Developing a Space Flight Simulator in Clojure

In 2017 I discovered the free of charge Orbiter 2016 space flight simulator which was proprietary at the time and it inspired me to develop a space flight simulator myself. I prototyped some rigid body physics in C and later in GNU Guile and also prototyped loading and rendering of Wavefront OBJ files. I used GNU Guile (a Scheme implementation) because it has a good native interface and of course it has hygienic macros. Eventually I got interested in Clojure because it has more generic multi-methods as well as fast hash maps and vectors. I finally decided to develop the game for real in Clojure. I have been developing a space flight simulator in Clojure for almost 5 years now. While using Clojure I have come to appreciate the immutable values and safe parallelism using atoms, agents, and refs.

In the beginning I decided to work on the hard parts first, which for me were 3D rendering of a planet, an atmosphere, shadows, and volumetric clouds. I read the OpenGL Superbible to get an understanding on what functionality OpenGL provides. When Orbiter was eventually open sourced and released unter MIT license here, I inspected the source code and discovered that about 90% of the code is graphics-related. So starting with the graphics problems was not a bad decision.

Software dependencies

The following software is used for development. The software libraries run on both GNU/Linux and Microsoft Windows.

  • Clojure the programming language
  • LWJGL provides Java wrappers for various libraries
    • lwjgl-opengl for 3D graphics
    • lwjgl-glfw for windowing and input devices
    • lwjgl-nuklear for graphical user interfaces
    • lwjgl-stb for image I/O and using truetype fonts
    • lwjgl-assimp to load glTF 3D models with animation data
  • Jolt Physics to simulate wheeled vehicles and collisions with meshes
  • Fastmath for fast matrix and vector math as well as spline interpolation
  • Comb for templating shader code
  • Instaparse to parse NASA Planetary Constant Kernel (PCK) files
  • Gloss to parse NASA Double Precision Array Files (DAF)
  • Coffi as a foreign function interface
  • core.memoize for least recently used caching of function results
  • Apache Commons Compress to read map tiles from tar files
  • Malli to add schemas to functions
  • Immuconf to load the configuration file
  • Progrock a progress bar for long running builds
  • Claypoole to implement parallel for loops
  • Midje for test-driven development
  • tools.build to build the project
  • clj-async-profiler Clojure profiler creating flame graphs
  • slf4j-timbre Java logging implementation for Clojure

The deps.edn file contains operating system dependent LWJGL bindings. For example on GNU/Linux the deps.edn file contains the following:

{:deps {; ...
        org.lwjgl/lwjgl {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl$natives-linux {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-opengl {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-opengl$natives-linux {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-glfw {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-glfw$natives-linux {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-nuklear {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-nuklear$natives-linux {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-stb {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-stb$natives-linux {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-assimp {:mvn/version "3.3.6"}
        org.lwjgl/lwjgl-assimp$natives-linux {:mvn/version "3.3.6"}}
        ; ...
        }

In order to manage the different dependencies for Microsoft Windows, a separate Git branch is maintained.

Atmosphere rendering

For the atmosphere, Bruneton’s precomputed atmospheric scattering was used. The implementation uses a 2D transmittance table, a 2D surface scattering table, a 4D Rayleigh scattering, and a 4D Mie scattering table. The tables are computed using several iterations of numerical integration. Higher order functions for integration over a sphere and over a line segment were implemented in Clojure. Integration over a ray in 3D space (using fastmath vectors) was implemented as follows for example:

(defn integral-ray
  "Integrate given function over a ray in 3D space"
  {:malli/schema [:=> [:cat ray N :double [:=> [:cat [:vector :double]] :some]] :some]}
  [{::keys [origin direction]} steps distance fun]
  (let [stepsize      (/ distance steps)
        samples       (mapv #(* (+ 0.5 %) stepsize) (range steps))
        interpolate   (fn interpolate [s] (add origin (mult direction s)))
        direction-len (mag direction)]
    (reduce add (mapv #(-> % interpolate fun (mult (* stepsize direction-len))) samples))))

Precomputing the atmospheric tables takes several hours even though pmap was used. When sampling the multi-dimensional functions, pmap was used as a top-level loop and map was used for interior loops. Using java.nio.ByteBuffer the floating point values were converted to a byte array and then written to disk using a clojure.java.io/output-stream:

(defn floats->bytes
  "Convert float array to byte buffer"
  [^floats float-data]
  (let [n           (count float-data)
        byte-buffer (.order (ByteBuffer/allocate (* n 4)) ByteOrder/LITTLE_ENDIAN)]
    (.put (.asFloatBuffer byte-buffer) float-data)
    (.array byte-buffer)))

(defn spit-bytes
  "Write bytes to a file"
  {:malli/schema [:=> [:cat non-empty-string bytes?] :nil]}
  [^String file-name ^bytes byte-data]
  (with-open [out (io/output-stream file-name)]
    (.write out byte-data)))

(defn spit-floats
  "Write floating point numbers to a file"
  {:malli/schema [:=> [:cat non-empty-string seqable?] :nil]}
  [^String file-name ^floats float-data]
  (spit-bytes file-name (floats->bytes float-data)))

When launching the game, the lookup tables get loaded and copied into OpenGL textures. Shader functions are used to lookup and interpolate values from the tables. When rendering the planet surface or the space craft, the atmosphere essentially gets superimposed using ray tracing. After rendering the planet, a background quad is rendered to display the remaining part of the atmosphere above the horizon.

Templating OpenGL shaders

It is possible to make programming with OpenGL shaders more flexible by using a templating library such as Comb. The following shader defines multiple octaves of noise on a base noise function:

#version 410 core

float <%= base-function %>(vec3 idx);

float <%= method-name %>(vec3 idx)
{
  float result = 0.0;
<% (doseq [multiplier octaves] %>
  result += <%= multiplier %> * <%= base-function %>(idx);
  idx *= 2;
<% ) %>
  return result;
}

One can then for example define the function fbm_noise using octaves of the base function noise as follows:

(def noise-octaves
  "Shader function to sum octaves of noise"
  (template/fn [method-name base-function octaves] (slurp "resources/shaders/core/noise-octaves.glsl")))

; ...

(def fbm-noise-shader (noise-octaves "fbm_noise" "noise" [0.57 0.28 0.15]))

Planet rendering

To render the planet, NASA Bluemarble data, NASA Blackmarble data, and NASA Elevation data was used. The images were converted to a multi resolution pyramid of map tiles. The following functions were implemented for color map tiles and for elevation tiles:

  • a function to load and cache map tiles of given 2D tile index and level of detail
  • a function to extract a pixel from a map tile
  • a function to extract the pixel for a specific longitude and latitude

The functions for extracting a pixel for given longitude and latitude then were used to generate a cube map with a quad tree of tiles for each face. For each tile, the following files were generated:

  • A daytime texture
  • A night time texture
  • An image of 3D vectors defining a surface mesh
  • A water mask
  • A normal map

Altogether 655350 files were generated. Because the Steam ContentBuilder does not support a large number of files, each row of tile data was aggregated into a tar file. The Apache Commons Compress library allows you to open a tar file to get a list of entries and then perform random access on the contents of the tar file. A Clojure LRU cache was used to maintain a cache of open tar files for improved performance.

At run time, a future is created, which returns an updated tile tree, a list of tiles to drop, and a path list of the tiles to load into OpenGL. When the future is realized, the main thread deletes the OpenGL textures from the drop list, and then uses the path list to get the new loaded images from the tile tree, load them into OpenGL textures, and create an updated tile tree with the new OpenGL textures added. The following functions to manipulate quad trees were implemented to realize this:

(defn quadtree-add
  "Add tiles to quad tree"
  {:malli/schema [:=> [:cat [:maybe :map] [:sequential [:vector :keyword]] [:sequential :map]] [:maybe :map]]}
  [tree paths tiles]
  (reduce (fn add-title-to-quadtree [tree [path tile]] (assoc-in tree path tile)) tree (mapv vector paths tiles)))

(defn quadtree-extract
  "Extract a list of tiles from quad tree"
  {:malli/schema [:=> [:cat [:maybe :map] [:sequential [:vector :keyword]]] [:vector :map]]}
  [tree paths]
  (mapv (partial get-in tree) paths))

(defn quadtree-drop
  "Drop tiles specified by path list from quad tree"
  {:malli/schema [:=> [:cat [:maybe :map] [:sequential [:vector :keyword]]] [:maybe :map]]}
  [tree paths]
  (reduce dissoc-in tree paths))

(defn quadtree-update
  "Update tiles with specified paths using a function with optional arguments from lists"
  {:malli/schema [:=> [:cat [:maybe :map] [:sequential [:vector :keyword]] fn? [:* :any]] [:maybe :map]]}
  [tree paths fun & arglists]
  (reduce (fn update-tile-in-quadtree
            [tree [path & args]]
            (apply update-in tree path fun args)) tree (apply map list paths arglists)))

Other topics

Solar system

The astronomy code for getting the position and orientation of planets was implemented according to the Skyfield Python library. The Python library in turn is based on the SPICE toolkit of the NASA JPL. The JPL basically provides sequences of Chebyshev polynomials to interpolate positions of Moon and planets as well as the orientation of the Moon as binary files. Reference coordinate systems and orientations of other bodies are provided in text files which consist of human and machine readable sections. The binary files were parsed using Gloss (see Wiki for some examples) and the text files using Instaparse.

Jolt bindings

The required Jolt functions for wheeled vehicle dynamics and collisions with meshes were wrapped in C functions and compiled into a shared library. The Coffi Clojure library (which is a wrapper for Java’s new Foreign Function & Memory API) was used to make the C functions and data types usable in Clojure.

For example the following code implements a call to the C function add_force:

(defcfn add-force
  "Apply a force in the next physics update"
  add_force [::mem/int ::vec3] ::mem/void)

Here ::vec3 refers to a custom composite type defined using basic types. The memory layout, serialisation, and deserialisation for ::vec3 are defined as follows:

(def vec3-struct
  [::mem/struct
   [[:x ::mem/double]
    [:y ::mem/double]
    [:z ::mem/double]]])


(defmethod mem/c-layout ::vec3
  [_vec3]
  (mem/c-layout vec3-struct))


(defmethod mem/serialize-into ::vec3
  [obj _vec3 segment arena]
  (mem/serialize-into {:x (obj 0) :y (obj 1) :z (obj 2)} vec3-struct segment arena))


(defmethod mem/deserialize-from ::vec3
  [segment _vec3]
  (let [result (mem/deserialize-from segment vec3-struct)]
    (vec3 (:x result) (:y result) (:z result))))

Performance

The clj-async-profiler was used to create flame graphs visualising the performance of the game. In order to get reflection warnings for Java calls without sufficient type declarations, *warn-on-reflection* was set to true.

(set! *warn-on-reflection* true)

Furthermore to discover missing declarations of numerical types, *unchecked-math* was set to :warn-on-boxed.

(set! *unchecked-math* :warn-on-boxed)

To reduce garbage collector pauses, the ZGC low-latency garbage collector for the JVM was used. The following section in deps.edn ensures that the ZGC garbage collector is used when running the project with clj -M:run:

{:deps {; ...
        }
 :aliases {:run {:jvm-opts ["-Xms2g" "-Xmx4g" "--enable-native-access=ALL-UNNAMED" "-XX:+UseZGC"
                            "--sun-misc-unsafe-memory-access=allow"]
                 :main-opts ["-m" "sfsim.core"]}}}

The option to use ZGC is also specified in the Packr JSON file used to deploy the application.

Building the project

In order to build the map tiles, atmospheric lookup tables, and other data files using tools.build, the project source code was made available in the build.clj file using a :local/root dependency:

{:deps {; ...
        }
 :aliases {; ...
           :build {:deps {io.github.clojure/tools.build {:mvn/version "0.10.10"}
                          sfsim/sfsim {:local/root "."}}
                   :ns-default build
                   :exec-fn all
                   :jvm-opts ["-Xms2g" "-Xmx4g" "--sun-misc-unsafe-memory-access=allow"]}}}

Various targets were defined to build the different components of the project. For example the atmospheric lookup tables can be build by specifying clj -T:build atmosphere-lut on the command line.

The following section in the build.clj file was added to allow creating an “Uberjar” JAR file with all dependencies by specifying clj -T:build uber on the command-line.

(defn uber [_]
  (b/copy-dir {:src-dirs ["src/clj"]
               :target-dir class-dir})
  (b/compile-clj {:basis basis
                  :src-dirs ["src/clj"]
                  :class-dir class-dir})
  (b/uber {:class-dir class-dir
           :uber-file "target/sfsim.jar"
           :basis basis
           :main 'sfsim.core}))

To create a Linux executable with Packr, one can then run java -jar packr-all-4.0.0.jar scripts/packr-config-linux.json where the JSON file has the following content:

{
  "platform": "linux64",
  "jdk": "/usr/lib/jvm/jdk-24.0.2-oracle-x64",
  "executable": "sfsim",
  "classpath": ["target/sfsim.jar"],
  "mainclass": "sfsim.core",
  "resources": ["LICENSE", "libjolt.so", "venturestar.glb", "resources"],
  "vmargs": ["Xms2g", "Xmx4g", "XX:+UseZGC"],
  "output": "out-linux"
}

In order to distribute the game on Steam, three depots were created:

  • a data depot with the operating system independent data files
  • a Linux depot with the Linux executable and Uberjar including LWJGL’s Linux native bindings
  • and a Windows depot with the Windows executable and an Uberjar including LWJGL’s Windows native bindings

When updating a depot, the Steam ContentBuilder command line tool creates and uploads a patch in order to preserve storage space and bandwidth.

Future work

Although the hard parts are mostly done, there are still several things to do:

  • control surfaces and thruster graphics
  • launchpad and runway graphics
  • sound effects
  • a 3D cockpit
  • the Moon
  • a space station

It would also be interesting to make the game modable in a safe way (maybe evaluating Clojure files in a sandboxed environment?).

Conclusion

You can find the source code on Github. Currently there is only a playtest build, but if you want to get notified, when the game gets released, you can wishlist it here.

Anyway, let me know any comments and suggestions.

Enjoy!

Updates

  • Submitted for discussion to Reddit here
  • See HackerNews discussion of this project here

Keyestudio Smart Home

Keyestudio Smarthome

A few months ago I bought a Keyestudio Smart Home, assembled it and tried to program it using the Arduino IDE. However I kept getting the following error when trying to upload a sketch to the board.

 avrdude: stk500_getsync() attempt 1 of 10: not in sync: resp=0x2e
 avrdude: stk500_getsync() attempt 2 of 10: not in sync: resp=0x2e
 avrdude: stk500_getsync() attempt 3 of 10: not in sync: resp=0x2e

Initially I thought it was an issue with the QinHeng Electronics CH340 serial converter driver software. After exchanging a few emails with keyestudio support however I was pointed out that the board type of my smart home version was not “Arduino Uno”. The box of the control board says “Keyestudio Control Board for ESP-32” and I had to install version 3.1.3 of the esp32 board software for being able to program the board. I.e. the Keyestudio IoT Smart Home Kit for ESP32 is not to be confused with the Keyestudio Smart Home Kit for Arduino.

The documentation for the Keyestudio smart home using ESP-32 is here. Also the correct version of the smart home sketches are here. Finally you can find many sample projects in the keyestudio blog. Note that in some cases you have to adapt the io pin numbers using the smart home documentation.

Many thanks to Keyestudio support for helping me to get it working.

Orbits with Jolt Physics

I want to simulate an orbiting spacecraft using the Jolt Physics engine (see sfsim homepage for details). The Jolt Physics engine solves difficult problems such as gyroscopic forces, collision detection with linear casting, and special solutions for wheeled vehicles with suspension.

The integration method of the Jolt Physics engine is the semi-implicit Euler method. The following formula shows how speed v and position x are integrated for each time step:

latex formula

The gravitational acceleration by a planet is given by:

latex formula

To test orbiting, one can set the initial conditions of the spacecraft to a perfect circular orbit:

latex formula

The orbital radius R was set to the Earth radius of 6378 km plus 408 km (the height of the ISS). The Earth mass was assumed to be 5.9722e+24 kg. For increased accuracy, the Jolt Physics library was compiled with the option -DDOUBLE_PRECISION=ON.

A full orbit was simulated using different values for the time step. The following plot shows the height deviation from the initial orbital height over time.

Orbits with symplectic Euler

When examining the data one can see that the integration method returns close to the initial after one orbit. The orbital error of the Euler integration method looks like a sine wave. Even for a small timestep of dt = 0.031 s, the maximum orbit deviation is 123.8 m. The following plot shows that for increasing time steps, the maximum error grows linearly.

Euler orbit deviation as a function of time step

For time lapse simulation with a time step of 16 seconds, the errors will exceed 50 km.

A possible solution is to use Runge Kutta 4th order integration instead of symplectic Euler. The 4th order Runge Kutta method can be implemented using a state vector consisting of position and speed:

latex formula

The derivative of the state vector consists of speed and gravitational acceleration:

latex formula

The Runge Kutta 4th order integration method is as follows:

latex formula

The Runge Kutta method can be implemented in Clojure as follows:

(defn runge-kutta
  "Runge-Kutta integration method"
  {:malli/schema [:=> [:cat :some :double [:=> [:cat :some :double] :some] add-schema scale-schema] :some]}
  [y0 dt dy + *]
  (let [dt2 (/ ^double dt 2.0)
        k1  (dy y0                0.0)
        k2  (dy (+ y0 (* dt2 k1)) dt2)
        k3  (dy (+ y0 (* dt2 k2)) dt2)
        k4  (dy (+ y0 (* dt  k3)) dt)]
    (+ y0 (* (/ ^double dt 6.0) (reduce + [k1 (* 2.0 k2) (* 2.0 k3) k4])))))

The following code can be used to test the implementation:

(def add (fn [x y] (+ x y)))
(def scale (fn [s x] (* s x)))

(facts "Runge-Kutta integration method"
       (runge-kutta 42.0 1.0 (fn [_y _dt] 0.0) add scale) => 42.0
       (runge-kutta 42.0 1.0 (fn [_y _dt] 5.0) add scale) => 47.0
       (runge-kutta 42.0 2.0 (fn [_y _dt] 5.0) add scale) => 52.0
       (runge-kutta 42.0 1.0 (fn [_y dt] (* 2.0 dt)) add scale) => 43.0
       (runge-kutta 42.0 2.0 (fn [_y dt] (* 2.0 dt)) add scale) => 46.0
       (runge-kutta 42.0 1.0 (fn [_y dt] (* 3.0 dt dt)) add scale) => 43.0
       (runge-kutta 1.0 1.0 (fn [y _dt] y) add scale) => (roughly (exp 1) 1e-2))

The Jolt Physics library allows to apply impulses to the spacecraft. The idea is to use Runge Kutta 4th order integration to get an accurate estimate of the speed and position of the spacecraft after the next time step. One can apply an impulse before running an Euler step so that the position after the Euler step matches the Runge Kutta estimate. A second impulse then is used after the Euler time step to also make the speed match the Runge Kutta estimate. Given the initial state (x(n), v(n)) and the desired next state (x(n+1), v(n+1)) (obtained from Runge Kutta) the formulas for the two impulses are as follows:

latex formula

The following code shows the implementation of the matching scheme using two speed changes in Clojure:

(defn matching-scheme
  "Use two custom acceleration values to make semi-implicit Euler result match a ground truth after the integration step"
  [y0 dt y1 scale subtract]
  (let [delta-speed0 (scale (/ 1.0 ^double dt) (subtract (subtract (:position y1) (:position y0)) (scale dt (:speed y0))))
        delta-speed1 (subtract (subtract (:speed y1) (:speed y0)) delta-speed0)]
    [delta-speed0 delta-speed1]))

The following plot shows the height deviations observed when using Runge Kutta integration.

Orbits with Runge Kutta 4th order

The following plot of maximum deviation shows that the errors are much smaller.

RK orbit deviation as a function of time step

Although the accuracy of the Runge Kutta matching scheme is higher, a loss of 40 m of height per orbit is undesirable. Inspecting the Jolt Physics source code reveals that the double-precision setting affects position vectors but is not applied to speed and impulse vectors. To test whether double precision speed and impulse vectors would increase the accuracy, a test implementation of the semi-implicit Euler method with Runge Kutta matching scheme was used. The following plot shows that the orbit deviations are now much smaller.

Orbits with Runge Kutta 4th order and double precision

The updated plot of maximum deviation shows that using double precision the error for one orbit is below 1 meter for time steps up to 40 seconds.

RK with double precision orbit deviation as a function of time step

I am currently looking into building a modified Jolt Physics version which uses double precision for speed and impulse vectors. I hope that I will get the Runge Kutta 4th order matching scheme to work so that I get an integrated solution for numerically accurate orbits as well as collision and vehicle simulation.

Update:

Jorrit Rouwé has informed me that he currently does not want to add support for double precision speed values. He also has more detailed information about using Jolt Physics for space simulation on his website.

I have managed to get a prototype working using the moving coordinate system approach. One can perform the Runge Kutta integration using double precision coordinates and speed vectors with the Earth at the centre of the coordinate system. The Jolt Physics integration then happens in a coordinate system which is at the initial position and moving with the initial speed of the spaceship. The first impulse of the matching scheme is applied and then the semi-implicit Euler integration step is performed using Jolt Physics with single precision speed vectors and impulses. Then the second impulse is applied. Finally the position and speed of the double precision moving coordinate system are incremented using the position and speed value of the Jolt Physics body. The position and speed of the Jolt Physics body are then reset to zero and the next iteration begins.

The following plot shows the height deviations observed using this approach:

Orbits using moving coordinate system

The maximum errors for different time steps are shown in the following plot:

Maximum errors with moving coordinate system as a function of time step

Akima splines

Recently I was looking for spline interpolation for creating curves from a set of samples. I knew cubic splines which are piecewise cubic polynomials fitted such that they are continuous up to the second derivative. I almost went ahead and implemented cubic splines using a matrix solver but then I found that the fastmath Clojure library already provides splines. The fastmath spline interpolation module is based on the interpolation module of the Java Smile library. I saved the interpolated samples to a text file and plotted them with Gnuplot.

(require '[fastmath.interpolation :as interpolation])
(use '[clojure.java.shell :only [sh]])

(def px [0 1 3 4 5 8 9])
(def py [0 0 1 7 3 4 6])
(spit "/tmp/points.dat" (apply str (map (fn [x y] (str x " " y "\n")) px py)))

(def cspline (interpolation/cubic-spline px py))
(def x (range 0 9.0 0.01))
(spit "/tmp/cspline.dat" (apply str (map (fn [x y] (str x " " y "\n")) x (map cspline x))))
(sh "gnuplot" "-c" "plot.gp" "/tmp/cspline.png" "/tmp/cspline.dat")
(sh "display" "/tmp/cspline.png")

I used the following Gnuplot script plot.gp for plotting:

set terminal pngcairo size 640,480
set output ARG1
set xlabel "x"
set ylabel "y"
plot ARG2 using 1:2 with lines title "spline", "/tmp/points.dat" using 1:2 with points title "points"

I used a lightweight configuration of the fastmath library without MKL and OpenBLAS. See following deps.edn:

{:deps {org.clojure/clojure {:mvn/version "1.12.1"}
        generateme/fastmath {:mvn/version "2.4.0" :exclusions [com.github.haifengl/smile-mkl org.bytedeco/openblas]}}}

The result is shown in the following figure. One can see that the spline is smooth and passes through all points, however it shows a high degree of oscillation:

cubic spline

However I found another spline algorithm in the fastmath wrappers: The Akima spline. The Akima spline needs at least 5 points and it first computes the gradient of the lines connecting the points. Then for each point it uses a weighted average of the previous and next slope value. The slope values are weighted using the absolute difference of the previous two slopes and the next two slopes, i.e. the curvature. The first and last two points use a special formula: The first and last point use the next or previous slope and the second and second last point use an average of the neighbouring slopes.

(require '[fastmath.interpolation :as interpolation])
(use '[clojure.java.shell :only [sh]])

(def px [0 1 3 4 5 8 9])
(def py [0 0 1 7 3 4 6])
(spit "/tmp/points.dat" (apply str (map (fn [x y] (str x " " y "\n")) px py)))

(def aspline (interpolation/akima-spline px py))
(def x (range 0 9.0 0.01))
(spit "/tmp/aspline.dat" (apply str (map (fn [x y] (str x " " y "\n")) x (map aspline x))))
(sh "gnuplot" "-c" "plot.gp" "/tmp/aspline.png" "/tmp/aspline.dat")
(sh "display" "/tmp/aspline.png")

Akima spline

So if you have a data set which causes cubic splines to oscillate, give Akima splines a try!

Enjoy!

Update: u/joinr showed, how you can use Clojupyter to quickly test a lot of splines.

Flight dynamics model for simulating Venturestar style spacecraft

sfsim space flight simulator screenshot

This is an informational post on how to simulate the physics of atmospheric flight of a Venturestar style single-stage-to-orbit space craft. My dad Gerhard Wedekind is an experienced aerodynamics engineer and I asked him to help with making the aerodynamics of the sfsim space flight simulator realistic to some extent. The information in this post is a write-up of relevant formulas and approximate data he obtained from numerical simulation and estimates from aerodynamics knowledge. The information provided in this article is for general informational purposes only and comes without any warranty, express or implied.

Simulation

Here are a few beautiful snapshots from simulation. The first one shows a Mach box for V = 2 Ma and α = 3°. Venturestar Mach Box for V = 2 Ma and α = 3°

The next one shows a Mach box for V = 4 Ma and α = 3°. Venturestar Mach Box for V = 4 Ma and α = 3°

Finally here is a distribution of the pressure difference between top and bottom of wing. Venturestar Pressure Difference

Coordinate systems

The following drawing shows the body coordinate system (xb, yb, zb) and the wind coordinate system (xw, yw, zw). The wind system is rotated against the body system so that the speed vector (in a stationary atmosphere) points in positive xw.

coordinate systems

Note that lift, drag, and side force are defined in the wind system and not in the body system.

  • A positive lift force points upwards (negative zw) in the wind system.
  • The drag force points backwards (negative xw) in the wind system.
  • A positive side force points starboard (positive yw) in the wind system.

Yaw, pitch, and roll moments on the other hand are specified in the body system.

A coordinate system transformation from body system to wind system can be performed using two angles:

  • α is the angle of attack
  • β is the sideslip angle

When transforming coordinates from body system to wind system, one first rotates by β (sideslip angle) about the body z axis (zb). Then one rotates by α (angle of attack) about the new y axis.

Dynamic pressure

The dynamic pressure q depends on air density ρ and speed V: latex formula

Air density (and temperature) as a function of height can be obtained from Hull’s book “Fundamentals of airplane flight mechanics”.

Forces

Drag consists of zero-lift drag and induced drag: latex formula

Zero-lift drag is computed using the zero-lift drag coefficient CD0 as well as dynamic pressure q and the reference area Sref: latex formula The zero-lift drag coefficient depends on the speed of the aircraft.

Induced drag is determined using the lift coefficient CL, the Oswald factor e, the aspect ratio Λ, as well as q and the reference area Sref. latex formula The Oswald factor e depends on the speed of the aircraft. The lift coefficient depends on the angle of attack α.

The aspect ratio Λ depends on wing span b and wing area S: latex formula

The lift L is computed using the lift coefficient CL, dynamic pressure q, and the reference area Sref: latex formula

The side force Y (and corresponding coefficient) is usually not important but we will look into it later in a future article.

Moments

The pitching moment M is computed using the pitching moment coefficient Cm, the dynamic pressure q, the reference area Sref, and the aerodynamic chord cbar: latex formula The pitching moment coefficient depends on the lift coefficient CL, the position of the neutral point XN, the centre of gravity xref. and the aerodynamic chord cbar: latex formula

The yawing moment N is the product of the yawing moment coefficient Cn, the dynamic pressure q, the reference area Sref, and half the wing span b: latex formula The yawing moment coefficient depends on the side slip angle β.

The rolling moment L (using the same symbol as lift for some reason) is the product of the rolling moment coefficient Cl, the dynamic pressure q, the reference area Sref, and half the wing span b: latex formula The rolling moment coefficient depends on the angle of attack α and the side slip angle β.

Data Sheet

Here are the parameters for the flight model above:

latex formula

Note that xref is defined in a coordinate system where x=0 is at the intersection of the inner leading edges (wing apex). The following picture also shows the position of the aerodynamic chord with length cbar. The center of gravity is at 25% of the aerodynamic chord.

coordinate system origin at intersection of leading edges

Tables

Here is a data table with information for determining the remaining coefficients depending on the airspeed in Mach (Ma). The table shows for each speed:

  • a factor to determine the lift coefficient CL
  • the position XN of the neutral point relative to the aerodynamic chord (note that the center of gravity xref is at the 25% mark of the aerodynamic chord)
  • the Oswald factor e
  • a factor to determine the rolling moment coefficient Cl
  • a factor to determine the yawing moment coefficient Cn
  • the zero-lift drag coefficient CD0

latex formula

The outlier of Clβα for V = 1.2 Ma (0.5971) should be ignored because the value was changing a lot with mesh resolution.

The speed of sound as a function of temperature T is according to Wikipedia: latex formula

For small values of α, the lift coefficient increases linearly with α (where α is specified in radians): latex formula

For small values of α and β, the rolling moment coefficient increases linearly with the product of α and β (where α and β are specified in radians). This is a particular behaviour of delta wing configurations. If there is side slip, the wings generate different amounts of lift causing a significant roll moment: latex formula

For small values of β, the yawing moment coefficient increases linearly with β (where β is specified in radians): latex formula

The following table shows for each speed:

  • the value for α at which the linear relation of CL and α breaks down
  • the maximum value of CL
  • the angle of attack where CL reaches its maximum
  • the drag coefficient for 90° angle of attack

latex formula

Near α=90°, the lift and drag coefficients behave as follows: latex formula

At hypersonic speeds (V/Ma=10.0), lift and induced drag coefficients behave as follows: latex formula

I.e. the coefficients are stabilising at hypersonic speeds!

Control surfaces

The following table shows parameters to determine different moments generated by control surfaces:

latex formula

The side force coefficient for a given rudder angle ζ is: latex formula The yawing moment coefficient for the rudder is: latex formula

The pitching moment coefficient for flaps δF (down is positive) is latex formula

The rolling moment coefficient for ailerons with angle ξ (positive: port aileron up, starboard aileron down) is: latex formula The yawing moment coefficient is latex formula

Angular damping

The formula for roll, pitch, and yaw damping moments (L, M, N) due to roll, pitch, and yaw rates (p, q, r) uses a coefficient matrix: latex formula The coefficients for V = 0 Ma are as follows. Note that damping moments are negligible for higher speeds. latex formula The last (10th) value in the table CLq is the change in the lift coefficient due to the pitch rate q.

Next steps

Using the information, the curves for a full range of angles and speeds need to be fitted and guessed in some places.

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