Pendulum Motion Analysis

OpenCV tracking · system identification · data fitting
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Acknowledgment

  • Eric Xie — tracking optimizations
  • Andy Wen — debugging frame processing issues

Color detection (cosine similarity)

Instead of HSV thresholding, the bob is detected using cosine similarity in RGB space. For reference bob color $$C_r=(R_r,G_r,B_r)$$ and pixel color $$C_p=(R_p,G_p,B_p)$$:

$$S = \frac{C_r \cdot C_p}{\|C_r\| \|C_p\|}$$

Pixels above a threshold are treated as bob pixels; we compute the centroid $$ (x_c, y_c) $$ over the mask.

Pivot estimation (circle fitting)

The pivot $$(x_0,y_0)$$ is estimated as the center of the fitted circle along the arc:

$$ E^* = \arg\min_{x_0, y_0} \sum_{i} \left( (x_i - x_0)^2 + (y_i - y_0)^2 - R^2 \right)^2 $$

Demonstration

Demonstration of the pendulum tracking algorithm.

Selected graphs

Exponential fitting graph
Exponential fitting of decay with quartic period fitting.
Period time analysis graph
Period time analysis of the pendulum at different angles.
Peak detection graph
Peak detection of the pendulum’s motion.

Code

Analysis scripts (e.g., dataAnalysis.py, expoFit.py, periodTime.py) are in the repository.