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edfas.org 19 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 22 NO. 2 PROPORTIONAL NAVIGATION IN THE DRAGONFLY MODEL As illustrated in Fig. 2, proportional navigation gen- erates straight-line trajectories when the prey does not change either speed or direction before capture. For each set of initial conditions (including dragonfly and target initial positions as well as dragonfly and target velocities), there is only one parallel navigation solution. The prey- image falls on a different eye-screen location for each of these trajectories but does not drift from the fovea as long as the model dragonfly follows proportional naviga- tion. Such interception trajectories can be implemented through foveation (not shown), but the position of the foveamust be pre-calculated (requiring knowledge of prey velocity and position) and set prior to each engagement. Fig. 4 Error correction based upon prey-image slippage. The model dragonfly uses prey-image slippage on the eye as an error signal for adjusting both interception trajectory and fovea location on the eye-screen. (a) Red stars and black circles are the location of prey and dragonfly (respectively) at each time step. Prey trajectory is the same as in Fig. 2b. For reference, the proportional navigation interception trajectory (also as in Fig. 2b) is provided in green. (b) Simulation time vs. fovea location on eye (x-coordinate only). Open circle and star indicate the initial and final positions of the fovea. (c) Range- vector correlation as a function of time for parallel navigation (green) and the model dragonfly with error correction (black circles). (b) (a) (c) Moreover, relatively small errors in calculating the fovea position resulted in an unsuccessful engagement (also not shown), suggesting that a man-made system relying upon such a strategy would be vulnerable to target eva- sive maneuvers. ERROR CORRECTION IN THE DRAGONFLY It should be noted that the biological dragonfly is inca- pable of adjusting the location of its fovea, yet still imple- ments a formof proportional navigation as it approaches its prey. [5] The final version of the model presented here (see Fig. 4) is foundedon the hypothesis that the biological dragonfly is using prey-image slippage as an error signal to determinewhether it is following proportional navigation. Figure 2 provides clues as to what that error signal could be. When the model dragonfly uses proportional navigation, no further maneuvers are required to inter- cept the prey and the prey-image remains fixed at the same position on the eye-screen. Therefore prey-image driftmay be treated as an error-signal, indicating that the dragonfly is no longer following the trajectory indicatedby parallel navigation. Using this error signal to implement proportional navigation would require that biological dragonflies search across potential trajectories as they are approaching their prey,maneuvering to find the trajectory that caused the least amount of prey-image slippage. [11] Data presentedbyMischiati et al. [5] (see Fig. 1) is consistent with this hypothesis.