Tracking attracting manifolds in flows

Abstract

This paper presents a collaborative control strategy designed to enable a team of robots to track attracting Lagrangian coherent structures (LCS) and unstable manifolds in two-dimensional flows. Tracking LCS in flows is important for many applications such as planning energy optimal paths in the ocean and for predicting the evolution of various physical and biological processes in the ocean. The proposed strategy which tracks attracting LCS and unstable manifolds in real-time through direct computation of the local finite time Lyapunov exponent field, does not require global information about the dynamics of the surrounding flow, and is based on local sensing, prediction, and correction. The collaborative control strategy is implemented on a team of robots and theoretical guarantees for the tracking and formation keeping strategies are presented. We demonstrate the performance of the tracking strategy in simulation using actual ocean flow data and experimental flow data generated in a tank. The strategy is validated experimentally using a team of micro autonomous surface vehicles in an actual fluid environment.

Appendix: Estimating the region of attraction

In this section, we describe the methodology used to estimate the region of attraction (RoA) of the boundary. This estimate for the RoA is used in Sect. 3.2 to set the desired group behavior of the sensing grid.

Fig. 12

a The red line shows the line segment perpendicular to the boundary through $P_c$. The length of the line is $2\varsigma$, b flow speed directed towards the boundary on the line segment. Two maximums on either side of the boundary, c maximums flow speeds are on the edges of the line, i.e., the line segment is not long enough to capture the RoA (Color figure online)

Even if we had an analytical description for the flow field, obtaining a closed form expression for the RoA is extremely difficult. In the case of a discrete representation of the flow field (in the form of sparse velocity measurements), we have no other option but to estimate the RoA numerically. To do this, we consider the flow velocities of a set of points along a perpendicular line segment across the boundary through $P_c$ (see Fig. 12a). The length of the line segment (l) is selected to be $2\times \varsigma$, so that it extends a distance of $\varsigma$ on either side of the boundary (note that $\varsigma$ is the standard deviation of the sensing grid). Due to the attractive nature of the boundary, the flow speed components on this line, towards the boundary, typically has the distribution shown in Fig. 12b. The flow speed towards the boundary increases on either side of the boundary as we move away from it and then tapers off as we approach the edge of the attracting region. Thus, as shown in Fig. 12b, we approximate RoA as,

$$\begin{aligned} RoA = max\left( \Vert \mathbf {x_{m1}}-\mathbf {x_c}\Vert ,\Vert \mathbf {x_{m2}}-\mathbf {x_c}\Vert \right) \end{aligned}$$

where $\mathbf {x_{m1}}, \mathbf {x_{m2}}$ are the points on the line on either side of the boundary having the maximum normal flow. If the maximum normal flows are at either end of the line, that indicates that the RoA is larger than l / 2, i.e., $RoA > \varsigma$ (see Fig. 12c). Therefore, on such instances we set

$$\begin{aligned} RoA = \alpha \times \varsigma \end{aligned}$$

where $\alpha >1$.