I’m now married!

We had a wonderful wedding, with loving friends and family members, many of which traveled pretty far to celebrate with us. It was a pretty small wedding, but it was a heartfelt one, and we put a lot of effort into things to make it the wedding we dreamed of. In this blog post I’ll cover some of the personal touches we were able to add.

TikZ Place Cards

Photo by @wendyandgeorgephoto

We had guests RSVP via a Google form, which asked them information like their name, which meal they wanted, and whether they had any food allergies. This produces a Google sheet, which I exported to CSV and loaded into a little Julia notebook. My notebook then typeset place cards for each guest, displaying their name, table assignment, meal choice, and any food allergies. This used TikZ via TikzPictures.jl, and inverted each image in order to get a fold-able card:

The script did some basic layout to produce paper sheets with multiple cards such that we could print them out with maximum effectiveness:

We had to address some corner cases, such as babies (which don’t have meals), guests with multiple allergies, and guests with long names.

We used the same approach to make a few other place cards for things like the desserts table:

Table Numbers

Each dining table had a number, and we needed a way to indicate which table was which. My partner had a fantastic idea, and made table signs with table numbers that included photos of us at that age. So, table 2 had photos of each of us at 2 years of age, table 3 had each of us at 3 years of age, etc:

Photo by @wendyandgeorgephoto

These were made in Keynote, and were printed out on photo paper and cut out by hand.

Succulent Centerpieces

We also wanted table centerpieces, but balked at the extreme price of floral arrangements. Again, my partner had a fantastic idea. She had been learning pottery, and decided that she would throw her own bowls, glaze and fire them, and then make succulent arrangements. She gradually worked on these bowls all year, learning to make bigger and better bowls as she progressed, and made some very beautiful works of art. Many of the succulents were clippings from my mom’s garden. The end result was amazing!

Photo by @wendyandgeorgephoto

Portable Piano

We wanted our best man to play the piano during the ceremony. The only problem was that we had to use a portable piano that wasn’t plugged into an electrical outlet.

Thankfully, we were able to use the best man’s electric keyboard, but said electric keyboard had to be modified to use a portable power source. After getting on YouTube, including watching a delightful video of a man playing music outdoors, we determined that building a battery was pretty straightforward. The piano takes 12v, so I ordered a 12v, 20 Ah battery.

The battery life in hours is its capacity (in amp-hours) divided by the current draw (in amps). The current draw is watts / volts, which for us is around 40 W / 12 v ~ 3.33 amps. Thus we expect the piano to last 20 Ah / 3.33 A ~ 6 hours. We figured that should be more than plenty.

I also ordered a charger, and a Yamaha piano wall wart. I cut the plug off of the wall wart and connected that, plus a fuse, to the battery:

(with lots of protective electric tape – don’t want to get shocked.)

We put the battery in the cardboard box it came in, spray painted it black, and 3d-printed a nice little cover:

We additionally 3d-printed a little wedge so that the battery could sit on the X-frame stand that supports the piano. Unfortunately, I forgot to get a photo of that.

And then there was the problem that the piano was initially dead after we borrowed it. After much troubleshooting, my partner’s internet investigations suggested that we replace one particular capacitor that was known to wear out on this brand of electric keyboard. So (with permission) we ordered it, took the piano apart, and replaced the capacitor.

Ring Box

We used the 3d printer for other things as well! My partner printed a wonderful ring box whose design she found:

Photo by @wendyandgeorgephoto

Our names are inlaid on the other side.

Homemade Desserts

You know what’s harder than making desserts for your own wedding? Transporting them to the wedding.

Enter – the Macaron Transport Vehicle (MTV). My partner is fantastic at baking macarons, and has spent something like 10 years perfecting the recipes. She baked the macarons, and also came up with the method for getting them to the venue.

Each MTV consisted of an Ikea cooler bag with a cooler-bag-bottom-sized ice pack. This could be loaded with trays of macarons, with 3d-printed spacers to allow for tray stacking. The macarons made it safe and sound, and cold!

Photo by @wendyandgeorgephoto

Making the macarons was a 3-day process. The fillings were made on Day 1, the shells on Day 2, and they were assembled on Day 3.

Wedding Favors

We wanted to have something for our guests to take home with them. We thought about giving out succulents, but figured those would be hard to travel with, and maybe guests wouldn’t want them. We figured that pretty much everyone likes chocolate.

There are services that let you make your own custom chocolate wrappers. We tried a sample or two and weren’t really blown away by the chocolate itself. So we decided to just buy chocolate bars that we liked and make our own basic sleeves:

These were made in Keynote, and were printed on heavier paper.

Funny story about the text on the back. I initially wrote something much longer, and my partner had the great idea to use ChatGPT to consolidate it a bit. It worked really well, with some minor manual adjustments.

We then tried asking it to add some chocolate puns. ChatGPT 3.5 was decidedly terrible at adding puns. It basically just inserted “choco” into the text in random places. Ah, well.

Website

Lastly, rather than use a wedding website service, I figured I had my own website already, and could just add to that. It was just a simple page based on the same HTML5 Up template that I use for my main website today. Per a friend’s suggestion, I used Leaflet and Open Street Maps to display the location of the venue and the hotel. This was also the page that guests could access the RSVP Google Form from.

Conclusion

We had a wonderful wedding, with wonderful guests and a lot of wholesome, positive energy. Things went incredibly smoothly, and the worst thing about it was how quickly it all flew by.

I think the little touches that we added made the wedding feel more personal.

I’ve run into subsumption architectures a few times now. They’re basically a way to structure a complicated decision-making system by letting higher-level reasoning modules control lower-level reasoning modules. Recently, I’ve been thinking about some parallels there with the transformer architecture that has dominated large-scale deep learning. This article is a brief overview of subsumption architectures and attempts to tie them in with attention models, with the hope to better understand both.

What is a Subsumption Architecture?

The Wikipedia entry on subsumption architecture describes it as an approach to robotics from the 80s and 90s that composes behavior into hierarchical modules. The modules operate in parallel, with higher-level modules providing inputs to lower-level modules, thereby being able to operate and reason at a higher level.

These architectures were introduced by Rodney Brooks (of Baxter fame, though also much more) and his colleagues in the 80s. Apparently he “started creating robots based on a different notion of intelligence, resembling unconscious mind processes”. Brooks argued that rather than having a monolithic symbolic reasoner that could deduce what to do for the system as a whole, a subsumption approach could allow each module to be tied closer to the inputs and outputs that it cared about, and thus conduct local reasoning. This argument is often tied together with his view that, rather than making a big symbolic model of the world, which is a hopeless endeavor, that “the world is its own best model” and submodules should be able to directly interact with the portion of the world they interact with.

Perhaps this is best described using an example. Consider a control system for a fixed-wing aircraft. A basic software engineering approach might architect the aircraft’s code to look something like this:

where, for most flight dynamics, we can decouple forward speed / pitching / climbing from yaw / rollFor a good overview, see Small Unmanned Aircraft by Beard and McLain. Unfortunately, we run into some complexity as we consider the behavior of the longitudinal controller. When taking off, we want full throttle and to maintain a fixed target pitch, when flying below our target altitude, we want to use full throttle and regulate airspeed with pitch control, when near our target altitude we want to control our altitude with pitch and control our airspeed with throttle, and when above our target altitude we want zero throttle and to regulate airspeed with pitch control:

As such, our longitudinal controller has quite a lot of internal complexity! Note that this is only for takeoff to cruise; landing would involve more logic. We want to control our aircraft with different strategies depending on the context.

Subsumption architectures are a means of tackling this sort of context-based control. Rather than programming a bunch of if/then statements in a single longitudinal control module, we can further break apart our controller and allow independent modules to operate concurrently, taking control when needed:

This architecture works by splitting our monolithic logic into independent modules. Each module can receive inputs from any number of upstream modules. For example, the throttle output module, which forwards the final throttle command to the motor, can receive direct throttle commands from the airspeed-via-throttle-controller, the descent controller, the climb controller, or the takeoff controller.

Each module thus needs a method for knowing which input to follow. There are multiple ways to achieve this, but the simplest is to assign different modules different priorities, and their outputs inherit those priorities. The highest propriety input into any given model would be the one acted upon. In our little example above, the cruise controller could just be running continuously, with its output being overwritten by the takeoff controller when it activates during takeoff.

We get our emergent takeoff behavior by allowing the takeoff controller, when it activates, to override the cruise controller’s output:

This modularization has some major advantages. First, modules are self-contained, and it is much easier to define their desired behavior and to test it. It is much easier to validate the behavior of a simple pitch controller than the complicated behavior of the monolithic longitudinal controller. Second, the self-contained nature means that it is much easier to extend behavior by swapping or implementing new modules. Good luck having a new member of the team hook up everything properly in a monolithic code base.

We can similarly take this approach further and take it farther up the stack, with our longitudinal and lateral controllers taking inputs from trajectory trackers, or waypoint followers, a loitering controller, or a collision-avoidance module. All of the same principles apply. Finally, it is easier to monitor and debug the behavior of a modular system, especially if we make the modules as stateless as possible. We can log their inputs and outputs, and then can (more) easily figure out what the system was doing at any given time.

The approach is not without its disadvantages. The most obvious change is that we now have a bunch of disparate modules that can all act at various times. In the monolithic case we can often guarantee (with if statements) that only certain modules are active at certain times, and curtail complexity that way. Another issue is added complexity in handoff – how do we handle the transitions between module activations? We don’t want big control discontinuities. Finally, we have the issue of choosing priority levels. For simple systems it may be possible to have a strict priority ordering, but in more complicated situations the relative ordering becomes trickier.

Subsumption architectures were originally invented to be able to easily introduce new behaviors to a robotic system. Our previous controller, for example, could be extended with multiple higher-level modules that use the same architectural approach to command the lower-level modules:

The running example is for a fixed-wing controller, but you could imagine the same approach being applied to a robot. The Wikipedia article suggests a higher-level exploration behavior that can call lower-level walk-around or avoid-object modules.

Unconscious Mind Processes

Brooks was originally motivated by subsumption architectures because it allowed for “unconscious mind processes”. By this I think Brooks meant that higher level controllers typically need not concern themselves with low level details, and could instead rely on lower level controllers to handle those details for them. In our example above, a trajectory follower can just direct the cruise controller to follow the trajectory, and does not need to think about elevator or throttle rates directly.

One might argue that claiming that this sort of an approach resembles how the human mind works is a stretch. We are all familiar with sensational articles making big claims on AI approaches. However, I think there are core ideas here that are in line with how human reasoning might come to be.

For example, Brooks argues that subsumption models have emergence. We don’t think of any of the lower-level modules as being particularly intelligent. A pitch controller is just a PID loop. However, the combined behavior of the symphony of higher-level modules knowing when to jump in and guide behavior causes the overall system to exhibit quite sophisticated reasoning. You, a human, also have some “dumb” low-level behavior – for example what happens when you put your hand on a hot stove top. Your higher-level reasoning can choose to think at the level of individual finger movements, but more often thinks about higher-level behavior like picking up a cup or how far you want to throw a baseball.

I think this line of reasoning bears more fruit when we consider the progress we have made with deep learning architectures. The field has more or less converged on a unified architecture – the transformer. Transformers depart from prior sequence learning architectures by doing away with explicit recurrence and convolution layers in favor of attention. Attention, it just so happens, is the process of deciding which of a set of inputs to pay attention to. Subsumption architectures are also very much about deciding which of a set of inputs to pay attention to.

We could restructure our subsumption architecture to use attention:

It would make a lot of sense for a robust deep learning model to have produced, inside of its layers, a diverse set of modules that can spring into action at appropriate moments and make use of other modules in lower levels. Such an outcome would have big wins in robustness (because lower-level modules become general tools used by higher-level modules), have the benefit of progressive learning (we’d refine our basic behaviors first, like when a child first learns how to walk before reasoning about where to walk to), and we’d be able to develop more nuanced behaviors later in life by building new higher-level modules.

Interestingly, this sort of emergence is the opposite of what a classical deep neural network typically produces. For perception, it has been shown that the first layers of a neural network produce high level features like edge detectors, and then the deeper you go, the more complicated the features get. Eventually, at the deepest levels, these neural networks have features for dogs or cats or faces or tails. For the subsumption architecture, the highest fidelity reasoning happens closest to the output, and the most abstract reasoning happens furthest from the output.

The Tribulations of Learning

Unfortunately, the deep-learning view of things runs into a critical issue – our current reliance on gradient information to refine our models.

First off, the way we have things structured now requires gradient information from the output (i.e., motor controllers) to propagate back up the system to whichever cause produced it. That works pretty well for the modules closest to the output, but modules further from the output can suffer from vanishing gradients.

Vanishing gradients affect any large model. Modern architectures include skip connections (a la ResNet) to help mitigate this effect. It helps some.

Second, these models are typically trained with reinforcement learning, and more specifically, with policy gradient methods. These are notorious for being high variance. It can take a very long time for large-scale systems to learn to generalize well. Apparently Alpha Go played 4.9 million times in order to train itself to play well, which is way more than any human master ever played. Less structured tasks, like robot grasping, seem to also require massive quantities of training time and data.

This second problem is more or less the main stickler of reinforcement learning. We know how to build big systems and train them if we are willing and able to burn a whole lot of cash to do so. But we haven’t figured out how to do it efficiently. We don’t even know how to leverage some sort of general pre-training. There are certainly projects that attempt this, it just isn’t there yet.

The third problem is more specific to the hierarchical module architecture. If we get sparse activation, where some modules are only affected by certain parent modules, then there are presumably a bunch of modules that currently do not contribute to the output at all. The back-propagated gradient to any inactive module will be zero. That is bad, because a module with zero gradient will not learn (and may degrade, if we have a regression term).

I think the path forward is to find a way past these three problems. I suspect that subsumption architectures have untapped potential in the space of reinforcement learning, and that these problems can be overcome. (Human babies learn pretty quickly, after all). There is something elegant about learning reliable lower-level modules that can be used by new higher level modules. That just seems like the way we’ll end up getting transfer learning and generalizable reasoning. We just need a way to get rewards from places other than our end-of-the-line control outputs.

Conclusion

So we talked about subsumption architectures, that’s cool. What did we really learn? Well, I hope that you can see the parallel that we drew between the subsumption architectures and transformers. Transformers have rapidly taken over all of large-scale deep learning, so they are something to pay attention to. If there are parallels to draw from them to a robotics architecture, then maybe that robotics architecture is worth paying attention to as well.

We also covered some of the primary pitfalls. These challenges more or less generally need to be overcome to figure out scalable / practical reinforcement learning, at least practical enough to actually apply to complex problems. I don’t provide specific suggestions here, but oftentimes laying out the problem helps us get a better perspective in actually solving it. I remain hopeful that we’ll figure it out.

A few weeks ago I got an email asking me about a game I worked on ages ago. I had to inform the sender that, unfortunately, my inspired dreams of creative development had not come to full fruition.

The message did get me thinking. What had I been doing back then, about 8 years ago? What problems did I run into? Why did I stop?

Now, I took a look at my old source code, and I’m pretty sure I stopped because of the mounting complexity of classes and cruft that happens in a growing codebase, especially when one has comparatively little idea of what one is doing. That, and I’m pretty sure this was around the time of my qualification exams, and well, I think that took over my life at the time.

Anyhow, I was thinking about this top-down 2D game, and the thoughts that kept circling around my brain centered around 2D character movement and collision. This was only magnified when I played some iPhone games to supplement my entertainment when I flew long distance over the holidays, and grew dismayed at some of the movement systems. I thought player movement was hard back then, and even today a fair number of games don’t really cut the mustard.

The Problem

So this post is borne out of mulling over this old problem of how to move a player around in a 2D game without them running into things. We want a few things:

• Nice continuous movement
• To not intersect static geometry
• To nicely slide along static geometry
• To not get hung up on edges, and nicely roll around them

And, as bonus objectives, we’d love to:

• Do all of these things efficiently
• Be able to handle non-tile maps with arbitrary geometry

The central issue with many games is that you have things you don’t want to run into. For example, below we have a screenshot from my old game-in-development Mystic Dave. We have a player character (Mystic Dave) and some objects (red) that we don’t want Dave to run into:

The player typically has a collision volume centered on their location. This collision volume is not allowed to intersect with any of the level collision geometry:

So what we’re really dealing with is moving this player collision volume around with the player, and making sure that it doesn’t intersect with anything.

Basic Player Movement

The player movement part is pretty simple. We can just look at whatever direction the player input is getting us to move in, and move ourselves in that direction:

void UpdatePlayer(Vec2 input_dir) {
   Vec2 pos_delta = input_dir * kPlayerStepSize;
   pos_player_ += pos_delta;
}

We’ll run into tuning problems later on if we don’t make this a function of the time step, so let’s go ahead and do that now:

void UpdatePlayer(Vec2 input_dir, float dt) {
   Vec2 pos_delta = input_dir * kPlayerStepSizePerTime * dt;
   pos_player_ += pos_delta;
}

This will give us some pretty basic movement:

That movement is a bit boring. It stutters around.

We can make it smoother by storing the player’s velocity and adding to that instead. We enforce a maximum velocity and apply some air friction to slow the player down as we stop our inputs:

void UpdatePlayer(Vec2 input_dir, float dt) {
   // Update the player's velocity
   vel_player_ += input_dir * kPlayerInputAccel * dt;
   
   // Clamp the velocity to a maximum magnitude
   float speed = Norm(vel_player_);
   if (speed > kPlayerMaxSpeed) {
       vel_player_ *= kPlayerMaxSpeed / norm;
   }

   // Update the player's position
   Vec2 pos_delta = vel_player_ * dt;
   pos_player_ += pos_delta;

   // Apply air friction
   vel_player_ *= kFrictionAir;
}

Alright, now we’re zooming around:

So how do we do the colliding part?

Naive Collision

The way I first approached player collision, way back in the day, was to say “hey, we have a top-down 2D tile-based game, so our collision objects are all squares, and we can just check for collision against squares”. A reasonable place to start.

Okay, so we have a rectangle for our player and consistently-sized squares for all of our static geometry. That can’t be too hard, right?

Checking for collisions between axis-aligned rectangles isn’t all that hard. You can use something like the separating axis theorem. A quick Google search gives you code to work with.

Our strategy now is to calculate the new player position, check it against our collision geometry for intersections, and to not move our player if that fails. If we do that, we find that we need (1) to zero out the player’s speed on collision, and (2) we still actually want to move the player, except we want to move them up against the colliding body:

Doing this part is harder. We can do it, but it takes some math. In the case of an axis-aligned rectangle, if we know which side our collision point is closest to, we can just shift our new position \(P’\) out of the closest face by half the player-rect’s width in that axis. The code looks something like:

void UpdatePlayer(Vec2 input_dir, float dt) {
   // Update the player's velocity, clamp, etc.
   ... same as before

   // Update the player's position
   Vec2 pos_delta = vel_player_ * dt;
   pos_player_ += pos_player_ + pos_delta;
   Box player_box_next = Box(pos_next, player_height, player_width);
   for (const Box& box : collision_objects_) {
      auto collision_info = CheckCollision(box, player_box_next)
      if (collision_info.collided) {
         // Collision!
         vel_player_ = Vec2(0,0);
         pos_player_ = box.center;
         if (collision_info.nearest_side == NORTH) {
            pos_player_ += Vec2(0, box.height/2 + player_height/2);
         } else if (collision_info.nearest_side == EAST) {
            pos_player_ += Vec2(box.width/2 + player_width/2, 0);
         } else if (collision_info.nearest_side == SOUTH) {
            pos_player_ -= Vec2(0, box.height/2 + player_height/2);
         } else if (collision_info.nearest_side == WEST) {
            pos_player_ -= Vec2(box.width/2 + player_width/2, 0);
         }
      }
   }

   // Apply air friction
   vel_player_ *= kFrictionAir;
}

So this works, but has some issues. First off, we’re iterating over all boxes. That’s a big inefficiency right off the bat, especially on a large level with a lot of filled tiles. That particular problem is pretty easy to avoid, typically by using a more efficient data structure for keeping track of our collision geometry, like a quad tree for general collision objects and searching around a simple tile-based index for something like Mystic Dave.

Second, we project out of the nearest face. This can lead to all sorts of glitches. If we start on one side of the box, but move such that our closest face is a different one, we get shunted out the wrong side:

This effect is particularly problematic when we have multiple collision bodies. The code above processes them all in series, which could be totally wrong. We want to collide with the first thing we’d hit first.

Alternatively we could use a while loop, and keep on checking for collisions after we get shunted around by our collision logic, but this could end up causing weird infinite loops:

After we process any collisions, we set the player’s speed to zero. This causes the player to just stop. There is no nice sliding along the surface, let alone nicely rolling around corners:

Thinking in Points

Okay, maybe we can do this differently. Let’s take a step back and refine our problem.

We have a point \(P\), which we are moving a small step to some \(P’\). We assume \(P\) is valid (the player is not colliding with anything).

We don’t want to move to \(P’\) if it collides with anything. Furthermore, we don’t want to move to \(P’\) if we collide with anything on the way there:

So far we’ve been talking about points (\(P\) and \(P’\)), but what complicates things are the bounding boxes around them. Can we do something to remove the boxes from the equation?

Let’s look at the set of points where \(P’\) collides with our collision volume:

We can change our strategy to simply test \(P’\), a single point, against this larger collision volume. By using it, we remove the complexity of the player’s bounding rectangle, and instead get to work directly with point-in-shape tests.

When checking for a collision, we just have to check whether the ray segment \(P\rightarrow P’\) intersects with the extended volumes, and if it does, only move the player up to the earliest intersection.

If we have multiple collision bodies, this approach will ensure that we stop at the right location:

So we are now working with extended volumes. These are obtained by using the Minkowski sum of a collision body with the player’s collision volume. The idea is quite general, and works for arbitrary polygons:

Computing the Minkowski sum of two polygons is actually pretty straightforward:

// Compute the Minkowski sum of two convex polygons P and Q, whose
// vertices are assumed to be in CCW order.
vector<Vec2i> MinkowskiSum(const vector<Vec2i>& P, const vector<Vec2i>& Q) {
    // Vertices (a,b) are a < b if a.y < b.y || (a.y == b.y && a.x < b.x)
    size_t i_lowest = GetLowestVertexIndex(P);
    size_t j_lowest = GetLowestVertexIndex(Q);

    std::vector<Vec2i> retval;

    size_t i = 0, j = 0;
    while (i < P.size() || j < Q.size()) {
        // Get p and q via cyclic coordinates
        // This uses mod to allow to indices to wrap around
        const Vec2i& p = GetCyclicVertex(P, i_lowest + i);
        const Vec2i& q = GetCyclicVertex(Q, j_lowest + j);
        const Vec2i& p_next = GetCyclicVertex(P, i_lowest + i + 1);
        const Vec2i& q_next = GetCyclicVertex(Q, j_lowest + j + 1);

        // Append p + q
        retval.push_back(p + q);

        // Use the cross product to gage polar angle.
        // This implementation works on integers to avoid floating point
        // precision issues. I just convert floats say, in meters, to
        // integers by multiplying by 1000 first so we maintain 3 digits
        // of precision.
        long long cross = Cross(p_next - p, q_next - q);

        if (cross >= 0 && i < P.size()) {
            i++;
        }
        if (cross <= 0 && j < Q.size()) {
            j++;
        }
    }

    return retval;
}

This post isn’t really about how the Minkowski sum works, so I won’t go too deep into it, but you basically store your polygon vertices in a counter clockwise order, start with a vertex on each polygon that is furthest in a particular direction, and then rotate that furthest direction around in a circle, always moving an index (or both if the directions match) to keep your indices matching that furthest direction.

Sliding and Rolling around Corners

Before we get too deep, let’s quickly cover how to slide along the collision face. Our code above was zeroing out the player’s velocity every time we hit a face. What we want to do instead is zero out the velocity into the face. That is, remove the component of the velocity that points into the face.

If we know the face unit normal vector \(u\), then we just project our velocity \(v\) into the direction perpendicular to \(u\).

Because running into things slows us down, we typically then multiply our resulting speed vector by a friction coefficient to mimic sliding friction.

// Lose all velocity into the face.
// This results in the vector along the face.
player_vel_ = VectorProjection(player_vel_, v_face);
player_vel_ *= player_friction_slide_;  // sliding friction

To roll around edges we can just trade out our rectangular player collision volume for something rounder. Ideally we’d use a circle, but a circle is not a polygon, so the Minkowski sum would result in a more awkward shape with circular segments. We can do that, but its hard.

We can instead approximate a circle with a polygon. I’m just going to use a diamond for now, which will give us some rolling-like behavior around box corners.

Thinking in Triangles

Now that we took that brief aside, let’s think again about what this Minkowski sum stuff means for the code we prototyped. We could maintain a list of these inflated collision objects and check our points against them every time step. That’s a bit better before, because we’re primarily doing point-in-polygon checks rather than polygon-polygon intersection, but it is still rather complicated, and we want to avoid having to repeatedly iterate over all objects in our level.

I had a big aha! moment that motivated this entire project, which elegantly avoids this list-of-objects problem and ties in nicely with a previous blog post.

What if we build a triangulation of our level that has edges everywhere the inflated collision polygons have edges, and we then mark all triangles inside said polygons as solid. Then all we have to do is keep track of the player’s position and which triangle they are currently in. Any time the player moves, we just check to see if they pass across any of the faces of their current triangle, and if they cross into a solid triangle, we instead move them up to the face.

TLDR: 2D player movement on a constrained Delaunay mesh with solid cells for Minkowksi-summed collision volumes.

To illustrate, here is the Mystic Dave screenshot with some inflated collision volumes:

And here is a constrained triangulation on that, with solid triangles filled in:

And here it all is in action:

We see the static collision geometry in red, the player’s collision volume in blue, and the inflated collision volumes in yellow. We render the constrained Delaunay triangulation, which has edges at the sides of all of the inflated collision volumes. For player movement we simply keep track of the player’s position, the player’s speed, and which triangle they are in (highlighted).

We don’t have to limit ourselves to tile-centered boxes for this. We don’t even have to limit ourselves to tile-aligned boxes:

Player movement now has to track both the player’s position and the triangle that the player is in. We can find the enclosing triangle during initialization, and can use the same techniques covered the Delaunay Triangulation post. I’m going to repeat that section here, since it is so relevant.

Suppose we start with some random face and extract its vertices A, B, and C. We can tell if a point P is inside ABC by checking whether P is to the left of the ray AB, to the left of the ray BC, and to the left of the ray CA:

Checking whether P is left of a ray AB is the same as asking whether the triangle ABP has a counter-clockwise ordering. Conveniently, the winding of ABP is given by the determinant of a matrix:

\[\det\left(\begin{bmatrix}A_x & A_y & 1 \\ B_x & B_y & 1 \\ P_x & P_y & 1\end{bmatrix}\right)\]

If this value is positive, then we have right-handed (counter-clockwise) winding. If it is negative, then we have left-handed (clockwise) winding. If it is zero, then our points are colinear.

If our current triangle has positive windings then it encloses our point. If any winding is negative, the we know we can cross over that edge to get closer to our point.

We simply iterate in this manner until we find our enclosing triangle. 

Every tick, when we update the player’s position, we run a similar procedure. We similarly are interested in whether the player’s new position has crossed any triangle edges.The basic concept is that, if the new position enters a new triangle, and the new triangle is solid, we need to prevent movement across the edge. If the new position enters a new triangle, and the new triangle is not solid, we can simply move the player and update the enclosing triangle.

Below we see the simplest case, where the move from P to P’ leads to a new triangle:

If we check the winding of CAP’ and find that it is negative, then we know that we have crossed AC.

There are a few corner cases that make simply accepting the new triangle the wrong thing to do. First off, a single move could traverse multiple triangles:

We need to make sure we iterate, and continue to check for edge transitions.

Second, we may cross multiple edges in the first triangle:

In the above example, the new point P’ is across both the line through BC and the line through CA. The order in which we choose to process these traversals matters, because some intermediate triangles may be solid.

The correct traversal order is to process them in the same order that the ray P -> P’ crosses them. In this case, we first cross CA:

Every time we cross a segment, we find the intersection with the edge and move our player up to that edge. If the new triangle is not solid, we can update our current triangle and iterate again, now with a segment from P” to P’.

If the new triangle is solid, we don’t update our triangle and we process a collision on the colliding face. This means removing all velocity into the face (perpendicular to it) and adding sliding friction along the face (parallel to it). We continue to iterate again, now with a reduced step in our new direction along the edge:

The code structure is thus a while loop that, in each iteration, checks for traversal across all three edges of the enclosing triangle, and retains the earliest collision (if any). If there is a collision, we move the player up to the crossed edge (P”) and potentially handle collision physics or update the enclosing triangle, and iterate again. If there is no collision, then we stay inside our current triangle and are done.

Finding the intersection point for a crossed segment (suppose it is AB) can be done as follows:

\[\begin{aligned}V & = B – A \\ W &= P – A \\ \alpha &= \frac{V \times W }{P’ \times V} \\ P^{\prime\prime} & = P + \alpha (P’ – P) \end{aligned}\]

The interpolant \(\alpha \in [0,1]\) tells us how far we’ve moved toward P’, so is a great way to track which edge we cross first. We first cross the edge with the lowest \(\alpha\).

The code for this is thus:

void UpdatePlayer(Vec2 input_dir, float dt) {
   // Update the player's velocity, clamp, etc.
   ... same as before

   // Update the player's position
   float dt_remaining = dt;
   while (dt_remaining > 0.0f) {
        Vec2f player_pos_delta = player_vel_ * dt_remaining;
        Vec2f player_pos_next = player_pos_ + player_pos_delta;

        // Exit if the step is too small, to avoid math problems.
        if (Norm(player_pos_delta) < 1e-4) {
            break;
        }

        // Check to see if we cross over any mesh edges
        QuarterEdge* qe_ab = qe_enclosing_triangle_->rot;
        QuarterEdge* qe_bc = qe_enclosing_triangle_->next->rot;
        QuarterEdge* qe_ca = qe_enclosing_triangle_->next->next->rot;

        // Get the vertices
        const auto& a = *(qe_ab->vertex);
        const auto& b = *(qe_bc->vertex);
        const auto& c = *(qe_ca->vertex);

        // The fraction of the distance we will move
        float interp = 1.0f; // (alpha)
        QuarterEdge* qe_new_triangle = nullptr;
        Vec2f v_face;

        if (GetRightHandedness(a, b, player_pos_next) < -1e-4) {
            // We would cross AB
            Vec2f v = b - a;
            Vec2f w = player_pos_ - a;
            float interp_ab = Cross(v, w) / Cross(player_pos_delta, v);
            if (interp_ab < interp) {
                interp = interp_ab;
                qe_new_triangle = qe_ab->rot;
                v_face = v;
            }
        }
        if (GetRightHandedness(b, c, player_pos_next) < -1e-4) {
            // We would cross BC
            Vec2f v = c - b;
            Vec2f w = player_pos_ - b;
            float interp_bc = Cross(v, w) / Cross(player_pos_delta, v);
            if (interp_bc < interp) {
                interp = interp_bc;
                qe_new_triangle = qe_bc->rot;
                v_face = v;
            }
        }
        if (GetRightHandedness(c, a, player_pos_next) < -1e-4) {
            // We would cross CA
            Vec2f v = a - c;
            Vec2f w = player_pos_ - c;
            float interp_ca = Cross(v, w) / Cross(player_pos_delta, v);
            if (interp_ca < interp) {
                interp = interp_ca;
                qe_new_triangle = qe_ca->rot;
                v_face = v;
            }
        }

        // Move the player
        // If we would have crossed any triangle, this merely moves
        // the player up to the edge instead
        player_pos_ += player_pos_delta * interp;
        dt_remaining *= (1.0f - interp);
        dt_remaining -= 1e-5;  // eps factor for numeric safety

        if (qe_new_triangle != nullptr) {
            // We would have crossed into a new triangle
            if (is_solid(qe_new_triangle)) {
                // The new triangle is solid, so do not change triangles
                // Process the collision
                player_vel_ = VectorProjection(player_vel_, v_face);
                player_vel_ *= player_friction_slide_; // sliding friction
            } else {
                // Accept the new triangle
                qe_enclosing_triangle_ = qe_new_triangle;
            }
        }
   }

   // Apply air friction
   vel_player_ *= kFrictionAir;
}

This code block isn’t drastically more complicated than what we had before. We get a lot for it – efficient collision physics against general static objects. There is a bit of repeated code, so you could definitely simplify this even further.

Conclusion

This post revisited the problem of 2D player collision against static geometry. We covered some basics of player movement, including integrating velocity over time, basic collision physics, and basic collision approaches based on geometric primitives. We used the Minkowski sum to inflate collision polygons, thereby changing polygon vs. polygon intersections into cheaper point vs. polygon intersections. We concluded by leveraging constrained Delaunay triangulations for our environment instead, which simplified the collision detection logic down to checking for when we cross enclosing triangle line segments.

Please note that I am not the first to come up with this. It turns out I independently discovered Nav Meshes. I since learned that they are used in motion planning for robotics and in path planning for games (notably, Starcraft II). I assume they are also used for collision physics in games, but I haven’t been able to verify that yet.

I implemented this demo in ROS 2 using C++ 20. This was a deviation from my previous posts, which are mostly created in Julia notebooks. This project had more game-like interactivity to it than I thought a Julia notebook could easily handle (I could be wrong) and I simply felt like I could get up and running faster with ROS 2.

The most complicated part to implement in this whole process turns out to be polygon unions. In order to get the constrained Delaunay triangulation to work, you want to union any overlapping collision polygons together. Polygon unions (and other Boolean operations between polygons, like clipping) is quite complicated. I tried to write my own, but ended up using Clipper2.

I hope you found this process interesting! I was a lot of fun to revisit.