The question is:
"The wheel kinematic constraints derived in section 3.2.3 of the book assume that
the wheel does not skid (equation 3.12) or slide (equation 3.13). These assumptions are unrealistic in many situations, such as on the surface of Mars or on the DARPA Grand Challenge course.
Let Z be a constant whose value is determined by the type of surface on which a robot is moving. When Z = 0, skidding and sliding are impossible. When Z is large, such as when the robot is driving on loose gravel, large amounts of sliding and skidding occur. For intermediate values of Z, intermediate amounts of sliding and skidding occur.
How would you modify equations 3.12 and 3.13 to account for skidding and sliding? What factors other than Z come into play and how? Factors to consider are the slope of the surface on which the robot travels and it’s orientation with respect to the slope, the robot’s acceleration ( ¨') and jerk (the derivative of acceleration). Hint: Think about what happens when you floor the gas in a powerful car when sitting still.
Your answer should consist of new versions of equations 3.12 and 3.13 and an explanation of how they were derived. It should be the case that when Z = 0 your equations are equivalent to equations 3.12 and 3.13."
A complete explanation of the kinematics is here cfar.umd.edu/~fer/cmsc828/classes/cse390-05-03.pdf
α is the angle that the wheel is rotated from the global x-axis
β is the steering angle
θ is the angle that the robot is rotated with respect to the global plane
ζ'_I = [x' y' θ']τ (robot motion)
R(θ) is the orthogonal rotation matrix
r is the radius of the wheel
ρ' is the velocity of the wheel
rolling constraint 3.12: [sin(α+β) -cos(α+β) (-l)cosβ] R(θ) ζ'_I - rρ' = 0
sliding constraint 3.13: [cos(α+β) sin(α+β) (l)sinβ] R(θ) ζ'_I = 0
The Attempt at a Solution
I s = the slope of the plane and c = the orientation of the robot with respect to the slope, I only changed the sliding constraint as follows:
[cos(α+β) sin(α+β) (l)sinβ] R(θ) ζ'_I -Z(s*c)= 0
but this is a total guess. I know the acceleration and jerk need to be somewhere, and I think I have to add Z to the rolling constraint as well, I just don't really know where. Can anyone wrap their head around this and point me in the right direction?