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Click (and drag) on the screen to place robots (red dots).
Click the Start rendezvous button to watch them converge to a single point.
To see a quick example of what it does, choose a preset scenario and click start.
More details in the description and features sections.


Red circles: Robots
Cyan circles: Dummy robots (only for MM curvature and when Merge close robots is selected, see below)
Bright Green square: Center of mass of robots at start
Dark Green square: Current center of mass of robots
Blue square: Center of mass of polygon (giving uniform density to edges)
Big Grey circle: Circumcircle of all robots (only for circumcenter scheme)
Grey dot: Circumcircle center (only for circumcenter scheme)
Grey paths: History of polygon over time
Graph in bottom: Area of the polygon over time

This applet simulates the robot rendezvous problem using three different schemes:
1. A circumcenter algorithm, in which all robots simple move to the center of the smallest circumcircle
containing all robots. (Least interesting to watch/simulate.)
2. A curve-shortening scheme analogous to Euclidean curve-shortening but for discrete points, using the
Menger-Melnikov curvature to evolve the polygon. 3. A linear curve-shortening scheme in which a robot moves toward the average position of its two neighbors.

3 integrator types for the linear and Menger-Melnikov curvature schemes: forward Euler, implicit, and Runge-Kutta (4th order).
The center of mass of the polygon is kept track of over time (one in terms of point masses, the other in terms of edges), as the Key describes.
The area of the polygon is recorded each run in the window at the bottom. Try the preset scenario Area increase
to see an example of a polygon of robots in which the area increases initially.
Two other interesting preset scenarios are included, Spiral and Snake.

Random formations of robots can be generated using the Generate n random robots button.
Availabe formations include:
1. A star formation, in which the angle of the ray from the center of mass to the robot increases with each subsequent robot in the polygon.
2. A convex polygon.
3. Completely random nonsimple polygon.
You can also convert a given formation to a star ordering with the Convert to star ordering button.

Checkboxes are available for recording the shape of the polygon every so often with a grey outline,
tracing the paths of each individual robot, and merging robots that get very close to each other in the MM curvature scheme.
(If you don't merge robots, this scheme starts to behave badly when robots get too close.)
Merged robots turn cyan, and their paths turn pink.

The mouse click action can be changed according to the radio buttons, and when dragging to create robots
a slider changes how frequently robots are placed.
The timestep size and number of steps run each time the screen is updated can also be changed with sliders,
and their current values are also displayed.
A timer keeps track of how much time passes in each run.

The four buttons at the top control the action of the simulation. Resetting goes back to how the formation was at time zero,
and removing all robots lets you start blank again.

This applet was made for a final project for the class Advanced Dynamics, MAE 542, at Princeton University.
The paper on which most of these methods are described can be found at

If you have any questions or comments, you can reach me at cfbrasz at gmail dot com
Last updated: 1/19/2010