Robots operating in flow-like environments benefit from a greater understanding of the characteristics of their surroundings. Flow-like environments, such as ocean and air currents, movements of vehicles and pedestrians, or transport of blood in the heart, are often complex and hard to model. However, robots conducting environmental monitoring, autonomous driving, crowd monitoring, or surgical procedures all require knowledge of these flow-like environments in their tasks. For robots to successfully operate in these situations, they need to be able to deduce important features of these environments and incorporate these features into their planning and decision-making. In this work, we propose a strategy for robots to learn high-level features in flow-like environments and develop planning frameworks based on these features.
Coherent sets are a global characteristic of turbulent flows that describe groups of particles that do not disperse very much and move together robustly . This means particles within coherent sets tend to stay within the sets, as in Fig. 1. Lagrangian coherent structures (LCS) are separatices that delineate dynamically distinct regions in dynamical systems . LCS provide different but complementary information when compared to coherent sets . Knowledge of coherent sets and LCS allows robots to compactly represent key characteristics of their environment and leverage them in application-specific ways. For example, surgical robots could understand vasculature conditions such as flow stagnation and separation during medical procedures . Aerial vehicles are affected by coherent features during takeoff and landing . Robots used for environmental monitoring may be interested in atmospheric and oceanic transport pathways that play a role in the movement of pollutants . Marine vehicles can plan energy efficient trajectories in the ocean by leveraging coherent structures [27, 24, 36] and maintaining sensors in their desired monitoring regions [26, 41].
There are many methods for studying coherent sets through transfer operators which provide a framework for analyzing the global behaviors of dynamical systems. These methods identify which geometric structures are the strongest barriers to global mixing [11, 15, 16, 17]. There are also methods for identifying LCS relying on detailed flow field data and integration over a parametrized time interval .
While the mentioned approaches elucidate coherence that can be used in planning and control of robots, neither take into account the complex interdependencies within the data sets. This presents the need for online algorithms for robotic applications, especially when the time parameterization of the system is unknown in the absence of prior models or historical data. In recent years, there has been a surge of machine learning approaches relying on kernels (or feature maps) and transfer operators to estimate dynamical systems and their key characteristics[42, 23, 7, 32, 30, 38, 31]. Kernels map the environmental data to a higher dimensional feature space with greater expressivity for learning models. To the best of our knowledge, these advances at the intersection of machine learning and dynamical systems have not been studied in a robotics context, despite the field’s growing interest in transfer operators [1, 2, 8, 13].
New advances in the detection of coherent sets leverage techniques from machine learning [32, 30]. Though these techniques highlight the expressiveness of data-driven, kernel-based methods, they still suffer from the same limitations in their assumptions about centralized, offline data, and time parameterization. Our work builds on these recent advances and develops strategies that allow robots to compute coherent sets online and contextualize them in the robots’ environment to increase awareness and facilitate decision-making.
We propose using the coherent set detection framework as a map for greater situational awareness for operating within flow-like environments that exhibit this notion of coherence. First, we provide the relevant background in dynamical systems and machine learning related to coherent sets and show how kernel methods can represent coherent sets from data. Next, we demonstrate the usefulness of computing coherent sets in an online setting with simulations and analysis of known benchmark. We apply these online methods for computing coherent sets for overhead data in an urban monitoring environment, where the coherent sets correspond to regions of interest for situational awareness. Lastly, we show how coherent sets in a fluid flow can be leveraged for energy-efficient navigation of surface vehicles.
Ii-a Dynamical Systems and Transfer Operators
Coherent sets are global properties of a dynamical system, extracted through constructing transfer operators. In this section, we discuss the connection between dynamical systems and transfer operators.
Define a discrete dynamical system as where is a measurable map. Transfer operators allow for a systematic study of the global properties of dynamical systems and alleviates computing the map , which is often difficult to model and estimate for unknown environments exhibiting complex, nonlinear dynamics. For example, the Koopman operator is an infinite dimensional linear transfer operator that fully captures the nonlinear dynamics of the system by acting on the space of scalar-valued functions operating on the observables of the system, . In terms of the dynamical systems map , the Koopman operator is also often written as .
Transfer operators can more generally be defined for stochastic dynamical systems to infer statistical aspects of a dynamical system. Define as a stochastic process on the state space .
is a stationary and ergodic Markov decision process. A transition density function,, is then defined by the relation for some measurable set .
Given a probability density, an observable of the system , and a specified lag time
, we can define the following transfer operators that provides information about iterated maps. Specifically, the Perron-Frobenius and Koopman transfer operators serve as adjoints to one another. Furthermore, the eigenvalues of transfer operators are real and their eigenfunctions form an orthogonal basis with respect to the corresponding scalar product. The Perron-Frobenius,, and Koopman, , transfer operators are defined as follows: where , while where .
Ii-B Kernel Methods
Instead of constructing models using data directly, kernel methods rely on transforming data into a feature representation that elucidates interesting, meaningful patterns. Here, we present the relevant background for kernel methods, which later serves as the foundation for estimating transfer operators.
Kernel functions are applied to elements of some space, , to measure the similarity between any pairs of elements. However, this similarity metric can be found by computing the inner product of features acting on elements of the input space in some high-dimensional, possibly infinite feature space in . Thus, kernels allow us to compare objects based on their features. Formally, for inputs , a feature map , and some valid inner product , a kernel is defined as
By using the relation in Eq. (1), commonly referred to as the kernel trick, operations in the high-dimensional feature space can be computed without ever explicitly computing the coordinates of and instead using a kernel function in inner product space. Remarkably, the inner product can be efficiently computed by the kernel function, alleviating the need to assess the high-dimensional coordinates, and
, while still receiving a metric for similarity between the vectorsand in the rich feature space .
Kernels on features spaces that are positive definite can be used to define a function on . The space of such functions , such that the evaluation of at can be represented as the inner product in feature space, is referred to as the Reproducing Kernel Hilbert Space (RKHS). An RKHS is a space of functions mapping to with two defining features:
These two properties show that in an RKHS the feature map of every point is in the feature space as in (2) and that the space exhibits the reproducing property as in (3). The reproducing property defines a functional evaluation at a point as the inner product between the function and evaluated feature map .
Let be the feature map associated with kernel defined on and be the feature map associated with kernel defined on . Then, define feature matrices as and , with Gram matrices and . is then the covariance operator and is the cross-covariance operator, which can be thought of as a nonlinear generalization of standard covariance and cross-covariance matrices . and can be estimated as
The covariance and cross-covariance operators are one of the most important and established tools in RKHS theory. These operators are used in the computation of kernel principal components analysis (PCA), the kernel Fisher discriminant, kernel partial least squares, the kernel canonical correlation analysis (CCA), and many other learning algorithms. Some learning algorithms relying on the use of (4) and (5), can be written in terms of their corresponding Gram matrices. For Gram matrices, only the inner products between and and and need to be computed. Thus, by computing Gram matrices on a finite number of data points, we can alleviate the need to explicitly compute the possibly infinite-dimensional covariance and cross-covariance operator. The ability to construct algorithms in the inner product space using kernels, and more specifically the kernel trick, lends them to be particularly useful in variety of learning scenarios . In the next section, we will lay the framework for use of transfer operator theory in the RKHS and later show how this expression in RKHS can be used as a learning technique for modeling coherent sets and planning with this awareness.
Ii-C Connecting Transfer Operators to Kernel Methods
Transfer operator approximation in the RKHS allows for the use of kernel-based methods, acting on feature maps of the data, while still capturing the properties of the transfer operators . Transfer operators can be expressed in terms of the covariance and cross-covariance operators defined on some RKHS . Assume the input and output space are the same, i.e., , then the kernels and resulting Hilbert spaces are also the same. In this scenario, we assume , meaning full-state observables are simply the states themselves.
The kernel Koopman operator is defined as
This value can be calculated empirically from training data. Similarly, the kernel Perron-Frobenius operator is defined as
with the empirical estimate of the Perron-Frobenius
We now have a representation of the transfer operators in terms of the covariance and cross-covariance operators that can be estimated using the kernel versions (4) and (5). Thus, we now have a framework unifying existing techniques for approximating transfer operators and their eigendecompositions in RKHS , as shown in Fig. 2. In our work, we can use additional techniques from statistics to extract the dominant characteristics from the transfer operator representation of the environment, specifically coherent sets.
Ii-D Kernel Canonical Correlation Analysis
Kernel CCA applied to the kernel transfer operators produces the coherent sets in the environment. The techniques outlined in this section were first presented in . Traditional CCA seeks to find two sets of basis such that the correlations between projections of two multidimensional variables and onto each set of basis vectors are maximized. Through the use of kernels, kernel CCA seeks to more generally find two nonlinear mappings and , for and , such that the correlation between the mappings is maximized.
Formally, the kernel CCA problem is
The solution to this problem is the largest eigenvalue, , corresponding to the following eigenvalue problem
In practice, we estimate covariance and cross covariance operators from finite samples of the environment, so eigenfunctions and are approximated from data as and .
In general, the inverses of the covariance and cross covariance operators do not exist. Thus, the regularized inverses are used, specifically we use and . Using these regularized inverses to solve (11) gives
By replacing the covariance and cross-covariance operators with their empirical estimates, the eigenvalue problem of the kernel canonical correlation analysis in Eq. (12) becomes
for . Instead of solving this eigenvalue problem directly, we can instead solve a surrogate eigenvalue problem
To solve for , we can compute
In practice, we will solve this eigenvalue problem to compute the coherent sets of a system. In order to compute the coherent sets, we first compute the dominant eigenfunctions, as in (14). Then, a -means clustering is applied to the dominant eigenfunctions to determine the coherent sets.
We will now show how the formulation of the eigenvalue problem as (13) is related to the detection of coherent sets, as presented in . Let be the forward operator, for some that is a reference density of interest. Given as the image density obtained by mapping the indicator function on forward in time. If we normalize with respect to , we can define a new operator and its adjoint
As per the properties of adjoint, we have that . Here, is the analogue of the reweighted Perron-Frobenius operator, or the Koopman operator with the roles of and reversed, and is the analogue of the Koopman operator. The operator is used to detect coherent sets. We want to show that the operator is equivalent to the operator in Eq. (12). With this equivalence, we can compute the coherent sets using the kernel representation of transfer operators and applying kernel CCA.
Given that , we have that . If we define the RKHS approximation of as . Then, the eigenvalue problem (12) can be reformulated as
The property of adjointness and for can be verified as follows:
This shows the eigenvalue problem for computing coherent sets and the eigenvalue problem for kernel CCA are equivalent.
Ii-E Online Computation of Coherent Sets
Robots collect data in real time and knowledge of the dynamics of the environment is limited; thus, we desire a way to detect the coherent sets online using the and solving the eigenvalue problem, estimated as Eq. (17). The operator can be represented as a function of covariance operators, and , and cross covariance operators, and . Instead of using the covariance and cross covariance operators directly, the eigenvalue problem is solved using a surrogate formulation (14). The surrogate formulation relies on Gram matrices and , where captures the correlations between the initial positions of the tracked objects and captures the correlations between the final positions of the objects. However, robots collect sensor measurements in real time. Other works assume knowledge of the fixed lag time , such that analyzing the operator with data at and computes the coherent sets . Our paper analyzes how coherence can be computed in an online setting with no prior assumptions about .
For coherence at several time instances, studies have shown it is sufficient to average the operators for all the different time instances and compute the dominant eigenfunctions of the resulting operator [17, 18, 5]. Let correspond to the Koopman operator and correspond to the Perron-Frobenius operator for the dynamics from to . Then, we are interested in the time-averaged quantity
Averaging the operator by using the representations of the operators in (12) and the corresponding surrogate eigenvalue problem in (14), we are interested, at each time step , the eigenvalue decomposition of
which can be rewritten as
The inner sum is computed as the standard optimal online averaging solution with the addition of more data. Then, the eigenvalue decomposition of the full resulting operator for the surrogate problem, Eq. (19), is computed at each time step. Finally, -means clustering is applied, as before.
Iii Simulations and Analysis
We present two examples of fluid flows, where the coherent sets are well-studied. We perform an analysis of our online method compared to offline methods.
Iii-a Time-Dependent Double Gyre
A simple model of the wind-driven, time-dependent double gyre flow, as in , is described by
For this paper, the parameters are set to . We simulate points uniformly sampled on grid . From time with step size , we use a differential equations solver based on an explicit Runge-Kutta (4,5) formula on the velocities to calculate the trajectories of the points, as shown in Fig. 3.
With a Gaussian radial basis function kernelfor , we compute the Gramian matrices for for the initial position of the particles and for the position of the particles at time , as in Eq. (19). In an online setting, only the inner sum needs to be recomputed as a moving average. The total product and its eigenvalue decomposition is performed. A -means clustering for is computed on the dominant eigenvalues of the estimated time-averaged matrix Eq. (18).
Iii-B Bickley Jet
The Bickley jet model is a prototypical model in the study of coherence that is a meandering zonal jet, flanked both above and below by counter rotating vertices. The Bickley jet is used as an idealized model for the Gulf Stream in the ocean and polar night jets in the atmosphere [10, 6].
The stream function for the Bickley jet model is
with . The velocities can be computed as and . We use scaled parameters from  resulting in .
We sample points uniformly on a grid . From time with step size , we calculate the trajectories of the points using a variable-step, variable-order Adams-Bashforth-Moulton solver of orders 1 to 13 on the velocity , as shown in Fig. 4. Some particles escape the grid , and the positions of these particles wrap around such that they stay within the grid. The true time lag parameter , as introduced in Sec. II-E, for this map is known to be for this simulation.
Iii-C Analysis of Online Coherent Sets Computation
We compare the online algorithm, using Eq. (19), to the offline algorithm, where we naively compute surrogate eigenvalues of the operator at each snapshot. We compare methods for external cluster validation, where evaluation is based on comparing the online and offline clustering to a known true clustering. The true clustering is found by constructing the operator for the true time parameter and using the -means algorithm for the known number of clusters over multiple random initializations of cluster centers and averaging the clusters. The results are shown in Table I. We use the Rand index adjusted for chance (RI), homogeneity (H), and the V-measure (V). The Rand index measures similarity of the assignments, homogeneity indicates if each cluster contains only members of a single class, and V-measure is a combination of homogeneity and completeness, where completeness indicates if all members of a given class are assigned to the same cluster. The evaluation metric is found by comparing clusters from each time step against the true clusters. This is then averaged across all time steps.
Across all metrics, the online method for the double gyre performs slightly worse than the offline method. In both the offline and online clustering, the clustering are qualitatively similar to the true clustering in size and shape. For the more complex and realistic Bickley jet, the online method outperforms the offline method across all metrics. In both scenarios, the online method is less sensitive to noise and invalid clustering, as shown in Fig. 5. The simulations and experiments conducted in this paper rely on the online method, which requires less knowledge about the environment and works better in complicated, real world scenarios due to its incorporation of data over time.
Iv Path Planning in Flow-like Environments with Aerial and Surface Vehicles 111Accompanying video can be found at https://youtu.be/D-4S6hnFr2E
In this section, we discuss how computing coherent sets in real time facilitates robots’ awareness. We then suggest how this awareness can be incorporated into a planning scheme.
Iv-a Aerial vehicle based crowd monitoring
Robots operating in dense environments, such as urban landscapes or roadways, rely on persistent monitoring of crowds for scene understanding and safe navigation. Much of the literature in this area surrounds tracking individual trajectories of pedestrians or other objects. However, understanding global information about pedestrian movement allows for greater awareness of high-level behaviors and facilitates informed planning. With aerial vehicles monitoring an environment overhead, we can collect trajectory information of pedestrians, which can be thought of as a flow-like environment. This information can be used within our framework to elucidate key features of the underlying persistent crowd behaviors.
Overhead data of pedestrians at a train station stop was collected from the th floor of an hotel in Bahanhofstr, Zurich . The original sequence is processed with a time step of . We found and analyzed the dataset using tools from OpenTraj . We compute coherent sets using trajectories of pedestrians over time steps, with a -degree polynomial kernel and -means clusters on eigenvalues.
In Fig. 6, we see coherent sets illustrate areas of congregation or high movement for pedestrians. Pedestrian data can be collected in real time to track coherent sets online, where sets would then be used to navigate through crowds, such as by avoiding the busy train platform, or deploy sensors for situational awareness, such as by monitoring the entrance for anomalous behavior or notable changes in coherence patterns.
Iv-B Energy-efficient trajectories for surface vehicles
Currents in rivers, oceans, and in the atmosphere are other examples of flow fields. While global information is complex and hard to analyze, local information does not provide an accurate description of environment features. Here, we leverage coherent sets from a single gyre to demonstrate the usefulness of path planning under awareness of these global features.
We create flows and deploy surface vehicles using the multirobot Coherent Structure Testbed (mCoSTe) which consists of a micro Autonomous Surface Vehicle (mASV) and a Multi Robot flow tank (MR tank). The MR tank is a tank of water equipped with an OptiTrack motion capture system providing localization of the mASV at . The mASV is a differential drive vehicle with a maximum forward speed of approximately . Two flow driving cylinders placed horizontally and rotating at approximately creates a single gyre flow in the MR tank, as shown in Fig 7.
We assume one or more robots are collecting velocity or trajectory data. This may be aerial vehicles collecting overhead imagery of the flow field or passive gliders directly sampling the velocity. This information is then processed in real time to compute the coherent sets. Instead of surface vehicles tracking and modeling individual trajectories within the flow field, the robot maintains the coherent sets as a global description of the dynamics of its environment. The coherent sets are calculated using an approximate flow model for the gyres in the tank for simulated particles over time steps using a Gaussian kernel with and -means clusters on eigenvalues. Then, the sets are used to construct energy-efficient paths. We plan paths from a specified start location to a goal location by picking waypoints near coherent sets, exerting control effort to travel to and escape from coherent sets, and remaining in coherent sets until the next waypoint is close.
The robot navigates the flow induced by a single gyre model and completes its mission when planning a trajectory based on the coherent set. During execution, the robot saves energy by turning off its motors when the boundary of the coherent set is reached and floating through the minimal dispersion region. However, a boat unaware of the coherent set drives into the gyre, causing the robot to become stuck and ultimately fail to reach its goal, as seen in Fig. 8. Modeling coherent sets gives the robot a descriptive framework to reason about flows in the environment, without maintaining detailed knowledge.
Robots operating in complex scenarios necessitates greater awareness of their environment. In flow-like environments, global dynamics may be more useful than tracking individual trajectories. Transfer operator theory provides a systematic framework for reasoning about global dynamics. Recent advances in machine learning and transfer operators combine powerful data representation with physical system knowledge. Here, we present a framework where coherent sets, defined by transfer operators, are learned with kernel methods, estimated online, and integrated into decision-making for robots. We foresee robotics applications incorporating awareness of environments in novel, meaningful ways with these connections. For example, our framework allows scene segmentation according to pedestrian movement instead of static objects and efficient navigation of strong flows with little information. Future work includes automated kernel function selection, a distributed framework for sensor information collected by multi-robot teams, development of more sophisticated decision-making schemes, and analysis in real environments.
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