Time Synchronization

Align data from multiple sources with different time bases.

The Problem

Different data sources use different time references:

Source	Time Base	Resolution
Sensor GPS	UTC (GNSS)	1 ms
Drone (DJI)	Internal tick	100 ms
Drone (ArduPilot)	Boot time	1 ms
Inclinometer	Internal RTC	10 ms
Camera	Frame index	33 ms (30fps)

Synchronization is essential for sensor fusion and accurate analysis.

Synchronizer Overview

flowchart LR
    A[GPS Reference]
    A --> S[NED Coordinates]

    B[Drone GPS]
    C[Litchi GPS]

    B --> S
    C --> S


    %% Vertical ordering: Drone above Litchi
    B
    C

    S --> O1[Time Offset: Reference to Litchi/Drone]

    C --> P[Pitch from Litchi]
    D[Inclinometer] --> P

    P --> O2[Time Offset: Inclinometer to Litchi/Drone]

    O2 --> O1

Mathematical Description of the Calculations

NED Position Correlation

The NED (North-East-Down) method correlates 3D position signals to find time offsets between GPS sources.

Step 1: LLA to ENU Conversion

Convert GPS coordinates to local East-North-Up (ENU) frame relative to reference point:

\[ \begin{aligned} \Delta\text{lat} &= \text{lat}_{\text{target}} - \text{lat}_{\text{ref}} \\ \Delta\text{lon} &= \text{lon}_{\text{target}} - \text{lon}_{\text{ref}} \\ \\ E &= R \cdot \Delta\text{lon} \cdot \cos(\text{lat}_{\text{ref}}) \\ N &= R \cdot \Delta\text{lat} \\ U &= \text{alt}_{\text{target}} - \text{alt}_{\text{ref}} \end{aligned} \]

where \(R = 6{,}371{,}000\) m (Earth radius)

Step 2: Interpolation to Common Timebase

Both GPS sources are interpolated to a common high-rate timebase (default 100 Hz):

\[ \begin{aligned} t_{\text{common}} &= [t_{\text{start}}, t_{\text{start}} + \Delta t, \ldots, t_{\text{end}}] \\ \Delta t &= \frac{1}{f_{\text{target}}} = \frac{1}{100\text{ Hz}} = 0.01\text{ s} \end{aligned} \]

Interpolated signals: - \(E_1(t), N_1(t), U_1(t)\) - Reference GPS in ENU - \(E_2(t), N_2(t), U_2(t)\) - Target GPS in ENU

Step 3: Cross-Correlation Per Axis

Cross-correlation for each spatial axis:

\[ R_{xy}[\tau] = \sum_{t} x(t) \cdot y(t + \tau) \]

Normalized correlation:

\[ \rho_{xy}[\tau] = \frac{R_{xy}[\tau]}{\sqrt{\sum_t x^2(t) \cdot \sum_t y^2(t)}} \]

Applied to each axis: - \(\rho_E[\tau]\) - East correlation - \(\rho_N[\tau]\) - North correlation
- \(\rho_U[\tau]\) - Up correlation

Step 4: Sub-Sample Peak Detection

Find sub-sample peak using parabolic interpolation around integer peak:

\[ \begin{aligned} i_{\text{peak}} &= \arg\max_i |\rho[\tau_i]| \\ y_1 &= |\rho[\tau_{i-1}]|, \quad y_2 = |\rho[\tau_i]|, \quad y_3 = |\rho[\tau_{i+1}]| \\ \\ \delta &= \frac{y_1 - y_3}{2(y_1 - 2y_2 + y_3)} \\ \tau_{\text{peak}} &= \tau_i + \delta \cdot \Delta t \end{aligned} \]

Step 5: Weighted Average Time Offset

Combine offsets from all three axes weighted by correlation strength:

\[ \begin{aligned} w_E &= |\rho_E[\tau_E]|, \quad w_N = |\rho_N[\tau_N]|, \quad w_U = |\rho_U[\tau_U]| \\ \\ \tau_{\text{offset}} &= \frac{w_E \cdot \tau_E + w_N \cdot \tau_N + w_U \cdot \tau_U}{w_E + w_N + w_U} \end{aligned} \]

Overall correlation quality:

\[ \rho_{\text{combined}} = \frac{w_E + w_N + w_U}{3} \]

Step 6: Spatial Offset After Time Alignment

After applying time offset, compute residual spatial offset:

\[ \begin{aligned} t_2' &= t_2 + \tau_{\text{offset}} \\ E_2'(t_1) &= \text{interp}(E_2, t_2', t_1) \\ \\ \Delta E &= \text{mean}(E_2'(t_1) - E_1(t_1)) \\ \Delta N &= \text{mean}(N_2'(t_1) - N_1(t_1)) \\ \Delta U &= \text{mean}(U_2'(t_1) - U_1(t_1)) \\ \\ d_{\text{spatial}} &= \sqrt{\Delta E^2 + \Delta N^2 + \Delta U^2} \end{aligned} \]

Pitch Correlation

Step 1: Interpolation to Common Timebase

Both pitch signals interpolated to high rate (100 Hz):

\[ \begin{aligned} \theta_1(t) &= \text{interp}(\theta_{\text{Litchi}}, t_{\text{Litchi}}, t_{\text{common}}) \\ \theta_2(t) &= \text{interp}(\theta_{\text{Inclinometer}}, t_{\text{Inclinometer}}, t_{\text{common}}) \end{aligned} \]

Step 2: Cross-Correlation

\[ R_{\theta\theta}[\tau] = \sum_{t} \theta_1(t) \cdot \theta_2(t + \tau) \]

Normalized:

\[ \rho_{\theta}[\tau] = \frac{R_{\theta\theta}[\tau]}{\sqrt{\sum_t \theta_1^2(t) \cdot \sum_t \theta_2^2(t)}} \]

Step 3: Sub-Sample Peak Detection

Same parabolic interpolation as NED method:

\[ \tau_{\text{pitch}} = \tau_{i_{\text{peak}}} + \delta \cdot \Delta t \]

Step 4: Chained Time Offset

Total time offset to align Inclinometer with GPS payload:

\[ \tau_{\text{Inclinometer→GPS}} = \tau_{\text{Inclinometer→Litchi}} + \tau_{\text{Litchi→GPS}} \]

This two-stage approach allows synchronization even when Inclinometer and GPS don't share overlapping spatial trajectories, using Litchi gimbal as an intermediate reference.

Physical Interpretation

• Time Offset (\(\tau\)): Seconds to add to target timestamp to align with reference
• Correlation (\(\rho\)): Quality metric, range [0, 1]
  • \(\rho > 0.9\): Excellent
  • \(0.7 < \rho < 0.9\): Good
  • \(\rho < 0.7\): Questionable
• Spatial Offset (\(d\)): Residual position error after time alignment (should be < 5m for good GPS)