A Kalman filter can do interesting things (like filtering poll results)

I recently learned about the Kalman filter and finally got to play around with it a little bit. Since I had a hard time figuring out how to get it to work, here’s a practical (but yet general) introduction with examples:

A Kalman filter works by matching a simulation model and measured data. For each data point, an estimation of the simulation model’s internal state is computed based on the estimate of the previous state. This works with noisy data and limited measurement signals (e.g. a model with 10 state variables but only 2 measurement signals, although there are obvious limitations here (the more and the better the sensor data, the better the results should be – there’s also some limit on observability).

So we have this interesting tool which does all these different things:

  • filtering noisy data, while taking knowledge (or assumptions) on the underlying dynamics into account
  • merge data from several different sensors into one signal (typical application: combine GPS and acceleration sensor data into one accurate position signal)
  • offer a prediction of a system’s future state
  • estimate internal parameters of a system (say a spring stiffness based on measured oscillations)

Another interesting use is that we might try two different simulation models on the same measurement and check which one does a better job at synchronizing to the measurement (I’ll do this in a very simple example below).

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Real-time simulations on an Arduino (double pendulum)

I got a small display compatible with my Due for christmas. And since I really wanted to see some arduino-in-the-loop simulations, I decided to use it for exactly this: real-time multibody simulations on the Ardunio and the results displayed on the tft.

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Speed comparisons for Arduino Uno/Nano, Due and Teensy 3.5

It’s been more than a year since I published my post on numerical integration on an Arduino. Since then, the post has been quite popular, recieving a steady stream of visitors (mostly via Google). When I originally wrote it, I only had an Arduino Uno at hand – since then I’ve added a couple of Nanos and lately an Arduino Due to my inventory and decided it would be interesting to do a couple of speed tests to see how they perform. The latest addition to my growing circus of microcontroller boards is a Teensy 3.5 board.

As I pointed out in the original post, numerical integration relies heavily on floating-point math – which is something the Arduino’s 8-bit processor is not particularly good at. The Due features a 32-bit processor, a clock frequency of 84 instead of 16 MHz and the possibility to use double (64 bit) instead of float (32 bit) as a data type – so I was curious to see how it would compare to the Arduino Uno. The Nano is supposed to have more or less the same characteristics as an Uno, but is a lot smaller and cheaper – see below for details.

Now added to the comparison, the Teensy 3.5 includes a 32-bit processor with 120 MHz clock speed and a FPU for speedier floating-point math.

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Object-oriented multibody simulation models in Matlab

What’s better suited for object-oriented programming (OOP) than building a simulation of a physical system? Each physical object (e.g. mass, spring etc.) can be implemented as an object and then we should be able to easily build and modify large models.

However, there are a lot of challenges to get there, including:

  • How to work with classes and objects at all (if you’re like me new to OOP)?
  • How to build a working object structure that adopts to the requirements of a typical ODE solver (which usually requires access to all state variables in the so-called state-space representation)
  • How to design classes for different object types in a way that allows them interact with each other to calculate and transmit forces and accelerations.

Getting there turned out to be quite interesting and I finally got a simple object-oriented multibody model to simulate in Matlab and GNU Octave. This post is a wrapup of all the stuff I learned to get there.

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Impulse-based dynamics: A simple and fast way to simulate contacts and constraints

So far, I’ve mostly been working on rotational mechanics – multibody models where each mass only has one rotational degree of freedom and two state variables (angle and angular velocity). Every now and then, I use some commercially available tools to simulate more complex 2d and 3d models – often being surprised by how slow simulation is. At the same time, I’ve been wondering a lot:

  • how I’d actually model these problems by hand – especially when additional boundary conditions and/or contacts are involved.
  • why some video games seem to be able to include complex dynamics with detailed contact and collisions involved, simulating in real time (while commercial multibody simulation often takes hours to calculate a few seconds of dynamic behavior).

An important answer to these questions is the topic of impulse-based dynamics. I stumbled upon it in some amazing literature from Prof. Jan Bender (see http://www.interactive-graphics.de/) and so far I’ve been experimenting a bit with the method myself. This post today features a small and practical introduction to modelling boundary conditions in impulse-based mechanics.

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How to solve differential equations on an arduino with the runge-kutta method

This is just a small update on my experiments with the Arduino. I implemented the runge-kutta-method for solving a multibody system a few weeks ago. So this is a working implementation of the standard 4th-order runge-kutta ODE (ordinary differential equations) solver for the arduino platform, something I haven’t seen elsewhere.

(Note: If you are just looking for the arduino sketch, you’ll find a link at the end of this blog post. If you are wondering about related questions or have trouble adopting it to your needs, please leave a comment.
Update in August 2016: See also my new post on achievable simulation rates with an Arduino Uno/Nano and Due)

My main goal was to get a better grip on simulation speeds. Why is my simulation so slow? is a question I’m really thinking about a lot. Since I wasn’t able to find a good example of an arduino sketch for ODE solving online, this may be interesting anyway.

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Viewing friction in dynamics as a hybrid system

It’s been almost a year since I wrote my post on how to model friction in a dynamic multibody system. Since then, I’ve reconsidered my ideas from the post a few times and finally found out that there’s a theory called “hybrid systems” that intersects with the problems described there.

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