1 Motion Planning for Small Formations of Autonomous Vehicles Navigating on Gradient Fields Shahab Kalantar, Uwe Zimmer Australian National University Research School for Information Science and Engineering and the Faculty for Engineering and Information Technology Autonomous Underwater Robotics Research Group Canberra, ACT 0200, Australia shahab.kalantar@rsise.anu.edu.au | uwe.zimmer@ieee.org Abstract – In this paper, we present a motion planning scheme for navigation of a contour-like formation of autonomous un- derwater vehicles on gradient fields and subsequent conver- gence to desired isoclines, inspired by evolution of closed planar curves. The basic evolution behaviour is modified to in- clude moving boundary points and incorporate safety con- straints on formation parameters. Also, the whole process is decomposed into a sequence of well-behaving states. As op- posed to the basic model, the regularized solution is character- ized by the maximum allowable curvature rather than balance of forces determined by fixed coefficients. Nevertheless, the proposed framework subsumes the original model. Blocking states and fairness are briefly discussed. I. INTRODUCTION The need to deploy large numbers of autonomous vehicles to safely monitor underwater phenomena, which are usually of considerable spatial extent, is currently a driving impetus for many research efforts. These monitoring tasks include, among others, characterizing diffusion processes (e.g., tem- perature and salinity [2]) through the evolution of their iso- clines, monitoring flows of one kind or another over iso- baths, identification of iso-tachs of flow fields, and delinea- tion of boundaries of plumes or biological concentrations. Due to inevitable uncertainties associated with measure- ments, any kind of imposed structure on the shape of vehicle formations can be of great help. Generally speaking, suita- ble formations will have to be deformable rather than rigid. They should have the capability to lose potential energy to get into the right shape and gain potential energy under the influence of ambient field gradients. Candidate formations are either dense networks (for area coverage [6],[7]) those resembling chains (for isocline tracking [5],[1],[8]) or both [3] (swarms with boundary). We will consider curved for- mations with open ends. The proposed strategy will ulti- mately be implemented on Serafina robots developed in our lab (figure 1). We will only consider the planar case where the robots are stabilized to navigate on an imaginary plane. In the following, we will precisely formulate the problem. Definition 1: A field is a continuously differentiable map- ping : , where denotes the real domain, is the domain of definition of , and is some time domain. A static field is time-invariant in the period of time , during which the field can be defined as : (we have extended to the whole plane for simplicity). The gra- dient of at is denoted by and defined as the vector , expressed in some inertial coordinate frame . Definition 2: For any fixed value , such that , a -level set (or isocline) of is defined by the equation . A level set is called a level curve if it is a single closed, simple and smooth ( ) curve : . We simplify the notation to . is parametrized by , where is the parametrization domain (such as ). For each such curve , at each point represented by , a Fer- renet-Serret frame can be defined, where is the unit tangent vector and the unit inward normal. As the only informa- tion available to the robots is the value and local gradient of the field, the measure of closeness to an iso-cline is defined over , for , rather than Euclidian metric , denoting the closest point on to . The -neigh- borhood of a curve is denoted and is defined as Definition 3: A planar formation is defined as a tuple , where denotes a collection of F F D T ⊗ ℜ → ℜ D ℜ 2 ⊂ F T T F ℜ 2 ℜ → Yaw Pitch Heave Plane of motion (x-y-plane) Yaw Heave Roll Surge z z y y x x Roll Surge Pitch Sway Sway Yaw Heave z y Pitch Sway T fl v T fl v T fl v T fl v T fr v T fr v T fr v T b v T b v T b l h T b l h T b r h T b r h T b r h ϒ ϒ ϒ ϒ figure 1: Serafina, the plane of motion on which the robots behave like non-holonomic devices, as sway is not actuated. Refer to [12] for more information. D F q ℜ 2 ∈ Fq () q ∇ ∂ ∂ x ⁄ Fq () ( ) ∂ ∂ x ⁄ Fq () ( ) , [ ] T ϒ xy , { } = F d inf q ℜ 2 ∈ Fq () F d sup q ℜ 2 ∈ Fq () ≤ ≤ F d F Fq () F d = C 2 γ F d F ℘ ℜ 2 → γ F d γ F d s ℘ ∈ ℘ ℜ ⊂ 01 , [ ] γ s ϒ γ s () N γ s () T γ s () , { } = T γ s () γ ˙ s () γ ˙ s () ⁄ = N γ s () T γ s () ⊥ = d 1 q γ F d , ( ) Fq () F d – = Im F ( ) FD ( ) = q ℜ 2 ∈ d 2 q γ F d , ( ) = q Qq γ F d , ( ) – Qq γ F d , ( ) γ F d q ε γ B ε γ () B ε γ () q ℜ 2 ∈ d 2 q γ F d , ( ) ε ≤ { } = Rt () R i { } ft () q R t () , , 〈 〉 = R i { }