多数据点中心计算与关联数据推断：自由落体测量设备技术问询

Great question—this is a classic signal processing + kinematics problem for free-fall setups, and your existing calibration data gives you a solid foundation to nail this. Let’s break this down step by step, with practical methods tailored to your setup:

Your calibration values (mean, standard deviation, threshold) are perfect for filtering out noise and identifying when the sphere is actively obscuring the sensor. Here’s how to use them:

• Define a trigger threshold: Use trigger_low = calib_mean - 2*calib_std (or your pre-set allowed threshold) to mark the start of a sphere pass. When a light reading first drops below this value, record the timestamp as t_start.
• Define an end threshold: Use trigger_high = calib_mean - 1*calib_std (a value closer to your unobscured baseline) to mark the end of the pass. When the light reading climbs back above this value, record t_end.
• This gives you a continuous segment of data points where the sphere is interacting with the sensor—ignore everything outside this window for your center calculation.

The "center" we care about is the exact moment the sphere’s midline crosses the sensor’s light path—this eliminates errors from the sphere’s top/bottom edges. Here are three robust methods, ordered by accuracy and complexity:

Method A: Weighted Time Average (Most Accurate) #

This method prioritizes data points where the sphere is obscuring the light most (i.e., the midpoint) by assigning higher weights to readings that deviate farthest from your calibrated baseline:

1. For each data point in your valid segment, calculate the obscuration deviation: deviation = calib_mean - light_value (positive because obscured readings are lower than the baseline).
2. Assign a weight proportional to this deviation—use weight = deviation for linear weighting, or weight = deviation^2 to amplify the midpoint’s influence.
3. Compute the weighted center time:

   t_center = sum(timestamp * weight for all points) / sum(weight for all points)
   

• This is ideal if your sensor has high sampling rate (100+ Hz) and you want to average out minor noise.

Method B: Valley Time Point (Simplest) #

If your sensor data is relatively clean, the darkest reading (lowest light_value) corresponds directly to the sphere’s midpoint passing the sensor:

1. Optional: Smooth your data with a sliding average (e.g., average 3-5 consecutive points) to reduce noise spikes.
2. Find the data point in your valid segment with the smallest light_value, then use its timestamp as t_center.

• Quick to implement, works well if you’ve already minimized sensor noise.

Method C: Median Time (Most Robust to Noise) #

If your readings have significant random noise, the median timestamp of your valid segment is a stable, low-variance alternative:

1. Collect all timestamps in your valid segment, sort them in order.
2. Pick the middle value as t_center (for even numbers of points, average the two middle timestamps).

• Great for low-cost sensors that produce noisy data, as it ignores outliers entirely.

Assuming your sensors are mounted vertically with a fixed, measured distance d between each pair:

• For each pair of adjacent sensors (e.g., Sensor 1 and Sensor 2), get their center times t1 and t2. The time difference is Δt = t2 - t1.
• Use the free-fall displacement equation to solve for g:

  d = v1*Δt + 0.5*g*(Δt)^2
  

  Where v1 is the sphere’s velocity as it passes Sensor 1.
• For better accuracy with 3+ sensors:
  1. Calculate the velocity at each sensor: v1 = d / Δt1 (from Sensor 1→2), v2 = d / Δt2 (from Sensor 2→3).
  2. Fit a linear regression to velocity vs. time (velocity increases linearly with time under gravity). The slope of this line is your measured g.

• Noise Reduction: Add a simple low-pass filter (like a 5-point sliding average) to your raw light readings before processing—this will make your center time calculations far more consistent.
• Calibration Validation: Test with a sphere of known diameter D. The time between t_start and t_end should roughly equal D/v (where v is the sphere’s speed). Use this to tweak your trigger thresholds if needed.
• Multi-Sphere Handling: If you’re testing multiple drops in quick succession, add a minimum time gap (e.g., 100ms) between detected events to avoid merging two separate sphere passes into one.

内容的提问来源于stack exchange，提问作者CoryG