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Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. To address this issue, this paper reconstructs the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein's lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step's objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we derive its natural gradient that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new iterative filter for nonlinear systems called Natural Gradient Gaussian Approximation filter, or NANO filter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. The estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like inequality across consecutive time steps.

Mobile robotics datasets are essential for research on robotics, for example for research on Simultaneous Localization and Mapping (SLAM). Therefore the ShanghaiTech Mapping Robot was constructed, that features a multitude high-performance sensors and a 16-node cluster to collect all this data. That robot is based on a Clearpath Husky mobile base with a maximum speed of 1 meter per second. This is fine for indoor datasets, but to collect large-scale outdoor datasets a faster platform is needed. This system paper introduces our high-speed mobile platform for data collection. The mapping robot is secured on the rear-steered flatbed car with maximum field of view. Additionally two encoders collect odometry data from two of the car wheels and an external sensor plate houses a downlooking RGB and event camera. With this setup a dataset of more than 10km in the underground parking garage and the outside of our campus was collected and is published with this paper.

In Gaussian Process (GP) dynamical model learning for robot control, particularly for systems constrained by computational resources like small quadrotors equipped with low-end processors, analyzing stability and designing a stable controller present significant challenges. This paper distinguishes between two types of uncertainty within the posteriors of GP dynamical models: the well-documented mathematical uncertainty stemming from limited data and computational uncertainty arising from constrained computational capabilities, which has been largely overlooked in prior research. Our work demonstrates that computational uncertainty, quantified through a probabilistic approximation of the inverse covariance matrix in GP dynamical models, is essential for stable control under computational constraints. We show that incorporating computational uncertainty can prevent overestimating the region of attraction, a safe subset of the state space with asymptotic stability, thus improving system safety. Building on these insights, we propose an innovative controller design methodology that integrates computational uncertainty within a second-order cone programming framework. Simulations of canonical stable control tasks and experiments of quadrotor tracking exhibit the effectiveness of our method under computational constraints.

Multi-object tracking (MOT) is an essential technique for navigation in autonomous driving. In tracking-by-detection systems, biases, false positives, and misses, which are referred to as outliers, are inevitable due to complex traffic scenarios. Recent tracking methods are based on filtering algorithms that overlook these outliers, leading to reduced tracking accuracy or even loss of the objects trajectory. To handle this challenge, we adopt a probabilistic perspective, regarding the generation of outliers as misspecification between the actual distribution of measurement data and the nominal measurement model used for filtering. We further demonstrate that, by designing a convolutional operation, we can mitigate this misspecification. Incorporating this operation into the widely used unscented Kalman filter (UKF) in commonly adopted tracking algorithms, we derive a variant of the UKF that is robust to outliers, called the convolutional UKF (ConvUKF). We show that ConvUKF maintains the Gaussian conjugate property, thus allowing for real-time tracking. We also prove that ConvUKF has a bounded tracking error in the presence of outliers, which implies robust stability. The experimental results on the KITTI and nuScenes datasets show improved accuracy compared to representative baseline algorithms for MOT tasks., Comment: IEEE Transactions on Intelligent Vehicles

Bayesian filtering serves as the mainstream framework of state estimation in dynamic systems. Its standard version utilizes total probability rule and Bayes' law alternatively, where how to define and compute conditional probability is critical to state distribution inference. Previously, the conditional probability is assumed to be exactly known, which represents a measure of the occurrence probability of one event, given the second event. In this paper, we find that by adding an additional event that stipulates an inequality condition, we can transform the conditional probability into a special integration that is analogous to convolution. Based on this transformation, we show that both transition probability and output probability can be generalized to convolutional forms, resulting in a more general filtering framework that we call convolutional Bayesian filtering. This new framework encompasses standard Bayesian filtering as a special case when the distance metric of the inequality condition is selected as Dirac delta function. It also allows for a more nuanced consideration of model mismatch by choosing different types of inequality conditions. For instance, when the distance metric is defined in a distributional sense, the transition probability and output probability can be approximated by simply rescaling them into fractional powers. Under this framework, a robust version of Kalman filter can be constructed by only altering the noise covariance matrix, while maintaining the conjugate nature of Gaussian distributions. Finally, we exemplify the effectiveness of our approach by reshaping classic filtering algorithms into convolutional versions, including Kalman filter, extended Kalman filter, unscented Kalman filter and particle filter.

Integral reinforcement learning (IntRL) demands the precise computation of the utility function's integral at its policy evaluation (PEV) stage. This is achieved through quadrature rules, which are weighted sums of utility functions evaluated from state samples obtained in discrete time. Our research reveals a critical yet underexplored phenomenon: the choice of the computational method -- in this case, the quadrature rule -- can significantly impact control performance. This impact is traced back to the fact that computational errors introduced in the PEV stage can affect the policy iteration's convergence behavior, which in turn affects the learned controller. To elucidate how computation impacts control, we draw a parallel between IntRL's policy iteration and Newton's method applied to the Hamilton-Jacobi-Bellman equation. In this light, computational error in PEV manifests as an extra error term in each iteration of Newton's method, with its upper bound proportional to the computational error. Further, we demonstrate that when the utility function resides in a reproducing kernel Hilbert space (RKHS), the optimal quadrature is achievable by employing Bayesian quadrature with the RKHS-inducing kernel function. We prove that the local convergence rates for IntRL using the trapezoidal rule and Bayesian quadrature with a Mat\'ern kernel to be $O(N^{-2})$ and $O(N^{-b})$, where $N$ is the number of evenly-spaced samples and $b$ is the Mat\'ern kernel's smoothness parameter. These theoretical findings are finally validated by two canonical control tasks.

Adaptive dynamic programming (ADP) for stochastic linear quadratic regulation (LQR) demands the precise computation of stochastic integrals during policy iteration (PI). In a fully model-free problem setting, this computation can only be approximated by state samples collected at discrete time points using computational methods such as the canonical Euler-Maruyama method. Our research reveals a critical phenomenon: the sampling period can significantly impact control performance. This impact is due to the fact that computational errors introduced in each step of PI can significantly affect the algorithm's convergence behavior, which in turn influences the resulting control policy. We draw a parallel between PI and Newton's method applied to the Ricatti equation to elucidate how the computation impacts control. In this light, the computational error in each PI step manifests itself as an extra error term in each step of Newton's method, with its upper bound proportional to the computational error. Furthermore, we demonstrate that the convergence rate for ADP in stochastic LQR problems using the Euler-Maruyama method is O(h), with h being the sampling period. A sensorimotor control task finally validates these theoretical findings.

Fluidic logic circuitry analogous to its electric counterpart could potentially provide soft robots with machine intelligence due to its supreme adaptability, dexterity, and seamless compatibility using state-of-the-art additive manufacturing processes. However, conventional microfluidic channel based circuitry suffers from limited driving force, while macroscopic pneumatic logic lacks timely responsivity and desirable accuracy. Producing heavy duty, highly responsive and integrated fluidic soft robotic circuitry for control and actuation purposes for biomedical applications has yet to be accomplished in a hydraulic manner. Here, we present a 3D printed hydraulic fluidic half-adder system, composing of three basic hydraulic fluidic logic building blocks: AND, OR, and NOT gates. Furthermore, a hydraulic soft robotic half-adder system is implemented using an XOR operation and modified dual NOT gate system based on an electrical oscillator structure. This half-adder system possesses binary arithmetic capability as a key component of arithmetic logic unit in modern computers. With slight modifications, it can realize the control over three different directions of deformation of a three degree-of-freedom soft actuation mechanism solely by changing the states of the two fluidic inputs. This hydraulic fluidic system utilizing a small number of inputs to control multiple distinct outputs, can alter the internal state of the circuit solely based on external inputs, holding significant promises for the development of microfluidics, fluidic logic, and intricate internal systems of untethered soft robots with machine intelligence., Comment: 26 pages, 14 figures

The accuracy of moving horizon estimation (MHE) suffers significantly in the presence of measurement outliers. Existing methods address this issue by treating measurements leading to large MHE cost function values as outliers, which are subsequently discarded. This strategy, achieved through solving combinatorial optimization problems, is confined to linear systems to guarantee computational tractability and stability. Contrasting these heuristic solutions, our work reexamines MHE from a Bayesian perspective, unveils the fundamental issue of its lack of robustness: MHE's sensitivity to outliers results from its reliance on the Kullback-Leibler (KL) divergence, where both outliers and inliers are equally considered. To tackle this problem, we propose a robust Bayesian inference framework for MHE, integrating a robust divergence measure to reduce the impact of outliers. In particular, the proposed approach prioritizes the fitting of uncontaminated data and lowers the weight of contaminated ones, instead of directly discarding all potentially contaminated measurements, which may lead to undesirable removal of uncontaminated data. A tuning parameter is incorporated into the framework to adjust the robustness degree to outliers. Notably, the classical MHE can be interpreted as a special case of the proposed approach as the parameter converges to zero. In addition, our method involves only minor modification to the classical MHE stage cost, thus avoiding the high computational complexity associated with previous outlier-robust methods and inherently suitable for nonlinear systems. Most importantly, our method provides robustness and stability guarantees, which are often missing in other outlier-robust Bayes filters. The effectiveness of the proposed method is demonstrated on simulations subject to outliers following different distributions, as well as on physical experiment data., Comment: 17 pages

The convergence of policy gradient algorithms in reinforcement learning hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. At the core of our results is the uniqueness of the stationary point of dLQR when it is observable, which is in a concise form of an observer-based controller with the optimal similarity transformation. These results shed light on designing efficient algorithms for general decision-making problems with partially observed information., Comment: arXiv admin note: substantial text overlap with arXiv:2201.09598

This paper proposes a primal-dual framework to learn a stable estimator for linear constrained estimation problems leveraging the moving horizon approach. To avoid the online computational burden in most existing methods, we learn a parameterized function offline to approximate the primal estimate. Meanwhile, a dual estimator is trained to check the suboptimality of the primal estimator during execution time. Both the primal and dual estimators are learned from data using supervised learning techniques, and the explicit sample size is provided, which enables us to guarantee the quality of each learned estimator in terms of feasibility and optimality. This in turn allows us to bound the probability of the learned estimator being infeasible or suboptimal. Furthermore, we analyze the stability of the resulting estimator with a bounded error in the minimization of the cost function. Since our algorithm does not require the solution of an optimization problem during runtime, state estimates can be generated online almost instantly. Simulation results are presented to show the accuracy and time efficiency of the proposed framework compared to online optimization of moving horizon estimation and Kalman filter. To the best of our knowledge, this is the first learning-based state estimator with feasibility and near-optimality guarantees for linear constrained systems.

The convergence of policy gradient algorithms hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. One of our core results is the uniqueness of the stationary point of dLQR when it is observable, which provides an optimality certificate for solving dynamic controllers using policy gradient methods. Moreover, we establish conditions under which dLQR and linear quadratic Gaussian control are equivalent, thus providing a unified viewpoint of optimal control of both deterministic and stochastic linear systems. These results further shed light on designing policy gradient algorithms for more general decision-making problems with partially observed information.

A suspension of nanoparticles with very low volume fraction is found to assemble into a macroscopic cellular phase under the collective influence of AC and DC voltages. Systematic study of this phase transition shows that it was the result of electrophoretic assembly into a two-dimensional configuration followed by spinodal decomposition into particle-rich walls and particle-poor cells mediated principally by electrohydrodynamic flow. This mechanistic understanding reveals two characteristics needed for a cellular phase to form, namely 1) a system that is considered two dimensional and 2) short-range attractive, long-range repulsive interparticle interactions. In addition to determining the mechanism underpinning the formation of the cellular phase, this work presents a method to reversibly assemble microscale continuous structures out of nanoscale particles in a manner that may enable the creation of materials that impact diverse fields including energy storage and filtration., Comment: 25 pages, 14 figures

State estimation is critical to control systems, especially when the states cannot be directly measured. This paper presents an approximate optimal filter, which enables to use policy iteration technique to obtain the steady-state gain in linear Gaussian time-invariant systems. This design transforms the optimal filtering problem with minimum mean square error into an optimal control problem, called Approximate Optimal Filtering (AOF) problem. The equivalence holds given certain conditions about initial state distributions and policy formats, in which the system state is the estimation error, control input is the filter gain, and control objective function is the accumulated estimation error. We present a policy iteration algorithm to solve the AOF problem in steady-state. A classic vehicle state estimation problem finally evaluates the approximate filter. The results show that the policy converges to the steady-state Kalman gain, and its accuracy is within 2 %.

H-infinity filter has been widely applied in engineering field, but copping with bounded noise is still an open problem and difficult to solve. This paper considers the H-infinity filtering problem for linear system with bounded process and measurement noise. The problem is first formulated as a zero-sum game where the dynamic of estimation error is non-affine with respect to filter gain and measurement noise. A nonquadratic Hamilton-Jacobi-Isaacs (HJI) equation is then derived by employing a nonquadratic cost to characterize bounded noise, which is extremely difficult to solve due to its non-affine and nonlinear properties. Next, a reinforcement learning algorithm based on gradient descent method which can handle nonlinearity is proposed to update the gain of reinforcement filter, where measurement noise is fixed to tackle non-affine property and increase the convexity of Hamiltonian. Two examples demonstrate the convergence and effectiveness of the proposed algorithm.