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Quantifying Motion Excitation for Metric Scale Observability in Monocular Visual-Inertial Odometry

Monocular visual-inertial odometry (VIO) cannot recover metric scale from vision alone; scale must be resolved through inertial measurements. We present a trajectory-dependent observability analysis showing that translational acceleration, produced by curvature, not constant-speed straight-line travel, is the fundamental source that couples scale to the inertial state. This relationship is formalized through the gravity-acceleration asymmetry in the IMU model, from which we derive rank conditions on the observability matrix and propose a lightweight excitation metric computable from raw IMU data. Controlled experiments on a differential-drive robot with a monocular camera and consumer-grade IMU validate the theory, with straight-line motion yielding 9.2% scale error, circular motion 6.4%, and figure-eight motion 4.8%, with excitation spanning four orders of magnitude. These results establish trajectory design as a practical mechanism for improving metric scale recovery.

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