Quantifying control effort of biological and technical movements: an information-entropy-based approach.

In biomechanics and biorobotics, muscles are often associated with reduced movement control effort and simplified control compared to technical actuators. This is based on evidence that the nonlinear muscle properties positively influence movement control. It is, however, open how to quantify the simplicity aspect of control effort and compare it between systems. Physical measures, such as energy consumption, stability, or jerk, have already been applied to compare biological and technical systems. Here a physical measure of control effort based on information entropy is presented. The idea is that control is simpler if a specific movement is generated with less processed sensor information, depending on the control scheme and the physical properties of the systems being compared. By calculating the Shannon information entropy of all sensor signals required for control, an information cost function can be formulated allowing the comparison of models of biological and technical control systems. Exemplarily applied to (bio-)mechanical models of hopping, the method reveals that the required information for generating hopping with a muscle driven by a simple reflex control scheme is only I=32 bits versus I=660 bits with a DC motor and a proportional differential controller. This approach to quantifying control effort captures the simplicity of a control scheme and can be used to compare completely different actuators and control approaches.

Semiclassical quantization by harmonic inversion: comparison of algorithms.

Harmonic inversion techniques have been shown to be a powerful tool for the semiclassical quantization and analysis of quantum spectra of both classically integrable and chaotic dynamical systems. Various computational procedures have been proposed for this purpose. Our aim is to find out which method is numerically most efficient. To this end, we summarize and discuss the different techniques and compare their accuracies by way of two example systems.

Semiclassical non-trace-type formulas for matrix-element fluctuations and weighted densities of states.

Densities of states weighted with the diagonal matrix elements of two operators A and B, i.e., rho(A,B)(E)= summation operator(n)delta(E-E(n)), cannot, in general, be written as a trace formula, and therefore no simple extension of semiclassical trace formulas is known for this case. However, from the high resolution analysis of quantum spectra in the semiclassical regime we find strong evidence that weighting the delta functions in the quantum mechanical density of states with the product of diagonal matrix elements, , is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic orbit means, (p)(p), of the classical observables A and B. Results are presented for the hydrogen atom in a magnetic field for both the chaotic and near-integrable regime, and for the circle billiard.

Semiclassical spectra and diagonal matrix elements by harmonic inversion of cross-correlated periodic orbit sums.

Semiclassical spectra weighted with products of diagonal matrix elements of operators A(alpha), i.e., g(alphaalpha')(E)= summation operator(n)/(E-E(n)), are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.