Yes, I believe I made an error in my derivation. Let me try again from scratch...
I tried again on paper and took a photo
shows the leaning motorcycle.
shows the balance of forces which apply torque on the center of mass with focal point where the tire touches the road.
shows the vector directions used in the equations (although I realized now I forgot to draw the up
direction which points opposite gravity and is not necessarily the same as the road normal N, since the road might be banked or going up/downhill).
shows some triangle geometry used to derive the fact that line segment AB
equals line segment BC
. Think of these segments as legs of a right triangle, and their values can be obtained by multiplying the hypotenuse with the dot-product (which provides sin(angle)
) of the correct unit vectors that point parallel to the adjacent side.
In the end I get this equation:
w = ( g / v) ( D(x)F (.) u ) / ( D(x)F (.) F(x)N )
= angular velocity of turn
= acceleration of gravity
= forward speed
= dorsal unit vector
= forward unit vector
= up unit vector (opposite gravity)
= normal of road plane (not necessarily the same as u)
After all of that I realize I made a slight mistake for the most general case where the road normal is not parallel to gravity. The direction of the component of gravitational force that is relevant to the turn balance is not parallel to gravity, but is parallel to the road normal N
. In other words: in the derivation you should replace u
u' = (u (.) N) N
is not a unit vector but includes a sin(angle)
factor from the dot product: u(.)N
As to the direction of the angular velocity: We're talking about a changing yaw of the motorcycle as it drives in a circle, NOT the roll of the bike which provides the angle theta, used in the force-balance equation. In other words, the game player controls the roll (via a joystick, or button mashing) and the result is the bike yaws about a circle.
Edited: supplied correct link to google shared photo.