# Which way should angles be measured given a coordinate frame, according to convention? And if going against convention what are the implications?s

Note: My application is not robotics, but purely theoretical. I have been suggested to ask this question on the robotics forum. The results of the project could easily be applied to robotics however and so I believe it is also suitable for this forum.

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.

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 competeness 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?