This article introduces an approach to facilitate cooperative exploration and mapping of large-scale, near-ground, underground, or indoor spaces via a novel integration framework for locally-dense agent map data. The effort targets limited Size, Weight, and Power (SWaP) agents with an emphasis on limiting required communications and redundant processing. The approach uses a unique organization of batch optimization engines to enable a highly efficient two-tier optimization structure. Tier I consists of agents which create and potentially share local maplets (local maps, limited in size) which are generated using Simultaneous Localization and Mapping (SLAM) map-building software and then marginalized to a more compact parameterization. Maplets are generated in an overlapping manner and used to estimate the transform and uncertainty between those overlapping maplets, providing an accurate and compact odometry or delta-pose representation between maplet's local frames. The delta poses can be shared between agents, and in cases where maplets have salient features (for loop closures), the compact representation of the maplet can also be shared. The second optimization tier consists of a global optimizer that seeks to optimize those maplet-to-maplet transformations, including any loop closures identified. This can provide an accurate global "skeleton" of the traversed space without operating on the high density point cloud. This compact version of the map data allows for scalable, cooperative exploration with limited communication requirements where most of the individual maplets, or low fidelity renderings, are only shared if desired.

Teams of small autonomous UAVs can be used to map and explore unknown environments which are inaccessible to teams of human operators in humanitarian assistance and disaster relief efforts (HA/DR). In addition to HA/DR applications, teams of small autonomous UAVs can enhance Warfighter capabilities and provide operational stand-off for military operations such as cordon and search, counter-WMD, and other intelligence, surveillance, and reconnaissance (ISR) operations. This paper will present a hardware platform and software architecture to enable distributed teams of heterogeneous UAVs to navigate, explore, and coordinate their activities to accomplish a search task in a previously unknown environment.

Integrating data from multiple cooperative robots can be important for expanding their individual capabilities. In an environmental mapping scenario, multiple ground robots map different local areas. Algorithm complexity on merging the maps to build a global map depends on the three factors: orientation, accuracy and scale of the maps. When the three factors are all unknown, the map merging becomes a challenging problem. In this paper, a new approach on merging of two maps with the three factors are unknown. The idea is to estimate the best shared-areas by means of rectangular features. The information of dimensions and connections of maximal empty rectangles allows the algorithms to match orientations and scales, also to find overlapping points. The advantage of this approach is that a map merging is accomplished without any location estimations between the robots. This paper explains the map-merging process with an example of a simple environment, and presents a result with a practical environment.

The fundamental solutions of the Tschauner–Hempel equations, which describe the motion of a deputy satellite relative to a chief satellite with arbitrary eccentricity, are interpreted geometrically as generalizations of the drifting two-by-one ellipse that describes relative motion in circular orbits. General solutions are formed by taking linear combinations of these fundamental solutions. The amplitudes of these fundamental solutions are proposed as a parameterization of relative motion in elliptic orbits. A simple maneuver scheme is also developed to achieve arbitrary desired changes in the fundamental-solution amplitudes.

T HE assumptions made in deriving the Hill–Clohessy–Wiltshire (HCW) equations for spacecraft relative motion are a chief satellite in a circular orbit, a deputy satellite in close proximity to the chief, and both satellites obeying Keplerian motion [1,2]. Additional descriptions have been derived to relax these assumptions. Specifically, analytical solutions have been developed for relative motion in elliptic orbits [3–5]. Other studies have considered nonlinear relative motion, see, e.g., [6,7]. Additional work has considered relative motion in the presence of non-Keplerian perturbations [8,9]. However, the HCW equations remain a popular starting point to describe relative motion, and this motivates a desire to understand the significance of the assumptions involved. Themain contribution of this Note is pedagogical in understanding the circularity assumption, proximity assumption, and their interrelation. This Note presents an intuitive approach to approximately generalize the HCW equations to chiefs in elliptic orbits. The approach defines a virtual chief as a circularized version of the actual chief. The motion of both the chief and deputy relative to the virtual chief can then be described by the HCWequations without violating the circularity assumption. The approach is equivalent to propagating the relative motion in a frame with constant angular velocity, and then transforming the solution into the chief’s local-vertical localhorizontal (LVLH) frame. This transformation becomes aLyapunov– Floquet transformation of the HCW solution, and it results in significant improvement in accuracy compared to the original HCW solution. Throughout this Note, orbital parameters of the deputy and virtual chief are indicated with subscripts D and VC, respectively, and parameters related to the chief’s orbit are left without subscripts. II. Background

This paper proposes a method for continuous-thrust control of satellite formations in elliptic orbits. A previously calculated Lyapunov-Floquet transformation relates the linearized equations of relative motion for elliptic chief orbits to the Hill-Clohessy-Wiltshire equations describing circular chiefs. Using a control law based on Lyapunov-Floquet theory, a time-varying feedback gain is computed that drives a deputy satellite toward rendezvous with an elliptic chief. This control law stabilizes the relative motion across a wide range of chief eccentricities.

No AccessSpecial Issue in Honor of Richard BattinUse of Cartesian-Coordinate Calibration for Satellite Relative-Motion ControlAndrew J. Sinclair, Ryan E. Sherrill and T. Alan LovellAndrew J. SinclairAerospace Engineering Department, Auburn University, Auburn, Alabama 36849*Associate Professor, Aerospace Engineering Department, 211 Davis Hall.Search for more papers by this author, Ryan E. SherrillEngineering & Program Management Group, Engility Corporation, Eglin Air Force Base, Florida 32542†Real-Time Simulation Engineer, Engineering & Program Management Group.Search for more papers by this author and T. Alan LovellAir Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, New Mexico 87117‡Research Aerospace Engineer, Air Force Research Laboratory.Search for more papers by this authorPublished Online:10 Oct 2014https://doi.org/10.2514/1.G000683SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Battin R. H., “Lambert’s Problem Revisited,” AIAA Journal, Vol. 15, No. 5, 1977, pp. 707–713. doi:https://doi.org/10.2514/3.60680 AIAJAH 0001-1452 LinkGoogle Scholar[2] Battin R. H., “Space Guidance Evolution—A Personal Narrative,” Journal of Guidance, Control, and Dynamics, Vol. 5, No. 2, 1982, pp. 97–110; JGCDDT 0162-3192 Erratum Published in Vol. 5, No. 3, 1982, p. 320. doi:https://doi.org/10.2514/3.19761 LinkGoogle Scholar[3] Battin R. H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Ed., AIAA, Reston, VA, 1999, Introduction. 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The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.

This paper investigates impulsive maneuver planning for relative satellite motion about either elliptical or circular reference orbits. Problems to intercept an arbitrary final position or reconfigure to an arbitrary final position and velocity, while minimizing the magnitude of the velocity changes, are considered. By specifying the maneuvers to use one or two impulses, respectively, the problem can be reduced to an unconstrained optimization problem for the maneuver times, without requiring solution for the primer vector. Results show the significance of the phasing of the maneuver times relative to the final time.

The relative motion between chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be described by linearized time-varying equations of motion. The linear time-invariant Hill-Clohessy-Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Three transformations derived from Lyapunov-Floquet theory and an integral-preserving transformation have previously been determined which relate the time-varying equations of motion to the HCW equations. These transformations are used as a method for selecting initial conditions to approximate elliptic relative motion using the HCW equations with far greater accuracy than the true initial conditions.

This paper proposes a method for continuous-thrust control for satellite formation flying in elliptic orbits. A previously calculated Lyapunov-Floquet transformation relates the linearized equations of relative motion for arbitrary chief eccentricity to the HCW equations. Using a control law based on Lyapunov-Floquet theory, a time-varying feedback gain is computed that drives a deputy satellite toward rendezvous with an eccentric chief. The control law stabilizes the relative motion across a wide range of chief eccentricities. By contrast, an associated constant gain control law is unstable for high eccentricities and uses more control effort for near-circular orbits.

The motion of a deputy satellite relative to a chief satellite can be described with either Cartesian coordinates or orbital-element differences. For close proximity, both descriptions can be linearized. An underappreciated fact is that the linearized descriptions are equivalent: the linearized transformation between the two solves the linearized dynamics. This suggests a calibrated initial condition for linearized Cartesian propagation that is related to the orbital-element differences by the linearized transformation. This calibration greatly increases the domain of validity of the linearized approximation, and provides far greater accuracy in matching the nonlinear solution over a larger range of separations.

The Tschauner-Hempel equations model the motion of a deputy satellite relative to a chief satellite with arbitrary eccentricity. They are linear non-autonomous differential equations with the chiefs true anomaly as the independent variable. Since they first appeared, numerous analytical solutions have been presented. This paper provides a focused review of some of these solutions: highlighting how they are related and their singularities. The fundamental solutions of the Tschauner-Hempel equations can be interpreted geometrically as generalizations of the drifting two-by-one ellipse that describes relative motion in circular orbits. General solutions are formed by taking linear combinations of these fundamental solutions.

The relative motion between chief and deputy satellites in close proximity in orbits of arbitrary eccentricity can be described by linearized time-varying equations of motion. The linear time-invariant Hill-Clohessy-Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. However, a Lyapunov-Floquet transformation relates the linearized equations of relative motion to the Hill-Clohessy-Wiltshire equations through a periodic coordinate transformation for any elliptic orbit. This transformation is based on the invariant form of the Tschauner-Hempel equations, and evaluates the Hill-Clohessy-Wiltshire equations at a virtual time.

The motion of a deputy satellite relative to a chief satellite in elliptic orbit can be modeled with linearized time-varying equations. The solutions to these equations are linear combinations of six fundamental solutions, with the proportionality constants representing integrals of the motion. These constants provide a geometric description of the relative motion, showing how it is composed from the six fundamental modes. This paper analyzes maneuver planning for relative motion using the fundamental-solution description. Impulsive burns by the deputy satellite in the radial, transverse, and cross-track directions are related to the changes in amplitude of each fundamental solution. This approach allows for intuitive, geometric maneuver planning for a broad class of formation-flying missions.

The Hill-Clohessy-Wiltshire equations are often used for preliminary mission design to model the relative motion between two satellites. The virtual-chief method described in this paper is a modification to the classic Hill-Clohessy-Wiltshire equations for chief satellites of non-zero eccentricity. A kinetics-based method for determining the initial conditions is also developed. Results compare the error and relative-motion trajectory of the Hill-Clohessy-Wiltshire and the virtual-chief method to Kepler's two-body motion to demonstrate the validity of this method.