ackerman II (calculation from steering geometry)

I had a big elaborate way to calculate ackerman using intersecting circles and a bunch of algebra that I programmed into excel.  Then today I saw the diagram a little differently and came up with a much simpler way to calculate this.  I spent 30 minutes at lunch while eating to replace about 4 hours of previous effort, and it works much better now.  Not sure whether that makes me happy or sad.

The goal here is to be able to find the steering angle as a function of the steering system geometry, and rack movement.  Then if I steer the rack say half an inch +y, I can calculate the wheel angle on each side of the car.  If I know both those angles and I assume the outside angle steers the car, I can figure out how much anti or pro ackerman I have.

Defining variables:

Here is a bird's eye diagram of a rear steer system on the driver's side to help that make sense:

We will make expressions from known dimensions to describe the position of:
The lower ball joint (point A)
The intersection of the hub face and perpendicular line to the tie rod axis (point B)
the the tie rod axis of rotation (point C)
the ball joint on the end of the rack (point D)
Note that we can calculate Dy0 which is the rack y position for alpha = 0.  Dx is mu for all rack positions.  We can then say that Dy = Dy0 + rack position we choose.

Note the shown coordinate system, where:

K1 and K2 describe the hub face and the steering arm offset (for ackerman)

R1 = the tie rod length

R2 = the fixed length from the point of rotation of the knuckle (A)to the point of rotation on the steering arm (C)

R3 = the distance from the point of rotation (A) on the knuckle to the end of the rack (D)

mu = how far back the rack is set

Finding the angles

delta = Asin(mu/R3)

law of cosines to get gamma:

gamma = Acos[(R1^2 - R2^2 - R3^2) /(-2*R2*R3)]

beta = atan(k2/k1)

Finally:

alpha = 90deg - delta - gamma - beta

Once we have these angles it is simple trig to show where each A,B,C,D are which will help us graphically picture what is going on as we will see in the next post.

Calculating Ackerman

I find lots of things talking about ackerman, but I haven't found much describing how to measure it so I had to make up a way:

We can assume that the outside wheel does the steering.  Since we know the wheelbase (B), and just found the angle the right wheel is at, we can find the distance (x).  We can do the same for the left wheel.  Then we can take (Rx-Lx)/Rx to express ackerman as a percentage.  If Lx is further out than Rx (as shown) we have anti ackerman. If it is further in board then it is pro-ackerman.

Next time I will show all this in excel as well as the ackerman dependance on a few of these variables.