paper: Analysis of “a torque-actuated module” used for strafing in an H-drive.

[RyanCahoon]

Is this assuming a value of θ that's fairly close to 90°? For small values of θ, it seems like assumption 1 for the B force no longer holds - the primary limitation on the normal force will come from the lever action of the module.

Here's my derivation for small values of θ:

• Rwheel - the radius of the wheel
• Rlever - the distance between the drive/pivot axle and the wheel axle
• θ - angle between the drive and wheel axle, relative to horizontal
• Rmoment - the distance between the drive axle and the contact point of the wheel on the ground
• ρ - the gear ratio between the drive axle and the wheel axle. (driven gear teeth)/(driving gear teeth)
• μ - coefficient of friction
• τ - motor torque about the drive axle
• N - normal component of the reactive force of the ground on the wheel
• F - frictional reactive force of the ground on the wheel
• Fmax - the maximum frictional force, μ*N using Coulomb friction

F = τ*ρ/Rwheel
Fmax = μ*N
N = τ/Rmoment*cos(θ)

Law of cosines:
Rmoment = sqrt(Rwheel^2+Rlever^2-2*Rwheel*Rlever*cos(90+θ))
= sqrt(Rwheel^2+Rlever^2+2*Rwheel*Rlever*sin(θ))

In order for the wheel not to slip, F <= Fmax

Thus, τ*ρ/Rwheel <= μ*τ/Rmoment*cos(θ)

ρ <= μ*Rwheel/Rmoment*cos(θ)
ρ <= μ*Rwheel*cos(θ)/sqrt(Rwheel^2+Rlever^2+2*Rwheel*Rlever*sin(θ))