ISO8855 toe and camber angle definition into rotation matrix form

[wlamers]

Dear forum members,

ISO8855 defines both the static toe angle and the camber angle as a rotation around one of the axis's of the vehicle coordinate system.

I would like to define a rotation matrix in which both angles are correctly assigned. So my starting point is having two scalar angle values, one for the toe angle and one for the camber.

The toe angle is defined as the angle between the wheel plane and the X_v axis about the Z_v axis. And the camber angle (epsilon_v) as the angle between the wheel plane and the Z_v axis about the X_v axis. The denominator 'v' stands for the vehicle axis system.

The problem is when you use a rotation matrix formulation that the second (and third) rotation is defined as a rotation in the coordinate system that results from the first rotation. And looking at the ISO8855 definition, you need to perform both rotations in the vehicle axes system. Which is not possible with an Euler angles/rotation matrix formulation directly. But I am sure one of you knows a simple solution to make such a formulation.

I am aware that the error you make is relatively small if you just do the second rotation in the second coordinate system (if you make the assumption that the angles are small), but I would to do it exactly.

 

[WolfHR]

I think the rotation problem could be used by using transformation tensor... But I'm not quite sure what exactly do you need this for- obtaining the wheel 'position' from given camber and toe, or obtaining camber and toe from wheel 'position'.

I've made a 3D model of SLA in which I calculated both toe and camber (along with many other parameters- scrub radius, mechanical trail, KPI, caster, &c), and assuming one knows the vector normal to the ground and wheel axis vector it can be pretty easily done... (if you need the process to go the other way around, I might also be of some help).

Incidentally, if I may ask- this camber definition IMHO should be applicable only with no steering input, or am I wrong? (it seems more logical that camber should be 'measured' when viewing the tyre from the front, not the vehicle)

 

[wlamers]

I need it the other way around indeed. I want to define a coordinate system in a multi-body system. I could of course simply use an euler axis representation, since the error I would back is small. In fact if have done it that way many times, but now I wasn't to do it exactly. It should not be that of a problem I think.

The definition is indeed such that no steer angle is applied. There is another definition, the inclination angle epsilon_w, which is the angle with respect to the road and between the wheel plane. But that's not the definition I need.

Calculating the angles from the simulation results is fairly simple, since you do simply two successive calculations using the instantanious velocity vectors.

Can you give me a hint how to use the transformation tensor?

 

[WolfHR]

You'll have to give me a bit of time to work out the model for it (hopefully I'll do it tonight). As for transformation tensors- they're used to transform coordinates and vectors from one coordinate system to another, and back. The transformations are quite simple (shown in the enclosed picture- first one is tensor notation, the other is more conventional array representation- v is a vector in old coord. system, v´ in new anf T is transformation tensor- or 3x3 matrix). There are two ways of looking at the transformation tensor- one is that Tij represents the angle of i axis of new coord. system and j axis of old one... The other is that new coord. system is defined by it's axis x, y and z, and then transformation tensor is made in the way depicted at the bottom of the picture.

I used my own coord. system, so I better check few things out- vehicle coord. system in SAE is x to front, z downwards and y defined by those two (to port, speaking off top of my head), and you want coordinate system where x points where the tyre points, z downwards in the wheel plane and y defined by those two)? Toe-in is negative in SAE?

 

[WolfHR]

I think this should be it, at least for the port wheel (starboard wheel should have different sign of the camber). *Ass*umptions I've made are that the ground is paralel to the xy plane of original coord. system and in the car coord. system axes are- x forward, y to starboard, z downwards.

Buuut, please have someone check this, etc, etc. The disclaimer is there for a reason- I'm prone to brainfades. The calculations were made in Mathcad (exported to pdf), and I've included 'unneccessary' equations describing how I got there- but the only relevant equations are those with := sign (others have equal sign in bold). Basically, you have 8 equations to calculate the transformation tensor.

Hope it helps.

 

[wlamers]

Thank you for your detailed explanation and derivation. I appreciate it. And sorry for the grammar mistakes in my earlier post.

The problem is that your derivation is equal to the Euler axis representation or rotation matrix form. Which is exactly the thing I used to do before, but then with a conventional (and in my opinion easier) rotation matrix derivation. I have included it in the attachment. You can have a look at it and use it yourself in the future if you like. I have noticed that element R13 and R31 have changed position. I did not look into the exact reason for that.

I will try to explain my problem a bit more. First by explaining the problem with the rotation matrix derivation. The problem is that you do a successive rotation of (in this case only) two angles. First you perform a rotation about the z-axis on the initial coordinate system (let’s call it CS1). This rotation is obviously the steer/toe angle. Then you end up with a new coordinate system (CS2). In this new CS2 you perform the second rotation about its x-axis (the x-axis of the CS2 and not the CS1). And end up with CS3, which has no third rotation needed. This is quite straight forward and maybe even intuitive, but conflicts with the ISO8855 definition, which states that both the toe angle and camber angle have to be rotated about the z and x-axis of the first (CS1) coordinate system respectively. To do so I am looking for a way to rotate the second coordinate system (CS2) around the x-axis of the first coordinate system (CS1).

I hope this explanation makes it a little bit more clear. I agree also that the ISO definition is a bit counterintuitive and also difficult to measure in reality on a vehicle.

Have you, or someone else, a clue how to achieve such a rotation?

 

[WolfHR]

You want to rotate the whole coord. system about an arbitrary axis (in this case x axis of the car coord. system)? No problem there, you'll just have to rotate each of the coord. system's axis using the below formula (but be sure to use vectors in the same coord. system).

 

[WolfHR]

And I forgot to mention that a must be a unit vector! |a|=1 Sorry for that...

 

[wlamers]

This does the job! I even found a way on Wolfram mathworld to make a rotation matrix representation of it using the Euler parameters:

http://mathworld.wolfram.com/RotationFormula.html

Thanks again WolfHR!

 

[WolfHR]

No prob, glad I could be of some help...

Interestingly, Wolfram site has slightly different formula (cosine multiplier)- v-(a*v)*a instead of (nxa)xn . I used this formula as a basis for my 3D suspension model in Mathcad (and flash)- and even though modeling the geometry was pretty straightforward I got into my thick skull I should also calculate kinematic and dynamic parameters (velocities and accelerations), and I can tell you that 2nd derivation of that formula is a b*tch, especially when applied twice. I'm somewhat relieved to see that this representation would've made for even more complicated formulae.