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colors list The result can be treated as true/false (false = 0, true = otherwise) or you can use the number returned if it is not zero as the distance to the color speci ed *rFeel() *rDFeel({ExprN}) Returns a number 0 to 31 according to the following logic: There are ve infrared sensors around the robot At 90 and 45 to the left and right of the robot and directly in the front The sensors are able to feel any object within a robot s radius ahead of the sensor (ignoring invisible colors) The number returned is a bitmap of the condition of the sensors If any of the sensors feels something then its bit position is set to 1, otherwise 0 So there are 5 bits MSB..LSB 00000.

We also have the useful double-angle formulas: Theorem 26.8: cos 2 = cos2 sin2 = 2 cos2 1 = 1 2 sin2 and sin 2 = 2 sin cos . For cos 2 , note that: (i) cos 2 = cos ( + ) = cos cos sin sin (ii) = cos2 sin2 = cos2 (1 cos2 ) = 2 cos2 1

The true anomaly is a function of time, and determining it is one of the more difficult steps in the calculations. The usual approach to determining proceeds in two stages. First, the mean anomaly M at time t is found. This is a simple calculation: M n(t Tp) (2.24)

The most signi cant bit (MSB) is the sensor 90 to the left, then 45 then 0 then 45 to the right, and the least signi cant (LSB) bit is 90 to the right So if the number is 01110 (14) then the sensors in front and 45 right and left are feeling objects and the rest do not Use bitwise operators (bAND, bOR, etc) to manipulate the result Colors on the invisible colors list are not considered objects The difference between rFeel() and rDFeel({ExprN}) is that rDFeel({ExprN}) will draw radials out from the sensors to show the range of feel of the sensors using the color speci ed in ExprN If you do not specify ExprN then the second color on the invisible colors list will be used This is a useful feature for debugging the rFeel() results.

Here, n is the mean motion, as previously defined in Eq. (2.8), and Tp is the time of perigee passage. The time of perigee passage Tp can be eliminated from Eq. (2.24) if one is working from the elements specified by NASA. For the NASA elements, M0 Therefore, Tp t0 M0 n (2.25) n(t0 Tp)

[by Theorem 26.6(a)] [by Theorem 26.1] [by (ii) and Theorem 26.1]

Substituting this in Eq. (2.24) gives M M0 n(t t0 ) (2.26)

Do not use rDFeel({ExprN}) unless you need to debug since it is signi cantly slower than rFeel() You must specify ExprN or have a valid color (not a oor color) in the second position in the invisible colors list The radial lines will be drawn using this ExprN color or the second color on the invisible colors list other wise the beams will not be visible *rBumper() *rDBumper({ExprN}) Returns a number 0 to 15 according to the following logic: There are four bumpers around the robot At the front covering an arc of 130 that is, from 65 to the left to 65 to the right There is also a bumper at the back just like the one in front, that is, from 115 to 245 .

Consistent units must be used throughout. For example, with n in degrees/day, time (t t0) must be in days and M0 in degrees, and M will then be in degrees.

To the left and to the right there are bumpers that cover an arc of 50 , that is, from 65 to 115 on the right and from 65 to 115 on the left The bumpers will close (turn on) if there is any object within 2 pixels of the robot s perimeter (ignores invisible colors) The number returned is a bit map of the condition of the bumpers If any of the bumpers is closed then its bit position is set to 1, otherwise 0 So there 4 bits MSB..LSB 0000 The least signi cant bit is the back bumper, then the right, front and then the left one If the number is 1110 (14) then the right, front and left bumpers are closed while the rear one is not Use bitwise operators (bAND, bOR, etc) to manipulate the result oor/line/beacon colors are ignored.