Computer-implemented symbolic differentiation using first-order retraction

A non-transitory computer-readable medium storing instructions which, when executed by a computing device, causes the computing device to perform operations comprising:

• accessing, at the computing device comprising processing circuitry and memory, an input element storage;
• accessing, from the memory, a symbolic perturbation vector in an input tangent space;
• retracting, using a symbolic computation engine of the computing device, the input element storage by the symbolic perturbation vector to generate a perturbed input element storage;
• accessing, from the memory, a symbolic expression for an output element storage as a function of the input element storage;
• substituting, in the symbolic expression using the symbolic computation engine, the perturbed input element storage for the input element storage;
• computing, using the symbolic computation engine, local coordinates for a perturbed output relative to an unperturbed output;
• generating, using the symbolic computation engine, a symbolic expression for a perturbed output element tangent vector as a function of input element storage and input tangent space perturbation;
• computing, using the symbolic computation engine, a Jacobian of output tangent vector elements with respect to an input perturbation;
• generating, using the symbolic computation engine, a symbolic expression for a tangent space Jacobian of the output element storage with respect to the input tangent space as a function of the input element storage and the input tangent space perturbation;
• setting, using the symbolic computation engine, the symbolic perturbation vector to zero to generate a symbolic expression for the tangent space Jacobian of the output element storage with respect to the input tangent space as a function of the input element storage;
• and
• controlling operation of an unmanned aerial vehicle based on the symbolic expression for the output element storage as the function of the input element storage, wherein the symbolic expression for the output element storage as the function of the input element storage represents a cost function for operating the unmanned aerial vehicle, wherein controlling the operation of the unmanned aerial vehicle comprises controlling the operation of the unmanned aerial vehicle to implement an optimization based on the cost function.

A computing device comprising:

• processing circuitry;
• and
• a memory storing instructions which, when executed by the processing circuitry, cause the processing circuitry to:
• access an input element storage;
• access, from the memory, a symbolic perturbation vector in an input tangent space;
• retract, using a symbolic computation engine of the computing device, the input element storage by the symbolic perturbation vector to generate a perturbed input element storage;
• access, from the memory, a symbolic expression for an output element storage as a function of the input element storage;
• substitute, in the symbolic expression using the symbolic computation engine, the perturbed input element storage for the input element storage;
• compute, using the symbolic computation engine, local coordinates for a perturbed output relative to an unperturbed output;
• generate, using the symbolic computation engine, a symbolic expression for a perturbed output element tangent vector as a function of input element storage and input tangent space perturbation;
• compute, using the symbolic computation engine, a Jacobian of output tangent vector elements with respect to an input perturbation;
• generate, using the symbolic computation engine, a symbolic expression for a tangent space Jacobian of the output element storage with respect to the input tangent space as a function of the input element storage and the input tangent space perturbation;
• set, using the symbolic computation engine, the symbolic perturbation vector to zero to generate a symbolic expression for the tangent space Jacobian of the output element storage with respect to the input tangent space as a function of the input element storage;
• and
• control operation of an unmanned aerial vehicle based on the symbolic expression for the output element storage as the function of the input element storage, wherein the symbolic expression for the output element storage as the function of the input element storage represents a cost function for operating the unmanned aerial vehicle, wherein controlling the operation of the unmanned aerial vehicle comprises controlling the operation of the unmanned aerial vehicle to implement an optimization based on the cost function.