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Distance estimation from vision is fundamental for a myriad of roboticapplications such as navigation, manipulation, and planning. Inspired by themammal's visual system, which gazes at specific objects, we develop two novelconstraints relating time-to-contact, acceleration, and distance that we callthe $\tau$-constraint and $\Phi$-constraint. They allow an active (moving)camera to estimate depth efficiently and accurately while using only a smallportion of the image. The constraints are applicable to range sensing, sensorfusion, and visual servoing.  We successfully validate the proposed constraints with two experiments. Thefirst applies both constraints in a trajectory estimation task with a monocularcamera and an Inertial Measurement Unit (IMU). Our methods achieve 30-70% lessaverage trajectory error while running 25$\times$ and 6.2$\times$ faster thanthe popular Visual-Inertial Odometry methods VINS-Mono and ROVIO respectively.The second experiment demonstrates that when the constraints are used forfeedback with efference copies the resulting closed loop system's eigenvaluesare invariant to scaling of the applied control signal. We believe theseresults indicate the $\tau$ and $\Phi$ constraint's potential as the basis ofrobust and efficient algorithms for a multitude of robotic applications.

TTCDist: Fast Distance Estimation From an Active Monocular Camera Using Time-to-Contact