Analysis and Design of a 1-DOF Leg for Walking Machines

and design of a leg, which is composed
by a Chebyshev mechanism and
pantograph. Its main characteristic is
to posses only 1-DOF, with many
advantages in terms of cost and
operation.
A leg design has been proposed to be
low-cost and easy in operation and it
has been also improved in its
mechanical design by adding two
articulated parallelograms that make
the foot of the robot always parallel
with respect to the ground.
III A KINEMATIC ANALYSIS
A Kinematic analysis has been carried
out in order to evaluate and simulate
performances and operations of the leg
system. A fixed reference system CXY
has been considered attached at point
C, as shown in Fig. 1. The position of
point B with respect to CXY frame can
be evaluated as a function of the input
crank angle α and kinematic
parameters of the Chebyshev
mechanism LEBDC in the form
X B = – a + m cosα + (c+f) cosα
Y B = – m sinα – (c+f) sinθ (1)
in which
2 2 2 1/2
−1
senα − (sen α + B − D )
θ = 2tan
B + D
(2)
Coefficients B and D can be obtained
by considering the closure equation of
the five-bar linkage CDBGM in Fig. 1.
Thus, one can obtain φ 2 and φ 3 angles
in the form
⎛
2 ⎞
−1
⎜ L1
− 4L1L3
⎟
ϕ2
= tan ⎜−
L2
−
⎟
2L1
⎝
⎠
⎛
2 ⎞
−1
⎜ K1
− 4K1K3
⎟
ϕ3
= tan ⎜−
K2
−
⎟
2K1
⎝
⎠ (3)
where
Figure 1
A kinematic scheme for the 1-DOF leg
L 1 = 2 X B z 2 – 2 X M z 2 + X 2 B + X 2 M +
z 2 2 + Y 2 B + Y 2 M – z 2 3 – 2 X B X M –
2 Y B Y M
L 2 = – 4 Y B z 2 +4 Y M z 2 (4)
L 3 = – 2 X B z 2 +2 X M z 2 + X 2 B + X 2 M +
z 2 2 + Y 2 2
B + Y M
K 1 = – 2 X B z 3 +2 X M z 3 + X B 2 + X M 2 +
z 3 2 + Y B 2 + Y M 2 – z 2 2 – 2 X B X M –
2 Y B Y M
K 2 = – 4 Y B z 3 +4 Y M z 3 (5)
K 3 = 2 X B z 3 – 2 X M z 3 + X B 2 + X M 2 +
z 3 2 + Y B 2 + Y M
2
Consequently, the transmission angles
γ and γ 2 shown in Fig. 1 can be
evaluated as γ =θ+φ 2 , and γ 2 =φ 2 +φ 3 .
The position of A with respect to the
fixed frame can be given as
X A = X B – (z 2 +z 4 ) cosφ 2 + (z 3 +z 5 ) cosφ 3
Y A = Y B – (z 2 +z 4 ) sinφ 2 – (z 3 +z 5 ) sinφ 3
(6)