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High-speed flight in GPS-denied environments is currently an important frontier in the research on autonomous flight of micro air vehicles. Autonomous drone races stimulate the advances in this area by representing a very challenging case with tight turns, texture-less floors, and dynamic spectators around the track. These properties hamper the use of standard visual odometry approaches and imply that the micro air vehicles will have to bridge considerable time intervals without position feedback. To this end, we propose an approach to trajectory estimation for drone racing that is computationally efficient and yet able to accurately estimate a micro air vehicle's state (including biases) and parameters based on sparse, noisy observations of racing gates. The key concept of the approach is to optimize unknown and difficult-to-observe state variables so that the observations of the racing gates best fit with the known control inputs, estimated attitudes, and the quadrotor dynamics and aerodynamics during a time window. It is shown that a gradient-descent implementation of the proposed approach converges $4 times quicker to (approximately) correct bias values than a state-of-the-art 15-state extended Kalman filter. Moreover, it reaches a higher accuracy, as the predicted end-point of an open-loop turn is on average only $20 cm away from the actual end-point, while the extended Kalman filter and the gradient descent method with kinematic model only reach an accuracy of $50 cm. Although the approach is applied here to drone racing, it generalizes to other settings in which a micro air vehicle may only have sparse access to velocity and/or position measurements.

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