# Circle-Tracking Atomic Motions

A Circle-Tracking Atomic Motion is an algorithm that tracks an oriented circle given with its center (*x _{c}, y_{c}*) and radius

*r*. If

*r*> 0, the circle is regarded as counterclockwise (CCW); if

*r*< 0, it is regarded as clockwise (CW). This motion adopts a feedback control algorithm with critical damping response; the vehicle exponentially converges to the circle.

There is another input parameter,

*smoothness, σ*. [6] Responsiveness of Tracking-Type Atomic Motions (on

*Characterizing Atomic Motions*page) discusses the role of the smoothness

*σ*.

The algorithms for line-tracking and circle-tracking Atomic Motions are somewhat similar because if the radius *r* of a circle becomes larger and larger, the circle converges to a line. The following seven motions showcase the capacity and programmability of Circle-Tracking Atomic Motions:

### [3.1] *Circle Medley* by *Science Robot*

This medley motion consists of a series of *Circle-Train* motions with 2 ≤ *n* ≤ 10. For each *n*, *Math Mind* tracks *n* circles aligned on a line, slightly detached from each other.

The orientations (CCW or CW) of adjacent circles are opposite.

This motion consists of 2*n*-1 circle tracking and 2*n*-2 switchings.

### [3.2] *Circle Train* with *n = 3* by the *Swan* robot

The *Swan* robot tracks three circles aligned on a line, slightly detached from each other.

*Science Robot* and the *Swan* robot are executing identical motions in [3.1] and [3.2],

demonstrating that Atomic Motions are hardware independent.

### [3.3] *Necklace* by *Science Robot*

*Math Mind* sequentially tracks 17 small circles, aligned on a big circle

and slightly detached from each other.

*Math Mind* visits each small circle

twice before it finishes the round trip;

notice that the number 17 is odd.

### [3.4] *Necklace (2)* by *Science Robot*

This motion is similar to the previous [3.3], except that the radii of small circles vary.

These variable radii make the movement

more complicated and more artistic.

This motion demonstrates the flexibility and programmability of Atomic Motions.

### [3.5] *Satellite* by the *Swan* robot

In this “Satellite” motion, the *Swan* robot applies Circle-Tracking Atomic Motions to create circular motions detecting the position of the nearest object (person) as the circle center.

When the robot detects another person

while tracking,

*Swan* switches the target circle

with an opposite turning orientation.

### [3.6] *Bubbles* by *Science Robot*

*Math Mind* initially tracks a CCW circle with a given constant radius.

A mouse click provides its new circle center. Then, the circle center continuously moves from the former center to the new one.

A user can change** **the turning orientation, from CCW to CW or from CW to CCW by clicking the *Switch Orientation* button.

### [3.7] Circular-Motion Creation through Mouse Clicking by *Science Robot*

A series of mouse clicks creates a path,

which *Math Mind* tracks.

A path segment to be tracked is *a circular arc*

determined by the latest three points.

If the path conflicts with an obstacle,

*Math Mind* automatically avoids it.

*Math Mind* reports a *driving-skill index*

at the end.

We discuss “*Driving Skills*” by Science Robot in Path Complexity and Driving Skills.

There we deal with the same motion from a different perspective.