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I am building a 2D Electric unicycle model and have a somewhat trivial question concerning convention when defining a coordinate system. The EUC is constrained to move upon flat ground, and a figure is given below.

Model of the proposed 2D EUC system

I want to define my coordinate system such that the $x$ axis is horizontal and positive to the right, and I want this axis to represent movement of the wheel. Since the model is 2D the wheel can only rotate in a single plane which will be plane that has an axis in $x$. Finally I would also find it be aesthetic if a positive angular velocity would result in a positive increase in the wheels $x$ position.

These conditions can only be satisfied if the axis not within the plane of the model points into the page, due to the right hand rule convention positive angular velocity would result in clockwise rotation of the wheel and therefore a positive change in $x$. Also following the right hand rule convention for defining axis', I thought it appropriate to define the $z$ axis upwards, the $y$ axis into the page and the $x$ axis horizontally to the right.

It then seemed logical to me to be measuring the angles of the chasiss and rider (and any angle of the system in general) from the $z$ axis clockwise to the $x$ axis. I convinced myself of this by drawing parallels between $x$, $y$, $z$ and $i$, $j$, $k$ where in a cross product positive values are receieved from $i \cdot j$, $j \cdot k$ and $k \cdot i$ and therefore angles in $x - y$ plane should be measure from $x$ to $y$, angles in $y - z$ plane should be measure from $y$ to $z$ and angles in $x - z$ should be measure from $z$ to $x$.

This all made alot of sense in my mind, but my supervisor informed me that angles must always be measured anticlockwise but he could not explain to me why.

I know this is a padentic question, but I am quite a padentic person and so :

Is my system defined correctly according to convention, will I run into any issues from defining angles to be measured clockwise (for completeness assume that imaginary values must be applicable to the system too).

If the system is incorrect and angles must be measured anticlokwise all the time then why?

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  • $\begingroup$ Good question for here, but possibly even better at robotics, since they do a lot of the kinematic stuff - robotics.stackexchange.com $\endgroup$
    – Pete W
    Mar 8 at 16:32
  • $\begingroup$ Your advisor is wrong. When using vector representation and not some grade school algebra, the convention is to read angles as positive from x to y, from y to z, and from z to x. So if z, not y, is up, you go clockwise. $\endgroup$
    – Phil Sweet
    Mar 8 at 19:55
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    $\begingroup$ @PeteW Thanks Pete, I have done as you said since I am interested in the implications in terms of imaginary numbers... I know that the convention is the way it is due to how imaginary numbers interact with real numbers. $\endgroup$ Mar 9 at 18:00

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It's called the corkscrew rule. Imagine a conventional corkscrew running along the positive direction of the axis. That's a positive rotation around that axis. Alternatively grasp the axis with your right hand, with the thumb pointing positive. Your fingers now indicate a positive rotation. That's called the right hand rule.

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Your trigonometry course probably used 2$ \pi$ or 360 degrees, counter clockwise (or "anti-clockwise" depending on your usage) starting from the positive x axis on a 2-coordinate cartesian plane. So this is what most people will be accustomed to.

In practice, define it however you want for ease of use and calculations.

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  • $\begingroup$ Thanks, I guess I just wanted some clarity and confirmation :) $\endgroup$ Mar 9 at 17:47

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