Navigation Functions with Time-varying Destination Manifolds in Star-worlds

Abstract

This paper formally constructs navigation functions with time-varying destinations on star worlds. The construction is based on appropriate diffeomorphic transformations and extends an earlier sphere-world formulation. A new obstacle modeling method is also introduced, reducing analytical complexity, and offering unified expressions of common classes of n-dimensional obstacles. The method allows for dynamic target tracking, and is validated through simulations and experiments.

Keywords: Navigation functions; dynamic environments; moving goal; target tracking.

Fig. 1:

An example of a potential field generated by a navigation function in a simple rectangular environment: (a) contour plot, and (b) three dimensional rendering.

Fig. 2:

How the purging transformation works. Intersecting star shapes form a parent-child hierarchy, and then the inverse of the transformation shown above draws the interior of the child within its parent, and maps the boundary of the child to the portion of the boundary of the parent which is in the overlap of the two shapes.

Fig. 3:

(a) an example of a star shape; all points on the boundary are “visible” from an interior point called the center, i.e., the ray from center to boundary does not intersect the boundary anywhere else. (b) the (inverse of a) star-to-sphere transformation is a bijective mapping that relates the boundary of a star to that of a sphere.

Fig. 4:

The pediatric rehabilitation clinical study environment.

Fig. 5:

(a) Environment layout for the first simulation study. (b) Environment layout for the second simulation study (left), and a zoomed view of the circled area (right).

Fig. 6:

Time-varying navigation functions of the first example: the resulting navigation function (left), the modeling star world navigation function after applying a purging transformation (middle), and the modeling sphere world navigation function after applying a star-to-sphere transformation (right).

Fig. 7:

Time-varying navigation functions of the second example from the resulting navigation function in star forests to its the modeling sphere world navigation function.

Fig. 8:

Simulations for different target movement and initial robot configurations. The paths of target and robot are shown in the figures on the upper row, and the evolution of the artificial potential v and robot-target d is shown, for each case, in the bottom row.

Fig. 9:

Contour plots of the time-varying navigation function at different instances of simulation time, as the target moves around in the workspace.

Fig. 10:

Gazebo simulation of a quadrotor (top right) intercepting a moving ground target (bottom center). The time-varying navigation function provides motion directions to the aerial vehicle.

Fig. 11:

Two experimental trials with a unicycle robot (larger robot) chasing a moving target (smaller robot).