# IBM Double Pendulum Chaotic Dataset — data/

## ⚠ DATA UNAVAILABLE — upstream CDN defunct (verified 2026-04-22)

`dax-cdn.cdn.appdomain.cloud` resolves as **NXDOMAIN**: IBM shut down the DAX
(Data Asset eXchange) platform and its CDN around 2023. The tarball URL in
`download.sh` is permanently dead. `data.csv` is **not present**.

**To reconstruct `data.csv` manually:**
1. Obtain the tarball `double-pendulum-chaotic.tar.gz` from an alternative
   source (e.g. a mirror, archive.org, or direct contact with the authors at
   IBM Research AI).
2. Place it at `benchmarks/dynamical_systems/ibm_double_pendulum/data/double-pendulum-chaotic.tar.gz`.
3. Re-run `ONLY_VIDEO=0 bash download.sh` — the script detects the local
   tarball and skips the download step.

---

## Source

IBM Data Asset eXchange (DAX), "Double Pendulum Chaotic" v2.0.1.

- Landing page: <https://developer.ibm.com/exchanges/data/all/double-pendulum-chaotic/>
- Project homepage: <https://ibm.github.io/double-pendulum-chaotic-dataset/>
- Tarball (primary): `https://dax-cdn.cdn.appdomain.cloud/dax-double-pendulum-chaotic/2.0.1/double-pendulum-chaotic.tar.gz`
- Paper: Asseman, Kornuta & Ozcan, *Learning beyond simulated physics*, NeurIPS 2018 MDSD workshop. <https://openreview.net/pdf?id=HylajWsRF7>
- License: **CDLA-Sharing-1.0** (<https://cdla.io/sharing-1-0/>)

Redistribution is permitted under CDLA-Sharing-1.0. We therefore rebuild
`data.csv` by downloading the upstream tarball on demand; the raw tarball and
the generated `data.csv` / `data_all_videos.csv` are gitignored per
`/data/yiming/real-sr/.gitignore`.

## Usage

```bash
./download.sh              # defaults to ONLY_VIDEO=0  -> data.csv is video 0
ONLY_VIDEO=5 ./download.sh # use video 5 as data.csv instead
ONLY_VIDEO=all ./download.sh  # make data.csv = all 21 videos concatenated
```

The script always writes `data_all_videos.csv` (full 21-video concatenation)
alongside the single-video `data.csv`.

## File layout (regenerated; gitignored)

- `double-pendulum-chaotic.tar.gz` — upstream tarball (~hundreds of MB).
- `dpc_raw/` — extracted tarball tree. Contains
  `.../original/dpc_dataset_csv/{0..20}.csv` (the 21 full-length marker-position
  CSVs we use) and `.../train_and_test_split/…` (IBM's 4-in / 200-out
  train/validation/test split, which we do **not** consume here).
- `data.csv` — benchmark entry data (one video by default).
- `data_all_videos.csv` — all 21 videos concatenated, with a `video_id` column.

## Upstream CSV format (per the Asseman 2018 notebook example)

Each `original/dpc_dataset_csv/<k>.csv` is **headerless**, whitespace-separated,
with 6 float columns per row:

    x_red  y_red  x_green  y_green  x_blue  y_blue

Rows are consecutive video frames at 400 Hz. Coordinates are in **image pixels
multiplied by 5** (IBM upscales the video 5× before pattern-matching to get
sub-pixel resolution; the multiplication is preserved in the stored CSV — see
Sec. 2.3 of Asseman 2018). Red = pivot (the fixed hinge), green = first datum
(tip of upper arm), blue = second datum (tip of lower arm).

## Derived columns we add

`download.sh` writes the following column order to `data.csv`:

| col | name             | role       | units      | source                                    |
|----:|------------------|-----------|------------|-------------------------------------------|
| 0   | `theta1_ddot_fd` | output    | rad/s²     | 2nd centred finite difference of `theta1` |
| 1   | `theta1`         | input     | rad        | `atan2(x_green - x_red, y_green - y_red)` |
| 2   | `theta2`         | input     | rad        | `atan2(x_blue - x_green, y_blue - y_green)` |
| 3   | `omega1`         | input     | rad/s      | 1st centred finite difference of `theta1` |
| 4   | `omega2`         | input     | rad/s      | 1st centred finite difference of `theta2` |
| 5   | `theta2_ddot_fd` | input     | rad/s²     | 2nd centred finite difference of `theta2` |
| 6   | `H_mpernorm`     | input     | J/kg (proxy) | see below — per-unit-mass Hamiltonian with `m1 = m2 = 1` |
| 7   | `x_red`          | input     | pixels × 5 | upstream (IBM)                            |
| 8   | `y_red`          | input     | pixels × 5 | upstream (IBM)                            |
| 9   | `x_green`        | input     | pixels × 5 | upstream (IBM)                            |
| 10  | `y_green`        | input     | pixels × 5 | upstream (IBM)                            |
| 11  | `x_blue`         | input     | pixels × 5 | upstream (IBM)                            |
| 12  | `y_blue`         | input     | pixels × 5 | upstream (IBM)                            |
| 13  | `frame_idx`      | input     | count      | row index within a video                  |
| 14  | `t_seconds`      | input     | s          | `frame_idx / 400`                         |
| 15  | `video_id`       | input     | int 0..20  | source-video index                        |

All upstream columns are preserved unchanged. The derived columns are
**documented finite-difference estimates** of the kinematic quantities — they
are not fits.

### Angle convention

With image coordinates (x right, y **down**), the downward vertical is the
direction of increasing y. We define

    theta1 = atan2(x_green - x_red , y_green - y_red)
    theta2 = atan2(x_blue  - x_green, y_blue  - y_green)

so that `theta = 0` when the arm hangs straight down, `theta > 0` when it swings
to the right of the image, and `theta` is measured independently for each arm
from the downward vertical. This matches the convention used in the simple
double-pendulum Lagrangian (Levien & Tan 1993, Wikipedia) consumed by
`formulas/hamiltonian.py`, `formulas/eom_theta1.py` and `formulas/eom_theta2.py`.

The paper's `alpha = atan2(...)` convention (from horizontal, for the first
arm only) and our convention differ only by a constant offset; either can be
recovered from the stored marker positions.

### Finite-difference conventions (defaults; not paper-prescribed)

The Asseman 2018 paper does not prescribe a derivative estimator. We use the
simplest defensible centred-difference scheme:

    omega_i  = (theta_{i+1} - theta_{i-1}) / (2 * dt)
    alpha_i  = (theta_{i+1} - 2 theta_i + theta_{i-1}) / dt^2
    dt       = 1 / 400 s

Endpoints are dropped. Angles are `np.unwrap`ed before differentiation so that
multi-revolution swings do not inject 2π artefacts. These O(dt²) schemes are
noise-amplifying by a factor O(1/dt) ≈ 400 for velocities and O(1/dt²) ≈ 1.6e5
for accelerations, so reference scores on `theta1_ddot_fd` / `theta2_ddot_fd`
are bounded from below by tracker noise, not by model fit. See
`description.md` for a full discussion.

### Hamiltonian-proxy caveat

The paper does **not** report the bob masses `m1`, `m2`. We therefore compute

    H_mpernorm = H(theta1, theta2, omega1, omega2; l1=0.091, l2=0.070, g=9.81,
                   m1 = 1, m2 = 1)

as a proxy. Absolute energy values are meaningful only up to an unknown scale;
formulas/hamiltonian.py matches this convention. A searcher that finds
`H(theta, omega; l1, l2, g) * k` for any constant `k > 0` has found the same
answer.

## Reproducibility notes

- IBM shut down the DAX platform (~2023); `dax-cdn.cdn.appdomain.cloud`
  returns NXDOMAIN as of 2026-04-22. The only recovery path is a manual
  copy of the tarball (see the "DATA UNAVAILABLE" notice at the top of
  this file).
- The CDLA-Sharing-1.0 license requires that any redistribution of the data,
  including derivatives, be under the same license and with attribution. This
  benchmark entry attributes Asseman, Kornuta & Ozcan (IBM Research AI, 2018).