## Thursday, December 29, 2011

### Pythagoras: Structural Update

Solving the Flex
After dealing with the flexing aluminum supports that allowed the whole robot to wobble for so long, I finally went ahead and bought steel tubing. The legs are also longer, allowing expansion of the drawing field. Unfortunately, they are also taller, so for now, the paper needs a little booster to get it to the optimal drawing height.

 Phone books to lift the paper to the pen

The tubing is connected by gusset plates, made of .03" galvanized steel.
 Slots in the plates for adjusting exact lengths

Now the bot is so sturdy I could probably stand on it and it would hold.

### Pythagoras Bitmap Mode: How It's Done

Introduction

Bitmap mode involves shading in the color of images, as opposed to just making the outlines. For some images, this makes vastly superior output compared to outlines. Faces are a prime example. When outlined, edges of shadows appear the same as edges of objects, causing a mess of lines to be drawn on a face. With shading, shadows and lower contrast features remain in the image, improving the familiarity of the face.
 Its my face!
The downside is that the output resolution is greatly reduced, and the time it takes to draw is greatly increased.

The Concept

How do I generate levels of grey scale when my pen only creates a line of constant darkness? The answer: squiggles. Each pixel forms a 2D square on the paper. The more that square is shaded in by the pen, the darker the pixel. This idea can be clearly seen in one of the earlier test pictures. Ramp function patterns are drawn on the paper to fill the square. Then I tested with square functions, then triangle waves.

Since the shading depends on the width of the pen stroke, the pixel size on paper becomes a factor. Smaller pixels make smoother borders around regions on the output image, but also decreases the number of squiggles that can be fit into a pixel before the pixel becomes solid dark. So at one extreme, the pixel is completely filled by a single stroke of the pen, while at the other extreme, the entire image is a single pixel.
Through testing, I found the best compromise between greyscale levels and output resolution is to have an image width of between 120 and 160 pixels, for a 7.5 inch (19 cm) square drawing area.

Tweaking Waveforms

First was experimenting with different waveforms. After creating ramp functions, I tried to do square waves. However, this created odd behavior whenever the squares shifted in and out of phase with the rows above and below them. It was quickly abandoned, apologies for the lack of pictures. What I settled on was triangle waves, which were always in phase like the ramp functions, yet more uniform like square waves, closer to the ideal of sine waves.

In an attempt to squeeze out more greyscale levels, I considered sharing squiggles across multiple pixels, up to four pixels. This way I could get even less dense pen strokes and therefore lighter colored intensities. The results were not so great though. This image allowed up to four combined pixels.
 The pixel combining can be observed near the center of the drawing.
The pixel combining created awkwardly shaded areas in an image. Even when combining less pixels.
 A series of test drawings under various brightness settings. Scribbling in the corner is unrelated.
Pixel combining didn't create evenly lighter areas, just awkward gaps. The solution to this is to use amplitude modulation. To make a lighter region, the amplitude of the squiggle is reduced, instead of the number of squiggles. This can be seen clearly below.
 Just below the center is an example of amplitude modulation of the squiggle.
This fills the gap in shading between one squiggle per pixel and no squiggle, making nicer transitions for lighter colors.

Color Mapping

Originally, color was mapped linearly to number of squiggles in each pixel. This caused for pretty dark images, as seen below.
 Can barely make out differences in the greys.
The first step to fixing this is to add gamma correction. Since human eyes are more sensitive to darker colors than lighter colors, color values in images are shifted toward the darker regions according to an exponential rule. This way, more bits are devoted toward the darker colors, instead of highlights that eyes cannot differentiate anyways.
To correct for gamma, simply raise the pixel value, mapped from 0 to 1, to the (1/2.2) power to get the actual color intensity. This has an effect of making the output image brighter. Technically, any image processing should be done after gamma correction, for proper mixing of pixel values. However, this will involve a lot of going between gamma encoded and decoded, so I let it slide. I do scaling before gamma correction, and gamma correct once, right before converting the pixel value to squiggles. This makes the picture a little brighter.
 The same picture, with gamma correction. A little brighter.
The next step in color mapping is to fix the mapping of intensity to squiggles. The mapping is not linear, and is depending on the size of the pixel. The mapping I got from experimenting is intensities 0.7 and less are linearly mapped to squiggle density. Between 0.85 and 0.7 intensity is the region for amplitude modulation of the squiggles. Testing was done with printing gradients and comparing the result to the computer image.

Software

The image processing software was written in Python, using the OpenCV libraries for image processing and simple GUIs. One slider lets the user change the output resolution until the aliasing looks alright, and a second slider lets the user change the gamma value from the default of 2.2, to change the brightness of an image.

There is a subtlety in the image preview. While the screen can display the full 256 values of greyscale, the robot can only print a much smaller range of greyscale, which varies as a function of pixel size. This can be especially misleading for lighter colored regions, since full white pixels cover color intensities of 0.85 and 0.7, and for darker colored regions, where seemingly differently colored regions are both shaded the same.

To fix this, the output pixel values are grouped into buckets based on the number of squiggles in that pixel. This way, the output image has the same number of greyscale levels as the printed image. With some funky mapping, the colors on screen are closer to the colors as printed. This way, it is much easier to adjust the brightness and scaling and predict the actual results. Following are screenshots of the program in action after bucketing. You can see how there are significantly less than 256 levels of grey.
 Default scaling of 120 pixels and gamma of 2.2 (220 / 100).
 Darker image.
 Brighter image.

Appendix

## Monday, December 26, 2011

### 3D Video of Pythagoras

While visiting at Lake Brantley High School, one of the students took a delightful video of Pythagoras in 3D as it sketched Pikachu in raster mode.

## Tuesday, December 6, 2011

### Pythagoras Bitmap Mode

Just finished cooking up a new drawing mode, letting Pythagoras draw raster images. A writeup will come soon, for now a bunch of drawings.

 First test
 Second test
 Third test
 Before gamma correction
 After some gamma correction. Teehee moire patterns.
 In progress

## Monday, December 5, 2011

### LightDial: Done!

I cooked up a quick Solidworks model of the clock. Protip: don't set your parts' appearances to transparent plastic until you're done. Basically working in wireframe the entire time.

The clock is planned to have a five inch diameter face, with two buttons and an RPG (rotary pulse generator) to adjust the time.

Using the Invention Studio's laser cutter, the pieces were cut out without hiccups. Holes were also cut out in the pieces for the LED leads. The LEDs were slotted in and wired together with a common cathode. This is when the true hairiness of 36 individual wires came into play, and we realized what we were getting ourselves into.

A digression: Through a random experiment, we discovered that a backlit clock face looks better than a frontlit clock face. Thus, we reversed the order of the clock rings, so instead of two LED rings and a face, it was a face and then two LED rings. This is why the actual build doesn't look like the Solidworks model.

Populating the board was easy enough. Until I realized the wire I have doesn't fit inside the connector, and the Molex connector crimper I can use is too big for the 1.25 mm pitch connectors I have. Begrudgingly, my lab partner and I decide in the interest of time to solder the LED wires directly onto the board. This we did.
 The horror! The horror!
Then I found out the LED drivers (the ones underneath all the wires) were soldered on backwards. Desolder and resolder all the wires!

There was a good thing that came out of all the soldering. In code, the lights were ordered B7G7R7B6G6R6...B0G0R0. In circuitry, the green and red channels were swapped, as in B7R7G7. Headaches were had all around, but they were wired in order to simplify programming.

After that, the major hardships were over.

 First light!
Wiring done, all that was left was a matter of coding. Using the SPI peripheral was a matter of setting registers. Time was kept by counting program cycles. I did add a watch crystal to the board, but due to time, I didn't bother enabling it, so the timekeeping accuracy is only good enough for short term use.

In terms of human interface, I used two buttons and a rotary pulse generator. Holding down one button and turning the RPG adjusts minutes, the other hours.

 Hour lights
And so, putting it all together,
 Hours and minutes. Woohoo!
 Rats nest

 Blue LEDs look funny on camera
Appendix
Source Code:
Eagle Design Files:
Solidworks Models:

## Friday, November 25, 2011

### Pythagoras v2 Drawings

Some drawings.
 Working out calibration. Third time's the charm!
 I like this face.
 Self Portrait
 Test of accuracy and capability. Nine hours to draw

## Monday, November 21, 2011

### Lightdial Update

The PCBs are in!
Mostly populated.
After wrangling with 0402 package SMD components, soldering these 1206 package components is such a relief.

### Fixing the Analog Signal

Routing an analog signal all over the place with ribbon cable and no filtering and no buffering made it extremely susceptible to any bit of noise, not to mention the voltage sag due to the loading of the ADC. Given that I'm using 5k pots and I need precision to roughly 0.3 degrees, the loading effects are quite significant. How to fix? Twisted wires, buffers, and some hardcore low pass filters.

 The new revised board

Twisted Wires
My friends advised I twist the wires to help reduce noise. To be honest, I'm not sure how it works with non-differential signals, but I supposed it was simple enough to implement, and it does clean up the rats nest. And so, twisted wires.

Low Pass Filtering

Nothing fancy like Chebyshev or Butterworth. Just a simple RC filter that does the trick. Using the formula 1/(2 * pi * R * C) = -3db frequency, a 3.9 kOhm resistor along with 2 uF capacitor, I get a cutoff frequency of around 20 Hz, which is plenty high enough for a mechanical system such as this. The schematic looks like this.
On the oscilloscope, it turns the input (black) into a nice smoothed line (red).
 Sorry, I don't have a legitimate oscope capture. I promise it looked like this!

To fix the loading problem by adding in some op-amp buffers. Using unity feedback, I boost the current supply from a couple microamps to a couple milliamps. Excellent for keeping a stable voltage. The schematic looks like this.

When I tried out my new circuit, the circuit seemed great. Until I hit the low ranges. Then the output shot straight up, as such.
And it does so at 1.3 V, suspiciously close to two diode voltages (0.7 volts). The problem is that op amps are normally not designed to operate near their rail voltages. Here I use +5 and 0 volts, with an input voltage of 0 to 3.3 volts. Thus, it craps out near zero volts.

The solution is some rail to rail op amps. I used some op amps from TI, the so called TLV2774 Quad 2.7-V High-Slew-Rate Rail-to-Rail Output Operational Amplifier. It is rail to rail, so it can operate near its supply voltages, has four op amps in one package, though I only need three, and has input bias current in the picoamps. Also, free samples!

The entire circuit looks like this, with incredibly stable output voltage.
Digital Filtering

In addition to the analog filtering, I implemented some (relatively) long working digital filter. It takes 24 samples at 100 microsecond intervals. It then drops the lower eight samples and upper eight samples, and averages the middle eight samples. Yeah filtering!

Precision in measuring voltage is absolutely essential. Unlike a standard cartesian robot, where a little imprecision creates a constant offset, incorrect calibration in the arms causes the robot to not draw on a plane, but on some curved surface (I don't exactly know what it looks like, as it is some complex surface).

And so, with this new precision, a new video of the drawing!
Comparison of the output, with version 1 on the left and 2 on the right.
 What an improvement!

## Sunday, November 6, 2011

### LightDial: A Bulbdial Cousin

For the final project for Georgia Tech's ECE 4175, by teammate and I decided to make a Bulbdial-like clock. It can be called an "artificial sundial" of sorts. Lights around a center post shine onto it, and the shadow of the post makes the hands of the clock. The original bulbdial does something smart, and uses charlieplexing to handle the large number of LEDs (for n pins, they can have n(n-1) LEDs running on them).

However, we have project requirements to fulfill. We have to use an SPI device as part of the design. So we thought might as well go all in and drive each channel independently with a massive shift register. We decided on Texas Instruments' 24 channel LED driver, the TLC5951. Two of them, so we can have 24 minute hand positions and 12 hour hand positions for a total of 36 independently driven LEDs. Sorry, no second hand, or we will go crazy laying out and wiring them all together.

The benefit of this design is that all LEDs can be lit at full power, for alarm clock functionality or something. The downside is that there are a lot of wires.

After some layout wrangling, its a board! The EAGLE files can be found in the links below.

And, that's about as far as I have gotten. Soon, once I check some things with my professor, I will send the board to fab.

Appendix

Design files: https://github.com/aaronbot3000/lightdial

## Wednesday, November 2, 2011

### Pythagoras: Moving the Steppers

How do I move the steppers now? I can no longer simply send a position, like in version one. Something else will have to be cooked up.

After much thought...

I decided on a method quite similar to how version one was controlled in that incremental sense. However, I will have to extend the "moving in increments" to the motors too. In a nutshell, the manipulator moves a small increment forward along the path. Through inverse kinematics, I then get a set of new arm angles. Given the current angles of the arm, I then calculate the number of stepper steps it will take for all the arms to reach the new angles. I then distribute the steps for all of the arms so that they all finish at the same time, providing a good enough approximation for straight line travel. Confusing, isn't it?

Example time!

Given the arms angles are all at zero degrees. The step size is five degrees. Every program cycle, an arm either steps or doesn't step.

Then, the manipulator moves. The requested arm angles are 10, 20, and 30 degrees for arms one, two, and three, respectively. This means it will take the steppers at least six steps to meet the requested arm angles (30 degrees difference, 5 degrees step, 30 / 5 = 6 steps). So this means each cycle, arm one moves 1.67 degrees, arm two 3.33 degrees, and arm three 5 degrees. But arms one and two move less than one stepper step size each cycle. What do?

We create an auxiliary set of arm angles. They represent the ideal positions of the arms, following the increments of the step even if they are less than a step size.

At the end of program cycle one, we have:
• Ideal:  Motor 1: 1.67 Motor 2: 3.33 Motor 3: 5
• Actual: Motor 1: 0    Motor 2: 0    Motor 3: 0
The difference for motor three is greater than or equal to one stepper step size, so it takes a step, its actual angle updates to 5. Continuing, the pattern,
• Ideal:  Motor 1: 3.33 Motor 2: 6.66 Motor 3: 10
• Actual: Motor 1: 0    Motor 2: 0    Motor 3: 5

• Ideal:  Motor 1: 3.33 Motor 2: 6.66 Motor 3: 10
• Actual: Motor 1: 0    Motor 2: 5    Motor 3: 10
Fast forwarding:
• Ideal:  Motor 1: 5.00 Motor 2: 10   Motor 3: 15
• Actual: Motor 1: 5    Motor 2: 10   Motor 3: 15

• Ideal:  Motor 1: 6.16 Motor 2: 13.3 Motor 3: 20
• Actual: Motor 1: 5    Motor 2: 10   Motor 3: 20

• Ideal:  Motor 1: 8.33 Motor 2: 16.6 Motor 3: 25
• Actual: Motor 1: 5    Motor 2: 15   Motor 3: 25

• Ideal:  Motor 1: 10.0 Motor 2: 20.0 Motor 3: 30
• Actual: Motor 1: 10   Motor 2: 20   Motor 3: 30

As you can see, all the motors reach their targets at the same time! However, how do you control motor speed then? The same as in version one, by using small increments of the manipulator. Since the algorithm is set to "consumes" at least one cycle, even if no stepper moves, by having very small increments the algorithm spends most of its time moving the manipulator and not much time moving the steppers.

Just the nasty details of actually implementing it. Its a good thing I already did it. Of course, the obligatory video:

The steppers here are moving slowly at 1/2 microstepping, so they are quite loud. But it is a good demonstration of the capability for different speeds, as well as a proof of concept.

There is one problem

Lack of experience with analog signals has come to get me. Noise is all over the place in the feedback from the potentiometers, so much that the manipulator home position differs by an inch between resets. That's not very good precision there. Coming up is the solution. Hint: Twisted wires, op-amp buffers, analog low pass filters supplementing hardcore digital filters.

## Saturday, October 29, 2011

### Pythagoras Version 2: Steppers!

The accuracy of servos was maddening. Since I couldn't tweak the control loop or control the velocity, going back and forth across a line would create two separate paths. It overshot corners like crazy, and made the manipulator oscillate. The search began for a better control.

1. Rip out the servos' circuitry and roll my own control loop
2. Rip out the servos' circuitry and use OpenServo's circuitry
3. Ditch servos completely and use stepper motors with gearing
With option 1, I would likely be implementing PID control with some sort of feedforward component. Option 2 was the least appealing, since I would have to adapt the circuitry to my servos. Also, they were pretty expensive. Option 3 I didn't seriously consider at first. Thinking about it, though, steppers have all the characteristics I want. I can easily control both velocity and position. All this without messing with control loops. Sounds pretty good.

The downside is that I have no absolute position reference to start the counting. But that's what potentiometers are for!

Design And Build

Learning from past mistakes, I made the upper-to-lower arm length ratio much smaller, to get the precision over a larger range on the XY plane, at the expense of Z traversal. Potentiometers were linked to the output shaft through gears. I cooked up something that looked like this.

I also rotated the supports and added tabs, so they stay straight better. Later, I would find that using a more flexible metal (6061 aluminum) in this way is a bad idea.

I had a pretty convoluted way of getting both gears and arms onto the stepper shaft. Since I needed the arms to be able to turn past the potentiometer gear, the gears needed to be separate and machined separately.  And I also needed a spacer to keep the arms from colliding with the potentiometer gear. The solution: A stack of parts screwed together: 1/8" thick arm, then 1/8" gear, then 1/4" combination spacer and set screw holder.

SolidWorks really crapped out solving the model. Since I kept the upper arms floating to check the range of the manipulator, I kept getting overdefined errors if I tweaked the manipulator a little too far. Moving the manipulator took some time for the computer to solve the model. But I managed. As always, the design files are linked below in the Appendix. Time to find the right motors!

Stepper Motors

After much Google-Fu, I setteled on some 3V, 1.6A, 233 oz-in geared stepper motors from RobotShop. They have dandy planetary gearboxes with very little backlash (< 1 degree), with a gear ratio of 57/11 to 1. Given 200 steps per revolution, after the gearbox, I can get 0.35 degrees per step before microstepping. With my current eight microsteps, I get 0.04375 degrees per step. MWAHAHAHA.

They look like this in real life.
But now to drive the motors from a microcontroller signal. I used Allegro's A4988 stepper drivers on Pololu's breakout board. They support 8-35V, up to 2 A of current, do really neat current control for accurate microstepping, and microstep 2, 4, 8, and 16 divisions.

What is microstepping? Stepper motors move in discrete steps, so to speak. Microstepping allows a motor to stop at various places between the halfway points, multiplying the number of steps a stepper makes per revolution. For example, my steppers are 200 steps per revolution. Microstep two divisions, and it becomes 400. Eight times, and I get 1600 steps per revolution. Of course, this comes at a cost of torque. More steps, less torque in the motor. Also, the microsteps are less accurate then the whole steps, but the loss is negligible in this case (3-5%). Since I'm driving a light load, the torque loss is acceptable too. More information on microstepping and how it's actually done on the Wikipedia page.

Cutting the Parts!

Cutting the parts in the Invention Studio at Georgia Tech lets me use a higher quality and thus better edges. Yay! Here is preliminary results. You can see the arm-gear-set screw holder stack on the stepper's output.
Cutting out the rest of the parts. I took the manipulator and ball joints from version 1.
The entire wiring for the drivers and pots was done point to point with small wire. Nasty, nasty stuff. I'm not  proud, but it did work.
All the control signals were wired down through a ribbon cable. Yes, this also includes the analog signals from the potentiometers. Running around the motors. In a ribbon cable parallel to switching digital logic signals. And no filtering, high impedance potentiometers (5 kOhm), and you basically get really really bad data. But I didn't know this now because I have a new control algorithm to write for the steppers.

Coming Up Next
• Updated control for steppers
• How to fix the crappy analog signal problem
• Movement!
Appendix
Solidworks models for version 2: https://github.com/aaronbot3000/deltadraw/tree/master/solidworks_models_v2