27
WALKING
ROBOT
CONSTRAINTS
87
27
Walking
Robot
Constraints
The
”Big
Pig”
project,
which
involves
a
pig-like
quadruped
robot,
wants
to
understand
how
to
control
the
device
when
it
operates
on
a
slippery
surface.
Viewed
from
above,
each
foot
is
at
a
location
described
by
the
vector
±
r
i
,
comprising
the
x
(positive
forward)
and
y
(positive
to
the
left)
coordinates
relative
to
the
center
of
mass;
the
vertical
direction
is
z
.
Clearly
static
stability
is
possible
only
if
the
centroid
is
contained
within
one
of
the
triangles
created
by
the
planted
foot
locations.
Hence,
slow
walking
can
be
achieved
by
moving
the
centroid
into
diﬀerent
triangle
sets,
and
moving
the
feet
in
turn.
Also
in
every
such
conﬁguration,
the
roll
and
pitch
moments
have
to
balance
to
zero.
Things
get
harder
during
actual
locomotion,
because
we
do
not
want
slippage
on
any
of
the
feet.
This
occurs
when
force
in
the
horizontal
plane
exceeds
a
critical
fraction
of
the
vertical
(normal)
force:
when
F
i,x
≥
μF
i,z
,wh
e
r
e
μ
is
the
Coulomb
friction
coeﬃcient.
A
similar
constraint
would
apply
of
course
in
the
y
-direction.
Here
we
take
on
a
simpliﬁed,
three-leg
problem
that
lays
out
some
very
practical
issues
in
large
and
slowly
moving
walking
robots.
Consider
the
statically
stable
condition
r
=
[[1.1,
0.8];[1.2,-0.7];[-1.5,-0.7]]
meters,
where
each
pair
gives
the
x
and
y
coordinates
of
three
planted
legs
(forward
left,
forward
right,
back
right).
The
weight
of
the
Big
Pig
is
4600
Newtons,
and
the
value
of
μ
is
0.12.
Along
with
the
non-slip
constraint
fo
r
eachfoo
t
inthe
x
-direction,
we
have
a
mechanical
design
constraint
that
the
x
force
magnitude
on
any
foot
cannot
exceed
250
Newtons.
Also,
since
a
foot
cannot
”pull”
on
the
ground,
each
F
i,z
has
to
be
positive.
This
is
a
total
of
nine
inequality
constraints.
There
are
also
three
equality
constraints.
First,
the
sum
of
the
vertical
forces
has
to
equal
the
vehicle
weight.
Second,
the
sum
of
moments
due
to
the
vertical
forces
in
the
pitch