I've seen a lot of papers define z-axis to be upside down when they define the inertial frame for the Earth. Is there any sensible explanation for that. To me, the more convenient way is that the z-axis should point to the up direction (i.e. to the sky).
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2 Answers
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I don't know what papers you are referring to, but in the normal definition of the Earth-Centered Inertial reference frame, the Z-axis is defined by the Earth's spin, and therefore points up through the North Pole. Unless you are standing on the North or South Pole, the Z-axis won't be vertical either up or down.
There are several other coordinate systems that may be relevant. Two common ones used near the Earth's surface are East-North-Up (ENU) and North-East-Down (NED). In the NED system, the Z-axis does indeed point down toward the Earth's center of mass. But ENU and NED are not generally considered inertial frames except over fairly short time periods.
It's also possibly you are referring to the coordinate system applied to an aircraft or ship. In this case it is common to define the X-axis out the nose, and the Y-axis out the right wing (or starboard side). In this case the Z-axis points down to complete a right-handed coordinate system.
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$\begingroup$ In this paper quad08topgun.groups.et.byu.net/documents/… they use NED, a typical representation in many papers for modelling quadrotor. $\endgroup$– CroCoCommented Feb 23, 2015 at 16:15
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$\begingroup$ Based on the paper, they are definitely using North as X so this is the right answer. No idea why, that just seems kinda funky, but to each his/her own! $\endgroup$– tpg2114Commented Feb 23, 2015 at 16:18
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$\begingroup$ A guess is that the z-axis is pointed downwards in order to eliminate the minus sign from equations that contain the gravitational acceleration. Also, the paper states (page 6) that it assumes a flat non-rotating earth as convenient approximation, therefore the earth's rotation and the direction of it probably play no part in choosing the axes. $\endgroup$– DK2AXCommented Feb 23, 2015 at 18:12
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$\begingroup$ @CroCro--Okay, I've read the paper. They use the NED coordinate system as the fixed "world" system, and the aircraft coord system I described in my last paragraph. When the aircraft's roll, pitch and yaw are all zero, the two coordinate systems align. Theirs is a common convention, arrived at by requiring a right-handed coordinate system with right-handed rotations. To be consistent with pilot lingo, "pitch" is defined as increasing nose-up, "roll" positive with right wing down, and "yaw" positive with nose to the right. To meet all those criteria, you are forced to define Z as down. $\endgroup$ Commented Feb 24, 2015 at 16:44
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If you are looking at the Earth's rotation from the outside, the direction of rotation would suggest a z-axis that points upwards to make a right-handed coordinate frame.
A possible explanation for the reversed z-axis could be, that when you are stationary on Earth, you see the sky move in the opposite direction; forming a right-handed coordinate frame again, the z-axis would be downwards then. Can you link to a paper you are referring to, to clarify what axis they are talking about?
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$\begingroup$ quad08topgun.groups.et.byu.net/documents/… $\endgroup$– CroCoCommented Feb 23, 2015 at 15:43