Foot forces (planar simplification, in fore/aft direction)

1
Preliminary foot force analysisNot checked
carefully for accuracy. Please do not circulate.
8/13/03 mrc

• Foot forces (planar simplification, in fore/aft
  direction)
• Assume a robot climbing a possibly irregular
  surface, inclined at angle ?. For purposes of
  obtaining some initial values, assume
• body length 20cm
• weight 1 Kg
• distance per stride 5cm
• step frequency f 4 steps/second (1
  step/half-period)
• speed 0.2 m/sec 1 bl/sec
• Variation in speed /- 20 gt fore/aft
  acceleration of roughly
• 0.20.2(2?f) 1 m/s2 or 1.1 Gs on vertical
  surface.
• (Call it ?G where ?? 1).
• So we expect shear tractions of 1.1Kg or greater,
  distributed unevenly among n2 or n3 feet. For
  more general analysis,
• DanielSantos analysis is addressing cases with
  internal forces, as a function of normal force,
  slope, foot placement and surface normal
  orientation. Current version is quasi-3D front a
  rear legs out of plane.

(http//www-cdr.stanford.edu/twiki/bin/view/Main/
CbotStaticModeling) --- the text in blue above
is hyperlinked to ClimbingRobot Twiki page ---
2
Preliminary foot force analysis contd.
One design implication concerns the location of
the center of mass (COM). In the figure at left,
the front foot normal force, fa is given by a
moment balance about the lower foot falf (lc
sin? d cos?)?mg 0 If the term in
parentheses becomes negative (COM too far back or
too far from wall surface) the front contact
force will be tensile (adhesion required). From
this, we see thatAdamPricketts LegoGuy device
works in part because lc is large and d is
small. (twiki/bin/view/Main/LegPlatformDevelopment
) The following slides show the results of some
quasistatic analysis in Matlab for this case as a
function of slope.

• Assume
• 2 legs, point contacts at feet
• Coulomb friction (m1, m2) at each foot
• arbitrary slope
• arbitrary external force, F_t, applied to body
  (in addition to gravity)
• Compute foot contact forces and internal force
  (compressive or tensile, along ground).

3
Cbot_v2 dimensions and notation in body
coordinatesSource http//www-cdr.stanford.edu/tw
iki/bin/view/Main/CbotStaticModeling(Download
and Extract Matlab files into a single directory
launch Matlab type cbot)
Upon launching cbot.m, one will get a graphical
window as shown on slide 5 with forms for
entering the robot physical parameters. The
dimensions are as labeled below and are measured
in a body-centered coordinate system, embedded at
the robots center of mass (COM). In addition to
gravity, there is a provision for an externally
applied thrust force, F_t. This force could be
the result of a fan, rocket or wings. It can also
be located at the front foot to simulate the
effect of some adhesive mechanism. The parameter
W is not shown below. It is the width of the
robot. With non-zero W, the plane of the front
leg will be moved W/2 to the left of the COM and
the plane of the rear leg will be moved W/2 to
the right of the COM.
f1
f2
Foot Contact Orientations
4
Explanation of following screen shot and results
The next slide shows a screen shot of the cbot.m
program. The physical parameters have been
entered at right and the corresponding stick
figure is shown in the upper left window. In this
case, we have assumed an external thrust of 1.5N
located at the front foot and pointing
perpendicularly into the wall. This could be the
effect of a fan or the effect of some adhesive
mechanism at the front foot. The upper center
plot shows the normal force components as a
function of slope. Note that the front foot does
not fall below 0 normal force, which would mean
adhesion is required. The lower left and center
plots shows the tangential forces, in the
fore-aft and lateral directions, respectively.
For each Angle of Inclination, the tangential
forces are computed to maximize the safety margin
with respect to friction constraints. This
approach can result in a compressive or tensile
internal force applied between the front and rear
foot contacts. In the lower left plot, we see a
kink corresponding to a change in which of the
friction constraints is active. The lower center
plot shows nothing interesting in this particular
configuration because W0, the robot has not been
given any width. The upper right plot shows the
feasible range of the internal body force and the
optimal choice. Note that the feasible range
tends to narrow as angle increases towards 90
degrees and, in this particular configuration,
slipping would occur at around 70 degrees, where
the feasible range disappears completely. The
lower right plot shows which friction constraints
are active at a particular angle. The angle can
be chosen using the slider located right beneath
the lower right plot. Because this version is
quasi-3D, there are 4 friction constraints
pertaining to each foot contact. Also, an upper
and lower limit on the magnitude of internal body
force have been added to the constraints. For
the friction constraints, the quadrant of the
active constraint will be highlighted in either
blue or red. Blue gt Less than constraint, Red
gt Greater than constraint. For the internal
body force magnitude constraints, a patch just
above the COM will be highlighted for the
compressive constraint and a patch just below the
COM will be highlighted for a tensile constraint.
5
Case1 1Kg robot, 1.5 N external thrust into wall
(e.g. from fan, wings or some adhesive
mechanism), m2 (sticky feet, perhaps with pins),
CG located near front and close to wall.
Feasible range of internal Body force
External Force Effects ofAdhesion or Wings
Front Foot Schematic
COM
Rear Foot Schematic
Change in Active Constraint
Current angle for viewing Active constraints
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Discussion

• The example on the last page shows typical
  results
• For a sprawled, 1 Kg robot at 1 G, we see normal
  and shear forces on the order of several N.
• A small amount of normal force at the front feet
  helps greatly when the coefficient of friction is
  high
• Initially, the rear foot is closer to slipping
  (because of the external thrust and because COM
  is near the front) but at higher slopes the front
  limb becomes unloaded and will slip first.
• Both front and rear limbs will, in general, have
  significant shear forces -- depending on the
  slope.
• At high slopes, the sign of the optimal internal
  force changes.
• Not inconsistent with what is seen in geckos (see
  next 2 slides copied from Bob Full Rise Intro).

• Next steps
• Examine transient Yaw torques (pseudo-dynamic).
• Examine effects of variationsin surface
  geometry, which result in different foot
  coordinate frame orientations.
• Examine different friction/adhesion models.

7
Gecko Ground Reaction Force (from R. Full Rise
Intro.ppt)
Level
Climbing
Fore
Hind
Fore-Aft
Fore
Hind
Lateral
Force (N)
Left

Right
Left
Right


Fore
Normal
Fore
Hind
Hind
8
Sprawled Posture Quadruped (R. Full RiSE Intro
ppt)
Fore
II
II
I

_
Force (N)
I
Hind
_
II
I
Time (s)
Sprawled Quadruped
9
Background slides on the approach used in Cbot.m

• The following slides are taken from Daniel
  Santos powerpoint presentation (cbot.ppt)
  explaining the approach taken in version 1 of the
  Cbot.m Matlab program.
• The main method is adapted from J. Kerr and B.
  Roth (IJRR 1986) grasp analysis paper.
• Note that these slides show the analysis for a
  planar robot.

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Equilibrium Equations
Tangential, Normal, Rotational Equilibrium
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Equilibrium Equations
Friction Constraints
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Solution
Decompose Homogenous and Particular Solution
Particular Solution set by External Forces
(Gravity and Thrust Vector)
Homogenous Solution set by Internal Body Forces
Internal Body Force Vector
Null Space comprises Internal Body Forces
13
Solution
Matrix Manipulation
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Choose Best Lambda
Feasible Range
?1- Direction
Optimal ?1
Boundary Conditions
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Solution

• 4 Unknowns, F1N, F1T, F2N, F2T
• 3 Equilibrium Equations
• Null Space and Internal Forces provide 4th
  Equation
• Use Internal Forces to Maximize Distance from
  Violation Any Constraints
• Solve

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Findings

• Ultimate Upper Limit on Slope Angle depends on µ
• Can Achieve Ultimate Upper Limit (based on a
  particular µ) w/o Adhesion by proper selection of
  Angle between C.G. and Rear Foot Contact

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Findings

• Better Results w/o Adhesion
• Performs worse w/ Adhesion at High Angles (less
  Thrust possible)
• Front Leg must Provide Thrust at Small Angles

• Better Results w/ Adhesion
• Requires Adhesion at All Angles
• Optimizes Rear Leg for Thrust Generation

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Findings
Ft
CG
Ft

• Optimal Thrust Vector Angle is 0 degrees (w/o
  Adhesion), independent of Front Leg Angle

• Optimal Thrust Vector Angle is 45 degrees (w/o
  Adhesion), independent of Rear Leg Angle

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Next Steps

• Change Constraints to incorporate
  FrictionAdhesion Model
• Go to 3D
• Examine effects of different foot orienation
• Examine hexapod (alternating tripod gait)

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Climbing Robot Static Modelling