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Most linear experimental design problems assume homogeneous variance, while the presence of heteroskedastic noise is present in many realistic settings. Let a learner have access to a finite set of measurement vectors that can be probed to receive noisy linear responses. We propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters. We demonstrate this method on two adaptive experimental design problems under heteroskedastic noise, fixed confidence transductive best-arm identification and level-set identification and prove the first instance-dependent lower bounds in these settings. Lastly, we construct near-optimal algorithms and demonstrate the large improvements in sample complexity gained from accounting for heteroskedastic variance in these designs empirically. 

Winged eVTOL aircraft’s ability to generate aerodynamic lift with wings and to create upward thrust with upward-facing rotors makes these vehicles capable of the kind of versatile flight needed in urban environments. Because of these vehicles’ aerodynamic complexities and their unique methods of producing thrusts and torques, control allocation is needed to determine how to distribute force and torque efforts across the aircraft’s actuators. However, current control allocation methods fail to properly represent the actuators’ complex dynamics and are unable to harness the full potential of these over-actuated vehicles. Current shortcomings include modeling rotors as linear effectors while the wide range of airspeeds experienced by eVTOL aircraft leads to significant nonlinearities in the thrust and torque achieved by each rotor. This means linear control allocation methods may consistently fail to produce desired thrusts and torques, which can inhibit the vehicle from tracking a trajectory at best, and at worst can cause the vehicle to stall and lose control. Additionally, current control allocation methods are often unable to prioritize low-energy actuators resulting in shorter battery life. We present a nonlinear control allocation method that considers a nonlinear rotor model, allows for prioritization of low-energy control surfaces over rotors, and reliably accounts for actuator saturation. Simulation results show a 90% reduction in high-airspeed trajectory tracking position error from a typical, linear least-squares pseudoinverse control allocation method while maintaining comparable energy use.