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A growing number of control problems are concerned with vehicles that have insufficient sensors to directly measure all system states and require active control actuation to observe all necessary states (e.g. underwater navigation, simultaneous localization and mapping, parameter identification). In this paper, we consider first-order nonholonomic systems in canonical form as a model system for exploring the development of active control that optimizes system observability characteristics. Observability analysis of this system shows that all nonholonomic states must be measured in the output, and actuation in a minimum of two control channels is required for observability. Analytical trajectories are derived that allow almost arbitrary placement of the observability gramian eigenvalues, which are shown to be inversely related to state estimation covariances. Simulation results show that the optimal trajectories provide four to five times faster estimator convergence and three times lower state estimate covariances than suboptimal trajectories.

The wings of the hawkmoth Manduca sexta are lined with mechanoreceptors called campaniform sensilla that encode wing deformations. During flight, the wings deform in response to a variety of stimuli, including inertial-elastic loads due to the wing flapping motion, aerodynamic loads, and exogenous inertial loads transmitted by disturbances. Because the wings are actuated, flexible structures, the strain-sensitive campaniform sensilla are capable of detecting inertial rotations and accelerations, allowing the wings to serve not only as a primary actuator, but also as a gyroscopic sensor for flight control. We study the gyroscopic sensing of the hawkmoth wings from a control theoretic perspective. Through the development of a low-order model of flexible wing flapping dynamics, and the use of nonlinear observability analysis, we show that the rotational acceleration inherent in wing flapping enables the wings to serve as gyroscopic sensors. We compute a measure of sensor fitness as a function of sensor location and directional sensitivity by using the simulation-based empirical observability Gramian. Our results indicate that gyroscopic information is encoded primarily through shear strain due to wing twisting, where inertial rotations cause detectable changes in pronation and supination timing and magnitude. We solve an observability-based optimal sensor placement problem to find the optimal configuration of strain sensor locations and directional sensitivities for detecting inertial rotations. The optimal sensor configuration shows parallels to the campaniform sensilla found on hawkmoth wings, with clusters of sensors near the wing root and wing tip. The optimal spatial distribution of strain directional sensitivity provides a hypothesis for how heterogeneity of campaniform sensilla may be distributed.

This work considers the optimal sensor placement problem for a general nonlinear system using the eigenvalues of the observability Gramian in the cost function. The problem is formulated as a mixed-integer convex optimization problem. Using the empirical observability Gramian, the input data to this optimization problem are computed from a simulation of the nonlinear system with no analytical model required. A piecewise linear approximation to the observability Gramian is proposed using special ordered sets of type two, allowing a coarser sensor location mesh and thus fewer binary variables and shorter solution times compared with standard gridded approaches. The solution methodology is applied to vortex estimation in the wake of a flapping airfoil using velocity sensors on the surface of the airfoil, which is modeled using unsteady potential flow and a Joukowski conformal mapping. Resulting optimal sensor sets are found near the trailing edge in pairs on the upper and lower surface. Although the observability Gramian is computed from the linearized system, the optimal sensor sets yield improved performance of an unscented Kalman filter estimating wake vortex structure on the nonlinear dynamics; the optimal sets outperform more than 99% of the sampled feasible sensor sets, thus validating the observability eigenvalues as a measure of nonlinear estimator performance. Nomenclature A, B, C = linearized system dynamics matrices a = Joukowski cylinder radius E = expectation operator F = Fisher information matrix f = nonlinear dynamics model H = linear measurement model h = nonlinear measurement model k = discrete time step k r = reduced frequency P = covariance matrix p = number of desired sensors r = number of candidate sensors s i = sensor location for sensor i u = control input u = control linearization deviation u 0 = control linearization trajectory v = measurement noise W = observability Gramiañ W = empirical observability Gramian W = interpolated observability Gramian W d = discrete-time observability Gramian w j = special ordered set weight vector for sensor j x = state vector x = state linearization deviation x 0 = state linearization trajectory

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