = dist - obstacle[3]


Once we have this, we can check it and generate $F_{psp}$ using the equation we defined above. The one part of that equation that wasn’t specified was exactly what $\frac{\partial \rho}{\partial \textbf{x}}$ was. Since it’s just the partial derivative of the distance to the target with respect to the closest point, we can calculate it as the normalized difference between the two points:

drhodx = (v - closest) / rho


Alright! Now we’ve found the closest point, and know the force we want to apply, from here it’s standard operational space procedure.

Transform the force into torques

As we all remember, the equation for transforming a control signal from operational space to involves two terms aside from the desired force. Namely, the Jacobian and the operational space inertia matrix:

$\textbf{u}_\textrm{psp} = \textbf{J}^T_{psp} \textbf{M}_{psp} \textbf{F}_{psp},$

where $\textbf{J}_{psp}$ is the Jacobian for the point of interest, $\textbf{M}_{psp}$ is the operational space inertia matrix for the point of interest, and $\textbf{F}_{psp}$ is the force we defined above.

Calculating the Jacobian for an unspecified point

So the first thing we need to calculate is the Jacobian for this point on the arm. There are a bunch of ways you could go about this, but the way I’m going to do it here is by building on the post where I used SymPy to automate the Jacobian derivation. The way we did that was by defining the transforms from the origin reference frame to the first link, from the first link to the second, etc, until we reached the end-effector. Then, whenever we needed a Jacobian we could string together the transforms to get the transform from the origin to that point on the arm, and take the partial derivative with respect to the joints (using SymPy’s derivative method).

As an example, say we wanted to calculate the Jacobian for the third joint, we would first calculate:

$^3_{\textrm{org}}\textbf{T} = ^0_{\textrm{org}}\textbf{T} \; ^1_0\textbf{T} \; ^2_1\textbf{T} \; ^3_2\textbf{T},$

where $^m_n\textbf{T}$ reads the transform from reference frame $m$ to reference frame $n$.

Once we have this transformation matrix, $^3_\textrm{org}\textbf{T}$, we multiply it by the point of interest in reference frame 3, which, previously, has usually been $\textbf{x} = [0, 0, 0]$. In other words, usually we’re just interested in the origin of reference frame 3. So the Jacobian is just

$\frac{\partial \; ^3_\textrm{org}\textbf{T} \textbf{x}}{\partial \textbf{q}}.$

what if we’re interested in some non-specified point along link 3, though? Well, using SymPy we set make $\textbf{x} = [x_0, x_1, x_2, 1]$ instead of $\textbf{x} = [0, 0, 0, 1]$ (recall the 1 at the end in these vectors is just to make the math work), and make the Jacobian function SymPy generates for us dependent on both $\textbf{q}$ and $\textbf{x}$, rather than just $\textbf{q}$. In code this looks like:

Torg3 = self._calc_T(name="3")
# transform x into world coordinates
Torg3x = sp.simplify(Torg3 * sp.Matrix(x))
J3_func = sp.lambdify(q + x, Torg3)


Now it’s possible to calculate the Jacobian for any point along link 3 just by changing the parameters that we pass into J3_func! Most excellent.

We are getting closer.

NOTE: This parameterization can significantly increase the build time of the function, it took my laptop about 4 hours. To decrease build time you can try commenting out the simplify calls from the code, which might slow down run-time a bit but significantly drops the generation time.

Where is the closest point in that link’s reference frame?

A sneaky problem comes up when calculating the closest point of each arm segment to the object: We’ve calculated the closest point of each arm segment in the origin’s frame of reference, and we need thew relative to each link’s own frame of reference. Fortunately, all we have to do is calculate the inverse transform for the link of interest. For example, the inverse transform of $^3_\textrm{org}\textbf{T}$ transforms a point from the origin’s frame of reference to the reference frame of the 3rd joint.

I go over how to calculate the inverse transform at the end of my post on forward transformation matrices, but to save you from having to go back and look through that, here’s the code to do it:

Torg3 = self._calc_T(name="3")
rotation_inv = Torg3[:3, :3].T
translation_inv = -rotation_inv * Torg3[:3, 3]
Torg3_inv = rotation_inv.row_join(translation_inv).col_join(
sp.Matrix([[0, 0, 0, 1]]))


And now to find the closest point in the coordinates of reference frame 3 we simply

x = np.dot(Torg3_inv, closest)


This x value is what we’re going to plug in as parameters to our J3_func above to find the Jacobian for the closest point on link 3.

Calculate the operational space inertia matrix for the closest point

OK. With the Jacobian for the point of interest we are now able to calculate the operational space inertia matrix. This code I’ve explicitly worked through before, and I’ll show it in the full code below, so I won’t go over it again here.

The whole implementation

You can run an example of all this code controlling the UR5 arm to avoid obstacles in VREP using this code up on my GitHub. The specific code added to implement obstacle avoidance looks like this:

# find the closest point of each link to the obstacle
for ii in range(robot_config.num_joints):
# get the start and end-points of the arm segment
p1 = robot_config.Tx('joint%i' % ii, q=q)
if ii == robot_config.num_joints - 1:
p2 = robot_config.Tx('EE', q=q)
else:
p2 = robot_config.Tx('joint%i' % (ii + 1), q=q)

# calculate minimum distance from arm segment to obstacle
# the vector of our line
vec_line = p2 - p1
# the vector from the obstacle to the first line point
vec_ob_line = v - p1
# calculate the projection normalized by length of arm segment
projection = (np.dot(vec_ob_line, vec_line) /
np.sum((vec_line)**2))
if projection < 0:
# then closest point is the start of the segment
closest = p1
elif projection > 1:
# then closest point is the end of the segment
closest = p2
else:
closest = p1 + projection * vec_line
# calculate distance from obstacle vertex to the closest point
dist = np.sqrt(np.sum((v - closest)**2))
# account for size of obstacle

if rho < threshold:
eta = .02
drhodx = (v - closest) / rho
Fpsp = (eta * (1.0/rho - 1.0/threshold) *
1.0/rho**2 * drhodx)

# get offset of closest point from link's reference frame
T_inv = robot_config.T_inv('link%i' % ii, q=q)
m = np.dot(T_inv, np.hstack([closest, [1]]))[:-1]
# calculate the Jacobian for this point
Jpsp = robot_config.J('link%i' % ii, x=m, q=q)[:3]

# calculate the inertia matrix for the
# point subjected to the potential space
Mxpsp_inv = np.dot(Jpsp,
np.dot(np.linalg.pinv(Mq), Jpsp.T))
svd_u, svd_s, svd_v = np.linalg.svd(Mxpsp_inv)
# cut off singular values that could cause problems
singularity_thresh = .00025
for ii in range(len(svd_s)):
svd_s[ii] = 0 if svd_s[ii] < singularity_thresh else \
1./float(svd_s[ii])
# numpy returns U,S,V.T, so have to transpose both here
Mxpsp = np.dot(svd_v.T, np.dot(np.diag(svd_s), svd_u.T))

u_psp = -np.dot(Jpsp.T, np.dot(Mxpsp, Fpsp))
if rho < .01:
u = u_psp
else:
u += u_psp


The one thing in this code I didn’t talk about is that you can see that if rho < .01 then I set u = u_psp instead of just adding u_psp to u. What this does is basically add in a fail safe take over of the robotic control saying that “if we are about to hit the obstacle forget about everything else and get out of the way!”.

Results

And that’s it! I really enjoy how this looks when it’s running, it’s a really effective algorithm. Let’s look at some samples of it in action.

First, in a 2D environment, where it’s real easy to move around the obstacle and see how it changes in response to the new obstacle position. The red circle is the target and the blue circle is the obstacle:

And in 3D in VREP, running the code example that I’ve put up on my GitHub implementing this. The example of it running without obstacle avoidance code is on the left, and running with obstacle avoidance is on the right. It’s kind of hard to see but on the left the robot moves through the far side of the obstacle (the gold sphere) on its way to the target (the red sphere):

And one last example, the arm dodging a moving obstacle on its way to the target.

The implementation is a ton of fun to play around with. It’s a really nifty algorithm, that works quite well, and I haven’t found many alternatives in papers that don’t go into path planning (if you know of some and can share that’d be great!). This post was a bit of a journey, but hopefully you found it useful! I continue to find it impressive how many different neat features like this can come about once you have the operational space control framework in place.

## Velocity limiting in operational space control

Recently, I was reading through an older paper on effective operational space control, talking specifically point to point control in operational space. The paper mentioned that even if you have a perfect model of the system, you’re going to run into trouble if you use just a basic PD formula to define your control signal in operational space:

$u_x = k_p (\textbf{x}^* - \textbf{x}) - k_v \dot{\textbf{x}},$

where $\textbf{x}$ and $\dot{\textbf{x}}$ are the system position and velocity in operational space, $\textbf{x}^*$ is the target position, and $k_p$ and $k_v$ are gains.

If you define your operational space control signal like this, and then translate this signal into joint torques (using, for example, methods discussed in other posts), you’re going to see a very non-straight trajectory emerge in larger movements as a result of “actuator saturation, and bandwidth and velocity limitations”. In the example of a standard robot, you might run into issues with your motors not being able to actually generate the torques that have been specified, the frequency of control and feedback might not be sufficient, and you could hit hard constraints on system velocity. The solution to this problem was presented in this 1987 paper by Dr. Oussama Khatib, and is pretty slick and very useful, so I thought I’d write it up here for any other unfortunate souls wandering around in ignorance. First though, here’s what it looks like to move large point to point distances without velocity limiting:

As you can see, the system isn’t moving in a straight line, which can be very aggravating if you’ve worked and reworked out the equations and double checked all your parameters, etc etc. A few things, first, when working with simulations it’s easy to forget how ridiculously fast this actually would be in real-time. Even though it takes a minute to simulate the above movement, in real-time, is happening over the course of 200ms. Taking that into account, this is pretty good. Also, there’s an obvious solution here, slow down your movement. The source of this problem is largely that all of the motors are not able to apply the torques specified and move at the required speed. Some of the motors have a lot less mass to throw around and will be able to move at the specified torques, but not all. Hence the not straight trajectory.

You can of course drop the gains on your PD signal, but that’s not really a great methodical solution. So, what can we do?

Well, if we rearrange the PD control signal specified above into

$u_x = k_v (\dot{\textbf{x}}^* - \dot{\textbf{x}}),$

where $\dot{\textbf{x}}^*$ is the desired velocity, we see that this signal can be interpreted as a pure velocity servo-control signal, with velocity gain $k_v$ and a desired velocity

$\dot{\textbf{x}}^* = \frac{k_p}{k_v}(\textbf{x}^* - \textbf{x})$.

When things are in this form, it becomes a bit more clear what we have to do: limit the desired velocity to be at most some specified maximum velocity of the end-effector, $V_\textrm{max}$. This value should be low enough that the transformation into joint torques doesn’t result in anything larger than the actuators can generate.

Taking $V_\textrm{max}$, what we want is to clip the magnitude of the control signal to be $V_\textrm{max}$ if it’s ever larger (in positive or negative directions), and to be equal to $\frac{kp}{kv}(\textbf{x}^* - \textbf{x})$ otherwise. The math for this works out such that we can accomplish this with a control signal of the form:

$\textbf{u}_\textbf{x} = -k_v (\dot{\textbf{x}} + \textrm{sat}\left(\frac{V_\textrm{max}}{\lambda |\tilde{\textbf{x}}|} \right) \lambda \tilde{\textbf{x}})$,

where $\lambda = \frac{k_p}{k_v}$ , $\tilde{\textbf{x}} = \textbf{x} - \textbf{x}^*$, and $\textrm{sat}$ is the saturation function, such that

$\textrm{sat}(y) = \left\{ \begin{array}{cc} |y| \leq 1 & \Rightarrow y \\ |y| > 1 & \Rightarrow 1 \end{array} \right.$

where $|y|$ is the absolute value of $y$, and is applied element wise to the vector $\tilde{\textbf{x}}$ in the control signal.

As a result of using this saturation function, the control signal behaves differently depending on whether or not $\dot{\textbf{x}}^* > V_\textrm{max}$:

$\textbf{u}_\textbf{x} = \left\{ \begin{array}{cc} \dot{\textbf{x}}^* \geq V_\textrm{max} & \Rightarrow -k_v (\dot{\textbf{x}} + V_\textbf{max} \textrm{sgn}(\tilde{\textbf{x}})) \\ \dot{\textbf{x}}^* < V_\textrm{max} & \Rightarrow -k_v \dot{\textbf{x}} + k_p \tilde{\textbf{x}} \end{array} \right.$

where $\textrm{sgn}(y)$ is a function that returns -1 if $y < 0$ and 1 if $y \geq 0$, and is again applied element-wise to vectors. Note that the control signal in the second condition is equivalent to our original PD control signal $k_p(\textbf{x}^* - \textbf{x}) - k_v \dot{\textbf{x}}$. If you’re wondering about negative signs make sure you note that $\tilde{\textbf{x}} = \textbf{x} - \textbf{x}^*$ and not $\textbf{x}^* - \textbf{x}$, as you might assume.

So now the control signal is behaving exactly as desired! Moves the system towards the target, but no faster than the specified maximum velocity. Now our trajectories look like this:

So those are starting to look a lot better! The first thing you’ll notice is that this is a fair bit slower of a movement. Well, actually, it’s waaaayyyy slower because the playback speed here is 4x faster than in that first animation, and this is a movement over 2s. Which has pros and cons, con: it’s slower, pro: it’s straighter, and you’re less likely to be murdered by it. When you move from simulations to real systems that latter point really moves way up the priority list.

Second thing to notice, the system seems to be minimising the error along one dimension, and then along the next, and then arrives at the target. What’s going on?  Because the error along each of the $(x,y,z)$ dimensions isn’t the same, when speed gets clipped along one of the dimensions you’re no longer going to be moving in a straight line directly to the target. To address this, we’re going to add a scaling term whenever clipping happens, such that you reduce the speed you move along each dimension by the same ratio, so that you’re still moving in a straight line.

It’s a liiiiittle bit more complicated than that, but not much. First, we’ll calculate the values being passed in to the saturation function for each $(x,y,z)$ dimension. We’ll then check to see if any of them are going to get clipped, and if there’s more than one that saturates we’ll find the one that is affected the most. After we’ve identified which dimension it is, we go through and calculate what the control signal would have been without velocity limiting, and what it will be now with velocity limiting. This scaling term tells us how much the control signal was reduced, and we can then use it to reduce the control signals of the other dimensions by the same amount. These other dimensions might still saturate, though, so we have to recalculate the saturation function for them once they’ve been scaled. Here’s what this all looks like in code:

# implement velocity limiting
lamb = kp / kv
x_tilde = xyz - target_xyz
sat = vmax / (lamb * np.abs(x_tilde))
scale = np.ones(3)
if np.any(sat < 1):
index = np.argmin(sat)
unclipped = kp * x_tilde[index]
clipped = kv * vmax * np.sign(x_tilde[index])
scale = np.ones(3) * clipped / unclipped
scale[index] = 1
u_xyz = -kv * (dx + np.clip(sat / scale, 0, 1) *
scale * lamb * x_tilde)


And now, finally, we start getting the trajectories that we’ve been wanting the whole time:

And finally we can rest easy, knowing that our robot is moving at a reasonable speed along a direct path to its goals. Wherever you’d like to use this neato ‘ish you should be able to just paste in the above code, define your vmax, kp, and kv values and be good to go!

## Using SymPy’s lambdify for generating transform matrices and Jacobians

I’ve been working in VREP with some of their different robot models and testing out force control, and one of the things that becomes pretty important for efficient workflow is to have a streamlined method for setting up the transform matrices and calculating the Jacobians for different robots. You do not want to be working these all out by hand and then writing them in yourself.

A solution that’s been working well for me (and is fully implemented in Python) is to use SymPy to set up the basic transform matrices, from each joint to the next, and then use it’s derivative function to calculate the Jacobian. Once this is calculated you can then use SymPy’s lambdify function to parameterize this, and off you go! In this post I’m going to work through an example for controlling VREP’s UR5 arm using force control. And if you’re just looking for code examples, you can find it all up on my GitHub.

Edit: Updated the code to be much nicer, added saving of calculated functions, and a null space controller.
Edit 2: Removed the simplify calls from the code, not worth the generation time increase (if execution is a great concern can generate the functions using cython).

Setting up the transform matrices

This is the part of this process that is unique to each arm. The goal is to set up a system so that you can specify your transforms for each joint (and to each centre-of-mass (COM) too, of course) and then be on your way. So here’s the UR5, cute thing that it is:

For the UR5, there are 6 joints, and we’re going to be specifying 9 transform matrices: 6 joints and the COM for three arm segments (the remaining arm segment COMs are centered at their respective joint’s reference frame). The joints are all rotary, with 0 and 4 rotating around the z-axis, and the rest all rotating around x.

So first, we’re going to create a class called robot_config. Then, to create our transform matrices in SymPy first we need to set up the variables we’re going to be using:

# set up our joint angle symbols (6th angle doesn't affect any kinematics)
self.q = [sp.Symbol('q%i'%ii) for ii in range(6)]
# segment lengths associated with each joint
self.L = np.array([0.0935, .13453, .4251, .12, .3921, .0935, .0935, .0935])


where self.q is an array storing all our joint angle symbols, and self.L is an array of all of the offsets for each joint and arm segment lengths.

Using these to create the transform matrices for the joints, we get a set up that looks like this:

# transform matrix from origin to joint 0 reference frame
self.T0org = sp.Matrix([[sp.cos(self.q[0]), -sp.sin(self.q[0]), 0, 0],
[sp.sin(self.q[0]), sp.cos(self.q[0]), 0, 0],
[0, 0, 1, self.L[0]],
[0, 0, 0, 1]])

# transform matrix from joint 0 to joint 1 reference frame
self.T10 = sp.Matrix([[1, 0, 0, -L[1]],
[0, sp.cos(-self.q[1] + sp.pi/2),
-sp.sin(-self.q[1] + sp.pi/2), 0],
[0, sp.sin(-self.q[1] + sp.pi/2),
sp.cos(-self.q[1] + sp.pi/2), 0],
[0, 0, 0, 1]])

# transform matrix from joint 1 to joint 2 reference frame
self.T21 = sp.Matrix([[1, 0, 0, 0],
[0, sp.cos(-self.q[2]),
-sp.sin(-self.q[2]), self.L[2]],
[0, sp.sin(-self.q[2]),
sp.cos(-self.q[2]), 0],
[0, 0, 0, 1]])

# transform matrix from joint 2 to joint 3
self.T32 = sp.Matrix([[1, 0, 0, L[3]],
[0, sp.cos(-self.q[3] - sp.pi/2),
-sp.sin(-self.q[3] - sp.pi/2), self.L[4]],
[0, sp.sin(-self.q[3] - sp.pi/2),
sp.cos(-self.q[3] - sp.pi/2), 0],
[0, 0, 0, 1]])

# transform matrix from joint 3 to joint 4
self.T43 = sp.Matrix([[sp.sin(-self.q[4] - sp.pi/2),
sp.cos(-self.q[4] - sp.pi/2), 0, -self.L[5]],
[sp.cos(-self.q[4] - sp.pi/2),
-sp.sin(-self.q[4] - sp.pi/2), 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])

# transform matrix from joint 4 to joint 5
self.T54 = sp.Matrix([[1, 0, 0, 0],
[0, sp.cos(self.q[5]), -sp.sin(self.q[5]), 0],
[0, sp.sin(self.q[5]), sp.cos(self.q[5]), self.L[6]],
[0, 0, 0, 1]])

# transform matrix from joint 5 to end-effector
self.TEE5 = sp.Matrix([[1, 0, 0, self.L[7]],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])


You can see a bunch of offsetting of the joint angles by -sp.pi/2 and this is to account for the expected 0 angle (straight along the reference frame’s x-axis) at those joints being different than the 0 angle defined in the VREP simulation (at a 90 degrees from the x-axis). You can find these by either looking at and finding the joints’ 0 position in VREP or by trial-and-error empirical analysis.

Once you have your transforms, then you have to specify how to move from the origin to each reference frame of interest (the joints and COMs). For that, I’ve just set up a simple function with a switch statement:

# point of interest in the reference frame (right at the origin)
self.x = sp.Matrix([0,0,0,1])

def _calc_T(self, name, lambdify=True):
""" Uses Sympy to generate the transform for a joint or link

name string: name of the joint or link, or end-effector
lambdify boolean: if True returns a function to calculate
the transform. If False returns the Sympy
matrix
"""

# check to see if we have our transformation saved in file
if os.path.isfile('%s/%s.T' % (self.config_folder, name)):
Tx = cloudpickle.load(open('%s/%s.T' % (self.config_folder, name),
'rb'))
else:
if name == 'joint0' or name == 'link0':
T = self.T0org
elif name == 'joint1' or name == 'link1':
T = self.T0org * self.T10
elif name == 'joint2':
T = self.T0org * self.T10 * self.T21
T = self.T0org * self.T10 * self.Tl21
elif name == 'joint3':
T = self.T0org * self.T10 * self.T21 * self.T32
T = self.T0org * self.T10 * self.T21 * self.Tl32
elif name == 'joint4' or name == 'link4':
T = self.T0org * self.T10 * self.T21 * self.T32 * self.T43
elif name == 'joint5' or name == 'link5':
T = self.T0org * self.T10 * self.T21 * self.T32 * self.T43 * \
self.T54
elif name == 'link6' or name == 'EE':
T = self.T0org * self.T10 * self.T21 * self.T32 * self.T43 * \
self.T54 * self.TEE5
Tx = T * self.x  # to convert from transform matrix to (x,y,z)

# save to file
cloudpickle.dump(Tx, open('%s/%s.T' % (self.config_folder, name),
'wb'))

if lambdify is False:
return Tx
return sp.lambdify(self.q, Tx)


So the first part is pretty straight forward, create the transform matrix, and then at the end to get the (x,y,z) position we just multiply by a vector we created that represents a point at the origin of the last reference frame. Some of the transform matrices (the ones to the arm segments) I didn’t specify above just to cut down on space.

The second part is where we use this awesome lambify function, which lets us turn the matrix we’ve defined into a function, so that we can pass in joint angles and get back out the resulting (x,y,z) position. There’s also the option to get the straight up SymPy matrix return, in case you need the symbolic form (which we will!).

NOTE: You can also see that there’s a check built in to look for saved files, and to just load those saved files instead of recalculating things if they’re available. This is because calculating some of these matrices and their derivatives takes a long, long time. I used the cloudpickle module to do this because it’s able to easily handle saving a whole bunch of weird things that makes normal pickle sour.

Calculating the Jacobian

So now that we’re able to quickly generate the transform matrix for each point of interest on the UR5, we simply take the derivative of the equation for each (x,y,z) coordinate with respect to each joint angle to generate our Jacobian.

def _calc_J(self, name, lambdify=True):
""" Uses Sympy to generate the Jacobian for a joint or link

name string: name of the joint or link, or end-effector
lambdify boolean: if True returns a function to calculate
the Jacobian. If False returns the Sympy
matrix
"""

# check to see if we have our Jacobian saved in file
if os.path.isfile('%s/%s.J' % (self.config_folder, name)):
(self.config_folder, name), 'rb'))
else:
Tx = self._calc_T(name, lambdify=False)
J = []
# calculate derivative of (x,y,z) wrt to each joint
for ii in range(self.num_joints):
J.append([])
J[ii].append(Tx[0].diff(self.q[ii]))  # dx/dq[ii]
J[ii].append(Tx[1].diff(self.q[ii]))  # dy/dq[ii]
J[ii].append(Tx[2].diff(self.q[ii]))  # dz/dq[ii]


Here we retrieve the Tx vector from our _calc_T function, and then calculate the derivatives. When calculating the Jacobian for the end-effector, this is all we need! Huzzah!

But to calculate the Jacobian for transforming the inertia matrices of each arm segment into joint space we’re going to need the orientation information added to our Jacobian as well. This we know ahead of time, for each joint it’s a 3D vector with a 1 on the axis being rotated around. So we can predefine this:

# orientation part of the Jacobian (compensating for orientations)
self.J_orientation = [
[0, 0, 1], # joint 0 rotates around z axis
[1, 0, 0], # joint 1 rotates around x axis
[1, 0, 0], # joint 2 rotates around x axis
[1, 0, 0], # joint 3 rotates around x axis
[0, 0, 1], # joint 4 rotates around z axis
[1, 0, 0]] # joint 5 rotates around x axis


And then we just fill in the Jacobians for each reference frame with self.J_orientation up to the last joint, and fill in the rest of the Jacobian with zeros. So e.g. when we’re calculating the Jacobian for the arm segment just past the second joint we’ll use the first two rows of self.J_orientation and the rest of the rows will be 0.

So this leads us to the second half of the _calc_J function:

        end_point = name.strip('link').strip('joint')
if end_point != 'EE':
end_point = min(int(end_point) + 1, self.num_joints)
# add on the orientation information up to the last joint
for ii in range(end_point):
J[ii] = J[ii] + self.J_orientation[ii]
# fill in the rest of the joints orientation info with 0
for ii in range(end_point, self.num_joints):
J[ii] = J[ii] + [0, 0, 0]

# save to file
cloudpickle.dump(J, open('%s/%s.J' %
(self.config_folder, name), 'wb'))

J = sp.Matrix(J).T  # correct the orientation of J
if lambdify is False:
return J
return sp.lambdify(self.q, J)


The orientation information is added in, we save the result to file, and a function that takes in the joint angles and outputs the Jacobian is created (unless lambdify == False in which case the SymPy symbolic form is returned.)

Then finally, two wrapper functions are added in to make creating / accessing these functions easier. First, define a couple of dictionaries

# create function dictionaries
self._T = {}  # for transform calculations
self._J = {}  # for Jacobian calculations


and then our wrapper functions look like this

def T(self, name, q):
""" Calculates the transform for a joint or link
name string: name of the joint or link, or end-effector
q np.array: joint angles
"""
# check for function in dictionary
if self._T.get(name, None) is None:
print('Generating transform function for %s'%name)
self._T[name] = self.calc_T(name)
return self._T[name](*q)[:-1].flatten()

def J(self, name, q):
""" Calculates the transform for a joint or link
name string: name of the joint or link, or end-effector
q np.array: joint angles
"""
# check for function in dictionary
if self._J.get(name, None) is None:
print('Generating Jacobian function for %s'%name)
self._J[name] = self.calc_J(name)
return np.array(self._J[name](*q)).T


So how you use this class (all of this is in a class) is to call these T and J functions with the current joint angles. They’ll check to see if the functions have already be created or stored in file, if they haven’t then the T and / or J functions will be created, then our wrappers do a bit of formatting to get them into the proper form (i.e. transposing or cropping), and return you your (x,y,z) or Jacobian!

NOTE: It’s a bit of a misnomer to have the function be called T and actually return to you Tx, but hey this is a blog. Lay off.

Calculating the inertia matrix in joint-space and force of gravity
Now, since we’re here we might as well also calculate the functions for our inertia matrix in joint space and the effect of gravity. So, define a couple more placeholders in our robot_config class to help us:

self._M = []  # placeholder for (x,y,z) inertia matrices
self._Mq = None  # placeholder for joint space inertia matrix function
self._Mq_g = None  # placeholder for joint space gravity term function


and then add in our inertia matrix information (defined in each link’s centre-of-mass (COM) reference frame)

# create the inertia matrices for each link of the ur5
self._M.append(np.diag([1.0, 1.0, 1.0,
self._M.append(np.diag([2.5, 2.5, 2.5,
self._M.append(np.diag([5.7, 5.7, 5.7,
self._M.append(np.diag([3.9, 3.9, 3.9,
self._M.append(np.diag([0.7, 0.7, 0.7,


and then using our equations for calculating the system’s inertia and gravity we create our _calc_Mq and _calc_Mq_g functions

def _calc_Mq(self, lambdify=True):
""" Uses Sympy to generate the inertia matrix in
joint space for the ur5

lambdify boolean: if True returns a function to calculate
the Jacobian. If False returns the Sympy
matrix
"""

# check to see if we have our inertia matrix saved in file
if os.path.isfile('%s/Mq' % self.config_folder):
Mq = cloudpickle.load(open('%s/Mq' % self.config_folder, 'rb'))
else:
# get the Jacobians for each link's COM
J = [self._calc_J('link%s' % ii, lambdify=False)

# transform each inertia matrix into joint space
# sum together the effects of arm segments' inertia on each motor
Mq = sp.zeros(self.num_joints)
Mq += J[ii].T * self._M[ii] * J[ii]

# save to file
cloudpickle.dump(Mq, open('%s/Mq' % self.config_folder, 'wb'))

if lambdify is False:
return Mq
return sp.lambdify(self.q, Mq)

def _calc_Mq_g(self, lambdify=True):
""" Uses Sympy to generate the force of gravity in
joint space for the ur5

lambdify boolean: if True returns a function to calculate
the Jacobian. If False returns the Sympy
matrix
"""

# check to see if we have our gravity term saved in file
if os.path.isfile('%s/Mq_g' % self.config_folder):
'rb'))
else:
# get the Jacobians for each link's COM
J = [self._calc_J('link%s' % ii, lambdify=False)

# transform each inertia matrix into joint space and
# sum together the effects of arm segments' inertia on each motor
Mq_g = sp.zeros(self.num_joints, 1)
for ii in range(self.num_joints):
Mq_g += J[ii].T * self._M[ii] * self.gravity

# save to file
cloudpickle.dump(Mq_g, open('%s/Mq_g' % self.config_folder,
'wb'))

if lambdify is False:
return Mq_g
return sp.lambdify(self.q, Mq_g)


and wrapper functions

def Mq(self, q):
""" Calculates the joint space inertia matrix for the ur5

q np.array: joint angles
"""
# check for function in dictionary
if self._Mq is None:
print('Generating inertia matrix function')
self._Mq = self._calc_Mq()
return np.array(self._Mq(*q))

def Mq_g(self, q):
""" Calculates the force of gravity in joint space for the ur5

q np.array: joint angles
"""
# check for function in dictionary
if self._Mq_g is None:
print('Generating gravity effects function')
self._Mq_g = self._calc_Mq_g()
return np.array(self._Mq_g(*q)).flatten()


and we’re all set!

Putting it all together

Now we have nice clean code to generate everything we need for our controller. Using the controller developed in this post as a base, we can replace those calculations with the following nice compact code (which also includes a secondary null-space controller to keep the arm near resting joint angles):

# calculate position of the end-effector
# derived in the ur5 calc_TnJ class
xyz = robot_config.T('EE', q)
# calculate the Jacobian for the end effector
JEE = robot_config.J('EE', q)
# calculate the inertia matrix in joint space
Mq = robot_config.Mq(q)
# calculate the effect of gravity in joint space
Mq_g = robot_config.Mq_g(q)

# convert the mass compensation into end effector space
Mx_inv = np.dot(JEE, np.dot(np.linalg.inv(Mq), JEE.T))
svd_u, svd_s, svd_v = np.linalg.svd(Mx_inv)
# cut off any singular values that could cause control problems
singularity_thresh = .00025
for i in range(len(svd_s)):
svd_s[i] = 0 if svd_s[i] < singularity_thresh else \
1./float(svd_s[i])
# numpy returns U,S,V.T, so have to transpose both here
Mx = np.dot(svd_v.T, np.dot(np.diag(svd_s), svd_u.T))

kp = 100
kv = np.sqrt(kp)
# calculate desired force in (x,y,z) space
u_xyz = np.dot(Mx, target_xyz - xyz)
# transform into joint space, add vel and gravity compensation
u = (kp * np.dot(JEE.T, u_xyz) - np.dot(Mq, kv * dq) - Mq_g)

# calculate our secondary control signal
# calculated desired joint angle acceleration
q_des = (((robot_config.rest_angles - q) + np.pi) %
(np.pi*2) - np.pi)
u_null = np.dot(Mq, (kp * q_des - kv * dq))

# calculate the null space filter
Jdyn_inv = np.dot(Mx, np.dot(JEE, np.linalg.inv(Mq)))
null_filter = (np.eye(robot_config.num_joints) -
np.dot(JEE.T, Jdyn_inv))
u_null_filtered = np.dot(null_filter, u_null)

u += u_null_filtered


And there you go!

You can see all of this code up on my GitHub, along a full example controlling a UR5 VREP model though reaching to a series of targets. It looks something pretty much like this (slightly better because this gif was made before adding in the null space controller):

Overhead of using lambdify instead of hard-coding

This was a big question that I had, because when I’m running simulations the time step is on the order of a few milliseconds, with the controller code called at every time step. So I reaaaally can’t afford a crazy overhead for using lambdify.

To test this, I used the handy Python timeit, which requires a bit awkward setup, but quite nicely calls the function a whole bunch of times (1,000,000 by default) and accounts for various other things going on that could affect the execution time.

I tested two sample functions, one simpler than the other. Here’s the code for setting up and testing the first function:

import timeit
import seaborn

# Test 1 ----------------------------------------------------------------------
print('\nTest function 1: ')
time_sympy1 = timeit.timeit(
stmt = 'f(np.random.random(), np.random.random())',
setup = 'import numpy as np;\
import sympy as sp;\
q0 = sp.Symbol("q0");\
l0 = sp.Symbol("l0");\
a = sp.cos(q0) * l0;\
f = sp.lambdify((q0, l0), a, "numpy")')
print('Sympy lambdify function 1 time: ', time_sympy1)

time_hardcoded1 = timeit.timeit(
stmt = 'np.cos(np.random.random())*np.random.random()',
setup = 'import numpy as np')
print('Hard coded function 1 time: ', time_hardcoded1)


Pretty simple, a bit of a pain in the sympy setup, but other than that not bad at all. The second function I tested was just a random collection of cos and sin calls that resemble what gets computed in a Jacobian:

l1*np.sin(q0 - l0*np.sin(q1)*np.cos(q2) - l2*np.sin(q2) - l0*np.sin(q1) + q0*l0)*np.cos(q0) + l2*np.sin(q0)


And here’s the results:

So it’s slower for sure, but again this is the difference in time after 1,000,000 function calls, so until some big optimization needs to be done using the SymPy lambdify function is definitely worth the slight gain in execution time for the insane convenience.

The full code for the timing tests here are also up on my GitHub.

## Dynamic movement primitives part 4: Avoiding obstacles – update

Edit: Previously I posted this blog post on incorporating obstacle avoidance, but after a recent comment on the code I started going through it again and found a whole bunch of issues. Enough so that I’ve recoded things and I’m going to repost it now with working examples and a description of the changes I made to get it going. The edited sections will be separated out with these nice horizontal lines. If you’re just looking for the example code, you can find it up on my pydmps repo, here.

Alright. Previously I’d mentioned in one of these posts that DMPs are awesome for generalization and extension, and one of the ways that they can be extended is by incorporating obstacle avoidance dynamics. Recently I wanted to implement these dynamics, and after a bit of finagling I got it working, and so that’s going to be the subject of this post.

There are a few papers that talk about this, but the one we’re going to use is Biologically-inspired dynamical systems for movement generation: automatic real-time goal adaptation and obstacle avoidance by Hoffmann and others from Stefan Schaal’s lab. This is actually the second paper talking about obstacle avoidance and DMPs, and this is a good chance to stress one of the most important rules of implementing algorithms discussed in papers: collect at least 2-3 papers detailing the algorithm (if possible) before attempting to implement it. There are several reasons for this, the first and most important is that there are likely some typos in the equations of one paper, by comparing across a few papers it’s easier to identify trickier parts, after which thinking through what the correct form should be is usually straightforward. Secondly, often equations are updated with simplified notation or dynamics in later papers, and you can save yourself a lot of headaches in trying to understand them just by reading a later iteration. I recklessly disregarded this advice and started implementation using a single, earlier paper which had a few key typos in the equations and spent a lot of time tracking down the problem. This is just a peril inherent in any paper that doesn’t provide tested code, which is almost all, sadface.

OK, now on to the dynamics. Fortunately, I can just reference the previous posts on DMPs here and don’t have to spend any time discussing how we arrive at the DMP dynamics (for a 2D system):

$\ddot{\textbf{y}} = \alpha_y (\beta_y( \textbf{g} - \textbf{y}) - \dot{\textbf{y}}) + \textbf{f},$

where $\alpha_y$ and $\beta_y$ are gain terms, $\textbf{g}$ is the goal state, $\textbf{y}$ is the system state, $\dot{\textbf{y}}$ is the system velocity, and $\textbf{f}$ is the forcing function.
As mentioned, DMPs are awesome because now to add obstacle avoidance all we have to do is add another term

$\ddot{\textbf{y}} = \alpha_y (\beta_y( \textbf{g} - \textbf{y}) - \dot{\textbf{y}}) + \textbf{f} + \textbf{p}(\textbf{y}, \dot{\textbf{y}}),$

where $\textbf{p}(\textbf{y}, \dot{\textbf{y}})$ implements the obstacle avoidance dynamics, and is a function of the DMP state and velocity. Now then, the question is what are these dynamics exactly?

Obstacle avoidance dynamics

It turns out that there is a paper by Fajen and Warren that details an algorithm that mimics human obstacle avoidance. The idea is that you calculate the angle between your current velocity and the direction to the obstacle, and then turn away from the obstacle. The angle between current velocity and direction to the obstacle is referred to as the steering angle, denoted $\varphi$, here’s a picture of it:

So, given some $\varphi$ value, we want to specify how much to change our steering direction, $\dot{\varphi}$, as in the figure below:

If we’re on track to hit the object (i.e. $\varphi$ is near 0) then we steer away hard, and then make your change in direction less and less as the angle between your heading (velocity) and the object is larger and larger. Formally, define $\dot{\varphi}$ as

$\dot{\varphi} = \gamma \;\varphi \;\textrm{exp}(-\beta | \varphi |),$

where $\gamma$ and $\beta$ are constants, which are specified as $1000$ and $\frac{20}{\pi}$ in the paper, respectively.

Edit: OK this all sounds great, but quickly you run into issues here. The first is how do we calculate $\varphi$? In the paper I was going off of they used a dot product between the $\dot{\textbf{y}}$ vector and the $\textbf{o} - \textbf{y}$ vector to get the cosine of the angle, and then use np.arccos to get the angle from that. There is a big problem with this here, however, that’s kind of subtle: You will never get a negative angle when you calculate this, which, of course. That’s not how np.arccos works, but it is still what we need so we will be able to tell if we should be turning left or right to avoid this object!

So we need to use a different way of calculating the angle, one that tells us if the object is in a + or – angle relative to the way we’re headed. To calculate an angle that will give us + or -, we’re going to use the np.arctan2 function.

We want to center things around the way we’re headed, so first we calculate the angle of the direction vector, $\dot{\textbf{y}}$. Then we create a rotation vector, R_dy to rotate the vector describing the direction of the object. This shifts things around so that if we’re moving straight towards the object it’s at 0 degrees, if we’re headed directly away from it, it’s at 180 degrees, etc. Once we have that vector, nooowwww we can find the angle of the direction of the object and that’s what we’re going to use for phi. Huzzah!

# get the angle we're heading in
phi_dy = -np.arctan2(dy[1], dy[0])
R_dy = np.array([[np.cos(phi_dy), -np.sin(phi_dy)],
[np.sin(phi_dy), np.cos(phi_dy)]])
# calculate vector to object relative to body
obj_vec = obstacle - y
# rotate it by the direction we're going
obj_vec = np.dot(R_dy, obj_vec)
# calculate the angle of obj relative to the direction we're going
phi = np.arctan2(obj_vec[1], obj_vec[0])


This $\dot{\varphi}$ term can be thought of as a weighting, telling us how much we need to rotate based on how close we are to running into the object. To calculate how we should rotate we’re going to calculate the angle orthonormal to our current velocity, then weight it by the distance between the object and our current state on each axis. Formally, this is written:

$\textbf{R} \; \dot{\textbf{y}},$

where $\textbf{R}$ is the axis $(\textbf{o} - \textbf{y}) \times \dot{\textbf{y}}$ rotated 90 degrees (the $\times$ denoting outer product here). The way I’ve been thinking about this is basically taking your velocity vector, $\dot{\textbf{y}}$, and rotating it 90 degrees. Then we use this rotated vector as a row vector, and weight the top row by the distance between the object and the system along the $x$ axis, and the bottom row by the difference along the $\textbf{y}$ axis. So in the end we’re calculating the angle that rotates our velocity vector 90 degrees, weighted by distance to the object along each axis.

So that whole thing takes into account absolute distance to object along each axis, but that’s not quite enough. We also need to throw in $\dot{\varphi}$, which looks at the current angle. What this does is basically look at ‘hey are we going to hit this object?’, if you are on course then make a big turn and if you’re not then turn less or not at all. Phew.

OK so all in all this whole term is written out

$\textbf{p}(\textbf{y}, \dot{\textbf{y}}) = \textbf{R} \; \dot{\textbf{y}} \; \dot{\varphi},$

and that’s what we add in to the system acceleration. And now our DMP can avoid obstacles! How cool is that?

Super compact, straight-forward to add, here’s the code:

Edit: OK, so not suuuper compact. I’ve also added in another couple checks. The big one is “Is this obstacle between us and the target or not?”. So I calculate the Euclidean distance to the goal and the obstacle, and if the obstacle is further away then the goal, set the avoidance signal to zero (performed at the end of the if)! This took care of a few weird errors where you would get a big deviation in the trajectory if it saw an obstacle past the goal, because it was planning on avoiding it, but then was pulled back in to the goal before the obstacle anyways so it was a pointless exercise. The other check added in is just to make sure you only add in obstacle avoidance if the system is actually moving. Finally, I also changed the $\gamma = 100$ instead of $1000$.

def avoid_obstacles(y, dy, goal):
p = np.zeros(2)

for obstacle in obstacles:
# based on (Hoffmann, 2009)

# if we're moving
if np.linalg.norm(dy) > 1e-5:

# get the angle we're heading in
phi_dy = -np.arctan2(dy[1], dy[0])
R_dy = np.array([[np.cos(phi_dy), -np.sin(phi_dy)],
[np.sin(phi_dy), np.cos(phi_dy)]])
# calculate vector to object relative to body
obj_vec = obstacle - y
# rotate it by the direction we're going
obj_vec = np.dot(R_dy, obj_vec)
# calculate the angle of obj relative to the direction we're going
phi = np.arctan2(obj_vec[1], obj_vec[0])

dphi = gamma * phi * np.exp(-beta * abs(phi))
R = np.dot(R_halfpi, np.outer(obstacle - y, dy))
pval = -np.nan_to_num(np.dot(R, dy) * dphi)

# check to see if the distance to the obstacle is further than
# the distance to the target, if it is, ignore the obstacle
if np.linalg.norm(obj_vec) > np.linalg.norm(goal - y):
pval = 0

p += pval
return p


And that’s it! Just add this method in to your DMP system and call avoid_obstacles at every timestep, and add it in to your DMP acceleration.

You hopefully noticed in the code that this is set up for multiple obstacles, and that all that that entailed was simply adding the p value generated by each individual obstacle. It’s super easy! Here’s a very basic graphic showing how the DMP system can avoid obstacles:

So here there’s just a basic attractor system (DMP without a forcing function) trying to move from the center position to 8 targets around the unit circle (which are highlighted in red), and there are 4 obstacles that I’ve thrown onto the field (black x’s). As you can see, the system successfully steers way clear of the obstacles while moving towards the target!

We must all use this power wisely.

Edit: Making the power yours is now easier than ever! You can check out this code at my pydmps GitHub repo. Clone the repo and after you python setup.py develop, change directories into the examples folder and run the avoid_obstacles.py file. It will randomly generate 5 targets in the environment and perform 20 movements, giving you something looking like this:

## Using VREP for simulation of force-controlled models

I’ve been playing around a bit with different simulators, and one that we’re a big fan of in the lab is VREP. It’s free for academics and you can talk to them about licences if you’re looking for commercial use. I haven’t actually had much experience with it before myself, so I decided to make a simple force controlled arm model to get experience using it. All in all, there were only a few weird API things that I had to get through, and once you have them down it’s pretty straight forward. This post is going to be about the steps that I needed to take to get things all set up. For a more general start-up on VREP check out All the code in this post and the model I use can be found up on my GitHub.

Getting the right files where you need them

As discussed in the remote API overview, you’ll need three files in whatever folder you want to run your Python script from to be able to hook into VREP remotely:

• remoteApi.dll, remoteApi.dylib or remoteApi.so (depending on what OS you’re using)
• vrep.py
• vrepConstants.py

You can find these files inside your VREP_HOME/programming/remoteApiBindings/python/python and VREP_HOME/programming/remoteApiBindings/lib/lib folders. Make sure that these files are in whatever folder you’re running your Python scripts from.

The model

It’s easy to create a new model to mess around with in VREP, so that’s the route I went, rather than importing one of their pre-made models and having some sneaky parameter setting cause me a bunch of grief. You can just right click->add then go at it. There are a bunch of tutorials so I’m not going to go into detail here. The main things are:

• Make sure joints are in ‘Torque/force’ mode.
• Make sure that joint ‘Motor enabled’ property is checked. The motor enabled property is in the ‘Show dynamic properties dialogue’ menu, which you find when you double click on the joint in the Scene Hierarchy.
• Know the names of the joints as shown in the Scene Hierarchy.

So here’s a picture:

where you can see the names of the objects in the model highlighted in red, the Torque/force selection highlighted in blue, and the Motor enabled option highlighted in green. And of course my beautiful arm model in the background.

Setting up the continuous server

The goal is to connect VREP to Python so that we can send torques to the arm from our Python script and get the feedback necessary to calculate those torques. There are a few ways to set up a remote connection in VREP.

The basic one is they have you add a child script in VREP and attach it to an object in the world that sets up a port and then you hit go on your simulation and can then run your Python script to connect in. This gets old real fast. Fortunately, it’s easy to automate everything from Python so that you can connect in, start the simulation, run it for however long, and then stop the simulation without having to click back and forth.

The first step is to make sure that your remoteApiConnections.txt file in you VREP home directory is set up properly. A continuous server should be set up by default, but doesn’t hurt to double check. And you can take the chance to turn on debugging, which can be pretty helpful. So open up that file and make sure it looks like this:

portIndex1_port                 = 19997
portIndex1_debug                = true
portIndex1_syncSimTrigger       = true


Once that’s set up, when VREP starts we can connect in from Python. In our Python script, first we’ll close any open connections that might be around, and then we’ll set up a new connection in:

import vrep

# close any open connections
vrep.simxFinish(-1)
# Connect to the V-REP continuous server
clientID = vrep.simxStart('127.0.0.1', 19997, True, True, 500, 5)

if clientID != -1: # if we connected successfully
print ('Connected to remote API server')


So once the connection is made, we check the clientID value to make sure that it didn’t fail, and then we carry on with our script.

Synchronizing

By default, VREP will run its simulation in its own thread, and both the simulation and the controller using the remote API will be running simultaneously. This can lead to some weird behaviour as things fall out of synch etc etc, so what we want instead is for the VREP sim to only run one time step for each time step the controller runs. To do that, we need to set VREP to synchronous mode. So the next few lines of our Python script look like:

    # --------------------- Setup the simulation

vrep.simxSynchronous(clientID,True)


and then later, once we’ve calculated our control signal, sent it over, and want the simulation to run one time step forward, we do that by calling

    # move simulation ahead one time step
vrep.simxSynchronousTrigger(clientID)


Get the handles to objects of interest

OK the next chunk of code in our script uses the names of our objects (as specified in the Scene Hierarchy in VREP) to get an ID for each which to identify which object we want to send a command to or receive feedback from:

    joint_names = ['shoulder', 'elbow']
# joint target velocities discussed below
joint_target_velocities = np.ones(len(joint_names)) * 10000.0

# get the handles for each joint and set up streaming
joint_handles = [vrep.simxGetObjectHandle(clientID,
name, vrep.simx_opmode_blocking)[1] for name in joint_names]

# get handle for target and set up streaming
_, target_handle = vrep.simxGetObjectHandle(clientID,
'target', vrep.simx_opmode_blocking)


Set dt and start the simulation

And the final thing that we’re going to do in our simulation set up is specify the timestep that we want to use, and then start the simulation. I found this in a forum post, and I must say whatever VREP lacks in intuitive API their forum moderators are on the ball. NOTE: To use a custom time step you have to also set the dt option in the VREP simulator to ‘custom’. The drop down is to the left of the ‘play’ arrow, and if you don’t have it set to ‘custom’ you won’t be able to set the time step through Python.

    dt = .01
vrep.simxSetFloatingParameter(clientID,
vrep.sim_floatparam_simulation_time_step,
dt, # specify a simulation time step
vrep.simx_opmode_oneshot)

# --------------------- Start the simulation

# start our simulation in lockstep with our code
vrep.simxStartSimulation(clientID,
vrep.simx_opmode_blocking)


Main loop

For this next chunk I’m going to cut out everything that’s not VREP, since I have a bunch of posts explaining the control signal derivation and forward transformation matrices.

So, once we’ve started the simulation, I’ve set things up for the arm to be controlled for 1 second and then for the simulation to stop and everything shut down and disconnect.

    while count < 1: # run for 1 simulated second

# get the (x,y,z) position of the target
_, target_xyz = vrep.simxGetObjectPosition(clientID,
target_handle,
-1, # retrieve absolute, not relative, position
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()
track_target.append(np.copy(target_xyz)) # store for plotting
target_xyz = np.asarray(target_xyz)

q = np.zeros(len(joint_handles))
dq = np.zeros(len(joint_handles))
for ii,joint_handle in enumerate(joint_handles):
# get the joint angles
_, q[ii] = vrep.simxGetJointPosition(clientID,
joint_handle,
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()
# get the joint velocity
_, dq[ii] = vrep.simxGetObjectFloatParameter(clientID,
joint_handle,
2012, # ID for angular velocity of the joint
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()


In the above chunk of code, I think the big thing to point out is that I’m using vrep.simx_opmode_blocking in each call, instead of vrep.simx_opmode_buffer. The difference is that you for sure get the current values of the simulation when you use blocking, and you can be behind a time step using buffer.

Aside from that the other notable things are I raise an exception if the first parameter (which is the return code) is ever not 0, and that I use simxGetObjectFloatParameter to get the joint velocity instead of simxGetObjectVelocity, which has a rotational velocity component. Zero is the return code for ‘everything worked’, and if you don’t check it and have some silly things going on you can be veeerrrrryyy mystified when things don’t work. And what simxGetObjectVelocity returns is the rotational velocity of the joint relative to the world frame, and not the angular velocity of the joint in its own coordinates. That was also a briefly confusing.

So the next thing I do is calculate u, which we’ll skip, and then we need to set the forces for the joint. This part of the API is real screwy. You can’t set the force applied to the joint directly. Instead, you have to set the target velocity of the joint to some really high value (hence that array we made before), and then modulate the maximum force that can be applied to that joint. Also important: When you want to apply a force in the other direction, you change the sign on the target velocity, but keep the force sign positive.

        # ... calculate u ...

for ii,joint_handle in enumerate(joint_handles):
# get the current joint torque
_, torque = \
vrep.simxGetJointForce(clientID,
joint_handle,
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()

# if force has changed signs,
# we need to change the target velocity sign
if np.sign(torque) * np.sign(u[ii]) < 0:
joint_target_velocities[ii] = \
joint_target_velocities[ii] * -1
vrep.simxSetJointTargetVelocity(clientID,
joint_handle,
joint_target_velocities[ii], # target velocity
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()

# and now modulate the force
vrep.simxSetJointForce(clientID,
joint_handle,
abs(u[ii]), # force to apply
vrep.simx_opmode_blocking)
if _ !=0 : raise Exception()

# move simulation ahead one time step
vrep.simxSynchronousTrigger(clientID)
count += dt


So as you can see we check the current torque, see if we need to change the sign on the target velocity, modulate the maximum allowed force, and then finally step the VREP simulation forward.

Conclusions

And there you go! Here’s an animation of it in action (note this is a super low quality gif and it looks way better / smoother when actually running it yourself):

All in all, VREP has been enjoyable to work with so far. It didn’t take long to get things moving and off the ground, the visualization is great, and I haven’t even scratched the surface of what you can do with it. Best of all (so far) you can fully automate everything from Python. Hopefully this is enough to help some people get their own models going and save a few hours and headaches! Again, the full code and the model are up on my GitHub.

Nits

• When you’re applying your control signal, make sure you test each joint in isolation, to make sure your torques push things in the direction you think they do. I had checked the rotation direction in VREP, but the control signal for both joints ended up needing to be multiplied by -1.
• Another nit when you’re building your model, if you use the rotate button from the VREP toolbar on your model, wherever that joint rotates to is now 0 degrees. If you want to set the joint to start at 45 degrees, instead double click and change Pos[deg] option inside ‘Joint’ in Scene Object Properties.

## Deep learning for control using augmented Hessian-free optimization

Traditionally, deep learning is applied to feed-forward tasks, like classification, where the output of the network doesn’t affect the input to the network. It is a decidedly harder problem when the output is recurrently connected such that network output affects the input. Because of this application of deep learning methods to control was largely unexplored until a few years ago. Recently, however, there’s been a lot of progress and research in this area. In this post I’m going to talk about an implementation of deep learning for control presented by Dr. Ilya Sutskever in his thesis Training Recurrent Neural Networks.

In his thesis, Dr. Sutskever uses augmented Hessian-free (AHF) optimization for learning. There are a bunch of papers and posts that go into details about AHF, here’s a good one by Andrew Gibiansky up on his blog, that I recommend you check out. I’m not going to really talk much here about what AHF is specifically, or how it differs from other methods, if you’re unfamiliar there are lots of places you can read up on it. Quickly, though, AHF is kind of a bag of tricks you can use with a fast method for estimating the curvature of the loss function with respect to the weights of a neural network, as well as the gradient, which allows it to make larger updates in directions where the loss function doesn’t change quickly. So rather than estimating the gradient and then doing a small update along each dimension, you can make the size of your update large in directions that change slowly and small along dimensions where things change quickly. And now that’s enough about that.

In this post I’m going to walk through using a Hessian-free optimization library (version 0.3.8) written by my compadre Dr. Daniel Rasmussen to train up a neural network to train up a 2-link arm, and talk about the various hellish gauntlets you need run to get something that works. Whooo! The first thing to do is install this Hessian-free library, linked above.

I’ll be working through code edited a bit for readability, to find the code in full you can check out the files up on my GitHub.

Build the network

Dr. Sutskever specified the structure of the network in his thesis to be 4 layers: 1) a linear input layer, 2) 100 Tanh nodes, 3) 100 Tanh nodes, 4) linear output layer. The network is connected up with the standard feedforward connections from 1 to 2 to 3 to 4, plus recurrent connections on 2 and 3 to themselves, plus a ‘skip’ connection from layer 1 to layer 3. Finally, the input to the network is the target state for the plant and the current state of the plant. So, lots of recursion! Here’s a picture:

The output layer connects in to the plant, and, for those unfamiliar with control theory terminology, ‘plant’ just means the system that you’re controlling. In this case an arm simulation.

Before we can go ahead and set up the network that we want to train, we also need to specify the loss function that we’re going to be using during training. The loss function in Ilya’s thesis is a standard one:

$L(\theta) = \sum\limits^{N-1}\limits_{t=0} \ell(\textbf{u}_t) + \ell_f(\textbf{x}_N),$
$\ell(\textbf{u}_t) = \alpha \frac{||\textbf{u}_t||^2}{2},$
$\ell_f(\textbf{x}_N) = \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2}$

where $L(\theta)$ is the total cost of the trajectory generated with $\theta$, the set of network parameters, $\ell(\textbf{u})$ is the immediate state cost, $\ell_f(\textbf{x})$ is the final state cost, $\textbf{x}$ is the state of the arm, $\textbf{x}^*$ is the target state of the arm, $\textbf{u}$ is the control signal (torques) that drives the arm, and $\alpha$ is a gain value.

To code this up using the hessianfree library we do:

from hessianfree import RNNet
from hessianfree.nonlinearities import (Tanh, Linear, Plant)
from hessianfree.loss_funcs import SquaredError, SparseL2

l2gain = 10e-3 * dt # gain on control signal loss
rnn = RNNet(
# specify the number of nodes in each layer
shape=[num_states * 2, 96, 96, num_states, num_states],
# specify the function of the nodes in each layer
layers=[Linear(), Tanh(), Tanh(), Linear(), plant],
# specify the layers that have recurrent connections
rec_layers=[1,2],
# specify the connections between layers
conns={0:[1, 2], 1:[2], 2:[3], 3:[4]},
# specify the loss function
loss_type=[
# squared error between plant output and targets
SquaredError(),
# penalize magnitude of control signal (output of layer 3)
SparseL2(l2gain, layers=[3])],
use_GPU=True)


Note that if you want to run it on your GPU you’ll need PyCuda and sklearn installed. And a GPU.

An important thing to note as well is that in Dr. Sustkever’s thesis when we’re calculating the squared error of the distance from the arm state to the target, this is measured in joint angles. So it’s kind of a weird set up to be looking at the movement of the hand but have your cost function in joint-space instead of end-effector space, but it definitely simplifies training by making the cost more directly relatable to the control signal. So we need to calculate the joint angles of the arm that will have the hand at different targets around a circle. To do this we’ll take advantage of our inverse kinematics solver from way back when, and use the following code:

def gen_targets(arm, n_targets=8, sig_len=100):
#Generate target angles corresponding to target
#(x,y) coordinates around a circle
import scipy.optimize

x_bias = 0
if arm.DOF == 2:
y_bias = .35
dist = .075
elif arm.DOF == 3:
y_bias = .5
dist = .2

# set up the reaching trajectories around circle
targets_x = [dist * np.cos(theta) + x_bias \
for theta in np.linspace(0, np.pi*2, 65)][:-1]
targets_y = [dist * np.sin(theta) + y_bias \
for theta in np.linspace(0, np.pi*2, 65)][:-1]

joint_targets = []
for ii in range(len(targets_x)):
joint_targets.append(arm.inv_kinematics(xy=(targets_x[ii],
targets_y[ii])))
targs = np.asarray(joint_targets)

# repeat the targets over time
for ii in range(targs.shape[1]-1):
targets = np.concatenate(
(np.outer(targs[:, ii], np.ones(sig_len))[:, :, None],
np.outer(targs[:, ii+1], np.ones(sig_len))[:, :, None]), axis=-1)
targets = np.concatenate((targets, np.zeros(targets.shape)), axis=-1)
# only want to penalize the system for not being at the
# target at the final state, set everything before to np.nan
targets[:, :-1] = np.nan

return targets


And you can see in the last couple lines that to implement the distance to target as a final state cost penalty only we just set all of the targets before the final time step equal to np.nan. If we wanted to penalize distance to target throughout the whole trajectory we would just comment that line out.

Create the plant

You’ll notice in the code that defines our RNN I set the last layer of the network to be plant, but that that’s not defined anywhere. Let’s talk. There are a couple of things that we’re going to need to incorporate our plant into this network and be able to use any deep learning method to train it. We need to be able to:

1. Simulate the plant forward; i.e. pass in input and get back the resulting plant state at the next timestep.
2. Calculate the derivative of the plant state with respect to the input; i.e. how do small changes in the input affect the state.
3. Calculate the derivative of the plant state with respect to the previous state; i.e. how do small changes in the plant state affect the state at the next timestep.
4. Calculate the derivative of the plant output with respect to its state; i.e. how do small changes in the current position of the state affect the output of the plant.

So 1 is easy, we have the arm simulations that we want already, they’re up on my GitHub. Number 4 is actually trivial too, because the output of our plant is going to be the state itself, so the derivative of the output with respect to the state is just the identity matrix.

For 2 and 3, we’re going to need to calculate some derivatives. If you’ve read the last few posts you’ll note that I’m on a finite differences kick. So let’s get that going! Because no one wants to calculate derivatives!

Important note, the notation in these next couple pieces of code is going to be a bit different from my normal notation because they’re matching with the hessianfree library notation, which is coming from a reinforcement learning literature background instead of a control theory background. So, s is the state of the plant, and x is the input to the plant. I know, I know. All the same, make sure to keep that in mind.

# calculate ds0/dx0 with finite differences
d_input_FD = np.zeros((x.shape[0], # number of trials
x.shape[1], # number of inputs
self.state.shape[1])) # number of states
for ii in range(x.shape[1]):
# calculate state adding eps to x[ii]
self.reset_plant(self.prev_state)
inc_x = x.copy()
inc_x[:, ii] += self.eps
self.activation(inc_x)
state_inc = self.state.copy()
# calculate state subtracting eps from x[ii]
self.reset_plant(self.prev_state)
dec_x = x.copy()
dec_x[:, ii] -= self.eps
self.activation(dec_x)
state_dec = self.state.copy()

d_input_FD[:, :, ii] = \
(state_inc - state_dec) / (2 * self.eps)
d_input_FD = d_input_FD[..., None]


Alrighty. First we create a tensor to store the results. Why is it a tensor? Because we’re going to be doing a bunch of runs at once. So our state dimensions are actually trials x size_input. When we then take the partial derivative, we end up with trials many size_input x size_state matrices. Then we increase each of the parameters of the input slightly one at a time and store the results, decrease them all one at a time and store the results, and compute our approximation of the gradient.

Next we’ll do the same for calculating the derivative of the state with respect to the previous state.

# calculate ds1/ds0
d_state_FD = np.zeros((x.shape[0], # number of trials
self.state.shape[1], # number of states
self.state.shape[1])) # number of states
for ii in range(self.state.shape[1]):
# calculate state adding eps to self.state[ii]
state = np.copy(self.prev_state)
state[:, ii] += self.eps
self.reset_plant(state)
self.activation(x)
state_inc = self.state.copy()
# calculate state subtracting eps from self.state[ii]
state = np.copy(self.prev_state)
state[:, ii] -= self.eps
self.reset_plant(state)
self.activation(x)
state_dec = self.state.copy()

d_state_FD[:, :, ii] = \
(state_inc - state_dec) / (2 * self.eps)
d_state_FD = d_state_FD[..., None]


Great! We’re getting closer to having everything we need. Another thing we need is a wrapper for running our arm simulation. It’s going to look like this:

def activation(self, x):
state = []
# iterate through and simulate the plant forward
# for each trial
for ii in range(x.shape[0]):
self.arm.reset(q=self.state[ii, :self.arm.DOF],
dq=self.state[ii, self.arm.DOF:])
self.arm.apply_torque(u[ii])
state.append(np.hstack([self.arm.q, self.arm.dq]))
state = np.asarray(state)

self.state = self.squashing(state)


This is definitely not the fastest code to run. Much more ideally we would put the state and input into vectors and do a single set of computations for each call to activation rather than having that for loop in there. Unfortunately, though, we’re not assuming that we have access to the dynamics equations / will be able to pass in vector states and inputs.

Squashing
Looking at the above code that seems pretty clear what’s going on, except you might notice that last line calling self.squashing. What’s going on there?

The squashing function looks like this:

def squashing(self, x):
index_below = np.where(x < -2*np.pi)
x[index_below] = np.tanh(x[index_below]+2*np.pi) - 2*np.pi
index_above = np.where(x > 2*np.pi)
x[index_above] = np.tanh(x[index_above]-2*np.pi) + 2*np.pi
return x


All that’s happening here is that we’re taking our input, and doing nothing to it as long as it doesn’t start to get too positive or too negative. If it does then we just taper it off and prevent it from going off to infinity. So running a 1D vector through this function we get:

This ends up being a pretty important piece of the code here. Basically it prevents wild changes to the weights during learning to result in the system breaking down. So the state of the plant can’t go off to infinity and cause an error to be thrown, stopping our simulation. But because the target state is well within the bounds of where the squashing function does nothing, post-training we’ll still be able to use the resulting network to control a system that doesn’t have this fail safe built in. Think of this function as training wheels that catch you only if you start falling.

With that, we no have pretty much all of the parts necessary to begin training our network!

Training the network

We’re going to be training this network on the centre-out reaching task, where you start at a centre point and reach to a bunch of target locations around a circle. I’m just going to be re-implementing the task as it was done in Dr. Sutskever’s thesis, so we’ll have 64 targets around the circle, and train using a 2-link arm. Here’s the code that we’ll use to actually run the training:

for ii in range(last_trial+1, num_batches):
# train a bunch of batches using the same input every time
# to allow the network a chance to minimize things with
# stable input (speeds up training)
err = rnn.run_batches(plant, targets=None,
max_epochs=batch_size,
optimizer=HessianFree(CG_iter=96, init_damping=100))

# save the weights to file, track trial and error
# err = rnn.error(inputs)
err = rnn.best_error
name = 'weights/rnn_weights-trial%04i-err%.3f'%(ii, err)
np.savez_compressed(name, rnn.W)


A quick aside: if you want to run this code yourself, get a real good computer, have an arm simulation ready, the hessianfree Python library installed, and download and run this train_hf.py file. (Note: I used version 0.3.8 of the hessianfree library, which you can install using pip install hessianfree==0.3.8) This will start training and save the weights into a weights/ folder, so make sure that that exists in the same folder as train_hf.py. If you want to view the results of the training at any point run the plot_error.py file, which will load in the most recent version of the weights and plot the error so far. If you want to generate an animated plot like I have below run gen_animation_plots.py and then the last command from my post on generating animated gifs.

Another means of seeing the results of your trained up network is to use the controller I’ve implemented in my controls benchmarking suite, which looks for a set of saved weights in the controllers/weights folder, and will load it in and use it to generate command signals for the arm by running it with

python run.py arm2_python ahf reach --dt=1e-2


where you replace arm2_python with whatever arm model you trained your model on. Note the --dt=1e-2 flag, that is important because the model was trained with a .01 timestep and things get a bit weird if you suddenly change the dynamics on the controller.

OK let’s look at some results!

Results

OK full discretion, these results are not optimizing the cost function we discussed above. They’re implementing a simpler cost function that only looks at the final state, i.e. it doesn’t penalize the magnitude of the control signal. I did this because Dr. Sutskever says in his thesis he was able to optimize with just the final state cost using much smaller networks. I originally looked at neurons with 96 neurons in each layer, and it just took forgoddamnedever to run. So after running for 4 weeks (not joking) and needing to make some more changes I dropped the number of neurons and simplified the task.

The results below are from running a network with 32 neurons in each layer controlling this 2-link arm, and took another 4-5 weeks to train up.

Hey that looks good! Not bad, augmented Hessian-free learning, not bad. It had pretty consistent (if slow) decline in the error rate, with a few crazy bumps from which it quickly recovered. Also take note that each training iteration is actually 32 runs, so it’s not 12,50-ish runs it’s closer to 400,000 training runs that it took to get here.

One biggish thing that was a pain was that it turns out that I only trained the neural network for reaching in the one direction, and when you only train it to reach one way it doesn’t generalize to reaching back to the starting point (which, fair enough). But, I didn’t realize this until I was took the trained network and ran it in the benchmarking code, at which point I was not keen to redo all of the training it took to get the neural network to the level of accuracy it was at under a more complicated training set. The downside of this is that even though I’ve implemented a controller that takes in the trained network and uses it to control the arm, to do the reaching task I have to just do a hard reset after the arm reaches the target, because it can’t reach back to the center, like all the other controllers. All the same, here’s an animation of the trained up AHF controller reaching to 8 targets (it was trained on all 64 above though):

Things don’t always go so smoothly, though. Here’s results from another training run that took around 2-3 weeks, and uses a different 2-link arm model (translated from Matlab code written by Dr. Emo Todorov):

What I found frustrating about this was that if you look at the error over time then this arm is doing as well or better than the previous arm at a lot of points. But the corresponding trajectories look terrible, like something you would see in a horror movie based around getting good machine learning results. This of course comes down to how I specified the cost function, and when I looked at the trajectories plotted over time the velocity of the arm is right at zero at the final time step, which it is not quiiiitte the case for the first controller. So this second network has found a workaround to minimize the cost function I specified in a way I did not intend. To prevent this, doing something like weighting the distance to target heavier than non-zero velocity would probably work. Or possibly just rerunning the training with a different random starting point you could get out a better controller, I don’t have a great feel for how important the random initialization is, but I’m hoping that it’s not all too important and its effects go to zero with enough training. Also, it should be noted I’ve run the first network for 12,500 iterations and the second for less than 6,000, so I’ll keep letting them run and maybe it will come around. The first one looked pretty messy too until about 4,000 iterations in.

Training regimes

Frustratingly, the way that you train deep networks is very important. So, very much like the naive deep learning network trainer that I am, I tried the first thing that pretty much anyone would try:

• run the network,
• update the weights,
• repeat.

This is what I’ve done in the results above. And it worked well enough in that case.

If you remember back to the iLQR I made a little while ago, I was able to change the cost function to be

$L(\theta) = \sum\limits^{N-1}\limits_{t=0} \ell(\textbf{u}_t) + \ell_f(\textbf{x}_N),$
$\ell(\textbf{u}_t, \textbf{x}_t) = \alpha \frac{||\textbf{u}_t||^2}{2} + \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2},$
$\ell_f(\textbf{x}_N) = \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2}$

(i.e. to include a penalty for distance to target throughout the trajectory and not just at the final time step) which resulted in straighter trajectories when controlling the 2-link arm. So I thought I would try this here as well. Sadly (incredibly sadly), this was fairly fruitless. The network didn’t really learn or improve much at all.

After much consideration and quandary on my part, I talked with Dr. Dan and he suggested that I try another method:

• run the network,
• record the input,
• hold the input constant for a few batches of weight updating,
• repeat.

This method gave much better results. BUT WHY? I hear you ask! Good question. Let me give giving explanation a go.

Essentially, it’s because the cost function is more complex now. In the first training method, the output from the plant is fed back into the network as input at every time step. When the cost function was simpler this was OK, but now we’re getting very different input to train on at every iteration. So the system is being pulled in different directions back and forth at every iteration. In the second training regime, the same input is given several times in a row, which let’s the system follow the same gradient for a few training iterations before things change again. In my head I picture this as giving the algorithm a couple seconds to catch its breath dunking it back underwater.

This is a method that’s been used in a bunch of places recently. One of the more high-profile instances is in the results published from DeepMind on deep RL for control and for playing Go. And indeed, it also works well here.

To implement this training regime, we set up the following code:

for ii in range(last_trial+1, num_batches):

# run the plant forward once
rnn.forward(input=plant, params=rnn.W)

# get the input and targets from above rollout
inputs = plant.get_vecs()[0].astype(np.float32)
targets = np.asarray(plant.get_vecs()[1], dtype=np.float32)

# train a bunch of batches using the same input every time
# to allow the network a chance to minimize things with
# stable input (speeds up training)
err = rnn.run_batches(inputs, targets, max_epochs=batch_size,
optimizer=HessianFree(CG_iter=96, init_damping=100))

# save the weights to file, track trial and error
# err = rnn.error(inputs)
err = rnn.best_error
name = 'weights/rnn_weights-trial%04i-err%.3f'%(ii, err)
np.savez_compressed(name, rnn.W)


So you can see that we do one rollout with the weights, then go in and get the inputs and targets that were used in that rollout, and start training the network while holding those constant for batch_size epochs (training sessions). From a little bit of messing around I’ve found batch_size=32 to be a pretty good number. So then it runs 32 training iterations where it’s updating the weights, and then saves those weights (because we want a loooottttt of check-points) and then restarts the loop.

Embarrassingly, I’ve lost my simulation results from this trial, somehow…so I don’t have any nice plots to back up the above, unfortunately. But since this is just a blog post I figured I would at least talk about it a little bit, since people might still find it useful if they’re just getting into the field like me. and just update this post whenever I re-run them. If I rerun them.

What I do have, however, are results where this method doesn’t work! I tried this with the simpler cost function, that only looks at the final state distance from the target, and it did not go so well. Let’s look at that one!

My guess here is basically that the system has gotten to a point where it’s narrowed things down in the parameter space and now when you run 32 batches it’s overshooting. It needs feedback about its updates after every update at this point. That’s my guess, at least. So it could be the case that for more complex cost functions you’d want to train it while holding the input constant for a while, and then when the error starts to plateau switch to updating the input after every parameter update.

Conclusions

All in all, AHF for training neural networks in control is pretty awesome. There are of course still some major hold-backs, mostly related to how long it takes to train up a network, and having to guess at effective training regimes and network structures etc. But! It was able to train up a relatively small neural network to move an arm model from a center point to 64 targets around a circle, with no knowledge of the system under control at all. In Dr. Sutskever’s thesis he goes on to use the same set up under more complicated circumstances, such as when there’s a feedback delay, or a delay on the outgoing control signal, and unexpected noise etc, so it is able to learn under a number of different, fairly complex situations. Which is pretty slick.

Related to the insane training time required, I very easily could be missing some basic thing that would help speed things up. If you, reader, get ambitious and run the code on your own machine and find out useful methods for speeding up the training please let me know! Personally, my plan is to next investigate guided policy search, which seems like it’s found a way around this crazy training time.

## Simultaneous perturbation vs finite differences for linear dynamics estimation and control signal optimization

Recently in my posts I’ve been using finite differences to approximate the gradient of loss functions and dynamical systems, with the intention of creating generalizable controllers that can be run on any system without having to calculate out derivatives beforehand. Finite differences is pretty much the most straight-forward way of approximating a gradient that there is: vary each parameter up and down (assuming we’re doing central differencing), one at a time, run it through your function and estimate the parameters effect on the system by calculating the difference between resulting function output. To do this requires 2 samples of the function for each parameter.

But there’s always more than one way to peel an avocado, and another approach that’s been used with much success is the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm, which was developed by Dr. James Spall (link to overview paper). SPSA is a method of gradient approximation, like finite differences, but, critically, the difference is that it varies all of the parameters at once, rather than one at a time. As a result, you can get an approximation of the gradient with far fewer samples from the system, and also when you don’t have explicit control over your samples (i.e. the ability to vary each parameter one at a time).

Given some function $\textbf{f}$ dependent on some set of parameters $\textbf{x}$, we’re used to finding the gradient of $\textbf{f}(\textbf{x})$ using FDSA (finite differences stochastic approximation) written in this form:

$\textbf{f}_x = \frac{\textbf{f}(\textbf{x} + \Delta \textbf{x}) - \textbf{f}(\textbf{x} - \Delta \textbf{x})}{2 \Delta \textbf{x}} = \Delta x^{-1} \frac{\textbf{f}(\textbf{x} + \Delta \textbf{x}) - \textbf{f}(\textbf{x} - \Delta \textbf{x})}{2},$

where $\Delta\textbf{x}$ is a perturbation to the parameter set, and the subscript on the left-hand side denotes the derivative of $\textbf{f}(\textbf{x})$ with respect to $\textbf{x}$.

And that’s how we’ve calculated it before, estimating the gradient of a single parameter at a time. But, we can rewrite this for a set perturbations $\Delta \textbf{X} = [\Delta\textbf{x}_0, ... , \Delta\textbf{x}_N]^T$:

$\textbf{f}_\textbf{x} = \Delta \textbf{X}^{-1} \textbf{F},$

where

$\textbf{F} = [\frac{\textbf{f}(\textbf{x} + \Delta \textbf{x}_0) - \textbf{f}(\textbf{x} - \Delta \textbf{x}_0)}{2}, ... , \frac{\textbf{f}(\textbf{x} + \Delta \textbf{x}_0) - \textbf{f}(\textbf{x}_N - \Delta \textbf{x}_N)}{2}]^T$,

which works as long as $\Delta\textbf{X}$ is square. When it’s not square (i.e. we have don’t have the same number of samples as we have parameters), we run into problems, because we can’t calculate $\Delta\textbf{X}^{-1}$ directly. To address this, let’s take a step back and then work forward again to get a more general form that works for non-square $\Delta \textbf{X}$ too.

By rewriting the above, and getting rid of the inverse by moving $\Delta\textbf{x}$ back to the other side, we have:

$\Delta\textbf{X} \; \textbf{f}_\textbf{x} = \textbf{F}$

Now, the standard trick to move a matrix that’s not square is to just make it square by multiplying it by its transpose to get a square matrix, and then the whole thing by the inverse:

$\Delta\textbf{X}^T \Delta\textbf{X} \; \textbf{f}_\textbf{x} = \Delta\textbf{X}^T \textbf{F}$

$(\Delta\textbf{X}^T \Delta\textbf{X})^{-1} (\Delta\textbf{X}^T \Delta\textbf{X}) \textbf{f}_\textbf{x} = (\Delta\textbf{X}^T \Delta\textbf{X})^{-1} \Delta\textbf{X}^T \textbf{F}$

$\textbf{f}_\textbf{x} = (\Delta\textbf{X}^T \Delta\textbf{X})^{-1} \Delta\textbf{X}^T \textbf{F}$

Alright! Now we’re comfortable with this characterization of gradient approximation using a form that works with non-square perturbation matrices.

Again, in FDSA, we only vary one parameter at a time. This means that there will only ever be one non-zero entry per row of $\Delta \textbf{X}$. By contrast, in SPSA, we vary multiple parameters, and so rows of $\Delta\textbf{X}$ will be just chalk full of non-zero entries.

Gradient approximation to estimate $\textbf{f}_\textbf{x}$ and $\textbf{f}_\textbf{u}$ for LQR control

Implementing this was pretty simple. Just had to modify the calc_derivs function, which I use to estimate the derivative of the arm with respect to the state and control signal, in my LQR controller code by changing from standard finite differences to simultaneous perturbation:

def calc_derivs(self, x, u):
"""" calculate gradient of plant dynamics using Simultaneous
Perturbation Stochastic Approximation (SPSA). Implemented
based on (Peters & Schaal, 2008).

x np.array: the state of the system
u np.array: the control signal
"""
# Initialization and coefficient selection
num_iters = 20

eps = 1e-4
delta_K = None
delta_J = None
for ii in range(num_iters):
# Generation of simultaneous perturbation vector
# choose each component from a Bernoulli +-1 distribution
# with probability of .5 for each +-1 outcome.
delta_k = np.random.choice([-1,1],
size=len(x) + len(u),
p=[.5, .5])
# Function evaluations
inc_x = np.copy(x) + eps * delta_k[:len(x)]
inc_u = np.copy(u) + eps * delta_k[len(x):]
state_inc = self.plant_dynamics(inc_x, inc_u)
dec_x = np.copy(x) - eps * delta_k[:len(x)]
dec_u = np.copy(u) - eps * delta_k[len(x):]
state_dec = self.plant_dynamics(dec_x, dec_u)

delta_j = ((state_inc - state_dec) /
(2.0 * eps)).reshape(-1)

# Track delta_k and delta_j
delta_K = delta_k if delta_K is None else \
np.vstack([delta_K, delta_k])
delta_J =  delta_j if delta_J is None else \
np.vstack([delta_J, delta_j])

f_xu = np.dot(np.linalg.pinv(np.dot(delta_K.T, delta_K)),
np.dot(delta_K.T, delta_J))
f_x = f_xu[:len(x)]
f_u = f_xu[len(x):]

return f_x.T , f_u.T


A couple notes about the above code. First, you’ll notice that the f_x and f_b matrices are both calculated at the same time. That’s pretty slick! And that calculation for f_xu is just a straight implementation of the matrix form of gradient approximation, where I’ve arranged things so that f_x is in the top part and f_u is in the lower part.

The second thing is that the perturbation vector delta_k is generated from a Bernoulli distribution. The reason behind this is that we want to have a bunch of different samples that pretty reasonably spread the state space and move all the parameters independently. Making each perturbation some distance times -1 or 1 is an easy way to achieve this.

Thirdly, there’s the num_iters variable. This is a very important variable, as it dictates how many random samples of our system we take before we estimate the gradient. I’ve found that to get this to work for both the 2-link arm and the more complex 3-link arm, it needs to be at least 20. Or else things explode and die horribly. Just…horribly.

OK let’s look at the results:

The first thing to notice is that I’ve finally discovered the Seaborn plotting package. The second is that SPSA does as well as FDSA.

You may ask: Is there any difference? Well, if we time these functions, on my lil’ laptop, for the 2-link arm it takes SPSA approximately 2.0ms, but it takes FDSA only 0.8ms. So for the same performance the SPSA is taking almost 3 times as long to run. Why? This boils down to how many times the system dynamics need to be sampled by each algorithm to get a good approximation of the gradient. For a 2-link arm, FDSA has 6 parameters ($\textbf{q}, \dot{\textbf{q}},$ and $\textbf{u}$) that it needs to sample twice (we’re doing central differencing), for a total of 12 samples. And as I mentioned above, the SPSA algorithm needs 20 samples to be stable.

For the 3-link arm, SPSA took about 3.1ms on average and FDSA (which must now perform 18 samples of the dynamics) still only 2.1ms. So number of samples isn’t the only cause of time difference between these two algorithms. SPSA needs to perform that a few more matrix operations, including a matrix inverse, which is expensive, while FDSA can calculate the gradient of each parameter individually, which is much less expensive.

OK so SPSA not really impressive here. BUT! As I discovered, there are other means of employing SPSA.

Gradient approximation to optimize the control signal directly

In the previous set up we were using SPSA to estimate the gradient of the system under control, and then we used that gradient to calculate a control signal that minimized the loss function (as specified inside the LQR). This is one way to use gradient approximation methods. Another way to use these methods is approximate the gradient of the loss function directly, and use that information to iteratively calculate a control signal that minimizes the loss function. This second application is the primary use of the SPSA algorithm, and is what’s described by Dr. Spall in his overview paper.

In this application, the algorithm works like this:

2. perturb input and simulate results
3. observe loss function and calculate gradient
4. update input to system
5. repeat to convergence

Because in this approach we’re iteratively optimizing the input using our gradient estimation, having a noisy estimate is no longer a death sentence, as it was in the LQR. If we update our input to the system with several noisy gradient estimates the noise will essentially just cancel itself out. This means that SPSA now has a powerful advantage over FDSA: Since in SPSA we vary all parameters at once, only 2 samples of the loss function are used to estimate the gradient, regardless of the number of parameters. In contrast, FDSA needs to sample the loss function twice for every input parameter. Here’s a picture from (Spall, 1998) that shows the two running against each other to optimize a 2D problem:

This gets across that even though SPSA bounces around more, they both reach the solution in the same number of steps. And, in general, this is the case, as Dr. Spall talks about in the paper. There’s also a couple more details of the algorithm, so let’s look at it in detail. Here’s the code, which is just a straight translation into Python out of the description in Dr. Spall’s paper:

# Step 1: Initialization and coefficient selection
max_iters = 5
converge_thresh = 1e-5

alpha = 0.602 # from (Spall, 1998)
gamma = 0.101
a = .101 # found empirically using HyperOpt
A = .193
c = .0277

delta_K = None
delta_J = None
u = np.copy(self.u) if self.u is not None \
else np.zeros(self.arm.DOF)
for k in range(max_iters):
ak = a / (A + k + 1)**alpha
ck = c / (k + 1)**gamma

# Step 2: Generation of simultaneous perturbation vector
# choose each component from a bernoulli +-1 distribution with
# probability of .5 for each +-1 outcome.
delta_k = np.random.choice([-1,1], size=arm.DOF, p=[.5, .5])

# Step 3: Function evaluations
inc_u = np.copy(u) + ck * delta_k
cost_inc = self.cost(np.copy(state), inc_u)
dec_u = np.copy(u) - ck * delta_k
cost_dec = self.cost(np.copy(state), dec_u)

gk = np.dot((cost_inc - cost_dec) / (2.0*ck), delta_k)

# Step 5: Update u estimate
old_u = np.copy(u)
u -= ak * gk

# Step 6: Check for convergence
if np.sum(abs(u - old_u)) < converge_thresh:
break


The main as-of-yet-unexplained parts of this code are the alpha, gamma, a, A, and c variables. What’s their deal?

Looking inside the loop, we can see that ck controls the magnitude of our perturbations. Looking a little further down, ak is just the learning rate. And all of those other parameters are just involved in shaping the trajectories that ak and ck follow through iterations, which is a path towards zero. So the first steps and perturbations are the biggest, and each successively becomes smaller as the iteration count increases.

There are a few heuristics that Dr. Spall goes over, but there aren’t any hard and fast rules for setting a, A, and c. Here, I just used HyperOpt to find some values that worked pretty well for this particular problem.

The FDSA version of this is also very straight-forward:

# Step 1: Initialization and coefficient selection
max_iters = 10
converge_thresh = 1e-5

eps = 1e-4
u = np.copy(self.u) if self.u is not None \
else np.zeros(self.arm.DOF)
for k in range(max_iters):

gk = np.zeros(u.shape)
for ii in range(gk.shape[0]):
# Step 2: Generate perturbations one parameter at a time
inc_u = np.copy(u)
inc_u[ii] += eps
dec_u = np.copy(u)
dec_u -= eps

# Step 3: Function evaluation
cost_inc = self.cost(np.copy(state), inc_u)
cost_dec = self.cost(np.copy(state), dec_u)

gk[ii] = (cost_inc - cost_dec) / (2.0 * eps)

old_u = np.copy(u)
# Step 5: Update u estimate
u -= 1e-5 * gk

# Step 6: Check for convergence
if np.sum(abs(u - old_u)) < converge_thresh:
break


You’ll notice that in both the SPSA and FDSA code we’re no longer sampling plant_dynamics, we’re instead sampling cost, a loss function I defined. From just my experience playing around with these algorithms a bit, getting the loss function to be appropriate and give the desired behaviour is definitely a bit of an art. It feels like much more of an art than in other controllers I’ve coded, but that could just be me.

The cost function that I’m using is pretty much the first thing you’d think of. It penalizes distance to target and having non-zero velocity. Getting the weighting between distance to target and velocity set up so that the arm moves to the target but also doesn’t overshoot definitely took a bit of trial and error, er, I mean empirical analysis. Here’s the cost function that I found worked pretty well, note that I had to make special cases for the different arms:

 def cost(self, x, u):
dt = .1 if self.arm.DOF == 3 else .01
next_x = self.plant_dynamics(x, u, dt=dt)
vel_gain = 100 if self.arm.DOF == 3 else 10
return (np.sqrt(np.sum((self.arm.x - self.target)**2)) * 1000 \
+ np.sum((next_x[self.arm.DOF:])**2) * vel_gain)


So that’s all the code, let’s look at the results!

For these results, I used a max of 10 iterations for optimizing the control signal. I was definitely surprised by the quality of the results, especially for the 3-link arm, compared to the results generated by a standard LQR controller. Although I need to note, again, that it was a fair bit of me playing around with the exact cost function to get these results. Lots of empirical analysis.

The two controllers generate results that are identical to visual inspection. However, especially in the 3-link arm, the time required to run the FDSA was significantly longer than the SPSA controller. It took approximately 140ms for the SPSA controller to run a single loop, but took FDSA on average 700ms for a single loop of calculating the control signal. Almost 5 times as long! For the same results! In directly optimizing the control signal, SPSA gets a big win over standard FDSA. So, if you’re looking to directly optimize over a loss function, SPSA is probably the way you want to go.

Conclusions

First off, I thought it was really neat to directly apply gradient approximation methods to optimizing the control signal. It’s something I haven’t tried before, but definitely makes sense, and can generate some really nice results when tuned properly. Automating the tuning is definitely I’ll be discussing in future posts, because doing it by hand takes a long time and is annoying.

In the LQR, the gradient approximation was best done by the FDSA. I think the main reasons for this is that in solving for the control signal the LQR algorithm uses matrix inverses, and any errors in the linear approximations to the dynamics are going to be amplified quite a bit. If I did anything less than 10-15 iterations (20 for the 3-link arm) in the SPSA approximation then things exploded. Also, here the SPSA algorithm required a matrix inverse, where the FDSA didn’t. This is because we only varied one parameter at a time in FDSA, and the effects of changing each was isolated. In the SPSA case, we had to consider the changes across all the variables and the resulting effects all at once, essentially noting which variables changed by how much and the changes in each case, and averaging. Here, even with the more complex 3-link arm, FDSA was faster, so I’m going to stick with it in my LQR and iLQR implementations.

In the direct control signal optimization SPSA beat the pants off of FDSA. It was almost 5 times faster for control of the 3-link arm. This was, again, because in this case we could use noisy samples of the gradient of the loss function and relied on noise to cancel itself out as we iterated. So we only needed 2 samples of the loss function in SPSA, where in FDSA we needed 2*num_parameters. And although this generated pretty good results I would definitely be hesitant against using this for any more complicated systems, because tuning that cost function to get out a good trajectory was a pain. If you’re interested in playing around with this, you can check out the code for the gradient controllers up on my GitHub.

## The iterative Linear Quadratic Regulator algorithm

A few months ago I posted on Linear Quadratic Regulators (LQRs) for control of non-linear systems using finite-differences. The gist of it was at every time step linearize the dynamics, quadratize (it could be a word) the cost function around the current point in state space and compute your feedback gain off of that, as though the dynamics were both linear and consistent (i.e. didn’t change in different states). And that was pretty cool because you didn’t need all the equations of motion and inertia matrices etc to generate a control signal. You could just use the simulation you had, sample it a bunch to estimate the dynamics and value function, and go off of that.

The LQR, however, operates with maverick disregard for changes in the future. Careless of the consequences, it optimizes assuming the linear dynamics approximated at the current time step hold for all time. It would be really great to have an algorithm that was able to plan out and optimize a sequence, mindful of the changing dynamics of the system.

This is exactly the iterative Linear Quadratic Regulator method (iLQR) was designed for. iLQR is an extension of LQR control, and the idea here is basically to optimize a whole control sequence rather than just the control signal for the current point in time. The basic flow of the algorithm is:

1. Initialize with initial state $x_0$ and initial control sequence $\textbf{U} = [u_{t_0}, u_{t_1}, ..., u_{t_{N-1}}]$.
2. Do a forward pass, i.e. simulate the system using $(x_0, \textbf{U})$ to get the trajectory through state space, $\textbf{X}$, that results from applying the control sequence $\textbf{U}$ starting in $x_0$.
3. Do a backward pass, estimate the value function and dynamics for each $(\textbf{x}, \textbf{u})$ in the state-space and control signal trajectories.
4. Calculate an updated control signal $\hat{\textbf{U}}$ and evaluate cost of trajectory resulting from $(x_0, \hat{\textbf{U}})$.
1. If $|(\textrm{cost}(x_0, \hat{\textbf{U}}) - \textrm{cost}(x_0, \textbf{U})| < \textrm{threshold}$ then we've converged and exit.
2. If $\textrm{cost}(x_0, \hat{\textbf{U}}) < \textrm{cost}(x_0, \textbf{U})$, then set $\textbf{U} = \hat{\textbf{U}}$, and change the update size to be more aggressive. Go back to step 2.
3. If $\textrm{cost}(x_0, \hat{\textbf{U}}) \geq$ $\textrm{cost}(x_0, \textbf{U})$ change the update size to be more modest. Go back to step 3.

There are a bunch of descriptions of iLQR, and it also goes by names like ‘the sequential linear quadratic algorithm’. The paper that I’m going to be working off of is by Yuval Tassa out of Emo Todorov’s lab, called Control-limited differential dynamic programming. And the Python implementation of this can be found up on my github in my Control repo. Also, a big thank you to Dr. Emo Todorov who provided Matlab code for the iLQG algorithm, which was super helpful.

Defining things

So let’s dive in. Formally defining things, we have our system $\textbf{x}$, and dynamics described with the function $\textbf{f}$, such that

$\textbf{x}_{t+1} = \textbf{f}(\textbf{x}_t, \textbf{u}_t),$

where $\textbf{u}$ is the input control signal. The trajectory $\{\textbf{X}, \textbf{U}\}$ is the sequence of states $\textbf{X} = \{\textbf{x}_0, \textbf{x}_1, ..., \textbf{x}_N\}$ that result from applying the control sequence $\textbf{U} = \{\textbf{u}_0, \textbf{u}_1, ..., \textbf{u}_{N-1}\}$ starting in the initial state $\textbf{x}_0$.

Now we need to define all of our cost related equations, so we know exactly what we’re dealing with.

Define the total cost function $J$, which is the sum of the immediate cost, $\ell$, from each state in the trajectory plus the final cost, $\ell_f$:

$J(\textbf{x}_0, \textbf{U}) = \sum\limits^{N-1}\limits_{t=0} \ell(\textbf{x}_t, \textbf{u}_t) + \ell_f(\textbf{x}_N).$

Letting $\textbf{U}_t = \{\textbf{u}_t, \textbf{u}_{t+1}, ..., \textbf{U}_{N-1}\}$, we define the cost-to-go as the sum of costs from time $t$ to $N$:

$J_t(\textbf{x}, \textbf{U}_t) = \sum\limits^{N-1}\limits_{i=t} \ell(\textbf{x}_i, \textbf{u}_i) + \ell_f(\textbf{x}_N).$

The value function $V$ at time $t$ is the optimal cost-to-go from a given state:

$V_t(\textbf{x}) = \min\limits_{\textbf{U}_t} J_t(\textbf{x}, \textbf{U}_t),$

where the above equation just says that the optimal cost-to-go is found by using the control sequence $\textbf{U}_t$ that minimizes $J_t$.

At the final time step, $N$, the value function is simply

$V(\textbf{x}_N) = \ell_f(\textbf{x}_N).$

For all preceding time steps, we can write the value function as a function of the immediate cost $\ell(\textbf{x}, \textbf{u})$ and the value function at the next time step:

$V(\textbf{x}) = \min\limits_{\textbf{u}} \left[ \ell(\textbf{x}, \textbf{u}) + V(\textbf{f}(\textbf{x}, \textbf{u})) \right].$

NOTE: In the paper, they use the notation $V'(\textbf{f}(\textbf{x}, \textbf{u}))$ to denote the value function at the next time step, which is redundant since $\textbf{x}_{t+1} = \textbf{f}(\textbf{x}_t, \textbf{u}_t)$, but it comes in handy later when they drop the dependencies to simplify notation. So, heads up: $V' = V(\textbf{f}(\textbf{x}, \textbf{u})$.

Forward rollout

The forward rollout consists of two parts. The first part is to simulating things to generate the $(\textbf{X}, \textbf{U}),$ from which we can calculate the overall cost of the trajectory, and find out the path that the arm will take. To improve things though we’ll need a lot of information about the partial derivatives of the system, calculating these is the second part of the forward rollout phase.

To calculate all these partial derivatives we’ll use $(\textbf{X}, \textbf{U})$. For each $(\textbf{x}_t, \textbf{u}_t)$ we’ll calculate the derivatives of $\textbf{f}(\textbf{x}_t, \textbf{u}_t)$ with respect to $\textbf{x}_t$ and $\textbf{u}_t$, which will give us what we need for our linear approximation of the system dynamics.

To get the information we need about the value function, we’ll need the first and second derivatives of $\ell(\textbf{x}_t, \textbf{u}_t)$ and $\ell_f(\textbf{x}_t, \textbf{x}_t)$ with respect to $\textbf{x}_t$ and $\textbf{u}_t$.

So all in all, we need to calculate $\textbf{f}_\textbf{x}$, $\textbf{f}_\textbf{u}$, $\ell_\textbf{x}$, $\ell_\textbf{u}$, $\ell_\textbf{xx}$, $\ell_\textbf{ux}$, $\ell_\textbf{uu}$, where the subscripts denote a partial derivative, so $\ell_\textbf{x}$ is the partial derivative of $\ell$ with respect to $\textbf{x}$, $\ell_\textbf{xx}$ is the second derivative of $\ell$ with respect to $\textbf{x}$, etc. And to calculate all of these partial derivatives, we’re going to use finite differences! Just like in the LQR with finite differences post. Long story short, load up the simulation for every time step, slightly vary one of the parameters, and measure the resulting change.

Once we have all of these, we’re ready to move on to the backward pass.

Backward pass

Now, we started out with an initial trajectory, but that was just a guess. We want our algorithm to take it and then converge to a local minimum. To do this, we’re going to add some perturbing values and use them to minimize the value function. Specifically, we’re going to compute a local solution to our value function using a quadratic Taylor expansion. So let’s define $Q(\delta \textbf{x}, \delta \textbf{u})$ to be the change in our value function at $(\textbf{x}, \textbf{u})$ as a result of small perturbations $(\delta \textbf{x}, \delta \textbf{u})$:

$Q(\delta \textbf{x}, \delta \textbf{u}) = \ell (\textbf{x} + \delta \textbf{x}, \textbf{u} + \delta \textbf{u}) + V(\textbf{f}(\textbf{x} + \delta\textbf{x}, \textbf{u} + \delta \textbf{u})).$

The second-order expansion of $Q$ is given by:

$Q_\textbf{x} = \ell_\textbf{x} + \textbf{f}_\textbf{x}^T V'_\textbf{x},$

$Q_\textbf{u} = \ell_\textbf{u} + \textbf{f}_\textbf{u}^T V'_\textbf{x},$

$Q_\textbf{xx} = \ell_\textbf{xx} + \textbf{f}_\textbf{x}^T V'_\textbf{xx} \textbf{f}_\textbf{x} + V'_\textbf{x} \cdot \textbf{f}_\textbf{xx},$

$Q_\textbf{ux} = \ell_\textbf{ux} + \textbf{f}_\textbf{u}^T V'_\textbf{xx} \textbf{f}_\textbf{x}+ V'_\textbf{x} \cdot \textbf{f}_\textbf{ux},$

$Q_\textbf{uu} = \ell_\textbf{uu} + \textbf{f}_\textbf{u}^T V'_\textbf{xx} \textbf{f}_\textbf{u}+ V'_\textbf{x} \cdot \textbf{f}_\textbf{uu}.$

Remember that $V' = V(\textbf{f}(\textbf{x}, \textbf{u}))$, which is the value function at the next time step. NOTE: All of the second derivatives of $\textbf{f}$ are zero in the systems we’re controlling here, so when we calculate the second derivatives we don’t need to worry about doing any tensor math, yay!

Given the second-order expansion of $Q$, we can to compute the optimal modification to the control signal, $\delta \textbf{u}^*$. This control signal update has two parts, a feedforward term, $\textbf{k}$, and a feedback term $\textbf{K} \delta\textbf{x}$. The optimal update is the $\delta\textbf{u}$ that minimizes the cost of $Q$:

$\delta\textbf{u}^*(\delta \textbf{x}) = \min\limits_{\delta\textbf{u}}Q(\delta\textbf{x}, \delta\textbf{u}) = \textbf{k} + \textbf{K}\delta\textbf{x},$

where $\textbf{k} = -Q^{-1}_\textbf{uu} Q_\textbf{u}$ and $\textbf{K} = -Q^{-1}_\textbf{uu} Q_\textbf{ux}.$

Derivation can be found in this earlier paper by Li and Todorov. By then substituting this policy into the expansion of $Q$ we get a quadratic model of $V$. They do some mathamagics and come out with:

$V_\textbf{x} = Q_\textbf{x} - \textbf{K}^T Q_\textbf{uu} \textbf{k},$

$V_\textbf{xx} = Q_\textbf{xx} - \textbf{K}^T Q_\textbf{uu} \textbf{K}.$

So now we have all of the terms that we need, and they’re defined in terms of the values at the next time step. We know the value of the value function at the final time step $V_N = \ell_f(\textbf{x}_N)$, and so we’ll simply plug this value in and work backwards in time recursively computing the partial derivatives of $Q$ and $V$.

Calculate control signal update

Once those are all calculated, we can calculate the gain matrices, $\textbf{k}$ and $\textbf{K}$, for our control signal update. Huzzah! Now all that’s left to do is evaluate this new trajectory. So we set up our system

$\hat{\textbf{x}}_0 = \textbf{x}_0,$

$\hat{\textbf{u}}_t = \textbf{u}_t + \textbf{k}_t + \textbf{K}_t (\hat{\textbf{x}}_t - \textbf{x}_t),$

$\hat{\textbf{x}}_{t+1} = \textbf{f}(\hat{\textbf{x}}_t, \hat{\textbf{u}}_t),$

and record the cost. Now if the cost of the new trajectory $(\hat{\textbf{X}}, \hat{\textbf{U}})$ is less than the cost of $(\textbf{X}, \textbf{U})$ then we set $\textbf{U} = \hat{\textbf{U}}$ and go do it all again! And when the cost from an update becomes less than a threshold value, call it done. In code this looks like:

if costnew < cost:
sim_new_trajectory = True

if (abs(costnew - cost)/cost) < self.converge_thresh:
break


Of course, another option we need to account for is when costnew > cost. What do we do in this case? Our control update hasn’t worked, do we just exit?

The Levenberg-Marquardt heuristic
No! Phew.

The control signal update in iLQR is calculated in such a way that it can behave like Gauss-Newton optimization (which uses second-order derivative information) or like gradient descent (which only uses first-order derivative information). The is that if the updates are going well, then lets include curvature information in our update to help optimize things faster. If the updates aren’t going well let’s dial back towards gradient descent, stick to first-order derivative information and use smaller steps. This wizardry is known as the Levenberg-Marquardt heuristic. So how does it work?

Something we skimmed over in the iLQR description was that we need to calculate $Q^{-1}_\textbf{uu}$ to get the $\textbf{k}$ and $\textbf{K}$ matrices. Instead of using np.linalg.pinv or somesuch, we’re going to calculate the inverse ourselves after finding the eigenvalues and eigenvectors, so that we can regularize it. This will let us do a couple of things. First, we’ll be able to make sure that our estimate of curvature ($Q_\textbf{uu}^{-1}$) stays positive definite, which is important to make sure that we always have a descent direction. Second, we’re going to add a regularization term to the eigenvalues to prevent them from exploding when we take their inverse. Here’s our regularization implemented in Python:


Q_uu_evals, Q_uu_evecs = np.linalg.eig(Q_uu)
Q_uu_evals[Q_uu_evals < 0] = 0.0
Q_uu_evals += lamb
Q_uu_inv = np.dot(Q_uu_evecs,
np.dot(np.diag(1.0/Q_uu_evals), Q_uu_evecs.T))


Now, what happens when we change lamb? The eigenvalues represent the magnitude of each of the eigenvectors, and by taking their reciprocal we flip the contributions of the vectors. So the ones that were contributing the least now have the largest singular values, and the ones that contributed the most now have the smallest eigenvalues. By adding a regularization term we ensure that the inverted eigenvalues can never be larger than 1/lamb. So essentially we throw out information.

In the case where we’ve got a really good approximation of the system dynamics and value function, we don’t want to do this. We want to use all of the information available because it’s accurate, so make lamb small and get a more accurate inverse. In the case where we have a bad approximation of the dynamics we want to be more conservative, which means not having those large singular values. Smaller singular values give a smaller $Q_\textbf{uu}^{-1}$ estimate, which then gives smaller gain matrices and control signal update, which is what we want to do when our control signal updates are going poorly.

How do you know if they’re going poorly or not, you now surely ask! Clever as always, we’re going to use the result of the previous iteration to update lamb. So adding to the code from just above, the end of our control update loop is going to look like:

lamb = 1.0 # initial value of lambda
...
if costnew < cost:
lamb /= self.lamb_factor
sim_new_trajectory = True

if (abs(costnew - cost)/cost) < self.converge_thresh:
break
else:
lamb *= self.lamb_factor
if lamb > self.max_lamb:
break


And that is pretty much everything! OK let’s see how this runs!

Simulation results

If you want to run this and see for yourself, you can go copy my Control repo, navigate to the main directory, and run

python run.py arm2 reach


or substitute in arm3. If you’re having trouble getting the arm2 simulation to run, try arm2_python, which is a straight Python implementation of the arm dynamics, and should work no sweat for Windows and Mac.

Below you can see results from the iLQR controller controlling the 2 and 3 link arms (click on the figures to see full sized versions, they got distorted a bit in the shrinking to fit on the page), using immediate and final state cost functions defined as:

l = np.sum(u**2)


and

pos_err = np.array([self.arm.x[0] - self.target[0],
self.arm.x[1] - self.target[1]])
l = (wp * np.sum(pos_err**2) + # pos error
wv * np.sum(x[self.arm.DOF:self.arm.DOF*2]**2)) # vel error


where wp and wv are just gain values, x is the state of the system, and self.arm.x is the $(x,y)$ position of the hand. These read as “during movement, penalize large control signals, and at the final state, have a big penalty on not being at the target.”

So let’s give it up for iLQR, this is awesome! How much of a crazy improvement is that over LQR? And with all knowledge of the system through finite differences, and with the full movements in exactly 1 second! (Note: The simulation speeds look different because of my editing to keep the gif sizes small, they both take the same amount of time for each movement.)

Changing cost functions
Something that you may notice is that the control of the 3 link is actually straighter than the 2 link. I thought that this might be just an issue with the gain values, since the scale of movement is smaller for the 2 link arm than the 3 link there might have been less of a penalty for not moving in a straight line, BUT this was wrong. You can crank the gains and still get the same movement. The actual reason is that this is what the cost function specifies, if you look in the code, only $\ell_f(\textbf{x}_N)$ penalizes the distance from the target, and the cost function during movement is strictly to minimize the control signal, i.e. $\ell(\textbf{x}_t, \textbf{u}_t) = \textbf{u}_t^2$.

Well that’s a lot of talk, you say, like the incorrigible antagonist we both know you to be, prove it. Alright, fine! Here’s iLQR running with an updated cost function that includes the end-effector’s distance from the target in the immediate cost:

All that I had to do to get this was change the immediate cost from

l = np.sum(u**2)


to

l = np.sum(u**2)
pos_err = np.array([self.arm.x[0] - self.target[0],
self.arm.x[1] - self.target[1]])
l += (wp * np.sum(pos_err**2) + # pos error
wv * np.sum(x[self.arm.DOF:self.arm.DOF*2]**2)) # vel error


where all I had to do was include the position penalty term from the final state cost into the immediate state cost.

Changing sequence length
In these simulations the system is simulating at .01 time step, and I gave it 100 time steps to reach the target. What if I give it only 50 time steps?

It looks pretty much the same! It’s just now twice as fast, which is of course achieved by using larger control signals, which we don’t see, but dang awesome.

What if we try to make it there in 10 time steps??

OK well that does not look good. So what’s going on in this case? Basically we’ve given the algorithm an impossible task. It can’t make it to the target location in 10 time steps. In the implementation I wrote here, if it hits the end of it’s control sequence and it hasn’t reached the target yet, the control sequence starts over back at t=0. Remember that part of the target state is also velocity, so basically it moves for 10 time steps to try to minimize $(x,y)$ distance, and then slows down to minimize final state cost in the velocity term.

In conclusion

This algorithm has been used in a ton of things, for controlling robots and simulations, and is an important part of guided policy search, which has been used to very successfully train deep networks in control problems. It’s getting really impressive results for controlling the arm models that I’ve built here, and using finite differences should easily generalize to other systems.

iLQR is very computationally expensive, though, so that’s definitely a downside. It’s definitely less expensive if you have the equations of your system, or at least a decent approximation of them, and you don’t need to use finite differences. But you pay for the efficiency with a loss in generality.

There are also a bunch of parameters to play around with that I haven’t explored at all here, like the weights in the cost function penalizing the magnitude of the cost function and the final state position error. I showed a basic example of changing the cost function, which hopefully gets across just how easy changing these things out can be when you’re using finite differences, and there’s a lot to play around with there too.

Implementation note
In the Yuval and Todorov paper, they talked about using backtracking line search when generating the control signal. So the algorithm they had when generating the new control signal was actually:

$\hat{\textbf{u}}_t = \hat{\textbf{u}}_t + \alpha\textbf{k}_t + \textbf{K}_t(\hat{\textbf{x}}_t - \textbf{x}_t)$

where $\alpha$ was the backtracking search parameter, which gets set to one initially and then reduced. It’s very possible I didn’t implement it as intended, but I found consistently that $\alpha = 1$ always generated the best results, so it was just adding computation time. So I left it out of my implementation. If anyone has insights on an implementation that improves results, please let me know!

And then finally, another thank you to Dr. Emo Todorov for providing Matlab code for the iLQG algorithm, which was very helpful, especially for getting the Levenberg-Marquardt heuristic implemented properly.

## Linear-Quadratic Regulation for non-linear systems using finite differences

One of the standard controllers in basic control theory is the linear-quadratic regulator (LQR). There is a finite-horizon case (where you have a limited amount of time), and an infinite-horizon case (where you don’t); in this post, for simplicity, we’re only going to be dealing with the infinite-horizon case.

The LQR is designed to handle a very specific kind of problem. First, it assumes you are controlling a system with linear dynamics, which means you can express them as

$\dot{\textbf{x}} = \textbf{A}\textbf{x} + \textbf{B}\textbf{u}$,

where $\textbf{x}$ and $\dot{\textbf{x}}$ are the state and its time derivative, $\textbf{u}$ is the input, and $\textbf{A}$ and $\textbf{B}$ capture the effects of the state and input on the derivative. And second, it assumes that the cost function, denoted $J$, is a quadratic of the form

$J = \int_0^{\infty} \left( (\textbf{x} - \textbf{x}^*)^T \textbf{Q} (\textbf{x} - \textbf{x}^*) + \textbf{u}^T \textbf{R} \textbf{u} \right) dt$

where $\textbf{x}^*$ is the target state, and $\textbf{Q} = \textbf{Q}^T \geq 0$ and $\textbf{R} = \textbf{R}^T \geq 0$ are weights on the cost of not being at the target state and applying a control signal. The higher $\textbf{Q}$ is, the more important it is to get to the target state asap, the higher $\textbf{R}$ is, the more important it is to keep the control signal small as you go to the target state.

The goal of the LQR is to calculate a feedback gain matrix $\textbf{K}$ such that

$\textbf{u} = -\textbf{K} \textbf{x},$

drives the system to the target. When the system is a linear system with a quadratic cost function, this can be done optimally. There is lots of discussion elsewhere about LQRs and their derivation, so I’m not going to go into that with this post. Instead, I’m going to talk about applying LQRs to non-linear systems, and using finite differences to do it, which works when you have a readily accessible simulation of the system on hand. The fun part is that by using finite differences you can get this to work without working out the dynamics equations yourself.

Using LQRs on non-linear systems

As you may have noticed, non-linear systems violate the first assumption of a linear quadratic regulator; that the system is linear. That doesn’t mean that we can’t apply it, it just means that it’s not going to be optimal. How poorly the LQR will perform depends on a few things, two important factors being how non-linear the system dynamics actually are, and how often you’re able to update the feedback gain matrix $\textbf{K}$. To apply LQR to non-linear systems we’re just going to close our eyes and pretend that the system dynamics are linear, i.e. they fit the form

$\dot{\textbf{x}} = \textbf{A}\textbf{x} + \textbf{B}\textbf{u}.$

We’ll do this by approximating the actual dynamics of the system linearly. We’ll then solve for our gain value $\textbf{K}$, generate our control signal for this timestep, and then re-approximate the dynamics again at the next time step and solve for $\textbf{K}$ from the new state. The more non-linear the system dynamics are, the less appropriate $\textbf{K}$ will be for generating our control signal $\textbf{u}$ as we move away from the state $\textbf{K}$ was calculated in; this is why update time of the LQR can become an important factor.

Using finite-differences to approximate system dynamics

An important question, then, is how do we find this system approximation? How can we calculate the $\textbf{A}$ and $\textbf{B}$ matrices that we then use to solve for $\textbf{K}$? If we know the dynamics of the system to be

$\dot{\textbf{x}} = f(\textbf{x}, \textbf{u})$,

then we can calculate

$\textbf{A} = \frac{\partial f(\textbf{x}, \textbf{u})}{\partial \textbf{x}}, \;\;\;\; \textbf{B} = \frac{\partial f(\textbf{x}, \textbf{u})}{\partial \textbf{u}}$.

If you’re going to try this for the 3-link arm, though, get out Mathematica. Do not try this by hand. If you disregard my warning and foolhardily attempt such a derivation you will regret, repent, and then appeal to Wolfram Alpha for salvation. These equations quickly become terrible and long even for seemingly not-so-complicated systems.

There are a few ways to skirt this. Here we’re going to assume that the system under control is a simulation, or that we at least have access to an accurate model, and use the finite differences method to compute these values. The idea behind finite differences is to approximate the rate of change of the function $f$ at the point $x$ by sampling $f$ near $x$ and using the difference to calculate $\dot{f}(x)$. Here’s a picture for a 1D system:

So here, our current state $x$ is the blue dot, and the red dots represent the sample points $x + \Delta x$ and $x - \Delta x$. We can then calculate

$\dot{f}(x) \approx \frac{f(x+\Delta x) - f(x-\Delta x)}{2\Delta x},$

and you can see the actual rate of change of $f$ at $x$ plotted in the blue dashed line, and the approximated rate of change calculated using finite differences plotted in the red dashed line. We can also see that the approximated derivative is only accurate near $x$ (the blue dot).

Back in our multi-dimensional system, to use finite differences to calculate the derivative with respect to the state and the input we’re going to vary each of the dimensions of the state and input by some small amount one at a time, calculating the effects of each one by one. Here’s a chunk of pseudo-code to hopefully clarify this idea:

eps = 1e-5
A = np.zeros((len(current_state), len(current_state))
for ii in range(len(current_state)):
x = current_state.copy()
x[ii] += eps
x_inc = simulate_system(state=x, input=control_signal)
x = current_state.copy()
x[ii] -= eps
x_dec = simulate_system(state=x, input=control_signal)
A[:,ii] = (x_inc - x_dec) / (2 * eps)

B = np.zeros((len(current_state), len(control_signal))
for ii in range(len(control_signal)):
u = control_signal.copy()
u[ii] += eps
x_inc = simulate_system(state=current_state, input=u)
u = control_signal.copy()
u[ii] -= eps
x_dec = simulate_system(state=current_state, input=u)
B[:,ii] = (x_inc - x_dec) / (2 * eps)


Now we’re able to generate our $\textbf{A}$ and $\textbf{B}$ matrices we have everything we need to solve for our feedback gain matrix $\textbf{K}$! Which is great.

Note on using finite differences in continuous vs discrete setup

Something that’s important to straighten out too is what exactly is returned by the simulate_system function in the code above. In the continuous case, your system is captured as

$\dot{\textbf{x}} = \textbf{A}\textbf{x} + \textbf{B}\textbf{u},$,

where in the discrete case your system is defined

$\textbf{x}(t+1) = \textbf{A}\textbf{x}(t) + \textbf{B}\textbf{u}(t).$

If you are calculating your feedback gain matrix $\textbf{K}$ using the continuous solution to the algebraic Riccati equation, then you need to be returning $\dot{\textbf{x}}(t)$. If you’re solving for $\textbf{K}$ using the discrete solution to the algebraic Riccati equation you need to return $\textbf{x}(t+1)$. This was just something that I came across as I was coding and so I wanted to mention it here in case anyone else stumbled across it!

Applying LQR to 2 and 3 link arm control

Alright! Let’s have a look at how the LQR does controlling non-linear systems. Below we have the control of a 2-link arm compared to a 3-link arm, and you can see the control of the 2-link arm is better. This is a direct result of the dynamics of a 3-link arm being significantly more complex.

Note on controlling at different timesteps

When I was first testing the LQR controller I expected the effects of different control update times to be a lot more significant than it was. As it turns out, for controlling a 3-link arm, there’s not really a visible difference in a controller that is updating every .01 seconds vs every .001 seconds vs every .0001 seconds. Let’s have a look:

Can’t even tell, eh? Fun fact, the simulation took 1 minute 30 seconds at .01 seconds time step and 45 minutes at .0001 seconds time step. The left-most animation is the .01 seconds and the right-most the .0001 seconds. But why is there seemingly so little difference? Well, this boils down to the dynamics of the 3-link arm changing actually pretty slowly. Below I’ve plotted just a few of the elements from the $\textbf{A}$, $\textbf{B}$, and $\textbf{K}$ matrices over .5 seconds of simulation time:

So, there are some obvious points where sampling the dynamics at a .01 time step is noticeably less accurate, but all in all there’s not a huuuggge difference between sampling at .01 and .0001 seconds. If you’re just watching the end-effector path it’s really not very noticeable. You can see how the elements of $\textbf{A}$ and $\textbf{B}$ are changing fairly slowly; this means that $\textbf{K}$ is going to be an effective feedback gain for a fair chunk of time. And the computational savings you get by sampling the dynamics and regenerating $\textbf{K}$ every .01 seconds instead of every .0001 seconds are pretty big. This was just another thing that I came across when playing around with the LQR, the take away being don’t just assume you need to update your system crazy often. You might get very comparable performance for much less computational cost.

Conclusions

All in all, the LQR controller is pretty neat! It’s really simple to set up, and generic. We don’t need any specific information about the system dynamics, like we do for effective operational space control (OSC). When we estimate the dynamics with finite differences, all need is a decent system model that we can sample. Again, the more non-linear the system, of course, the less effective a LQR will be. If you’re interested in playing around with one, or generating the figures that I show above, the code is all up and running on my Github for you to explore.