Systems in nature are stochastic as well as nonlinear. In traditional applications, engineered filters aim to minimize the stochastic effects caused by process and measurement noise. Conversely, a previous study showed that the process noise can reveal the observability of a system that was initially categorized as unobservable when deterministic tools were used. In this paper, we develop a stochastic framework to explore observability analysis and sensor placement. This framework allows for direct studies of the effects of stochasticity on optimal sensor placement and selection to improve filter error covariance. Numerical results are presented for sensor selection that optimizes stochastic empirical observability in a bioinspired setting.

Organisms and machines must use measured sensory cues to estimate unknown information about themselves or their environment. Cleverly applied sensor motion can be exploited to enrich the quality of sensory data and improve estimation. However, a major barrier to modeling such active sensing problems is the lack of empirical, yet rigorous, tools for quantifying the relationship between movement and estimation performance. Here, we introduce "BOUNDS: Bounding Observability for Uncertain Nonlinear Dynamic Systems". BOUNDS can discover patterns of sensor motion that increase information and reduce uncertainty in either real or simulated data. Crucially, it is suitable for high dimensional and partially observable nonlinear systems with sensor noise. We demonstrate BOUNDS through a case study on how flying insects estimate wind properties, showing that specific active sensing motifs improve estimation. Additionally, we present a framework to refine sporadic estimates from active sensing. When combined with an artificial neural network, we show that the information gained via active sensing in real Drosophila flight trajectories is suitable for precise wind direction estimation. Collectively, our work will help decode active sensing in organisms and inform the design of estimation algorithms for machines. ### Competing Interest Statement The authors have declared no competing interest.

Given a set of measurements, observability characterizes the distinguishability of a system's initial state, whereas constructability focuses on the final state in a trajectory. In the presence of process and/or measurement noise, the Fisher information matrices with respect to the initial and final states-equivalent to the stochastic observability and constructability Gramians-bound the performance of corresponding estimators through the Cram & eacute;r-Rao inequality. This letter establishes a connection between stochastic observability and constructability of discrete-time linear systems and provides a more numerically robust way for calculating the stochastic observability Gramian. We define a dual system and show that the dual system's stochastic constructability is equivalent to the original system's stochastic observability, and vice versa. This duality enables the interchange of theorems and tools for observability and constructability. For example, we use this result to translate an existing recursive formula for the stochastic constructability Gramian into a formula for recursively calculating the stochastic observability Gramian for both time-varying and time-invariant systems, where this sequence converges for the latter. Finally, we illustrate the robustness of our formula compared to existing (non-recursive) formulas through a numerical example.

View Video Presentation: https://doi.org/10.2514/6.2023-1809.vid Nonlinear estimation of inertial parameters (mass, center of mass and the inertia matrix entries) of a rigid body with six degrees of freedom is a challenging problem given the analytical complexity of the nonlinear system dynamics. Using Lie algebraic observability methods, we show that an appropriate, nonzero net force is needed to be able to estimate all the parameters. Using the parameters of an airliner in 2D we then compute the empirical observability Gramian for three excitation (input) profiles to study the effects on the system observability. Finally, we show that the performance of an extended Kalman filter is consistent with the numerical observability analysis results.

The neural encoding by biological sensors of flying insects, which prefilters stimulus data before sending them to the central nervous system in the form of voltage spikes, enables sensing capabilities that are computationally low cost while also being highly robust to noise. This process, which can be modeled as the composition of a linear moving average filter and a nonlinear decision function, inspired the work reported here to improve engineered sensing performance by maximizing the observability of particular neural-inspired composite measurement functions. We first present a tool to determine the observability of a linear system with measurement delay (the first element of the composition), and then use a Lie algebraic observability approach to study nonlinear autonomous systems with output delay (the second element of the composition). The Lie algebraic tools are then extended to address overall observability of systems with composite outputs as in the neural encoder model we adopt. The analytical outcomes are supported using the empirical observability Gramian, and optimal sensor placement on a bioinspired wing model is performed using metrics based on the empirical Gramian.

A dynamical system is observable if there is a one-to-one mapping from the system's measured outputs and inputs to all of the system's states. Analytical and empirical tools exist for quantifying the (full state) observability of linear and nonlinear systems; however, empirical tools for evaluating the observability of individual state variables are lacking. Here, a new empirical approach termed Empirical Individual State Observability (E-ISO) is developed to quantify the level of observability of individual state variables. E-ISO first builds an empirical observability matrix via simulation, then applies convex optimization to efficiently determine the subset of its rows required to estimate each state variable individually. Finally, (un)observability measures for these subsets are calculated to provide independent estimates of the observability of each state variable. Multiple example applications of E-ISO on linear and nonlinear systems are shown to be consistent with analytical results. Broadly, E-ISO will be an invaluable tool both for designing active sensing control laws or optimizing sensor placement to increase the observability of individual state variables for engineered systems, and analyzing the trajectory decisions made by organisms.

Neural encoding plays an important role in animal sensing. The process can be modeled as the composition of a linear filter and a nonlinear decision function. In systems theory, this kind of mechanism is termed measurement with memory. The focus of this paper is the observability analysis of systems with memory. First, we present an observability analysis tool for discrete-time linear systems and show that an unobservable system cannot be made observable using a linear neural encoding model with memory. Then nonlinear autonomous systems with memory are studied using a Lie algebraic approach. Third, we use the empirical observability Gramian for observability of non-autonomous nonlinear systems with memory and apply the results to the analysis of the flapping wing of the hawkmoth Manduca sexta, which has mechanoreceptors called campaniform sensilla to provide data for rapid maneuvers by encoding deformations. Optimal sensor placement is determined using metrics based on empirical Gramians, and the results are compared with the memoryless case. Finally, encoding model parameters are examined individually in terms of observability metrics.

Underwater gliders are often used to perform in-situ measurements of oceanographic systems. In particular, there have been a number of recent efforts to use their state estimation capabilities in order to infer physical oceanographic process dynamics, such as modeling currents and eddies. Such inference requires that the dead-reckoning systems used on the vehicles, typically hand-tuned Extended Kalman Filters (EKFs), to be tuned consistently, which can be tedious and burdensome, especially across large numbers of trials. In this work, we describe a method to automatically tune state estimation hyper-parameters used in underwater glider EKFs, that does not require ground truth estimates. This automation is achieved by taking advantage of the normalized innovation squared (NIS) metric, which can be used inside of the objective function of various optimization methods. In this paper, we demonstrate its validity in simulated environments and provide initial results for its use on fielded glider data.

Trajectory planning is one of the most studied topics in robotics. Among several methods, a sampling-based method, Rapidly-exploring Randomized Tree (RRT) algorithm, has become popular over the last two decades due to its computational efficiency. However, the RRT method does not suggest an exact way to obtain a smooth trajectory along the viapoints given by itself. In this paper, we present an approach using a timeoptimal trajectory planning algorithm, specifically for robotic manipulators without using inverse kinematics. After the trajectory smoothing with cubic splines in an environment with obstacles considering not only velocity and acceleration but also jerk constraints; the study is simulated on a six degrees of freedom humanoid robot arm model and always finds a solution successfully if there is a feasible one.