3/30/07 TreeRobotDesign

It appears that a coef. of friction of 2 is about the best we can get in short term. See bottom of page for how this affects things... -mrc June 1 2004

## Convex (or concave) surface analysis

Suppose the robot is climbing on a smooth, gently curved surface. We are looking down at the front of the robot and we see two front limbs in contact with the wall. Suppose we have a materials on the feet that provide little or no adhesion, but a large effective coefficient of friction.
The question is: If we apply lateral forces, how much net pull into the wall can we sustain by virtue of the convexity of the surface?

Nomenclature:

• = the width of the robot
• = radius of curvature of the convex surface
• = half of the angle spanned by the robot
• = net pull-in force that helps prevent pitchback
• = lateral (internal or "pinch") force applied by robot legs
• = normal force at the contact
• = tangential force at the contact

Equations:

• if on the verge of slipping, then , where = coefficient of friction

So,
, and . Then and

Some example numbers letting w=1 (stance = unit width) and radius of curvature as multiples of w:

w       r        theta    mu    gamma     alpha     fl    fp
1.000  1.000     0.524    5     0.197     0.326     5     1.691
1.000  1.500     0.340    5     0.197     0.142     5     0.717
1.000  2.000     0.253    5     0.197     0.055     5     0.277
1.000  4.000     0.125   10     0.100     0.026     5     0.128
1.000  6.000     0.083   15     0.067     0.017     5     0.084
1.000  8.000     0.063   20     0.050     0.013     5     0.063
1.000  1.000     0.524   2      0.464    0.060      5     0.300

-- MarkCutkosky - 12 May 2004

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