https://github.com/shelepuginivan/double-pendulum
Double pendulum simulation
https://github.com/shelepuginivan/double-pendulum
animation double-pendulum-simulation manim-animations numerical-analysis simulation
Last synced: 5 days ago
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Double pendulum simulation
- Host: GitHub
- URL: https://github.com/shelepuginivan/double-pendulum
- Owner: shelepuginivan
- Created: 2026-03-13T02:01:40.000Z (4 months ago)
- Default Branch: main
- Last Pushed: 2026-03-15T06:24:10.000Z (4 months ago)
- Last Synced: 2026-03-15T15:06:43.125Z (4 months ago)
- Topics: animation, double-pendulum-simulation, manim-animations, numerical-analysis, simulation
- Language: C
- Homepage:
- Size: 585 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Double Pendulum
Simulation of [Double pendulum](https://en.wikipedia.org/wiki/Double_pendulum)
with configurable parameters.
## Simulation
Build the simulation program:
```sh
make
```
The compiled binary is written to `./build/double-pendulum`.
Both double pendulum system and simulation are configured via environment
variables.
| Variable | Description | Value | Default |
| -------------------------- | ------------------------------------------ | -------- | ------- |
| `DP_SYSTEM_M1` | Mass of the first weight | _double_ | 3 |
| `DP_SYSTEM_M2` | Mass of the second weight | _double_ | 3 |
| `DP_SYSTEM_L1` | Length of the first rod | _double_ | 2 |
| `DP_SYSTEM_L2` | Length of the second rod | _double_ | 1 |
| `DP_SYSTEM_PHI1` | Initial angle of the first rod | _double_ | π |
| `DP_SYSTEM_PHI2` | Initial angle of the second rod | _double_ | π/2 |
| `DP_SYSTEM_G` | Gravitational acceleration | _double_ | 9.81 |
| `DP_SYSTEM_DT` | Time delta in seconds | _double_ | 1e-4 |
| `DP_SYSTEM_DT_MIN` | Min time delta for embedded methods | _double_ | 1e-4 |
| `DP_SYSTEM_DT_MAX` | Max time delta for embedded methods | _double_ | 1e-4 |
| `DP_SYSTEM_ATOL_PHI1` | Absolute tolerance of the first angle | _double_ | 1e-6 |
| `DP_SYSTEM_RTOL_PHI1` | Relative tolerance of the first angle | _double_ | 1e-4 |
| `DP_SYSTEM_ATOL_PHI2` | A. tol. of the second angle | _double_ | 1e-6 |
| `DP_SYSTEM_RTOL_PHI2` | R. tol. of the second angle | _double_ | 1e-4 |
| `DP_SYSTEM_ATOL_OMEGA1` | A. tol. of the first angular velocity | _double_ | 1e-6 |
| `DP_SYSTEM_RTOL_OMEGA1` | R. tol. of the first angular velocity | _double_ | 1e-4 |
| `DP_SYSTEM_ATOL_OMEGA2` | A. tol. of the second angular velocity | _double_ | 1e-6 |
| `DP_SYSTEM_RTOL_OMEGA2` | R. tol. of the second angular velocity | _double_ | 1e-4 |
| `DP_SYSTEM_ERR_MIN_FACTOR` | Min dt scaling factor for embedded methods | _double_ | 0.25 |
| `DP_SYSTEM_ERR_MAX_FACTOR` | Max dt scaling factor for embedded methods | _double_ | 4.0 |
| Variable | Description | Value | Default |
| ------------------------ | ------------------------------ | -------------------------------------------------------- | ------- |
| `DP_SIMULATION_DURATION` | Simulation duration in seconds | _double_ | 30 |
| `DP_SIMULATION_METHOD` | ODE computation method | "ralston", "RK4", "RK3/8" (= "RK38"), "DOPRI5", "DOPRI8" | "RK4" |
| `DP_SIMULATION_OUTPUT` | Output CSV file | _path_ or "-" for stdout | "-" |
The resulting CSV file contains rows with coordinates of weights
(`x1,y1,x2,y2`).
For explicit Runge-Kutta methods (`ralston`, `RK4`, `RK3/8`) the timespan
between rows is `DP_SYSTEM_DT`, for embedded (`DOPRI5`, `DOPRI8`) it is scaled
after each step, based on computation error and system parameters. The scaling
factor is clamped between `DP_SYSTEM_ERR_MIN_FACTOR` and
`DP_SYSTEM_ERR_MAX_FACTOR`, and the time delta itself is clamped between
`DP_SYSTEM_DT_MIN` and `DP_SYSTEM_DT_MAX`.
## Chaos plot
> [!note]
> The graphics are generated using Python, managed with
> [uv](https://docs.astral.sh/uv/). Make sure it is installed and working.
Chaos plot demonstrates how two systems with nearly identical settings diverge
over time. It consists of coordinate points of the second weight.
```sh
# Generate two sets of coordinates with slightly different system conditions.
DP_SYSTEM_PHI1=3.14159265358979323846 DP_SIMULATION_OUTPUT=1.csv ./build/double-pendulum
DP_SYSTEM_PHI1=3.141592653589 DP_SIMULATION_OUTPUT=2.csv ./build/double-pendulum
# Generate chaos plot.
# uv run ./graphics/chaos_plot.py [csv 1] [csv 2] [csv 3] [...]
uv run ./graphics/chaos_plot.py 1.csv 2.csv
```
The following environment variables are used for configuration.
| Variable | Description | Value | Default |
| -------------------- | ------------------------------------------------ | ------------------------ | ----------------------- |
| `DP_CHAOS_FRAMERATE` | Number of frames per second sampled for plotting | _int_ | 2 |
| `DP_CHAOS_COLORS` | Which colors to use for coordinates points | _comma-separated colors_ | "red,blue,green,orange" |
| `DP_CHAOS_OUTPUT` | Output image file | _path_ | chaos.png |
In addition, `DP_SYSTEM_DT` is used with `DP_CHAOS_FRAMERATE` to sample
coordinates for plotting.
## Animation
[Manim](https://www.manim.community/) animation of the double pendulum system.
```sh
DP_ANIMATION_DATASET=data.csv uv run manim -qh ./graphics/animation.py DoublePendulum
```
Environment variables are as follows.
| Variable | Description | Value | Default |
| ------------------------ | ----------------------------------------------------- | ------- | ------- |
| `DP_ANIMATION_FRAMERATE` | Frame rate of the animation | _int_ | 24 |
| `DP_ANIMATION_DATASET` | CSV file to animate data from | _path_ | — |
| `DP_ANIMATION_TRAIL` | Duration of the trail of the second weight in seconds | _float_ | 0.5 |
Similar to chaos plot, `DP_SYSTEM_DT` is used with `DP_ANIMATION_FRAMERATE` for
sampling of coordinates.
Radius of the weight points are controlled by `DP_SYSTEM_M1` and `DP_SYSTEM_M2`
env variables (`DEFAULT_DOT_RADIUS` constant is used by default).
You can also overlay multiple animations with transparency using [FFmpeg](https://ffmpeg.org/):
```sh
DP_SIMULATION_DURATION=60 DP_SYSTEM_PHI1=3.14159265358979323846 DP_SIMULATION_OUTPUT=1.csv ./build/double-pendulum
DP_SIMULATION_DURATION=60 DP_SYSTEM_PHI1=3.141592653589 DP_SIMULATION_OUTPUT=2.csv ./build/double-pendulum
DP_ANIMATION_DATASET=1.csv uv run manim -o 1.mp4 -qh ./graphics/animation.py DoublePendulum
DP_ANIMATION_DATASET=2.csv uv run manim -o 2.mp4 -qh ./graphics/animation.py DoublePendulum
# Overlay multiple videos using transparency filter.
# Adjust the scale as needed.
ffmpeg \
-i ./media/videos/animation/1080p24/1.mp4 -i ./media/videos/animation/1080p24/2.mp4 \
-filter_complex " \
[0:v]setpts=PTS-STARTPTS, scale=1920x1080[top]; \
[1:v]setpts=PTS-STARTPTS, scale=1920x1080, \
format=yuva420p,colorchannelmixer=aa=0.5[bottom]; \
[top][bottom]overlay=shortest=1" \
-acodec libvo_aacenc -vcodec libx264 out.mp4
```
## ODE computation method comparison
Finally, I've implemented comparison of point positions across different
simulations. For instance, this can be used to plot relative error graphs for
various ODE solvers:
```sh
# Generate CSV files for each supported method
for method in "ralston" "RK4" "RK38" "DOPRI5" "DOPRI8"; do
DP_SIMULATION_METHOD=$method DP_SIMULATION_OUTPUT=${method}.csv ./build/double-pendulum
done
# uv run ./graphics/methods.py [csv 1] [csv 2] [csv 3] [...]
# The first CSV file is the reference
uv run ./graphics/methods.py DOPRI8.csv DOPRI5.csv RK38.csv RK4.csv ralston.csv
```
Configuration:
| Variable | Description | Value | Default |
| ------------------- | ------------------ | ------- | ----------- |
| `DP_METHODS_OUTPUT` | Output image file | _path_ | methods.png |
As usual, `DP_SYSTEM_DT` is used for configuration as well. In this case, it
controls the values of X-axis.