Spiking neural networks take control

Back in September I had an review article in Science Robotics published, discussing new work from Abadia et al, 2021 titled A cerebellar-based solution to the nondeterministic time delay problem in robotic control. In my review I talk about the parallel’s between the brain inspired approach Abadia et al used to create a neural circuit that can effectively control a compliant, highly nonlinear robotic system in the face of variable feedback delay, and the history and application of convolutional neural networks. The basic premise is, to no one’s surprise I’m sure, that there is still much to learn from the brain in building effective artificial neural networks. If you’d like to read the article, you can now check it out here: Spiking neural networks take control.

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ABR Control supports Mujoco!

https://github.com/abr/abr_control

And it has for a while, actually. ‘Oh great’, you may be thinking, ‘but I can never afford the Mujoco license, I’m not a tenured professor!’

Well fear not, friends! Mujoco recently became free for all to use! You can now download their public license and be on your merry way. The associated downside is there’s also no more official support from Dr. Emo Todorov for paid subscribers on the forum. This, I’m sure, is unrelated to my barrage of questions over the years (really I’ve only posted 6 times so it actually probably is unrelated…probably).

And ABR Control has all your favorite examples written up for Mujoco! We’ve got the all the classics:

• operational space position control,
• operational space orientation control,
• operational space position AND orientation control,
• nonlinear adaptive control,
• control of a ball joint arm,
• path planning with force control,
• path planning with inverse kinematics,
• and many more!

You can find the full set of examples in the docs/examples folder of the repository.

If you’re not familiar with Mujoco, I highly recommend checking it out. What’s extra nice about Mujoco is that all of the simulation information used to compute the forward dynamics is available (we access it using the mujoco-py package through our mujoco interface file), which means our control is exact. For simulators like CoppelliaSim we don’t have access to this information, so we’re calculating the values ourselves, and there’s often a slight discrepancy. With Mujoco, however, if you set up a floating controller you can bet your bottom dollar that arm will float.

The other benefit of Mujoco, of course, is that it’s super fast, and handles contact dynamics well. Great for robotics applications. Also the visualizations are really slick:

The operational space position and orientation controller running with the path planner running with the Jaco2 arm from Kinova model. The red rectangle is the final target, and the blue rectangle is the path planner target.

I’m very excited that the Mujoco license is free, and I encourage everyone to download it and start playing around. It’s by far the best robotics simulator I’ve used, and it runs fast enough that you can use it to calculate everything you need for controlling a physical robot arm as well (which we’ve also done with the above Jaco2 arm)! The Jaco2, UR5, and a simple geometric arm are included in the ABR Control package, and if you want to have your own arm in there you can follow this previous post about how to build your own model.

Workshop talk – Adaptive neurorobotics using Nengo and the Loihi

I gave a talk last month at the HALO (Harware and Algorithms for Learning On-achip) workshop at ICCAD (the International Conference on Computed Aided Design). It’s goes over the work from our paper that came out in Frontiers in Neurorobotics in October, titled: Nengo and Low-Power AI Hardware for Robust, Embedded Neurorobotics. I’ve put it up on YouTube, and am sharing it here, now! I recommend watching it at 2x speed, as I talk pretty slowly (I’ve been told it’s a good ASMR voice).

Building models in Mujoco

Mujoco is an awesome simulation tool. You’re probably familiar with it from it’s use in the OpenAI gym, or from it featuring in articles and videos on model predictive control and robots learning to walk research. If you’re not familiar with it, you should check it out! Mujoco provides super fast dynamics simulation with a focus on contact dynamics. It’s especially useful for simulating robotic arms and gripping tasks.

There is a growing model repository, but it’s not unlikely you’re going to want to build your own model. It turns out this is relatively easy in Mujoco. There are two parts to defining a model: 1) The STL files, which are 3D models of the robot components, and 2) the XML file, which specifies the kinematic and dynamic relationships in the model. For STL manipulation, we use the program SketchUp because it is freely available with both offline and online versions.

This post will start by going over some settings in SketchUp assuming you haven’t used it before. We’ll then walk through how to set up the STL files for the individual components of the robot model (assuming that you already have the 3D modeling completed), steps for generating a Mujoco XML robot description file, and go over a couple of the more common problems that come up in the process.

If you don’t have the 3D modeling aleady completed, it’s worth doing a quick search around online to see if someone else has already made it because there are a lot out there not posted officially on any model repo. Github is a good source, and so is https://www.traceparts.com/en. We’re not going to cover any 3D modeling how-to here.

Shout out to my colleague Pawel Jaworski who worked through this process and wrote up the first of this document!

Sketchup Settings – Set model units on import

Before opening your STL, in the open file window select your file and click on the Options button next to Import. Select the units that the model was defined in. If you import your object and cannot see it, it is likely that the units selected during the import were incorrect and the object is simply too small to see.

Skechup Settings – Set model units for export

Mujoco uses the units specified in your STL models. ABR Control uses metres, and things generally are easiest when everyone is using the same units. To change your units to meters, go to Model Info > Length Units > 0.0m.

Sketchup Settings – Measurement precision

Before doing any measuring in SketchUp, change your measurement precision in Model Info > Units. precision to the largest number of digits to make sure you get accurate measurements (which you will need when building the Mujoco XML), the default in the online version is 1 significant digit.

Sketchup Settings – Disable snapping

If you are like me then auto-snapping is often a huge pet peeve. To disable it, go to Model Info > Units and unclick Length snapping and Angle snapping.

Sketchup Settings – Xray mode

It can be helpful to set the default view to xray mode so you can see inside the model when manipulating the components. To do this go to Styles > Default Styles > X-ray on the right hand side.

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Generating component STL files

A common starting point for modeling is that you have a 3D model of the full robot. When this is the case, the first step in building a Mujoco model is to generate separate STL files for each of the components of the robot that you want to be able to move independently. For each of these component STL files, we want the point where it connects to the joint to be at the origin (0, 0, 0). We’ve found that this streamlines the process of building the model in the XML and makes it much easier to correctly specify the inertial properties.

If your 3D model is already broken up into each dynamic component (i.e. each arm section that moves independently) then you can skip to the section on centering at the origin.

Exporting components to separate STLs

Make sure you have the model unlocked. To do so, choose the Select tool and highlight the entire model. If the outline is red, it is locked. To unlock it, right click and choose Unlock.

Locked

Unlocked

Unlocking the UR5 model in SketchUp

To get each component piece: Select the entire arm, deselect your component with Ctrl + Shift + Click, and then delete the rest. Go to the folder icon in the top left and select Export > STL. When exporting your model, make sure to not save your model in ASCII format, choose Binary. Repeat this until you have exported every part as its own STL.

How to save an individual component from a complete robot model

Positioning your object at the origin

For each component, identify where it will connect to the previous component, i.e. where the joint will attach it to the rest of the robot. We want to set up the STL such that this point is at the origin. By doing this we simplify building the XML file later. For the base of the robot this would simply be the bottom of the link, centered.

The easiest way to move the object to the origin is to click a point on the object with the move tool, type [0, 0, 0], and hit enter. This will move the selected point to the origin.

If your object is asymmetrical or you want to center on a point that is difficult to select with your mouse, you can use the measurement tool to set some guidelines. Setting two intersecting lines can help greatly in finding the center of an object (see dotted lines in figure).

Using dotted black guidelines to select an off-center point

Measure the offset distance for the next link

Once the component is appropriately placed at the origin, the next thing we need is the distance to the next attachment point from the origin.

The screenshots below show an example of drawing a verticle guideline to make selecting the next attachment point easy.

Making a verticle reference line that extends beyond the point of interest

Selecting the attachment point of the next component to measure the offset distance

When the next attachment point is selected, the Measurements box will display its coordinates. We will need to know all of the offset distances for each component for building the XML file, so make sure you take note!

Some measuring tips

• Press and hold a directional button while measuring to lock an axis. The up arrow locks the z (blue) axis, the left arrow locks the y (green) axis, and the right arrow locks the x (red) axis.
• The measuring tape line changes color to match an axis colors when it is parallel to any axis.
• Notes on how to use the Tape Measure tool effectively

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Building your model XML

The easiet way to put together your robot is to build your model up one joint and component at a time, running the simulation to make sure that it works as expected, and then adding the next piece. You will be tempted to assemble a bunch of the model all at once, but this is the siren’s song. You must be strong and resist!

For testing, we created the basic script below, which uses the ABR Control library. To check alignment you can press the space bar to pause the simulation, if you want to make the arm to move then you can change the script to send in some small value instead of zeros.

import mujoco_py as mjp
import glfw
import numpy as np

from abr_control.interfaces.mujoco import Mujoco
from abr_control.arms.mujoco_config import MujocoConfig

robot_config = MujocoConfig(xml_file='example.xml', folder='.')
interface = Mujoco(robot_config, dt=0.001)
interface.connect()

try:
    while True:
        if interface.viewer.exit:
            glfw.destroy_window(interface.viewer.window)
            break

        interface.send_forces(np.zeros(robot_config.N_JOINTS))

finally:
    interface.disconnect()

To run this script with your model, place it in the same directory as your XML file, with a folder called ‘meshes’ that has all of your STLs. If you call your XML file something other than ‘example.xml’ be sure to change that in the script above.

XML model parts

When you’re building your XML file, you’re going to want to have the Mujoco XML Reference constantly open. It’s super detailed and thorough. You can also check out this basic template file on my GitHub which shows the different parts of the file (you will need to fill this out with your own STLs and measurements).

Inside the standard definition tags, the main parts that we’ll be using are:

• <asset> – where you import your STLs using the <mesh> tag
• <worldbody> – where all of the objects in the simulation are defined using <body> tags, including lights, the floor, and of course your robot
• <acutator> – where you specify which joints defined in your robot can be actuated. The ordering of these joints is important, it should be the same as their order in the kinematic tree (i.e. start with joint0 up to the last joint.)

Body building

The meat of the file is of course in the sequence of <body> tags that define the robot model. Each section of the robot looks like this:

<body name="link_n" pos="0 0 0">
    <joint name="joint_n-1" axis="0 0 0" pos="0 0 0"/>
    <geom name="link_n" type="mesh" mesh="shoulder" pos="0 0 0" euler="0 0 0"/>
    <inertial pos="0 0 0" mass="0.01" diaginertia="0 0 0"/>

    <!-- nest the next joint here -->
</body>

Set body position and geoms
Set the offset from the previous body in the pos attribute on the <body> tag, instead of on the geoms. On the <joint> you can then set pos="0 0 0" which we’ve found to help simplify debugging later on.

In each body section you can have several <joint>s and <geom>s. Geoms defined on the same body will be fused together. If you have specific inertia properties for several geoms that are fused together, you will have to create a <body> for each to be able to instantiate their own <inertial> tag. Otherwise, it’s recommended to put them all in the same <body> to optimize simulation speed.

Orientation and inertia
You may need to rotate your STLs to align them properly with the rest of the robot as you build. You can do this either in the <body> or <geom> tags. If you are using an <inertial> tag for this body, we recommend you do this using the euler parameter inside the <geom> tag, instead of inside the <body> tag. If you specify rotation in the <body> tag, you also need to apply the same rotation to your <inertia> parameters, which complicates things.

If you don’t provide an <inertial> tag, the inertia properties will be inferred from the geom.

Contype and conaffinity
If you have geoms that you don’t want to collide with other parts in the model, you can set the contype and conaffinity parameters on the geom tags. This can be handy if you have a tightly-fit 3D model an run into issues with friction, we do this on the UR5 model in the ABR Control library.

End-effector tag for ABR Control
If you’re going to use the ABR Control library operational space controller, you’ll need to add a tag <body name="EE" pos="0 0 0"> at the point of the robot that you want to control. Usually the hand.

Once you’ve added on your robot body segment, save the XML and run the above python script to test out the set up. You will likely need to do some fine tuning by iteratively adjusting the parameters and viewing the model in Mujoco. We’ve found commenting out all of the joints except the joint of interest in the XML file makes assessment much easier.

Template XML file

<mujoco model="example">
    <!-- set some defaults for units and lighting -->
    <compiler angle="radian" meshdir="meshes"/>

    <!-- import our stl files -->
    <asset>
        <mesh file="base.STL" />
        <mesh file="link1.STL" />
        <mesh file="link2.STL" />
    </asset>

    <!-- define our robot model -->
    <worldbody>
        <!-- set up a light pointing down on the robot -->
        <light directional="true" pos="-0.5 0.5 3" dir="0 0 -1" />
        <!-- add a floor so we don't stare off into the abyss -->
        <geom name="floor" pos="0 0 0" size="1 1 1" type="plane" rgba="1 0.83 0.61 0.5"/>
        <!-- the ABR Control Mujoco interface expects a hand mocap -->
        <body name="hand" pos="0 0 0" mocap="true">
            <geom type="box" size=".01 .02 .03" rgba="0 .9 0 .5" contype="2"/>
        </body>

        <!-- start building our model -->
        <body name="base" pos="0 0 0">
            <geom name="link0" type="mesh" mesh="base" pos="0 0 0"/>
            <inertial pos="0 0 0" mass="0" diaginertia="0 0 0"/>

            <!-- nest each child piece inside the parent body tags -->
            <body name="link1" pos="0 0 1">
                <!-- this joint connects link1 to the base -->
                <joint name="joint0" axis="0 0 1" pos="0 0 0"/>

                <geom name="link1" type="mesh" mesh="link1" pos="0 0 0" euler="0 3.14 0"/>
                <inertial pos="0 0 0" mass="0.75" diaginertia="1 1 1"/>

                <body name="link2" pos="0 0 1">
                    <!-- this joint connects link2 to link1 -->
                    <joint name="joint1" axis="0 0 1" pos="0 0 0"/>

                    <geom name="link2" type="mesh" mesh="link2" pos="0 0 0" euler="0 3.14 0"/>
                    <inertial pos="0 0 0" mass="0.75" diaginertia="1 1 1"/>

                    <!-- the ABR Control Mujoco interface uses the EE body to -->
                    <!-- identify the end-effector point to control with OSC-->
                    <body name="EE" pos="0 0.2 0.2">
                        <inertial pos="0 0 0" mass="0" diaginertia="0 0 0" />
                    </body>
                </body>
            </body>
        </body>
    </worldbody>

    <!-- attach actuators to joints -->
    <actuator>
        <motor name="joint0_motor" joint="joint0"/>
        <motor name="joint1_motor" joint="joint1"/>
    </actuator>

</mujoco>

If you want to use the above template XML file with the example Mujoco simulation script withot having to create your own STLs, you can download the meshes folder from ABR Control’s Jaco 2 model to get going. The placement of things is wildly inaccurate but it’ll get you going. You can also checkout the abr_control/arms/jaco2/jaco2.xml and abr_control/arms/ur5/ur5.xml files for more examples.

Hopefully this is enough to get you started, if you run into questions you can post below or on the Mujoco forums. Happy modeling!

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Troubleshooting

SketchUp – I’m importing my model but I can’t see it

It is likely that the units selected during the import were incorrect and the object is simply too small to see. When opening your STL, in the open file window select your file and click on the options button next to Import to change the units.

Mujoco – My arm isn’t moving, or moves slightly and stops

In this case you likely have collisions between links. You can either add a small gap between links (make sure to account for this shift in subsequent links and joints), or you can use the contype and conaffinity tags to set up the model such that collisions between the two components are not calculated.

For example, in our UR5 model, we set the touching geoms between links to have different contype and conaffinity values so that they don’t scrape against one another and prevent movement.

Mujoco – Parts of my model are spinning around wildly

This usually arises from being instantiated in contact with another part of the model. Sometimes it looks like there is clearly no contact between different model segments, but in fact there is because of how the contact dynamics are calculated.

Only convex shapes are supported. To see the shapes being used to calculate the contact dynamics, press F1 while the model is running. 

• 

  How the STL looks when it’s loaded, lots of space between the neck and jaw

• 

  The actual shapes that are used for calculating contact dynamics.

There is no space between the neck and jaw. To address this, you need break things up into multiple component STL files and then stitch them together in the XML. For example, in the above skeleton you would need to break it up into skull and spine STLs. NB: I will not be answering any questions about why a skeleton was used in this example.

Force control of task-space orientation

So you want to use force control to control the orientation of your end-effector, eh? What a noble endeavour. I, too, wished to control the orientation of the end-effector. While the journey was long and arduous, the resulting code is short and quick to implement. All of the code for reproducing the results shown here is up on my GitHub and in the ABR Control repo. 

Introduction

There are numerous resources that introduce the topic of orientation control, so I’m not going to do a full rehash here. I will link to resources that I found helpful, but a quick google search will pull up many useful references on the basics. 

When describing the orientation of the end-effector there are three different primary methods used: Euler angles, rotation matrices, and quaternions. The Euler angles , and denote roll, pitch, and yaw, respectively. Rotation matrices specify 3 orthogonal unit vectors, and I describe in detail how to calculate them in this post on forward transforms. And quaternions are 4-dimensional vectors used to describe 3 dimensional orientation, which provide stability and require less memory and compute but are more complicated to understand.

Most modern robotics control is done using quaternions because they do not have singularities, and it is straight forward to convert them to other representations. Fun fact: Quaternion trajectories also interpolate nicely, where Euler angles and rotation matrices do not, so they are used in computer graphics to generate a trajectory for an object to follow in orientation space.

While you won’t need a full understanding of quaternions to use the orientation control code, it definitely helps if anything goes wrong. If you are looking to learn about or brush up on quaternions, or even if you’re not but you haven’t seen this resource, you should definitely check out these interactive videos by Grant Sanderson and Ben Eater. They have done an incredible job developing modules to give people an intuition into how quaternions work, and I can’t recommend their work enough. There’s also a non-interactive video version that covers the same material.

In control literature, angular velocity and acceleration are denoted and . It’s important to remember that is denoting a velocity, and not a position, as we work through things or it could take you a lot longer to understand than it otherwise might…

How to generate task-space angular forces

So, if you recall a post long ago on Jacobians, our task-space Jacobian has 6 rows:

In position control, where we’re only concerned about the position of the hand, the angular velocity dimensions are stripped out of the Jacobian so that it’s a 3 x n_joints matrix rather than a 6 x n_joints matrix. So the orientation of the end-effector is not controlled at all. When we did this, we didn’t need to worry or know anything about the form of the angular velocities.

Now that we’re interested in orientation control, however, we will need to learn up (and thank you to Yitao Ding for clarifying this point in comments on the original version of this post). The Jacobian for the orientation describes the rotational velocities around each axis with respect to joint velocities. The rotational velocity of each axis “happens” at the same time. It’s important to note that this is not the same thing as Euler angles, which are applied sequentially.

To generate our task-space angular forces we will have to generate an orientation angle error signal of the appropriate form. To do that, first we’re going to have to get and be able to manipulate the orientation representation of the end-effector and our target.

Transforming orientation between representations

We can get the current end-effector orientation in rotation matrix form quickly, using the transformation matrices for the robot. To get the target end-effector orientation in the examples below we’ll use the VREP remote API, which returns Euler angles.

It’s important to note that Euler angles can come in 12 different formats. You have to know what kind of Euler angles you’re dealing with (e.g. rotate around X then Y then Z, or rotate around X then Y then X, etc) for all of this to work properly. It should be well documented somewhere, for example the VREP API page tells us that it will return angles corresponding to x, y, and then z rotations.

The axes of rotation can be static (extrinsic rotations) or rotating (intrinsic rotations). NOTE: The VREP page says that they rotate around an absolute frame of reference, which I take to mean static, but I believe that’s a typo on their page. If you calculate the orientation of the end-effector of the UR5 using transform matrices, and then convert it to Euler angles with axes='rxyz' you get a match with the displayed Euler angles, but not with axes='sxyz'.

Now we’re going to have to be able to transform between Euler angles, rotation matrices, and quaternions. There are well established methods for doing this, and a bunch of people have coded things up to do it efficiently. Here I use the very handy transformations module from Christoph Gohlke at the University of California. Importantly, when converting to quaternions, don’t forget to normalize the quaternions to unit length.

from abr_control.arms import ur5 as arm
from abr_control.interfaces import VREP
from abr_control.utils import transformations

robot_config = arm.Config()
interface = VREP(robot_config)
interface.connect()

feedback = interface.get_feedback()
# get the end-effector orientation matrix
R_e = robot_config.R('EE', q=feedback['q'])
# calculate the end-effector unit quaternion
q_e = transformations.unit_vector(
    transformations.quaternion_from_matrix(R_e))

# get the target information from VREP
target = np.hstack([
    interface.get_xyz('target'),
    interface.get_orientation('target')])
# calculate the target orientation rotation matrix
R_d = transformations.euler_matrix(
    target[3], target[4], target[5], axes='rxyz')[:3, :3]
# calculate the target orientation unit quaternion
q_d = transformations.unit_vector(
    transformations.quaternion_from_euler(
        target[3], target[4], target[5],
        axes='rxyz'))  # converting angles from 'rotating xyz'

Generating the orientation error

I implemented 4 different methods for calculating the orientation error, from (Caccavale et al, 1998), (Yuan, 1988) and (Nakinishi et al, 2008), and then one based off some code I found on Stack Overflow. I’ll describe each below, and then we’ll look at the results of applying them in VREP.

Method 1 – Based on code from StackOverflow

Given two orientation quaternion and , we want to calculate the rotation quaternion that takes us from  to :

To isolate , we right multiply by the inverse of . All orientation quaternions are of unit length, and for unit quaternions the inverse is the same as the conjugate. To calculate the conjugate of a quaternion, denoted here, we negate either the scalar or vector part of the quaternion, but not both.

Great! Now we know how to calculate the rotation needed to get from the current orientation to the target orientation. Next, we have to get from a set of target Euler angle forces. In A new method for performing digital control system attitude computations using quaternions (Ickes, 1968), he mentions mention that

For control purposes, the last three elements of the quaternion define the roll, pitch, and yaw rotational errors…

So you can just take the vector part of the quaternion and use that as your desired Euler angle forces.

# calculate the rotation between current and target orientations
q_r = transformations.quaternion_multiply(
    q_target, transformations.quaternion_conjugate(q_e))

# convert rotation quaternion to Euler angle forces
u_task[3:] = ko * q_r[1:] * np.sign(q_r[0])

NOTE: You will run into issues when the angle is crossed where the arm ‘goes the long way around’. To account for this, use q_r[1:] * np.sign(q_r[0]). This will make sure that you always rotate along a trajectory < 180 degrees towards the target angle. The reason that this crops up is because there are multiple different quaternions that can represent the same orientation.

The following figure shows the arm being directed from and to the same orientations, where the one on the left takes the long way around, and the one on the right multiplies by the sign of the scalar component of the quaternion as specified above.

• 

• 

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Method 2 – Quaternion feedback from Resolved-acceleration control of robot manipulators: A critical review with experiments (Caccavale et al, 1998)

In section IV, equation (34) of this paper they specify the orientation error to be calculated as

where is the rotation matrix for the end-effector, and is the vector part of the unit quaternion that can be extracted from the rotation matrix

.

To implement this is pretty straight forward using the transforms.py module to handle the representation conversions:

# From (Caccavale et al, 1997)
# Section IV - Quaternion feedback
R_ed = np.dot(R_e.T, R_d)  # eq 24
q_ed = transformations.quaternion_from_matrix(R_ed)
q_ed = transformations.unit_vector(q_ed)
u_task[3:] = -np.dot(R_e, q_ed[1:])  # eq 34

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Method 3 – Angle/axis feedback from Resolved-acceleration control of robot manipulators: A critical review with experiments (Caccavale et al, 1998)

In section V of the paper, they present an angle / axis feedback algorithm, which overcomes the singularity issues that classic Euler angle methods suffer from. The algorithm defines the orientation error in equation (45) to be calculated

,

where and are the scalar and vector part of the quaternion representation of 

Where is the rotation matrix representing the desired orientation and is the rotation matrix representing the end-effector orientation. The code implementation for this looks like:

# From (Caccavale et al, 1997)
# Section V - Angle/axis feedback
R_de = np.dot(R_d, R_e.T)  # eq 44
q_ed = transformations.quaternion_from_matrix(R_de)
q_ed = transformations.unit_vector(q_ed)
u_task[3:] = -2 * q_ed[0] * q_ed[1:]  # eq 45

From playing around with this briefly, it seems like this method also works. The authors note in the discussion that it may “suffer in the case of large orientation errors”, but I wasn’t able to elicit poor behaviour when playing around with it in VREP. The other downside they mention is that the computational burden is heavier with this method than with quaternion feedback.

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Method 4 – From Closed-loop manipulater control using quaternion feedback (Yuan, 1988) and Operational space control: A theoretical and empirical comparison (Nakanishi et al, 2008)

This was the one method that I wasn’t able to get implemented / working properly. Originally presented in (Yuan, 1988), and then modified for representing the angular velocity in world and not local coordinates in (Nakanishi et al, 2008), the equation for generating error (Nakanishi eq 72):

where and are the scalar and vector components of the quaternions representing the end-effector and target orientations, respectively, and is defined in (Nakanishi eq 73):

My code for this implementation looks like:

S = np.array([
    [0.0, -q_d[2], q_d[1]],
    [q_d[2], 0.0, -q_d[0]],
    [-q_d[1], q_d[0], 0.0]])

u_task[3:] = -(q_d[0] * q_e[1:] - q_e[0] * q_d[1:] +
               np.dot(S, q_e[1:]))

If you understand why this isn’t working, if you can provide a working code example in the comments I would be very grateful.

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Generating the full orientation control signal

The above steps generate the task-space control signal, and from here I’m just using standard operational space control methods to take u_task and transform it into joint torques to send out to the arm. With possibly the caveat that I’m accounting for velocity in joint-space, not task space. Generating the full control signal looks like:

 
# which dim to control of [x, y, z, alpha, beta, gamma]
ctrlr_dof = np.array([False, False, False, True, True, True])

feedback = interface.get_feedback()
# get the end-effector orientation matrix
R_e = robot_config.R('EE', q=feedback['q'])
# calculate the end-effector unit quaternion
q_e = transformations.unit_vector(
    transformations.quaternion_from_matrix(R_e))

# get the target information from VREP
target = np.hstack([
    interface.get_xyz('target'),
    interface.get_orientation('target')])
# calculate the target orientation rotation matrix
R_d = transformations.euler_matrix(
    target[3], target[4], target[5], axes='rxyz')[:3, :3]
# calculate the target orientation unit quaternion
q_d = transformations.unit_vector(
    transformations.quaternion_from_euler(
        target[3], target[4], target[5],
        axes='rxyz'))  # converting angles from 'rotating xyz'

# calculate the Jacobian for the end effectora
# and isolate relevate dimensions
J = robot_config.J('EE', q=feedback['q'])[ctrlr_dof]

# calculate the inertia matrix in task space
M = robot_config.M(q=feedback['q'])

# calculate the inertia matrix in task space
M_inv = np.linalg.inv(M)
Mx_inv = np.dot(J, np.dot(M_inv, J.T))
if np.linalg.det(Mx_inv) != 0:
    # do the linalg inverse if matrix is non-singular
    # because it's faster and more accurate
    Mx = np.linalg.inv(Mx_inv)
else:
    # using the rcond to set singular values < thresh to 0
    # singular values < (rcond * max(singular_values)) set to 0
    Mx = np.linalg.pinv(Mx_inv, rcond=.005)

u_task = np.zeros(6)  # [x, y, z, alpha, beta, gamma]

# generate orientation error
# CODE FROM ONE OF ABOVE METHODS HERE

# remove uncontrolled dimensions from u_task
u_task = u_task[ctrlr_dof]

# transform from operational space to torques and
# add in velocity and gravity compensation in joint space
u = (np.dot(J.T, np.dot(Mx, u_task)) -
     kv * np.dot(M, feedback['dq']) -
     robot_config.g(q=feedback['q']))

# apply the control signal, step the sim forward
interface.send_forces(u)

The control script in full context is available up on my GitHub along with the corresponding VREP scene. If you download and run both (and have the ABR Control repo installed), then you can generate fun videos like the following:

• 

  Method 1 – Stack overflow code

• 

  Method 2 – Caccavale quaternion

• 

  Method 3 – Caccavale angle/axis

• 

  Method 4 – Yuan / Nakanishi

Here, the green ball is the target, and the end-effector is being controlled to match the orientation of the ball. The blue box is just a visualization aid for displaying the orientation of the end-effector. And that hand is on there just from another project I was working on then forgot to remove but already made the videos so here we are. It’s set to not affect the dynamics so don’t worry. The target changes orientation once a second. The orientation gain for these trials is ko=200 and kv=np.sqrt(600).

The first three methods all perform relatively similarly to each other, although method 3 seems to be a bit faster to converge to the target orientation after the first movement. But it’s pretty clear something is terribly wrong with the implementation of the Yuan algorithm in method 4; brownie points for whoever figures out what!

Controlling position and orientation

So you want to use force control to control both position and orientation, eh? You are truly reaching for the stars, and I applaud you. For the most part, this is pretty straight-forward. But there are a couple of gotchyas so I’ll explicitly go through the process here.

How many degrees-of-freedom (DOF) can be controlled?

If you recall from my article on Jacobians, there was a section on analysing the Jacobian. It comes down to two main points: 1) The Jacobian specifies which task-space DOF can be controlled. If there is a row of zeros, for example, the corresponding task-space DOF (i.e. cannot be controlled. 2) The rank of the Jacobian determines how many DOF can be controlled at the same time.

For example, in a two joint planar arm, the variables can be controlled, but cannot be controlled because their corresponding rows are all zeros. So 3 variables can potentially be controlled, but because the Jacobian is rank 2 only two variables can be controlled at a time. If you try to control more than 2 DOF at a time things are going to go poorly. Here are some animations of trying to control 3 DOF vs 2 DOF in a 2 joint arm:

• 

  controlling 3 DOF

• 

  controlling 2 DOF

How to specify which DOF are being controlled?

Okay, so we don’t want to try to control too many DOF at once. Got it. Let’s say we know that our arm has 3 DOF, how do we choose which DOF to control? Simple: You remove the rows from you Jacobian and your control signal that correspond to task-space DOF you don’t want to control.

To implement this in code in a flexible way, I’ve chosen to specify an array with 6 boolean elements, set to True if you want to control the corresponding task space parameter and False if you don’t. For example, if you to control just the parameters, you would set ctrl_dof = [True, True, True, False, False, False].

We then strip the Jacobian and task space control signal down to the relevant rows with J = robot_config.('EE', q)[ctrlr_dof] and u_task = (current - target)[ctrlr_dof]. This means that both current and target must be 6-dimensional vectors specifying the current and target values, respectively, regardless of how many dimensions we’re actually controlling.

Generating a position + orientation control signal

The UR5 has 6 degrees of freedom, so we’re able to fully control the task space position and orientation. To do this, in the above script just ctrl_dof = np.array([True, True, True, True, True, True]), and there you go! In the following animations the gain values used were kp=300, ko=300, and kv=np.sqrt(kp+ko)*1.5. The full script can be found up on my GitHub.

• 

  Method 1 – Stack overflow code

• 

  Method 2 – Caccavale quaternion

• 

  Method 3 – Caccavale angle/axis

• 

  Method 4 – Yuan / Nakanishi

NOTE: Setting the gains properly for this task is pretty critical, and I did it just by trial and error until I got something that was decent for each. For a real comparison, better parameter tuning would have to be undertaken more rigorously. 

NOTE: When implementing this minimal code example script I ran into a problem that was caused by the task-space inertia matrix calculation. It turns out that using np.linalg.pinv gives very different results than np.linalg.inv, and I did not realise this. I’m going to have to explore this more fully later, but basically heads up that you want to be using np.linalg.inv as much as possible. So you’ll notice in the above code I check the dimensionality of Mx_inv and first try to use np.linalg.inv before resorting to np.linalg.pinv.

NOTE: If you start playing around with controlling only one or two of the orientation angles, something to keep in mind: Because we’re using rotating axes, if you set up False, False, True then it’s not going to look like of the end-effector is lining up with the of the target. This is because and weren’t set first. If you generate a plot of the target orientations vs the end-effector orientations, however, you’ll see that you are in face reaching the target orientation for .

In summary

So that’s that! Lots of caveats, notes, and more work to be done, but hopefully this will be a useful resource for any others embarking on the same journey. You can download the code, try it out, and play around with the gains and targets. Let me know below if you have any questions or enjoyed the post, or want to share any other resources on force control of task-space orientation.

And once again thanks to Yitao Ding for his helpful comments and corrections on the original version of this article.

Building a spiking neural model of adaptive arm control

About a year ago I published the work from my thesis in a paper called ‘A spiking neural model of adaptive arm control’. In this paper I presented the Recurrent Error-driven Adaptive Control Hierarchy (REACH) model. The goal of the model is to begin working towards reproducing behavioural level phenomena of human movement with biologically plausible spiking neural networks.

To do this, I start by using three methods from control literature (operational space control, dynamic movement primitives, and non-linear adaptive control) to create an algorithms level model of the motor control system that captures behavioural level phenomena of human movement. Then I explore how this functionality could be mapped to the primate brain and implemented in spiking neurons. Finally, I look at the data generated by this model on the behavioural level (e.g. kinematics of movement), the systems level (i.e. analysis of populations of neurons), and the single-cell level (e.g. correlating neural activity with movement parameters) and compare/contrast with experimental data.

By having a full model framework (from observable behaviour to neural spikes) is to have a more constrained computational model of the motor control system; adding lower-level biological constraints to behavioural models and higher-level behavioural constraints to neural models.

In general, once you have a model, the critical next step is to generating testable predictions that can be used to discriminate between other models with different implementations or underlying algorithms. Discriminative predictions allow us to design experiments that can gather evidence in favour or against different hypotheses of brain function, and provide clues to useful directions for further research. Which is the real contribution of computational modeling.

So that’s a quick overview of the paper; there are quite a few pages of supplementary information that describe the details of the model implementation, and I provided the code and data used to generate the data analysis figures. However, code to explicitly run the model on your own has been missing. As one of the major points of this blog is to provide code for furthering research, this is pretty embarrassing. So, to begin to remedy this, in this post I’m going to work through a REACH framework for building models to control a two-link arm through reaching in a line, tracing a circle, performing the centre-out reaching task, and adapting online to unexpected perturbations during reaching imposed by a joint-velocity based forcefield.

This post is directed towards those who have already read the paper (although not necessarily the supplementary material). To run the scripts you’ll need Nengo, Nengo GUI, and NengoLib all installed. There’s a description of the theory behind the Neural Engineering Framework, which I use extensively in my Nengo modeling, in the paper. I’m hoping that between that and code readability / my explanations below that most will be comfortable starting to play around with the code. But if you’re not, and would like more resources, you can check out the Getting Started page on the Nengo website, the tutorials from How To Build a Brain, and the examples in the Nengo GUI.

You can find all of the code up on my GitHub.

NOTE: I’m using the two-link arm (which is fully implemented in Python) instead of the three-link arm (which has compile issues for Macs) both so that everyone should be able to run the model arm code and to reduce the number of neurons that are required for control, so that hopefully you can run this on you laptop in the event that you don’t have a super power Ubuntu work station. Scaling this model up to the three-link arm is straight-forward though, and I will work on getting code up (for the three-link arm for non-Mac users) as a next project.

Implementing PMC – the trajectory generation system

I’ve talked at length about dynamic movement primitives (DMPs) in previous posts, so I won’t describe those again here. Instead I will focus on their implementation in neurons.

def generate(y_des, speed=1, alpha=1000.0):
    beta = alpha / 4.0

    # generate the forcing function
    forces, _, goals = forcing_functions.generate(
        y_des=y_des, rhythmic=False, alpha=alpha, beta=beta)

    # create alpha synapse, which has point attractor dynamics
    tau = np.sqrt(1.0 / (alpha*beta))
    alpha_synapse = nengolib.Alpha(tau)

    net = nengo.Network('PMC')
    with net:
        net.output = nengo.Node(size_in=2)

        # create a start / stop movement signal
        time_func = lambda t: min(max(
            (t * speed) % 4.5 - 2.5, -1), 1)

        def goal_func(t):
            t = time_func(t)
            if t <= -1:
                return goals[0]
            return goals[1]
        net.goal = nengo.Node(output=goal_func, label='goal')

        # -------------------- Ramp ---------------------------
        ramp_node = nengo.Node(output=time_func, label='ramp')
        net.ramp = nengo.Ensemble(
            n_neurons=500, dimensions=1, label='ramp ens')
        nengo.Connection(ramp_node, net.ramp)

        # ------------------- Forcing Functions ---------------
        def relay_func(t, x):
            t = time_func(t)
            if t <= -1:
                return [0, 0]
            return x
        # the relay prevents forces from being sent on reset
        relay = nengo.Node(output=relay_func, size_in=2)

        domain = np.linspace(-1, 1, len(forces[0]))
        x_func = interpolate.interp1d(domain, forces[0])
        y_func = interpolate.interp1d(domain, forces[1])
        transform=1.0/alpha/beta
        nengo.Connection(net.ramp, relay[0], transform=transform,
                         function=x_func, synapse=alpha_synapse)
        nengo.Connection(net.ramp, relay[1], transform=transform,
                         function=y_func, synapse=alpha_synapse)
        nengo.Connection(relay, net.output)

        nengo.Connection(net.goal[0], net.output[0],
                         synapse=alpha_synapse)
        nengo.Connection(net.goal[1], net.output[1],
                         synapse=alpha_synapse)
    return net

The generate method for the PMC takes in a desired trajectory, y_des, as a parameter. The first thing we do (on lines 5-6) is calculate the forcing function that will push the DMP point attractor along the desired trajectory.

The next thing (on lines 9-10) is creating an Alpha (second-order low-pass filter) synapse. By writing out the dynamics of a point attractor in Laplace space, one of the lab members, Aaron Voelker, noticed that the dynamics could be fully implemented by creating an Alpha synapse with an appropriate choice of tau. I walk through all of the math behind this in this post. Here we’ll use that more compact method and project directly to the output node, which improves performance and reduces the number of neurons.

Inside the PMC network we create a time_func node, which is the pace-setter during simulation. It will output a linear ramp from -1 to 1 every few seconds, with the pace set by the speed parameter, and then pause before restarting.

We also have a goal node, which will provide a target starting and ending point for the trajectory. Both the time_func and goal nodes are created and used as a model simulation convenience, and proposed to be generated elsewhere in the brain (the basal ganglia, why not? #igotreasons #provemewrong).

The ramp ensemble is the most important component of the trajectory generation system. It takes the output from the time_func node as input, and generates the forcing function which will guide our little system through the trajectory that was passed in. The ensemble itself is nothing special, but rather the function that it approximates on its outgoing connection. We set up this function approximation with the following code:

        domain = np.linspace(-1, 1, len(forces[0]))
        x_func = interpolate.interp1d(domain, forces[0])
        y_func = interpolate.interp1d(domain, forces[1])
        transform=1.0/alpha/beta
        nengo.Connection(net.ramp, relay[0], transform=transform,
                         function=x_func, synapse=alpha_synapse)
        nengo.Connection(net.ramp, relay[1], transform=transform,
                         function=y_func, synapse=alpha_synapse)
        nengo.Connection(relay, net.output)

We want the forcing function be generated as the signals represented in the ramp ensemble moves from -1 to 1. To achieve this, we create interpolation functions, x_func and y_func, which are set up to generate the forcing function values mapped to input values between -1 and 1. We pass these functions into the outgoing connections from the ramp population (one for x and one for y). So now when the ramp ensemble is representing -1, 0, and 1 the output along the two connections will be the starting, middle, and ending x and y points of the forcing function trajectory. The transform and synapse are set on each connection with the appropriate gain values and Alpha synapse, respectively, to implement point attractor dynamics.

NOTE: The above DMP implementation can generate a trajectory signal with as many dimensions as you would like, and all that’s required is adding another outgoing Connection projecting from the ramp ensemble.

The last thing in the code is hooking up the goal node to the output, which completes the point attractor implementation.

Implementing M1 – the kinematics of operational space control

In REACH, we’ve modelled the primary motor cortex (M1) as responsible for taking in a desired hand movement (i.e. target_position - current_hand_position) and calculating a set of joint torques to carry that out. Explicitly, M1 generates the kinematics component of an OSC signal:

In the paper I did this using several populations of neurons, one to calculate the Jacobian, and then an EnsembleArray to perform the multiplication for the dot product of each dimension separately. Since then I’ve had the chance to rework things and it’s now done entirely in one ensemble.

Now, the set up for the M1 model that computes the above function is to have a single ensemble of neurons that takes in the joint angles, , and control signal , and outputs the function above. Sounds pretty simple, right? Simple is good.

Let’s look at the code (where I’ve stripped out the comments and some debugging code):

def generate(arm, kp=1, operational_space=True,
             inertia_compensation=True, means=None, scales=None):

    dim = arm.DOF + 2

    means = np.zeros(dim) if means is None else means
    scales = np.ones(dim) if scales is None else scales
    scale_down, scale_up = generate_scaling_functions(
        np.asarray(means), np.asarray(scales))

    net = nengo.Network('M1')
    with net:
        # create / connect up M1 ------------------------------
        net.M1 = nengo.Ensemble(
            n_neurons=1000, dimensions=dim,
            radius=np.sqrt(dim),
            intercepts=AreaIntercepts(
                dimensions=dim,
                base=nengo.dists.Uniform(-1, .1)))

        # expecting input in form [q, x_des]
        net.input = nengo.Node(output=scale_down, size_in=dim)
        net.output = nengo.Node(size_in=arm.DOF)

        def M1_func(x, operational_space, inertia_compensation):
            """ calculate the kinematics of the OSC signal """
            x = scale_up(x)
            q = x[:arm.DOF]
            x_des = kp * x[arm.DOF:]

            # calculate hand (dx, dy)
            if operational_space:
                J = arm.J(q=q)

                if inertia_compensation:
                    # account for inertia
                    Mx = arm.Mx(q=q)
                    x_des = np.dot(Mx, x_des)
                # transform to joint torques
                u = np.dot(J.T, x_des)
            else:
                u = x_des

                if inertia_compensation:
                    # account for mass
                    M = arm.M(q=q)
                    u = np.dot(M, x_des)

            return u

        # send in system feedback and target information
        # don't account for synapses twice
        nengo.Connection(net.input, net.M1, synapse=None)
        nengo.Connection(
            net.M1, net.output,
            function=lambda x: M1_func(
                x, operational_space, inertia_compensation),
            synapse=None)

    return net

The ensemble of neurons itself is created with a few interesting parameters:

        net.M1 = nengo.Ensemble(
            n_neurons=1000, dimensions=dim,
            radius=np.sqrt(dim),
            intercepts=AreaIntercepts(
                dimensions=dim, base=nengo.dists.Uniform(-1, .1)))

Specifically, the radius and intercepts parameters.

Setting the intercepts

First we’ll discuss setting the intercepts using the AreaIntercepts distribution. The intercepts of a neuron determine how much of state space a neuron is active in, which we’ll refer to as the ‘active proportion’. If you don’t know what kind of functions you want to approximate with your neurons, then you having the active proportions for your ensemble chosen from a uniform distribution is a good starting point. This means that you’ll have roughly the same number of neurons active across all of state space as you do neurons that are active over half of state space as you do neurons that are active over very small portions of state space.

By default, Nengo sets the intercepts such that the distribution of active proportions is uniform for lower dimensional spaces. But when you start moving into higher dimensional spaces (into a hypersphere) the default method breaks down and you get mostly neurons that are either active for all of state space or almost none of state space. The AreaIntercepts class calculates how the intercepts should be set to achieve the desire active proportion inside a hypersphere. There are a lot more details here that you can read up on in this IPython notebook by Dr. Terrence C. Stewart.

What you need to know right now is that we’re setting the intercepts of the neurons such that the percentage of state space for which any given neuron is active is chosen from a uniform distribution between 0% and 55%. In other words, a neuron will maximally be active in 55% of state space, no more. This will let us model more nonlinear functions (such as the kinematics of the OSC signal) with fewer neurons. If this description is clear as mud, I really recommend checking out the IPython notebook linked above to get an intuition for what I’m talking about.

Scaling the input signal

The other parameter we set on the M1 ensemble is the radius. The radius scales the range of input values that the ensemble can represent, which is by default anything inside the unit hypersphere (i.e. hypersphere with radius=1). If the radius is left at this default value, the neural activity will saturate for vectors with magnitude greater than 1, leading to inaccurate vector representation and function approximation for input vectors with magnitude > 1. For lower dimensions this isn’t a terrible problem, but as the dimensions of the state space you’re representing grow it becomes more common for input vectors to have a norm greater than 1. Typically, we’d like to be able to, at a minimum, represent vectors with any number of dimensions where any element can be anywhere between -1 and 1. To do this, all we have to do is calculate the norm of the unit vector size dim, which is np.sqrt(dim) (the magnitude of a vector size dim with all elements set to one).

Now that we’re able to represent vectors where the input values are all between -1 and 1, the last part of this sub-network is scaling the input to be between -1 and 1. We use two scaling functions, scale_down and scale_up. The scale_down function subtracts a mean value and scales the input signal to be between -1 and 1. The scale_up function reverts the vector back to it’s original values so that calculations can be carried out normally on the decoding. In choosing mean and scaling values, there are two ways we can set these functions up:

1. Set them generally, based on the upper and lower bounds of the input signal. For M1, the input is where is the control signal in end-effector space, we know that the joint angles are always in the range 0 to (because that’s how the arm simulation is programmed), so we can set the means and scales to be for . For a mean of zero is reasonable, and we can choose (arbitrarily, empirically, or analytically) the largest task space control signal we want to represent accurately.
2. Or, if we know the model will be performing a specific task, we can look at the range of input values encountered during that task and set the means and scales terms appropriately. For the task of reaching in a straight line, the arm moves in a very limited state space and we can set the mean and we can tune these parameter to be very specific:

                                means=[0.6, 2.2, 0, 0],
                                scales=[.25, .25, .25, .25]
   

The benefit of the second method, although one can argue it’s parameter tuning and makes things less biologically plausible, is that it lets us run simulations with fewer neurons. The first method works for all of state space, given enough neurons, but seeing as we don’t always want to be simulating 100k+ neurons we’re using the second method here. By tuning the scaling functions more specifically we’re able to run our model using 1k neurons (and could probably get away with fewer). It’s important to keep in mind though that if the arm moves outside the expected range the control will become unstable.

Implementing CB – the dynamics of operational space control

The cerebellum (CB) sub-network has two components to it: dynamics compensation and dynamics adaptation. First we’ll discuss the dynamics compensation. By which I mean the term from the OSC signal.

Much like the calculating the kinematics term of the OSC signal in M1, we can calculate the entire dynamics compensation term using a single ensemble with an appropriate radius, scaled inputs, and well chosen intercepts.

def generate(arm, kv=1, learning_rate=None, learned_weights=None,
             means=None, scales=None):
    dim = arm.DOF * 2

    means = np.zeros(dim) if means is None else means
    scales = np.ones(dim) if scales is None else scales
    scale_down, scale_up = generate_scaling_functions(
        np.asarray(means), np.asarray(scales))

    net = nengo.Network('CB')
    with net:
        # create / connect up CB --------------------------------
        net.CB = nengo.Ensemble(
            n_neurons=1000, dimensions=dim,
            radius=np.sqrt(dim),
            intercepts=AreaIntercepts(
                dimensions=dim,
                base=nengo.dists.Uniform(-1, .1)))
        # expecting input in form [q, dq, u]
        net.input = nengo.Node(output=scale_down,
                               size_in=dim+arm.DOF+2)
        cb_input = nengo.Node(size_in=dim, label='CB input')
        nengo.Connection(net.input[:dim], cb_input)
        # output is [-Mdq, u_adapt]
        net.output = nengo.Node(size_in=arm.DOF*2)

        def CB_func(x):
            """ calculate the dynamic component of OSC signal """
            x = scale_up(x)
            q = x[:arm.DOF]
            dq = x[arm.DOF:arm.DOF*2]

            # calculate inertia matrix
            M = arm.M(q=q)
            return -np.dot(M, kv * dq)
        # connect up the input and output
        nengo.Connection(net.input[:dim], net.CB)
        nengo.Connection(net.CB, net.output[:arm.DOF],
                         function=CB_func, synapse=None)

I don’t think there’s anything noteworthy going on here, most of the relevant details have already been discussed…so we’ll move on to the adaptation!

Implementing CB – non-linear dynamics adaptation

The final part of the model is the non-linear dynamics adaptation, modelled as a separate ensemble in the cerebellar sub-network (a separate ensemble so that it’s more modular, the learning connection could also come off of the other CB population). I work through the details and proof of the learning rule in the paper, so I’m not going to discuss that here. But I will restate the learning rule here:

where are the neural decoders, is the learning rate, is the neural activity of the ensemble, and is the joint space control signal sent to the arm. This is a basic delta learning rule, where the decoders of the active neurons are modified to push the decoded function in a direction that reduces the error.

The adaptive ensemble can be initialized either using saved weights (passed in with the learned_weights paramater) or as all zeros. It is important to note that setting decoders to all zeros means that does not mean having zero neural activity, so learning will not be affected by this initialization.

        # dynamics adaptation------------------------------------
        if learning_rate is not None:
            net.CB_adapt = nengo.Ensemble(
                n_neurons=1000, dimensions=arm.DOF*2,
                radius=np.sqrt(arm.DOF*2),
                # enforce spiking neurons
                neuron_type=nengo.LIF(),
                intercepts=AreaIntercepts(
                    dimensions=arm.DOF,
                    base=nengo.dists.Uniform(-.5, .2)))

            net.learn_encoders = nengo.Connection(
                net.input[:arm.DOF*2], net.CB_adapt,)

            # if no saved weights were passed in start from zero
            weights = (
                learned_weights if learned_weights is not None
                else np.zeros((arm.DOF, net.CB_adapt.n_neurons)))
            net.learn_conn = nengo.Connection(
                # connect directly to arm so that adaptive signal
                # is not included in the training signal
                net.CB_adapt.neurons, net.output[arm.DOF:],
                transform=weights,
                learning_rule_type=nengo.PES(
                    learning_rate=learning_rate),
                synapse=None)

            nengo.Connection(net.input[dim:dim+2],
                             net.learn_conn.learning_rule,
                             transform=-1, synapse=.01)
    return net

We’re able to implement that learning rule using Nengo’s prescribed error-sensitivity (PES) learning on our connection from the adaptive ensemble to the output. With this set up the system will be able to learn to adapt to perturbations that are functions of the input (set here to be ).

The intercepts in this population are set to values I found worked well for adapting to a few different forces, but it’s definitely a parameter to play with in your own scripts if you’re finding that there’s too much or not enough generalization of the decoded function output across the state space.

One other thing to mention is that we need to have a relay node to amalgamate the control signals output from M1 and the dynamics compensation ensemble in the CB. This signal is used to train the adaptive ensemble, and it’s important that the adaptive ensemble’s output is not included in the training signal, or else the system quickly goes off to positive or negative infinity.

Implementing S1 – a placeholder

The last sub-network in the REACH model is a placeholder for a primary sensory cortex (S1) model. This is just a set of ensembles that represents the feedback from the arm and relay it on to the rest of the model.

def generate(arm, direct_mode=False, means=None, scales=None):
    dim = arm.DOF*2 + 2  # represents [q, dq, hand_xy]

    means = np.zeros(dim) if means is None else means
    scales = np.ones(dim) if scales is None else scales
    scale_down, scale_up = generate_scaling_functions(
        np.asarray(means), np.asarray(scales))

    net = nengo.Network('S1')
    with net:
        # create / connect up S1 --------------------------------
        net.S1 = nengo.networks.EnsembleArray(
            n_neurons=50, n_ensembles=dim)

        # expecting input in form [q, x_des]
        net.input = nengo.Node(output=scale_down, size_in=dim)
        net.output = nengo.Node(
            lambda t, x: scale_up(x), size_in=dim)

        # send in system feedback and target information
        # don't account for synapses twice
        nengo.Connection(net.input, net.S1.input, synapse=None)
        nengo.Connection(net.S1.output, net.output, synapse=None)

    return net

Since there’s no function that we’re decoding off of the represented variables we can use separate ensembles to represent each dimension with an EnsembleArray. If we were going to decode some function of, for example, q0 and dq0, then we would need an ensemble that represents both variables. But since we’re just decoding out f(x) = x, using an EnsembleArray is a convenient way to decrease the number of neurons needed to accurately represent the input.

Creating a model using the framework

The REACH model has been set up to be as much of a plug and play system as possible. To generate a model you first create the M1, PMC, CB, and S1 networks, and then they’re all hooked up to each other using the framework.py file. Here’s an example script that controls the arm to trace a circle:

def generate():
    kp = 200
    kv = np.sqrt(kp) * 1.5

    center = np.array([0, 1.25])
    arm_sim = arm.Arm2Link(dt=1e-3)
    # set the initial position of the arm
    arm_sim.init_q = arm_sim.inv_kinematics(center)
    arm_sim.reset()

    net = nengo.Network(seed=0)
    with net:
        net.dim = arm_sim.DOF
        net.arm_node = arm_sim.create_nengo_node()
        net.error = nengo.Ensemble(1000, 2)
        net.xy = nengo.Node(size_in=2)

        # create an M1 model -------------------------------------
        net.M1 = M1.generate(arm_sim, kp=kp,
                             operational_space=True,
                             inertia_compensation=True,
                             means=[0.6, 2.2, 0, 0],
                             scales=[.5, .5, .25, .25])

        # create an S1 model -------------------------------------
        net.S1 = S1.generate(arm_sim,
                             means=[.6, 2.2, -.5, 0, 0, 1.25],
                             scales=[.5, .5, 1.7, 1.5, .75, .75])

        # subtract current position to get task space direction
        nengo.Connection(net.S1.output[net.dim*2:], net.error,
                         transform=-1)

        # create a trajectory for the hand to follow -------------
        x = np.linspace(0.0, 2.0*np.pi, 100)
        PMC_trajectory = np.vstack([np.cos(x) * .5,
                                    np.sin(x) * .5])
        PMC_trajectory += center[:, None]
        # create / connect up PMC --------------------------------
        net.PMC = PMC.generate(PMC_trajectory, speed=1)
        # send target for calculating control signal
        nengo.Connection(net.PMC.output, net.error)
        # send target (x,y) for plotting
        nengo.Connection(net.PMC.output, net.xy)

        net.CB = CB.generate(arm_sim, kv=kv,
                             means=[0.6, 2.2, -.5, 0],
                             scales=[.5, .5, 1.6, 1.5])

    model = framework.generate(net=net, probes_on=True)
    return model

In line 50 you can see the call to the framework code, which will hook up the most common connections that don’t vary between the different scripts.

The REACH model has assigned functionality to each area / sub-network, and you can see the expected input / output in the comments at the top of each sub-network file, but the implementations are open. You can create your own M1, PMC, CB, or S1 sub-networks and try them out in the context of a full model that generates high-level movement behaviour.

Running the model

To run the model you’ll need Nengo, Nengo GUI, and NengoLib all installed. You can then pull open Nengo GUI and load any of the a# scripts. In all of these scripts the S1 model is just an ensemble that represents the output from the arm_node. Here’s what each of the scripts does:

1. a01 has a spiking M1 and CB, dynamics adaptation turned off. The model guides the arm in reaching in a straight line to a single target and back.
2. a02 has a spiking M1, PMC, and CB, dynamics adaptation turned off. The PMC generates a path for the hand to follow that traces out a circle.
3. a03 has a spiking M1, PMC, and CB, dynamics adaptation turned off. The PMC generates a path for the joints to follow, which moves the hand in a straight line to a target and back.
4. a04 has a spiking M1 and CB, dynamics adaptation turned off. The model performs the centre-out reaching task, starting at a central point and reaching to 8 points around a circle.
5. a05 has a spiking M1 and CB, and dynamics adaptation turned on. The model performs the centre-out reaching task, starting at a central point and reaching to 8 points around a circle. As the model reaches, a forcefield is applied based on the joint velocities that pushes the arm as it tries to reach the target. After 100-150 seconds of simulation the arm has adapted and learned to reach in a straight line again.

Here’s what it looks like when you pull open a02 in Nengo GUI:

I’m not going to win any awards for arm animation, but! It’s still a useful visualization, and if anyone is proficient in javascript and want’s to improve it, please do! You can see the network architecture in the top left, the spikes generated by M1 and CB to the right of that, the arm in the bottom left, and the path traced out by the hand just to the right of that. On the top right you can see the a02 script code, and below that the Nengo console.

Conclusions

One of the most immediate extensions (aside from any sort of model of S1) that comes to mind here is implementing a more detailed cerebellar model, as there are several anatomically detailed models which have the same supervised learner functionality (for example (Yamazaki and Nagao, 2012)).

Hopefully people find this post and the code useful for their own work, or at least interesting! In the ideal scenario this would be a community project, where researchers add models of different brain areas and we end up with a large library of implementations to build larger models with in a Mr. Potato Head kind of fashion.

You can find all of the code up on my GitHub. And again, this code all should have been publicly available along with the publication. Hopefully the code proves useful! If you have any questions about it please don’t hesitate to make a comment here or contact me through email, and I’ll get back to you as soon as I can.

ABR Jaco repo public release!

https://github.com/abr/abr_jaco2

We’ve been working with Kinova’s Jaco arm with joint torque sensors for the last year or so as part of our research at Applied Brain Research, and we put together a fun adaptive control demo and got to show it to Justin Trudeau. As you might have guessed based on previous posts, the robotic control used force control. Force control is available on the Jaco, but the API that comes with the arm has much too slow an update time for practical use for our application (around 100Hz, if I recall correctly).

So part of the work I did with Pawel Jaworski over the last year was to write an interface for force control to the Jaco arm that had a faster control loop. Using Kinova’s low level API, we managed to get things going at about 250Hz, which was sufficient for our purposes. In the hopes of saving other people the trouble of having to redo all this work to begin to be able to play around with force control on the Kinova, we’ve made the repo public and free for non-commercial use. It’s by no means fully optimized, but it is well tested and hopefully will be found useful!

The interface was designed to plug into our ABR Control repo, so you’ll also need that installed to get things working. Once both repos are installed, you can either use the controllers in the ABR Control repo or your own. The interface has a few options, which are shown in the following demo script:

import abr_jaco2
from abr_control.controllers import OSC

robot_config = abr_jaco2.Config()
interface = abr_jaco2.Interface(robot_config)
ctrlr = OSC(robot_config)
# instantiate things to avoid creating 200ms delay in main loop
zeros = np.zeros(robot_config.N_LINKS)
ctrlr.generate(q=zeros, dq=zeros, target=zeros(3))
# run once outside main loop as well, returns the cartesian
# coordinates of the end effector
robot_config.Tx('EE', q=zeros)

interface.connect()
interface.init_position_mode()
interface.send_target_angles(robot_config.INIT_TORQUE_POSITION)

target_xyz = [.57, .03 .87]  # (x, y, z) target (metres)
interface.init_force_mode()

while 1:
    # returns a dictionary with q, dq
    feedback = interface.get_feedback() 
    # ee position
    xyz = robot_config.Tx('EE', q=q, target_pos = target_xyz)
    u = ctrlr.generate(feedback['q'], feedback['dq'], target_xyz)
    interface.send_forces(u, dtype='float32')

    error = np.sqrt(np.sum((xyz - TARGET_XYZ[ii])**2))

    if error < 0.02:
        break

# switch back to position mode to move home and disconnect
interface.init_position_mode()
interface.send_target_angles(robot_config.INIT_TORQUE_POSITION)
interface.disconnect()

You can see you have the option for position control, but you can also initiate torque control mode and then start sending forces to the arm motors. To get a full feeling of what is available, we’ve got a bunch of example scripts that show off more of the functionality.

Here are some gifs feature Pawel showing the arm operating under force control. The first just shows compliance of normal operational space control (on the left) and an adaptation example (on the right). In both cases here the arm is moving to and trying to maintain a target location, and Pawel is pushing it away.

You can see that in the adaptive example the arm starts to compensate for the push, and then when Pawel lets go of the arm it overshoots the target because it’s compensating for a force that no longer exists.

So it’s our hope that this will be a useful tool for those with a Kinova Jaco arm with torque sensors exploring force control. If you end up using the library and come across places for improvement (there are many), contributions are very appreciated!

Also a big shout out to the Kinova support team that provided many hours of support during development! It’s an unusual use of the arm, and their engineers and support staff were great in getting back to us quickly and with useful advice and insights.

ABR Control repo public release!

https://github.com/abr/abr_control

Last August I started working on a new version of my old control repo with a resolve to make something less hacky, as part of the work for Applied Brain Research, Inc, a startup that I co-founded with others from my lab after most of my cohort graduated. Together with Pawel Jaworski, who comprises other half of ABR’s motor team, we’ve been building up a library of useful functions for modeling, simulating, interfacing, and controlling robot arms.

Today we’re making the repository public, under the same free for non-commercial use that we’ve released our Nengo neural modelling software on. You can find it here: ABR_Control

It’s all Python 3, and here’s an overview of some of the features:

• Automates generation of functions for computing the Jacobians, joint space and task space inertia matrices, centrifugal and Coriolis effects, and Jacobian derivative, provided each link’s inertia matrix and the transformation matrices
• Option to compile these functions to speedy Cython implementations
• Operational space, joint space, floating, and sliding controllers provided with PyGame and VREP example scripts
• Obstacle avoidance and dynamics adaptation add-ons with PyGame example scripts
• Interfaces with VREP
• Configuration files for one, two, and three link arms, as well as the UR5 and Jaco2 arms in VREP
• Provides Python simulations of two and three link arms, with PyGame visualization
• Path planning using first and second order filtering of the target and example scripts.

Structure

The ABR Control library is divided into three sections:

1. Arm descriptions (and simulations)
2. Robotic system interfaces
3. Controllers

The big goal was to make all of these interchangeable, so that to run any permutation of them you just have to change which arm / interface / controller you’re importing.

To support a new arm, the user only needs to create a configuration file specifying the transforms and inertia matrices. Code for calculating the necessary functions of the arm will be symbolically derived using SymPy, and compiled to C using Cython for efficient run-time execution.

Interfaces provide send_forces and send_target_angles functions, to apply torques and put the arm in a target state, as well as a get_feedback function, which returns a dictionary of information about the current state of the arm (joint angles and velocities at a minimum).

Controllers provide a generate function, which take in current system state information and a target, and return a set of joint torques to apply to the robot arm.

VREP example

The easiest way to show it is with some code examples. So, once you’ve cloned and installed the repo, you can open up VREP and the jaco2.ttt model in the abr_control/arms/jaco2 folder, and to control it using an operational space controller you would run the following:

import numpy as np
from abr_control.arms import jaco2 as arm
from abr_control.controllers import OSC
from abr_control.interfaces import VREP

# initialize our robot config for the ur5
robot_config = arm.Config(use_cython=True, hand_attached=True)

# instantiate controller
ctrlr = OSC(robot_config, kp=200, vmax=0.5)

# create our VREP interface
interface = VREP(robot_config, dt=.001)
interface.connect()

target_xyz = np.array([0.2, 0.2, 0.2])
# set the target object's position in VREP
interface.set_xyz(name='target', xyz=target_xyz)

count = 0.0
while count < 1500:  # run for 1.5 simulated seconds
    # get joint angle and velocity feedback
    feedback = interface.get_feedback()
    # calculate the control signal
    u = ctrlr.generate(
        q=feedback['q'],
        dq=feedback['dq'],
        target_pos=target_xyz)
    # send forces into VREP, step the sim forward
    interface.send_forces(u)

    count += 1
interface.disconnect()

This is a minimal example of the examples/VREP/reaching.py code. To run it with a different arm, you can just change the from abr_control.arms import as line. The repo comes with the configuration files for the UR5 and a onelink VREP arm model as well.

PyGame example

I’ve also found the PyGame simulations of the 2 and 3 link arms very helpful for quickly testing new controllers and code, as an easy low overhead proof of concept sandbox. To run the threelink arm (which runs in Linux and Windows fine but I’ve heard has issues in Mac OS), with the operational space controller, you can run this script:

import numpy as np
from abr_control.arms import threelink as arm
from abr_control.interfaces import PyGame
from abr_control.controllers import OSC

# initialize our robot config
robot_config = arm.Config(use_cython=True)
# create our arm simulation
arm_sim = arm.ArmSim(robot_config)

# create an operational space controller
ctrlr = OSC(robot_config, kp=300, vmax=100,
            use_dJ=False, use_C=True)

def on_click(self, mouse_x, mouse_y):
    self.target[0] = self.mouse_x
    self.target[1] = self.mouse_y

# create our interface
interface = PyGame(robot_config, arm_sim, dt=.001, 
                   on_click=on_click)
interface.connect()

# create a target
feedback = interface.get_feedback()
target_xyz = robot_config.Tx('EE', feedback['q'])
interface.set_target(target_xyz)

try:
    while 1:
        # get arm feedback
        feedback = interface.get_feedback()
        hand_xyz = robot_config.Tx('EE', feedback['q'])

        # generate an operational space control signal
        u = ctrlr.generate(
            q=feedback['q'],
            dq=feedback['dq'],
            target_pos=target_xyz)

        new_target = interface.get_mousexy()
        if new_target is not None:
            target_xyz[0:2] = new_target
        interface.set_target(target_xyz)

        # apply the control signal, step the sim forward
        interface.send_forces(u)

finally:
    # stop and reset the simulation
    interface.disconnect()

The extra bits of code just set up a hook so that when you click on the PyGame display somewhere the target moves to that point.

So! Hopefully some people find this useful for their research! It should be as easy to set up as cloning the repo, installing the requirements and running the setup file, and then trying out the examples.

If you find a bug please file an issue! If you find a way to improve it please do so and make a PR! And if you’d like to use anything in the repo commercially, please contact me.

Deriving a robot’s transform matrices

While doing some soul searching recently, I realised that in previous posts I’ve glossed over actually deriving transform matrices, and haven’t discussed a methodical way of going about it. If you’ve ever tried working out transforms you know they can be a pain if you don’t know have a set process to follow. Although I’ve given a bunch of examples in previous posts, this post is intended to clear up any confusion that crops up when you’re trying to work out the transforms for your own robot. This is far from an absolute guide covering all the cases, but hopefully it gives a firm enough footing that you can handle the rest.

Note: I’m only going to be dealing with revolute joints in this post. Linear joints are easy (just put the joint offset in the translation part of the transform matrix) and spherical joints are horrible death beyond the scope of this post.

First thing’s first, quick recap of what transform matrices actually are. A transform matrix changes the coordinate system (reference frame) a point is defined in. We’re using the notation to denote a transformation matrix that transforms a point from reference frame 1 to reference frame 0. To calculate this matrix, we described the transformation from reference frame 0 to reference frame 1.

To make things easier, we’re going to break up each transform matrix into two parts. Unfortunately I haven’t found a great way to denote these two matrices, so we’re going to have to use additional subscripts:

1. : accounting for the joint rotation, and
2. : accounting for static translations and rotations.

So the matrix accounts for transformations that involve joint angles, and the matrix accounts for all the static transformations between reference frames.

Step 1: Account for the joint rotation

When calculating the transform matrix from a joint to a link’s centre-of-mass (COM), note that the reference frame of the link’s COM rotates along with the angle of the joint. Conversely, the joint’s reference frame does not rotate with the joint angle. Look at this picture!

In these pictures, we’re looking at the transformation from joint i’s reference frame to the COM for link i. denotes the joint angle of joint i. In the first image (on the left) the joint angle is almost 90 degrees, and on the right it’s closer to 45 degrees. Keeping in mind that the reference frame for COM i changes with the joint angle, and the reference frame for joint i does not, helps make deriving the transform matrices much easier.

So, to actually account for joint rotation, all you have to do is determine which axis the joint is rotating around and create the appropriate rotation matrix inside the transform.

For rotations around the x axis:

For rotations around the y axis:

For rotations around the z axis:

And there you go! Easy. The should be one of the above matrices unless something fairly weird is going on. For the arm in the diagram above, the joint is rotating around the axis, so

Step 2: Account for translations and any static axes rotations

Once the joint rotation is accounted for, translations become much simpler to deal with: Just put joint i at 0 degrees and the offset from the origin of reference frame i to i+1 is your translation.

So for the above example, this is what the system looks like when joint i is at 0 degrees:

where I’ve also added labels to the and axes for clarity. Say that the COM is at , then that is what you set the translation values to in the .

Also we need to note that there is a rotation in the axes between reference frames (i.e. and do not point in the same direction for both axes sets). Here the rotation is 90 degrees. So the rotation part of the transformation matrix should be a 90 degree rotation.

Thus, the transformation matrix for our static translations and rotations is

Step 3: Putting them together

This is pretty easy, to compute the full transform from joint i to the COM i you just string together the two matrices derived above:

where I’m using simplified notation on the left-hand side because we don’t need the full notation for clarity in this simple example.

So when you’re transforming a point, you left multiply by :

which is not great notation, but hopefully conveys the idea. To get our point into the reference frame of joint i, we left multiply the point as defined in the reference frame of COM i by .

So there you go! That’s the process for a single joint to COM transform. For transforms from COM to joints it’s even easier because you only need to account for the static offsets and axes rotations.

Examples

It’s always nice to have examples, so let’s work through the deriving the transform matrices for the Jaco2 arm as it’s set up in the VREP. The process is pretty straight-forward, with really the only tricky bit converting from the Euler angles that VREP gives you to a rotation matrix that we can use in our transform.

But first thing’s first, load up VREP and drop in a Jaco2 model.

NOTE: I’ve renamed the joints and links to start at 0, to make indexing in / interfacing with VREP easier.

We’re going to generate the transforms for the whole arm piece by piece, sequentially: origin -> link 0 COM, link 0 COM -> joint 0, joint 0 -> link 1 COM, etc. I’m going to use the notation and to denote link i COM and joint i, respectively, in the transform sub and superscripts.

The first transform we want then is for origin -> link 0 COM. There’s no joint rotation that we need to account for in this transform, so we only have to look for static translation and rotation. First click on link 0, and then

• ‘Object/item shift’ button from the toolbar,
• the ‘Position’ tab,
• select the ‘Parent frame’ radio button,

and it will provide you with the translation of link 0’s COM relative to its parent frame, as shown below:

So the translation part of our transformation is .

The rotation part we can see by

• ‘Object/item rotate’ button from the toolbar,
• the ‘Orientation’ tab,
• select the ‘Parent frame’ radio button,

Here, it’s not quite as straight forward to use this info to fill in our transform matrix. So, if we pull up the VREP description for their Euler angles we see that to generate the rotation matrix from them you perform:

where the matrix is a rotation matrix around the axis (as described above). For this works out to no rotation, as you may have guessed. So then our first transform is

For the second transform matrix, from link 0 to joint 0, again we only have to account for static translations and rotations. Pulling up the translation information:

we have the translation .

And this time, for our rotation matrix, there’s a flip in the axes so we have something more interesting than no rotation:

we have Euler angles . This is also a good time to note that the angles are provided in degrees, so let’s convert those over, giving us approximately . Now, calculating our rotation matrix:

which then gives us the transform matrix

And the last thing I’ll show here is accounting for the joint rotation. In the transform from joint 0 to link 1 we have to account for the joint rotation, so we’ll break it down into two matrices as described above. To generate the first one, all we need to know is which axis the joint rotates around. In VREP, this is almost always the axis, but it’s good to double check in case someone has built a weird model. To check, one easy way is to make the joint visible, and the surrounding arm parts invisible:

You can do this by

• double clicking on the joint in the scene hierarchy to get to the Scene Object Properties window,
• selecting the ‘Common’ tab,
• selecting or deselecting check boxes from the ‘Visibility’ box.

As you can see in the image, this joint is indeed rotating around the axis, so the first part of our transformation from joint 0 to link 1 is

The second matrix can be generated as previously described, so I’ll leave that, and the rest of the arm, to you!

If you’d like to see a full arm example, you can check out the tranforms for the UR5 model in VREP 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:

where is the distance to the target, is the threshold distance to the target at which point the avoidance function activates, is the partial derivative of the distance to the target with respect to a given point on the arm, and 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 , and gets huge as the obstacle nears the object (as ). Using and 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 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:

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

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

which can be rewritten

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

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 using the equation we defined above. The one part of that equation that wasn’t specified was exactly what 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:

where is the Jacobian for the point of interest, is the operational space inertia matrix for the point of interest, and 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:

where reads the transform from reference frame to reference frame .

Once we have this transformation matrix, , we multiply it by the point of interest in reference frame 3, which, previously, has usually been . In other words, usually we’re just interested in the origin of reference frame 3. So the Jacobian is just

what if we’re interested in some non-specified point along link 3, though? Well, using SymPy we set make instead of (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 and , rather than just . 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 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
    rho = dist - obstacle_radius

    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.