Sign of the self-aligning moment

[moorepants]

This is a figure from the Pacejka handbook:

This is the sign convention I'm (trying) to use in the model here:

My sign convention defines the forces and moments acting on the tire with a positive sense that aligns with the wheel's respective "yaw" reference frame. For the rear wheel this is the A frame. So this means:

• -inf < Fz < 0, i.e. Fz is always negative when the wheel is in contact with the ground
• positive slip angle means that the velocity vector of the contact point moving in the ground plane has a positive $\hat{a}_y$ component, i.e. if $\alpha_r > 0$ then $^N\bar{v}^{n_d} \cdot \hat{a}_y > 0$.

For my convention if the slip angle is positive, the lateral force Fy should be negative. And comparing to Pacejka's diagram if the slip angle is positive the self-aligning moment Mz should be positive, which would tend to align the wheel with the velocity vector.

Using the model at 42a5f8096 (Steering torque = 0, kick = 100 N, Mz = 0.0, nonlinear.), these are the results:

This shows a negative Fry for a positive alpha_r. For the front there are some initial relaxation length dynamics, likely due to the q4 being not exact zero (have to set to 1e-14 to avoid a divide by zero).

Now with [master be85bce] Steer torque = 0, Kick = 0, nonlinear Fy and Mz active with lateral force opposing slip direction and Mz aligning with slip direction. where I think the signs are correct you get:

The bicycle fails over to the right and the slip angle is always negative, giving a positive Fz and negative Mz. I would expect the bicycle's self stability to be retained.

Now if I [master 64cc016] Steer torque = 0, Kick = 0, nonlinear Fy and Mz active with lateral force opposing slip direction and Mz opposing the slip direction. oddly the bicycle doesn't fall over as quickly.

Now with [master b058eed] Steer torque = 0, Kick = 0, nonlinear Fy and Mz active with lateral force opposing slip direction and Mz = 0. It seems the lateral force isn't negative of the slip angle for the rear.

[moorepants]

@GabrieleDello have a look at this. I'll add more as I investigate.

[moorepants]

I added a tire force/moment plotter in [master 34a1a6f] Back to Mz = 0 and added plotter for the tire force curves.

[moorepants]

What range of slip angles is Pajecka's model valid?

[moorepants]

Making the duration of the lateral force pulse very small allows 1000N to be reached:

[moorepants]

Pacejka's book has this sign convention diagram in the appendices:

I'm using the SAE convention (other than my Fz is still opposite, plan to fix).

[moorepants]

Pajecka presents his linear equations in this form: which correspond to his sign convention: His stiffnesses in 1.5 are all positive quantities.

So, that means that the lateral force has the opposite direction of the lateral velocity and that the self aligning moment tends to rotate the wheel in the same direction as the slip angle. A positive camber angle (roll to the right) will cause a lateral force to the left and a self-aligning moment that tends to rotate the wheel in the opposite direction of the slip angle.

So, for the SAE sign convention that I use in this model where Fz is always negative, the linear equations take this form:

$$ Fy = (C\alpha \alpha + C\phi \phi) F_z $$

$$ Mz = (-C{M\alpha} \alpha + C{M\phi} \phi) F_z $$

which is the same as Pajecka's except for the sign change due to $F_z < 0$.

This produces this graph for our model:

[moorepants]

Note that correcting the sign of the self aligning moment causes our steer controller used in the icsc abstract to no longer stabilize the bike after the perturbation ($C_{M\phi} = 0$ as we originally had in the abstract):

[moorepants]

@GabrieleDello I believe I have all the signs correct now. I'm closing this. Let me know if you agree/disagree.