Robotics kinematics help, rotation, velocity, acceleration, jerk

[ravenfurie]

Homework Statement

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."

Homework Equations

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?

 

[KataKoniK]

Thank you for bringing up this important point about the wheel kinematic constraints. You are correct in recognizing that the assumptions made in equations 3.12 and 3.13 may not hold in all situations. To account for skidding and sliding, we can modify these equations as follows:

Rolling constraint with skidding and sliding: [sin(α+β) -cos(α+β) (-l)cosβ] R(θ) ζ'_I - rρ' - Z(s*cos(θ-α)) = 0

Sliding constraint with skidding and sliding: [cos(α+β) sin(α+β) (l)sinβ] R(θ) ζ'_I - Z(s*sin(θ-α)) = 0

Here, Z represents the constant that takes into account the type of surface on which the robot is moving. As you mentioned, when Z = 0, skidding and sliding are not possible. When Z is large, we can expect to see more skidding and sliding.

In addition to Z, there are other factors that come into play when considering skidding and sliding. These include the slope of the surface on which the robot is traveling and its orientation with respect to the slope. This is taken into account in the modified equations by including the term Z(s*cos(θ-α)) in the rolling constraint and Z(s*sin(θ-α)) in the sliding constraint. This term represents the force that is exerted on the wheel due to the slope and orientation of the robot on the surface. The robot's acceleration and jerk also play a role in determining the amount of skidding and sliding that may occur. This is why we see the terms for acceleration (ρ') and jerk (ζ'_I) still present in the modified equations.

To derive these modified equations, we can start with the original equations 3.12 and 3.13 and add in the terms for skidding and sliding. We can then use the fact that when Z = 0, these equations should be equivalent to the original equations, and solve for the additional terms. This will give us the modified equations as shown above.

I hope this helps in understanding how to account for skidding and sliding in the wheel kinematic constraints. Please let me know if you have any further questions or if you need clarification on any part of this explanation.


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