26 Jun 2020

# Toolbox-Plane Update - Part 2: Simulation, Si and More

As a result of the Corona pandemic i had a lot of time available during the period between the winter and the summer semesters. Some of this time was spent by implementing new features for our Toolbox-Plane flight computer. The flight computer is a Rasperry Pi which runs all high-level aspects of the plane such as sensors fusion, route planning and some parts of the feedback-control. Additionally it is the centre of our communication network: the flight computer communications with the flight controller, the power distribution board, the primary remote and the base station.

This is part 2 of the update, covering our new airframe, our library for providing unit safe code and our simulation environment.

## New Airframe

Our old airframe, the Mini Talon was bought in the spring of 2017 and testing, especially in the early stages with manual control, involved lots of crashes and hard landings. Thus we had to fix the airplane multiple times, in the end it consisted primarily of hot glue and duct tape. This greatly reduced the reliability of the plane and made testing difficult. Additionally, during the initial build, we decided to glue to wings onto the fuselage, this made transport difficult. Furthermore the space inside the plane became quite limited and access to the components was very limited. Thus we decided to switch to a new, larger airframe. We decided on the FX-79 Buffalo flying wing, which provides ample space on the inside. Due to the corona situation we where not able to assemble the complete plane and test the plane, this will be done as soon as we have access to our local makerspace.

## Si and C++-20

All of the flight computer code is written in C++ to be able to write fast but also safe software. To add another layer of safety, beyond simple type safety, we implemented our own unit-library. This library requires for every type to have an associated unit and then checks, during compile time, if an operation is valid and of what type the result is. For example: adding a number with unit meter to a number of unit second is malformed and will throw a compiler error, on the other hand multiplying a number of unit meter with a number of unit second is completly valid and will yield a number with unit meter*second.

As code:

auto m = 1_meter;
auto s = 1_second;

auto error = m + s; // Compiler error
auto correct = m * s; // Compiles, m has the correct type


To achieve this a template class Si<m, kg, s, A, K, MOL, CD, T> is used. The first seven template arguments are the exponents of the respective units, the last argument is the underlying numeric type. The units used above (with underlying type float) are represented as:

using meter = Si<1, 0, 0, 0, 0, 0, 0, float>;
using second = Si<0, 0, 1, 0, 0, 0, 0, float>;


to simplify type conversions all scalar values are represented by Si types as well:

using scalar = Si<0, 0, 0, 0, 0, 0, 0, float>;


This makes the interaction of scalar values with values with unit easy and intuitive, but there are some problems that arise when interacting with libraries which do not use Si, especially when converting from and to Si units.

For example the following expression is malformed:

scalar s = 1.0F;


simply because the constructor of Si is marked explicit to avoid unintentional conversions for non scalar type.

Additionally there are problems when converting the other way, assuming a function void f(float) which is provided by a library, in this case the expression:

scalar s{1.0F};
f(s);


is malformed as well because Si::operator T() is explicit for the same reasons as above.

So what we want to achieve is for both the constructor and the operator T() to be only explicit if and only if one of the exponents is not equal to zero. Luckily for us C++-20 added a nice little feature, often refered to as explicit(bool) (P0892R2), which makes it possible to enable/disable the explicit based on (compile-time) predicate.

Using this feature we can improve our functions, for example for the constructor:

static constexpr bool isScalar = (m==0 && kg == 0 && s == 0 && A == 0 && K == 0 && MOL == 0 && CD == 0);

constexpr explicit(!isScalar) Si(T val) noexcept;


for this we require a recent compiler with (at least partial) support for C++-20. When using GCC this means at least version 9 (see en.cppreference.com/w/cpp/compiler_support). As this compiler is not available on the current version of Raspbian GCC was compiled from source.

For the full documentation of the library see the github repository: teamspatzenhirn/SI.

## Simulation

By choosing a flying wing instead of a more conventional plane design we lost two degrees of freedom when designing our controller: the ailerons and elevators are combined and there is no yaw control. Thus there was the necessity to update our controller. The general concept remains the same for now: the flightcontroller runs the controller for pitch and roll and the flightcomputer does the trajectory planning and controls the heading, altitude and speed. This cascaded PID controller is easy to implement, fast during execution and provides reasonably good results with minor tuning. A point that is especially important considering we have only limited testing opportunities.

To extend our testing capabilitites, especially now where can not test at all i decided to extend our simulator used for controller tuning to be able to find a good tune for the first tests without being able to fly the plane. Another nice advantage of the simulator is that we can quickly test new controller types without having to risk the plane. Obviously this all depends on a good simulator which reflects the reality as good as possible. Thus is decided to port our current simulator from Simulink to python and extend it by using a better model.

### Modelling

To make modelling easier the model was split into multiple subsystems: the inputs are first split, so to say demultiplexed, for the different axis. Then most of the dynamics is handled for the axis separatly, only in the last step the results of the subsystems are joined to form the complete plane state.

The model is based on the forces which are generated by the actuators (flaps and the motor). As the plane is modelled as a pointmass, thus first all forces need to be converted to equivalent forces (include torque forces) which influence this point mass. From these forces the acceleration (both translation and rotational), velocity and positions can be calculated.

### Parameter Estimation

Most of the parameters used for the model can be measured directly, for example the dimensions and the weight. For other the frontal and wing area, which are required for drag and lift calculations, a picture with a known scale has been taken. The area can then be measured as pixels in this image and then converted using this scale.

The moment of inertia can be calculated when measuring the angular position over time when torque is applied. I.e. a known weight is placed at a known position on the wing of the plane. The plane hangs freely and thus is able to rotate. When recording this rotation the angle at each time step can be calculated, from these angles the angular velocity and the angular acceleration can be estimated. Given the angular acceleration and the known torque the moment of inertia can be calculated.