Thursday, November 29, 2018

KSP: Laythe Colony Part 2, The Robotic Fleet and Launch Window #1

In honor of the successful Mars InSight landing this week, I thought I'd do a progress report on my long-term KSP mission to get as many Kerbals off Kerbin by Year 2, Day 0 as possible. Part 1 sets up the premise and the main strategy. In this Part 2, I throw about 1,000 tons of robotic hardware at Jool during the first available launch window, with hopes that at least some of it winds up in a single spot on the surface of Laythe as the seed for a colony.

The busy 1000m/s on-ramp to Jool Transfer Orbit.

The Robotic Fleet

For the first launch window, I decided to send only uncrewed vehicles to feel out the Jool transfer orbit, the details of maneuvering within the Jool system, and the landing procedures at Laythe. The robotic fleet consists of three types of ship: Triple Relay Satellites (RS3), Laythe Rovers (LR1), and Habitats (HAB1). Each one has a different function crucial to settling a remote colony.


Does this thing get HBO?
These are the smallest and lightest ships, but critical to this first remote-controlled mission phase. One of the relatively new realism additions to KSP is that uncrewed vehicles need to have a line-of-sight communication path back to Kerbin, or to a ship with a crew, in order to maneuver. To achieve this requires lining up a bunch of relay satellites around Kerbin and at other useful locations in the system. 

Each RS3 assembly carries three small relay satellites with their own ion drives. In addition to the ones already parked around Kerbin, two sets of three are on their way out to Kerbin L4 and L5 stations and then eventually other equally spaced points in the orbit. Three more sets are heading out to an intermediate orbit between Kerbin and Jool. And four sets of three are in the fleet heading for Jool, to set up a network around Laythe.

The start of the mission's comms network.


Practice driving on Kerbin.
The Laythe Rovers are giant 30 ton workhorses. The main function of these 8WD crawlers is to seek out ore to mine and make fuel on Laythe. They each have two large drills, a refinery, and a huge fuel storage tank. They can dock with a parked space plane to refuel it, which is critical for sustaining a link between the Laythe surface and hardware/habitats in orbit.

Landing the rovers is a four-step process. They come packaged in an aero shell with a heat shield, so the initial descent involves just surviving with the heat shield pointed in the right direction. After some time, the drag on the large heat shield flips the package around and the heat shield itself becomes a supersonic air brake, with the aero shell protecting the rover. Once subsonic, the heat shield and fairing are discarded and set of parachutes further slows and rights the rover. Lastly, a set of four rockets slows it to a safe velocity just in time for touchdown.

Step 5 is to quickly deploy the solar panels and drive out of the way of falling fairing debris.


Home, sweet home.
The Kerbals can live for extended periods of time in orbit, but having a home base on the Laythe surface will be important for long-term survival. In order to facilitate construction, the surface habitats are themselves rovers with roughly the same chassis as the LR1s. They land the same way and, once on the surface, can drive to each other. This will be important, since the landing target might hundreds of square kilometers.

The habitats are extremely modular. They can be individual homes for a single Kerbal family, including single-passenger mini-rover parked in front. Or, they can be docked together indefinitely to form a larger base, thanks to a central hallway section with docking ports on either end. The slight angle of the hallways allows them to fit inside the aero shell.

How much fuel to bring?

The LR1 and HAB1 landed payloads are both around 28 tons, about half the mass of my first Laythe lander. (That lander had to be heavy in order to have enough fuel to get back off of Laythe, a task to be handled by space planes this time around.) In that mission, two identical ships flew independently to Laythe with an average of about 2500m/s of fuel-burning Δv. But, they also made heavy use of aerocapture at both Jool and Laythe. Without that, it would take something more like 4360m/s to get from low Kerbin orbit to low Laythe orbit, according to the amazing KSP Subway Map.

With a known Δv requirement, figuring out how much fuel to bring is simple. With the 800s specific impulse of the LV-Ns, the minimum wet to dry mass ratio is:
So, the ships need to carry about 3/4 ton of fuel for every one ton of dry mass. Not too bad, since the same heavy lifter that carries up the packaged landers can carry up an equivalent mass of LV-N engine, fuel tank, and liquid fuel. Thus, each robotic lander requires two separate launches:

First, a heavy lift booster hauls the lander payload in its aero shell into orbit.
These are the unsung heroes of the mission, relentlessly hauling all the more exciting hardware into orbit.
Next, a second booster brings up a propulsion module, with LV-Ns and a lot of liquid fuel.
Just remember to check yo' staging...
The two meet in orbit and create a transport ship with a wet to dry mass ratio of about 1.775, for a Δv of about 4500m/s.

Rendezvous between an LR1 lander package and its propulsion module.
A total Δv of 4500m/s is cutting it a bit close, but they would only need a small amount of aerobraking or Tylo gravity assist to gain back a comfortable margin. There's also a good amount of RCS fuel on board that can be dumped (in a prograde or retrograde fashion) near the end of the trip if it's not needed. Additionally, the RS3 ships have a lot of fuel to spare if their propulsion modules can be swapped onto the more thirsty landers nearer to Laythe. The wet to dry mass ratio of the fleet as a whole has a comfortable margin.

Launch Window #1

The first Jool launch window happens around Day 190 in-game. (The simple and more complex online calculators both agree to within a few days).  Up to that point, I spent time refining the landers and practicing the landings on Kerbin. But once the designs were locked, the push began to assemble the fleet in orbit. 

The practical limit on fleet size is how many ships can be juggled during the actual launch window. In order to boost the wet to dry mass ratio, these ships have the two-engine version of the propulsion module, which gives them a somewhat low thrust to weight ratio. The ~2000m/s ejection burn had to be split into two parts: one into a 10-day elliptical orbit and a second to escape onto the final Jool transfer. Even still, the burns were 10 minutes each, so the ships had to be spaced out so they would reach their final periapsis burn at reasonable intervals.

Also, it would be nice if they didn't hit the Mun on their way in.
After the final burn, the transfer takes over two Kerbin years, meaning Kerbin will have been destroyed by the time the first ship even arrives at Jool. The lander designs can never be tweaked, and the crewed fleet will have to set out with no guarantee that there will be a base waiting. No pressure.

Lots of empty space to cross now.

661 Days Remain

With the first 18 ships on their way to Jool and then Laythe to set up base, the priority shifts to getting Kerbals off-planet. This means mass-producing and then filling the immense colony ships, which are the most intricate builds I have attempted in KSP yet.

More to come...

Monday, September 24, 2018

TinyCross: Electron Herding Unit

My ultralight electric crosskart kit has arrived:

By "kit" I mean the water jet-cut plates and the 80/20 extrusion. There's still some custom machining to do, but most of the kart is designed as plate-and-standoff structure, so it does feel a lot like a kit build. I want to start putting it together, but also realize I am way behind on the electrical side and should get some PCBs on order before I really dive in to the build.

Since it's 4WD with independent motors, I want two dual controllers, front and rear. One of my first three-phase controllers was a dual layout, using IXYS GWM100-01X1 three-phase bridge modules. These modules have excellent cooling, thanks to their huge isolated heat sink interface that can be mounted directly to the chassis (with thermal paste). And while the GWM100-01X1 is undersized for this project, the newer MTI200WX75GD looks spectacular. So let's begin.

3ph v2.2: MTI200WX75GD High-Power Upgra...crap.

They're not available. Despite being in stock for a while, they seem to have disappeared. I mean if you want, you can still get one, but the big distributors have stopped stocking them and the little distributors are pains in the ass. Some day they will return, and I'll be ready for that day, but until then I need some other way of herding electrons.

Can I Just Stuff Some FETs Somewhere?

Sometimes having an extra constraint, even if silly, is good for making progress on things. It makes optimization easier by removing degrees of freedom. So in this case my silly constraint is that any alternative MOSFET configuration must fit entirely within the footprint of the MTI, so they could be swapped without changing the board design. 

I played with some DirectFET and Power SO8 layout ideas before settling on just one monstrous FDMT80080DC per leg. They're similar in many ways to the MTI, with comparable resistance, gate charge, and current ratings. They also have top-side cooling that, although smaller in surface area and not isolated, could sink heat to the chassis fairly well. Using just one per leg yields a tight and tidy layout that fits within the 12mm span of the eight MTI pads allotted to each phase. I've also left solderable tracks for adding extra conductors if needed.

Extending this constraint further, I decided the 12mm strips should fit an entire phase: FETs, gate drive, local capacitance, current sense, and output.

Copy-paste six total for a dual controller. A header in the middle eats current/voltage sensor signals and spits back gate drive inputs. The gate drive is the same one I use on everything: anti-parallel optos with dead-time RC. It's low part-count, easy to bootstrap the high side, tri-stateable, and guarantees dead-time in hardware. And as far as the microcontroller knows, it's just driving some LEDs. What more could you ask for?

Cross Karts, Not Current

For current sense, the ACS781 offers an impressively tiny 100A (150A peak) surface-mount Hall effect current sensor. The current sensing range can be extended by diverting some of the current around the sensor. I've had issues with nearby large currents coupling into small Hall effect sensors before, so in this case I've taken extra care to keep non-sense currents flowing perpendicular to the sensitive axis of the ACS781:

The sensors are aligned so that their sensitive axis is along the board X axis (left to right). Except for the sense leg, all nearby current is routed along the board Y axis (top to bottom). This is especially important for the sense and bypass current flowing immediately adjacent to the next sensor over, since cross-coupling is harder to deal with than a slight change in sensitivity on a single phase. Similarly, phase wires exiting vertically (Z axis) should not produce much field in the sensitive direction. Hopefully.

The Rest of the Panel

I've done separate logic/power boards for a while, but this is my first controller designed as a multi-board panel. In this case I wanted the power board, logic board, and two motor sensor boards all in one. OSH Park has a four-layer prototyping service by the square inch (no panelizing penalty), and some of the most beautiful-looking boards I've ever gotten have come from them. They do quantity three for everything, which is perfect since it gives me two controllers and a spare.

The logic board sits on the gate drive / sensor interface, with a single STM32 running the show. It turned out a lot denser than I thought. While it's more common to do a full schematic first and then layout and route, I tend to make nicer boards if I route as I go, doing a few nets at a time and planning the placement and pinout with the routing in mind. And sometimes you just get lucky and find an open path on the layer you need.

It's All So Clean Until You Add Wires

Since the dual controllers are on the kart centerline and the motors are in the corners, I need to run about 700mm of motor phase and sensor signal wires together to each corner, and I know how that story usually ends. I'll use shielded cable for the signal wires and try to keep them separated as much as possible, but I have a new trick to try for making the Hall sensor signals more robust. For starters, they will run through a buffer on the sensor side, to decrease the impedance. (Normally, they are just open-drain with pull-up resistors.) The real trick is on the logic board side, though, where they will be sent through this:

The signals are fed into a three-channel optocoupler, which means noise would have to be able to inject mA-level current into the signal to change its state. But the input side of the optocoupler also references only the phase signals, not a local or even remote ground. Common-mode noise, even on the order of Volts, can't change the state. It's a sort-of three-phase differential signal.

The Big Picture

It's been interesting to revist the idea of a dual controller. I'm honestly not sure how my first one worked at all, although I think the robustness of the IXYS bricks and opto gate drive helped me out a lot there. This one feels a lot more routine, and I'll consider it a success if it just works without any fuss, like Twitch X's drive. Speaking of Twitch X, there's a really neat consequence of independent front-wheel drive that might make for some interesting software parallels down the road...

Back to the build for now.

Saturday, August 25, 2018

TinyCross: An Ultralight Electric Crosskart

It's time for a new go-kart!

I'm going for something a little bigger and more capable than tinyKart, but with the same ultralight genetics. This time, the inspiration is from something I didn't even know existed until recently, called a crosskart. The best way to understand the appeal of a crosskart is just to watch this video:

Now, there's no way an ultralight electric version will be as exciting as that, but if I can get to an end result that's about 33% crosskart, 33% downhill racer, and 33% electric scooter, I'll be more than satisfied. So, here's the concept:

TinyCross chassis and single corner module.
The design follows the water jet-cut plate and 80/20 construction that's been my go-to for rapid and adjustable construction. There are still a few machined parts, mostly turned shafts, but a large portion of the chassis can just be "printed" on a couple sheets of aluminum.

The 0.25in aluminum plate components to be cut.
Long-time readers may recognize that I've switched over to new CAD software. The days of free SolidWorks are over, so I'm using Fusion 360 now. Rather than arguing with people about the best ECAD/MCAD software, I tend to just try the different packages until I understand their fundamental strengths and weaknesses. I've got maybe 100 hours in Fusion 360 now and the biggest difference to SolidWorks I've noticed is the assembly constraint framework: SolidWorks "mates" are refined, purposeful, and user-friendly. The "Width Mate", for example, is sublime. Fusion 360 "joints" are raw and don't inherently guide you to the good practice of constraining your CAD the same way as your real assembly. But, they have a general flexibility that I find very powerful and time-saving.

As I've done more and more projects, I spend way more time in the CAD than actually building. This design has been almost a year in the works before I felt confident enough to order plates. I'm still not sure it'll turn into exactly what I want, but here are some of the details that I have been working on:

Drive Module x4

Drive block with motor and brakes.
The powertrain is entirely contained in a ~4kg block that is repeated in each of the four corners. It uses the same motors as tinyKart, Alien Power System 6374-170 outrunners, but with different gearing for the larger 12.5in scooter wheels. With four of them, I expect around 8-10kW peak output. I've carried over the offset bearing  belt tensioner that worked so well on tinyKart. The brakes are the same as well, but I managed to fit the caliper in the middle of the belt loop to keep everything in-line.

Suspension and Steering

Front suspension and steering geometry.
Suspension separates a crosskart from a go-kart in my mind. I wanted something that can handle speed bumps, potholes, gravel, dirt, snow, and other surfaces that would (and did) tear tinyKart apart. Now, I don't have any experience designing vehicle suspension, so I took a reasonable first guess on the geometry but left room for tweaking later. The shocks are mountain bike air shocks, which are both very light and easily adjustable. The total suspension travel is about 5in.

Because the suspension takes up so much space, I wound up moving the steering linkage to the very front of the kart to make more legroom. This turned out to be a major headache since the Ackermann geometry is backwards. I was able to get back to a reasonable geometry by using a slightly more complex linkage. Minimizing bump steer from suspension travel was also an interesting new challenge for me.


Box chassis to add a little stiffness.
tinyKart's chassis was truly flat-packable, but as a result it twisted a lot. This actually worked in its favor somewhat, keeping all four wheels on the ground in lieu of suspension. But here, I was willing to go vertical and make a bit more of a box structure to keep the chassis stiff. Note that there are many stiffening spacers between the thinner bottom plates that I've left out of the model to keep the assembly smaller.

I still want to be able to separate the chassis into front an rear sections for transport with as few fasteners as possible. Removing the two side rails and then sliding the rear chassis out of the front chassis plate sandwich seems reasonable.


I've settled on motors and batteries (12S/20Ah LiPo), but haven't started any of the controller work yet. I'd like to do an update version of the 3ph "Duo" v2.1 controller, one in front and one in back, but based on the newer IXYS MTI200WX75GD-SMD, which is a pretty awesome little part. I need to run the math though, since the original design was never meant for these power levels.

I'm also planning a simple board for the steering wheel, to host a display, trigger throttle, and some knobs or switches for quick setup. Not sure how far I will dive into independent torque control, but I do plan to link everything together with CAN to leave that possibility open.

Power/Weight Budget

I'm shooting for about 34kg / 75lb for the kart without batteries. With the full 12S/20Ah it will be about 40kg / 88lbs. This is about 50% heavier than tinyKart, but still light enough to manipulate up onto a table to work on, or into a vehicle for transport. Plus, most of the weight of a go-kart is the rider, so with double the power it should have plenty of acceleration to spare.

Parts should be coming in this week so next post will have some real pictures!

Sunday, August 6, 2017

KSP: Laythe Colony, Part 1

A while back, I read a book called Seveneves, by Neal Stephenson, and found it to be immensely entertaining. The premise is that, for unknown reasons, the moon explodes and humanity has two years to get off the surface of the planet before the pieces crash into it. What I particularly enjoyed is that it's set in the near future, so we only have the tools we have now to work with. One of those tools is the International Space Station, which becomes the hub of a permanent space habitat. With no budget constraints and no point in risk aversion, a convincing amount of hacking manages to throw lots of hardware into space in a short period of time. And it falls on a few people to figure out what to do with all of it, since no viable long-term mission plan is proposed by ground leadership, who have other concerns. (Just go read it!)

You know who else has no budget constraint and no concept of risk aversion? The Kerbals! I decided a fun challenge would be to see how many Kerbals I can get off-planet in two (Kerbin) years, with the ultimate goal being to land them on Laythe and set up a colony. Laythe's atmosphere, while not directly breathable by Kerbals, has enough oxygen for jet combustion. And there is plenty of water.

Like, seriously a lot of water.
It's been a while since my Kerbals first carried out a mission to Laythe and back. Making a precision landing on Laythe's sparse, sand dune-covered islands was a challenge, especially with a top-heavy vertical lander. In order to establish a colony, Kerbals and equipment will have to meet at a single location on the surface of Laythe, so a better approach is needed. For the passengers, this means riding down on one of these:

This space plane uses four CR-7 R.A.P.I.E.R engines, hybrid engines that switch from air-breathing to closed-cycle at altitude. So, it takes off like a jet, builds up as much speed and altitude as it can, then switches to rocket propulsion to reach orbit. KSP's aero model was entirely redone for v1.0 and onward, making atmospheric flight both more realistic and more difficult. But this plane can still ferry six Kerbals into Low Kerbin Orbit (LKO) with a bit of fuel to spare.

It's also surprisingly sporty.
The Kerbals can't ride space planes all the way to Laythe, though. Well, they can if you build one like this. But I can barely get my six-passenger one into orbit so I'll need a more suitable living space for the passengers on the long cruise to Laythe. This led me to build my first true space station in LKO:

Finally a place to park all my space planes...
I wanted it to be modular and symmetric so it would be easier to add a propulsion stage for the long journey. This meant docking two space planes, and after some thought I decided they should be back-to-back to minimize the inertia on the long axis. (Any guesses as to why I would want to do that?) The planes are docked with two ports each for precise alignment and more rigidity.

The T-shaped station is actually launched into orbit in two pieces. The docking module attaches to the space planes, while the crossbar of the T is the main passenger module, where the Kerbals will ride out the long trip. The passenger module will become the hub of a much larger ship once propulsion is added on opposite the docking module.

Getting pieces this large into orbit is the work of a new two-stage heavy lifter.

No, the first stage is not recoverable.
The first stage is powered by five RE-M3 Mainsail engines, and includes all eight side-mounted fuel tanks and the bottom central fuel tank. The second stage is just a single tank and Mainsail. Together, they can hurl about 50t into orbit. Except for the space planes, every piece of hardware that goes to Laythe will start out mounted to the top of one of these ascent stages.

And there are many more pieces of hardware required to make the trip. The main strategy will be to send a crapload of unmanned stuff (rovers, habitats, and utility vehicles) to Laythe during the first launch window (~184d in) and then follow up with a fleet of passenger ships, based on the station above, during the second launch window (~1yr, 225d in). But, since KSP has added an element of realism in that unmanned craft need communications to be remote controlled, a network of relay satellites is also needed. All of this also needs efficient propulsion for the long leg of the trip.

More to come...

Sunday, July 23, 2017

Link Update (Again) and Bonus Link Trig!

Once again, all my documentation permalinks got broken when Google Drive stopped supporting static URL hosting. But, I've transferred everything over to AWS now and gone through updating links accordingly on all the static project pages and a few select posts, especially this one. Hopefully these permalinks last a bit longer! Here's the full directory: I may transfer more stuff over to there as I go, including maybe using it for hosting new static project pages. Still learning my way around AWS.

Speaking of links, here's an odd bit of trigonometry to solve last post's linkage mystery:

For Twitch X, I mentioned that rather than solving the trig for the geometry of the linkage that connects the two sets of diagonal wheels together (the sin/cos link, as I called it), I came up with a weird exponential parametric equation for the two angles:
x = [sin(θ1)]^K = 1 - [cos(θ2)]^K
The value x ranges from 0 to 1 as θ1 and θ2 both sweep from 0º to 90º, albeit on different trajectories. It makes an excellent feedback variable for the controller, since it can be derived from a simple weighted average of the two linkage angle sensors (after trig and exponent). And conveniently, x = 0.5 is where the wheel sets are perpendicular, for omni mode. For the link lengths used on Twitch X (L1 = 1.00in, L3 = 7.50in), I experimentally derived K = joke. But I wasn't ever convinced this was an exact solution, and as it turns out it isn't.

A few pages of trig later, this is the actual parametric solution:
L3*[cos(θ2) + sin(θ1) - 1] = L1*sin(θ1 + θ2)
This one's a lot less convenient from a control standpoint: since θ1 and θ2 are on both sides of the equation it's not obvious what to do with the wheels based on measurements of the linkage angles, at least not without further math. But this is the geometrically accurate solution.

What's amazing is how close the other parametric equation is. As it turns out, the max error is just over 1º and the value K = 1.23456789 is actually a pretty good one in terms of the average error over the full range from 0 to 90º:

None of this matters as far as Twitch X control software is concerned - the exponential approximation with θ1 and θ2 separated is still far better for the application. But solving the mystery of the sin/cos link was really interesting.

Thursday, June 16, 2016

Twitch X - Servoless Linkage Drive

I've got a new bot.

I'm not ready to call it 100% finished yet, but, most importantly, it drives!

It can pull off the trickiest bit of linkage drive maneuvering, Translation While axis-swITCHing, which proves it is possible to control all four degrees of freedom at the same time with just the four motors. That's a much better ratio of actuators to degrees of freedom than Twitch, Jr, which relied on two giant servos to steer the linkages. To steal a term from another Twitch, it's a "holonom-ish" drivetrain: able to fully control all three of its planar degrees of freedom, sort-of, in some cases, with an extra degree of freedom just for fun (and for generating more pushing power in a particular direction when desired). In reality, it doesn't have a practical advantage over some other drivetrains, but I've driven lots of different types of robots and this is by far the most fun.

I've Missed Going Home with Aluminum Chips in My Hair

The build was relatively quick and easy.

It helps when most of your parts are topologically similar waterjet-cut plates.
There were a few minor issues...can you spot the one in this picture? (Not the small linkages. Those are just from Twitch, Jr. for comparison.)
 The main actual fabrication required was making a few turned parts on my tinyLathe.

Poor tinyLathe.
These were the posts and keyed hubs for the wheels. The posts hold the thrust and radial bearings on which the drive units swivel. They're straight from tinyKart's steering system, so they should be very overkill for this robot.

Other than that, there was a just a lot of finish-drilling and countersinking. Oh, and some sketchy sheet metal bending that came out surprisingly well (using 5052 aluminum instead of 6061, for better forming properties).

The dimensional accuracy of the build wasn't quite as good as I was hoping, mostly due to lazy machining on my part. The top and bottom plate don't quite drop on with a satisfying zero-force slip fit, but with the bearings it really doesn't matter much. The main mechanical problem I ran into was not leaving any clearance for the rounded-off linkage ends to the inside surface of the motor mounting blocks. This led to a bit of binding that I thought was due to frame alignment issues but in fact was easily solved with a belt sander.

By virtue of careful design and forethought complete luck, I actually mitigated one of the biggest deficiencies of Twitch X over Twitch, Jr., the relative difficulty of changing out wheels. As it turns out, because of the way wheel closeouts are shaped, it's just barely possible to put on and take off a wheel without removing the top and bottom plates. This is a huge win because the top and bottom plates have the most hardware, the trickiest alignment, and are attached to the two linkage position-sensing potentiometers (so, taking them off would usually require re-calibration).

You might also notice the magnetic hubs. More on these in a later post, I think.

It's also possible to access most of the linkage shoulder screws from the wheel wells by rotating the linkages to different positions, so if one comes loose it doesn't necessarily require taking the whole robot apart to fix it. The center of the robot is also relatively accessible (for adjusting a linkage pot or soldering a motor lead, for example) thanks to the lack of giant servos in the middle and the fact that the battery and controller are outside of the central section.

It's so...empty.
The Mystery of the Sin/Cos Link

Where the two servos would be are just two potentiometers, one attached to each diagonal linkage. In Twitch, Jr., the diagonal linkages are independent, but in order for servoless linkage drive to work, they need to be tied together (to reduce the total number of degrees of freedom to four). I was a bit naive in naming the link that ties them together the sin/cos link, thinking that it just caused one to sweep out asin(x) while the other swept out acos(1-x), or something like that. The actual trajectory is not that simple.

If you can figure out the function f, you will win a cookie.

In what might be a first for this blog, I actually don't have an analytical solution for it. I'm sure one exists, but I think it would be a messy bit of trig. Through some random guesswork and not wanting to rename the linkage, I found that the following parameterization is very nearly perfect:
x = [sin(θ1)]^K = 1 - [cos(θ2)]^K
The exponent K is determined by the geometry of the linkage and can be found by fitting to CAD-solved angles. For Twitch X, it's somewhere around 1.23456789. (I'm not joking.) You can have an extra cookie if you can explain this parameterization and how the exponent can be derived from the geometry. I actually haven't worked this out. (For the purpose of verification, L1 = 1.00in and L3 = 7.50in on Twitch X.)

The parameter x is actually extremely convenient for controlling the linkage degree of freedom. It's easy to measure θ1 and θ2 using the pots, but it would be awkward to control one or the other. As the linkage sweeps through its range of motion, the sensitivity of the two angles to wheel rotation changes. Near the ends of travel, one angle is barely changing. By converting both angles to x and taking a weighted average based on the sensitivity, which is itself a function of x, a much better control variable is made available, one that ranges from 0.0 to 1.0 as the linkage degree of freedom sweeps from full forward to full sideways.

One other really nice thing about the parameter x is that at 0.5, the wheels are perpendicular. This is true for any exponent K. This gives a simple target for the linkage controller to get into the traditional diamond-layout omnidirectional drive configuration. Because of the weird geometry, the wheels are not at 45º angles to the chassis in this mode the way they were on Twitch, Jr. But, they are perpendicular to each other, which is the necessary condition for properly-constrained driving with four omniwheels. The driving coordinate system is actually rotated about 10º from the chassis at this point.

At x = 0.5, any exponent will give a linkage angle sum of 90º...very convenient.

I was mistaken in my last post when I said that it was possible to write a continuous mixer for all the in-between states that aren't linkage angle sums of {0º, 90º, 180º}, corresponding to the {forward, omni, sideways} driving. As it turns out these are all over-constrained and don't have a roll-without-slipping solution for wheel rotational velocity. (And I'm not talking about sideways slipping, I mean true tangential slipping.) So, I used a three-state mixer similar to Twitch, Jr. to handle the three driving modes. The pot-derived x parameter determines which state it's in. 

Twitch Drive

The quad H-bridge board I designed for Twitch came in and went together pretty easily.

I don't know yet if it's worthy of being it's own separate thing, but I really do like the layout and the modularity of the design. Besides the four H-bridges, there's some power conversion, an STM32F3 microcontroller, an MPU-6050 IMU, and headers for an XBee. The optocoupled gate drive works the same way it always does: without any drama.

Mmmmm, free deadtime.
The main issues I had were with relatively low-quality boards causing some soldering mishaps requiring blue wire micro-surgery. I've already ordered some spares from my absolute favorite board place, OSHPark, so if this one eventually dies I have some really nice ones to replace it. The board doesn't quite fit the way I wanted it to in the front wedge. It was supposed to mount vertically, but the capacitors and wiring take up too much space. I spent several hours trying to figure out how to modify the chassis to fit it either vertically or horizontally before I realized that I don't have to do either...

Yes, I felt stupid.
Since I'm only using the vertical gyro anyway, it doesn't matter which way the board is oriented. The extra trig just goes into the rotation controller's gain scaling anyway. Also, this mounting allows me to put some padding around the board to protect against impacts and vibration a bit more. The batteries will still go in the opposite wedge, and all the wiring runs in a tidy channel down the middle of the bottom plate.

Control and Controllers

As I mentioned, Twitch X has four degrees of freedom, all of which can be controlled independently by the four actuators. The "mixer" handles assigning wheel velocities based on the outputs of four degree of freedom controllers. In general, all four wheels are involved in each degree of freedom:

Driving forward and turning are the obvious ones and are the same as any other 4WD "tank steer" or "skid steer" robot. Driving sideways requires that the linkage be moved to the sideways position and then it's the same as driving forward (although two of the wheels have reversed their "forward" direction). Omnidirectional drive in the x = 0.5 "diamond" state also has a well-known mixer. The last degree of freedom is moving the linkage itself. This is accomplished by driving pairs of wheels against each other. (Which pairs depends on which way you want the linkage to move.)

Might help picture it...

Forward and sideways translation are easy enough to control manually, so the mixer just forwards commands for these directly to the correct wheels, depending on the linkage position. The other two degrees of freedom are much better handled by a closed-loop controller. For rotation, the vertical gyro is used in a feedback loop to control an exact rate of rotation, commanded by the driver. This helps keep the bot straight even if the wheels slip a little, and is crucial to this type of drivetrain. Likewise, the linkage degree of freedom is feedback-controlled off of the x parameter, as measured by the two potentiometers.

Everyone asks what the operator interface is like - I use a Playstation 4 controller with the following layout:

I don't know why I chose this layout originally, but I've been training on it since Twitch, Jr. and have the maneuvers all in muscle memory. The coolest tricks are ones involving all four degrees of freedom at the same time, like Translation While axis swITCHing, where the bot travels in a straight line but rotates and changes linkage orientations on the fly.

There are still some improvements to be made - I haven't gotten around to finishing the magnetic wheel hubs yet or really tying down all the loose parts and getting it ready to take abuse. But I also enjoy driving robots even more so than I do building them, so I couldn't resist doing some test drives of the new servoless system as soon as it was functional.