Environmental contours are an established method in probabilistic engineering design, especially in ocean engineering. The contours help engineers to select the environmental states which are appropriate for structural design calculations. Defining an environmental contour means enclosing a region in the variable space which corresponds to a certain return period. However, there are multiple definitions and methods to calculate an environmental contour for a given return period. Here, we analyze the established approaches and present a new concept which we call highest density contour (HDC). We define this environmental contour to enclose the highest density region (HDR) of a given probability density. This region occupies the smallest possible volume in the variable space among all regions with the same included probability, which is advantageous for engineering design. We perform the calculations using a numerical grid to discretize the original variable space into a finite number of grid cells. Each cell's probability is estimated and used for numerical integration. The proposed method can be applied to any number of dimensions, i.e. number of different variables in the joint probability model. To put the highest density contour method in context, we compare it to the established inverse first-order reliability method (IFORM) and show that for common probability distributions the two methods yield similarly shaped contours. In multimodal probability distributions, however, where IFORM leads to contours which are dificult to interpret, the presented method still generates clearly defined contours.Comment: preprint of the accepted version, 14 pages, 10 figure

Flapping wings enable flying animals and biomimetic robots to generate elevated aerodynamic forces. Measurements that demonstrate this capability are based on experiments with tethered robots and animals, and indirect force calculations based on measured kinematics or airflow during free flight. Remarkably, there exists no method to measure these forces directly during free flight. Suchin vivorecordings in freely behaving animals are essential to better understand the precise aerodynamic function of their flapping wings, in particular during the downstroke versus upstroke. Here, we demonstrate a new aerodynamic force platform (AFP) for non-intrusive aerodynamic force measurement in freely flying animals and robots. The platform encloses the animal or object that generates fluid force with a physical control surface, which mechanically integrates the net aerodynamic force that is transferred to the earth. Using a straightforward analytical solution of the Navier–Stokes equation, we verified that the method is accurate. We subsequently validated the method with a quadcopter that is suspended in the AFP and generates unsteady thrust profiles. These independent measurements confirm that the AFP is indeed accurate. We demonstrate the effectiveness of the AFP by studying aerodynamic weight support of a freely flying birdin vivo. These measurements confirm earlier findings based on kinematics and flow measurements, which suggest that the avian downstroke, not the upstroke, is primarily responsible for body weight support during take-off and landing.

Tiger beetles are fast diurnal predators capable of chasing prey under closed-loop visual guidance. We investigated this control system using statistical analyses of high-speed digital recordings of beetles chasing a moving prey dummy in a laboratory arena. Correlation analyses reveal that the beetle uses a proportional control law in which the angular position of the prey relative to the beetle's body axis drives the beetle's angular velocity with a delay of about 28 ms. The proportionality coefficient or system gain, 12 s⁻¹, is just below critical damping. Pursuit simulations using the derived control law predict angular orientation during pursuits with a residual error of about 7°. This is of the same order of magnitude as the oscillation imposed by the beetle's alternating tripod gait, which was not factored into the control law. The system delay of 28 ms equals a half-stride period, i.e. the time between the touch down of alternating tripods. Based on these results, we propose a physical interpretation of the observed control law: to turn towards its prey, the beetle on average exerts a sideways force proportional to the angular position of the prey measured a half-stride earlier.