3
12
1
2
3
21
12
1
;
;
cos
sin;
cc
a
db
bd
b
bh
bh
d
y
z
=
−
=
+
=
−
2
2
3
21
12
3
21
2
cos(
);
;
;
b
x
d
da
da
h
ha
ah
=
−
=
−
=
+
1
2
2
2
2
cos(
)
sin(
);
sin(
);
a
y
z
a
x
=
−
+
−
=−
−
2
1
2
2
sin;
sin(
)
cos(
);
c
d
x
b
y
z
=
=
−
−
−
1
2
12
12
cos
sin;
sin;
;
c
c
c
h
y
z
h
x
ab
ba
=
+
=
=
−
x
2
, y
2
, z
2
– coordinates of the
contact point of the second leg of the robot with the displacement surface; x
c
, y
c
, z
c
are
the coordinates of the center of gravity of the robot; α, φ – angles of inclination to the
horizon of surfaces on which the robot's legs rest; ψ is the angle of inclination of the
central axis of the robot passing through its center of gravity G (see Fig.5). Then, from
the same system of equilibrium equations, we find the remaining unknown reactions
N
1
and the frictional forces Q
1y
:
1
1
4
4
1
5
6
;
,
y
N
Q
Ga
Nh
Q
Gh
Nh
=
+
−
=
+
(6)
where also for the compactness of writing variables is defined:
4
3
3
4
3
3
5
3
3
6
3
3
cos
cos(
)
sin(
);
cos
cos(
)
sin(
);
sin
cos(
)
sin(
);
sin
cos(
)
sin(
).
a
a
d
h
b
h
h
d
a
h
h
b
=
−
−
−
−
=
−
−
+
−
=
−
−
+
−
=
−
−
−
−
For the stability of the robot, the frictional forces of each of its legs must not
exceed the boundary values:
1
1
2
2
1
2
;
;
0;
0,
yy
Q
N
Q
N
N
N
(7)
where μ – the coefficient of friction of the grip of the robot's leg with the surface
along which the robot moves. Substituting in expression (7) the expressions above the
found reactions of forces (5) and (6), we find limitations for the technological load of
the robot taking into account the forces acting on it:
1
4
2
3
1
2
4
3
2
3
3
1
4
5
3
3
6
4
0
;
0
;
(
)
(
)
;
.
Q
Ga
Q
Ga
N
N
N
N
h
b
Q
G
a
d
Q
G
a
h
N
N
h
b
h
h
++
+
−
+
−
++
(8)
Among the values of the reaction N of the technological load calculated in
accordance with conditions (8), we choose the largest, which simultaneously satisfies
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