This paper plans quadrotor trajectories so that the state estimator can actually pin down its own calibration states: accelerometer and gyroscope biases and the position of the GPS antenna relative to the IMU. Energy-minimising planners tend to fly paths along which these states are unobservable. The authors score a trajectory by the smallest singular value of an approximate local observability Gramian, built from a Taylor expansion of the sensor model with higher-order Lie derivatives, so it also sees states that never appear directly in the measurement. Trajectories are differentially flat piecewise polynomials, optimised with SQP under limits on thrust-to-weight ratio (at most 1.5), angular velocity (at most π rad/s) and angular acceleration (at most 5π rad/s²).
Findings
- Simulated quadrotor with GPS (0.2 m noise) and IMU, fused in an iterated error-state EKF; statistics over 50 runs per trajectory.
- For the GPS-IMU offset, the optimised trajectory gave almost 2x lower final RMSE than every baseline (figure-eight, star, random near the physical limits, covariance-trace optimisation). Bias estimates converged comparably across the best four methods.
- The covariance-trace baseline needed about 13 hours to optimise against about 10 minutes for this method, the ~80x speed-up in the abstract.
- On a 50 s waypoint task, integrated position RMSE was 3.69 ± 0.31 m against 6.06 ± 1 m for a minimum-snap trajectory, and the GPS-IMU offset RMSE was 0.85 ± 0.22 m against 3.40 ± 0.83 m.
Limits
- Simulation only, one sensor suite, one vehicle model. Real-UAV tests are listed as future work.
- The measure is local: it captures weak local observability around the planned path.
- It assumes accurate measurements are available and ignores what the environment offers to sensors.
- The optimised path is longer and more complex than minimum snap, which costs time and energy (authors admit).
Relevance
Unobservable directions are where faults hide: a constant bias that the trajectory never excites produces no residual, whether it comes from a drifting sensor or a slow GPS spoofing attack. Observability analysis says which states a residual-based detector can see at all, and this paper shows that how the robot moves changes the answer. A follow-up by the same authors appeared at RSS 2017; see robust estimation.