Abstract: Systems and methods for operating an unmanned aerial vehicle (UAV). A method includes capturing a raw video stream input using one or more camera sensors of the UAV. The method includes quantifying characteristics of motion of the UAV as commanded by a user of the UAV. The method includes quantifying characteristics of total motion of the one or more camera sensors including the motion commanded by the user. The method includes subtracting the characteristics of the motion commanded by the user from the characteristics of the total motion to generate a residual. The method includes applying electronic image stabilization to the raw video stream based on the residual to generate a stabilized video stream output. Accounting for unintended UAV motion in EIS separate and apart from the total motion being experienced by the UAV can significantly enhance a UAV user's video viewing experience.

Abstract: A calibration of an unmanned aerial vehicle is performed without the use of a magnetometer. The unmanned aerial vehicle generates a first acceleration vector in a navigation frame of reference and a second acceleration vector in a GPS frame of reference. The unmanned aerial vehicle estimates a heading of the unmanned aerial vehicle based on the first acceleration vector and the second acceleration vector. The unmanned aerial vehicle performs a calibration based on the estimated heading of the unmanned aerial vehicle.

Abstract: A computer accesses an input element storage and an output element storage. The computer accesses a symbolic expression for output element storage as a function of the input element storage. The computer computes, using a symbolic computation engine of the computer, a symbolic expression for the tangent space Jacobian of the output element storage with respect to an input tangent space. The computer outputs the computed expression.

Abstract: A computer accesses a first symbolic expression for an output matrix as a function of an input matrix at a computing device comprising processing circuitry and memory. The computer computes a first Jacobian of the input matrix with respect to an input tangent space. The computer computes a second Jacobian of the output matrix with respect to the input matrix. The computer computes a third Jacobian of an output tangent space with respect to the input matrix. The computer applies symbolic matrix multiplication to the first Jacobian, the second Jacobian, and the third Jacobian to obtain a second symbolic expression for the output tangent space with respect to the input tangent space. The computer provides a representation of the second symbolic expression, the second symbolic expression representing a computed tangent-space Jacobian.

Abstract: A computer accesses a first symbolic expression for an output value as a function of an input value. The computer computes a first symbolic Jacobian of the input value with respect to an input tangent space from a symbolic Lie group definition. The computer computes a second symbolic Jacobian of the output value with respect to the input value. The computer computes a third symbolic Jacobian of an output tangent space with respect to the input value from the symbolic Lie group definition. The computer applies symbolic matrix multiplication to the first symbolic Jacobian, the second symbolic Jacobian, and the third symbolic Jacobian to obtain a second symbolic expression for the output tangent space with respect to the input tangent space. The computer provides a representation of the second symbolic expression.

Abstract: A computer of an unmanned aerial vehicle (UAV) accesses, from a memory unit, a problem definition comprising cost functions associated with travel of the UAV. The computer causes movement of the UAV based on the cost functions. The computer adjusts one or more of the cost functions during a flight of the UAV. The computer causes further movement of the UAV based on the adjusted one or more of the cost functions.