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.