Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

The relative order and inverses of recurrent networks

Differential geometry is used to investigate the structure of neural-network-based control systems. The key aspect is relative order—an invariant property of dynamic systems. Finite relative order allows the specification of a minimal architecture for a recurrent network. Any system with finite relative order has a left inverse. It is shown that a recurrent network with finite relative order has a local inverse that is also a recurrent network with the same weights. The results have implications for the use of recurrent networks in the inverse-model-based control of nonlinear systems.

Relative order defines a topology for recurrent networks

This paper uses techniques from control theory in the analysis of trained recurrent neural networks. Differential geometry is used as a framework, which allows the concept of relative order to be applied to neural networks. Any system possessing finite relative order has a left-inverse. Any recurrent network with finite relative order also has an inverse, which is shown to be a recurrent network.

A novel satellite mission concept for upper air water vapour, aerosol and cloud observations using integrated path differential absorption LiDAR limb sounding

We propose a new satellite mission to deliver high quality measurements of upper air water vapour. The concept centres around a LiDAR in limb sounding by occultation geometry, designed to operate as a very long path system for differential absorption measurements. We present a preliminary performance analysis with a system sized to send 75 mJ pulses at 25 Hz at four wavelengths close to 935 nm, to up to 5 microsatellites in a counter-rotating orbit, carrying retroreflectors characterized by a reflected beam divergence of roughly twice the emitted laser beam divergence of 15 µrad. This provides water vapour profiles with a vertical sampling of 110 m; preliminary calculations suggest that the system could detect concentrations of less than 5 ppm. A secondary payload of a fairly conventional medium resolution multispectral radiometer allows wide-swath cloud and aerosol imaging. The total weight and power of the system are estimated at 3 tons and 2,700 W respectively. This novel concept presents significant challenges, including the performance of the lasers in space, the tracking between the main spacecraft and the retroreflectors, the refractive effects of turbulence, and the design of the telescopes to achieve a high signal-to-noise ratio for the high precision measurements. The mission concept was conceived at the Alpbach Summer School 2010.