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Precise steady turning conditions

by Dale Lukas Peterson — last modified Apr 28, 2011 07:50 PM

Below are some precise numerical values for stable steady turns to the right with steer torque to the right.  Lean and steer are defined analogous to the benchmark coordinates.  The pitch I report below is defined as the pitch coordinate in the benchmark paper plus the head angle (lambda), which is why the pitch values are non-zero, but are very close to \(\pi/10\) (after all these turns are near the origin, where pitch changes very little as a function of lean and steer).  Steer torque, front wheel rate and velocity are also reported.  

The next four numbers give an indication of how well the following four constraints are satisfied:

  • Front wheel height (configuration constraint on lean, pitch and steer), should be zero
  • NH[i] for i=1,2,3 are the three components of the velocity of the front wheel contact point.  For no slip rolling these should all be identically zero. NH[1] and NH[2] represent the no-slip constraints, NH[3] is effectively the differentiated holonomic constraint.  All three of these should be zero if the rolling constraints for both wheels are to be satisfied.
 
Configuration:
Lean \(\phi\) = 0.08726646259971647
Pitch \(\theta\) = 0.3139899180509261
Steer \(\delta\) = 0.0249931148885184
Constraints (should be zero): 
FW Contact Height = 1.994931997373328e-17
NH[0] = 8.881784197001252e-16
NH[1] = -3.122502256758253e-17
NH[2] = 2.220446049250313e-16
Steer torque = 0.002386180613544564
Front wheel rate = -17.27895470227879
Velocity = 6.047634145797575
Eigenvalues:
lambda_0 = -0.0008960513685150272 + 0j
lambda_1 = -1.565399502765081 + 5.940516021028886j
lambda_2 = -1.565399502765081 - 5.940516021028886j
lambda_3 = -16.11615742578447 + 0j
 
Configuration:
Lean  = 0.08726646259971647
Pitch = 0.3139889374771921
Steer = 0.025132741228678
Constraints (should be zero): 
FW Contact Height = 1.908195823574488e-17
NH[0] = 0
NH[1] = 7.979727989493313e-17
NH[2] = 2.220446049250313e-16
Steer torque = 0.001432842907858944
Front wheel rate = -17.23146391878335
Velocity = 6.031012371574174
Eigenvalues:
lambda_0 = -0.003439017988870276 + 0j
lambda_1 = -1.554891032061446 + 5.918052698264985j
lambda_2 = -1.554891032061446 - 5.918052698264985j
lambda_3 = -16.08145923521178 + 0j
 
Configuration:
Lean  = 0.08726646259971647
Pitch = 0.3139879565265826
Steer = 0.0252723675688376
Constraints (should be zero): 
FW Contact Height = 7.719519468096792e-17
NH[0] = -8.881784197001252e-16
NH[1] = -5.204170427930421e-17
NH[2] = -4.440892098500626e-16
Steer torque = 0.0004796830975322131
Front wheel rate = -17.18436903579476
Velocity = 6.014529162528164
Eigenvalues:
lambda_0 = -0.006003419792954939 + 0j
lambda_1 = -1.544442434016426 + 5.895751882573691j
lambda_2 = -1.544442434016426 - 5.895751882573691j
lambda_3 = -16.0470623299768 + 0j
 
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