IEEE Robotics & Automation Magazine - March 2016 - 77

G p = 11 and G d = 1.8, in Method III, c = 0.01 and G p = 0.1,
G d = 0.017 for u e0 , while G p = 10, G d = 1.7 for u s0 . In this article, the control parameters are manually tuned.
We use the dimensionless Froude number Fr (defined as
Fr = V/ gl , where V is the velocity, g is the gravitational
acceleration, and l is the leg length) to measure the speed.
The dimensionless Froude number is commonly used to evaluate the velocity of passivity-based bipedal walking [6], [26],
[27]. Bipedal walkers with different leg lengths can be compared fairly by employing the Froude number.
In the experiments of speed control, we studied the stepping variation and continuous variation of the desired speed.
In the case of stepping variation of the desired speed, the
biped model first performs stable walking at an Fr of 0.157.
At the end of the third step, the desired Fr changes to 0.207.
The transition starts at the fourth step and the two control
parameters u e0 and u s0 begin to vary according to the control
law in (11). Figure 6(a) shows the speeds of each step for the

0.22
0.2
0.19
0.18
0.17
Method-I
Method-II
Method-III
Desired Speed

0.16
0.15
0

6
8 10 12
Step Number

0.6
0.55

0.45
0

Step Number
(b)
Figure 6. The speed control of the simulated model with three
different methods. (a) Speed variation. (b) Step length variation.
The speed is measured by the dimensionless Froude number Fr.
The step length is normalized by the leg length. The gray dotteddashed line in (a) represents the target speed. The three solid
lines describe the speeds or the step lengths of each step of the
three control methods, respectively. The desired Fr is changed
from 0.157 to 0.207 at the end of the third step. Method-I,
Method-II, and Method-III represent changing only u e0 , changing
only u 0s , and changing both u e0 and u 0s , respectively.

0.55
0.5
0.45
0.4
0.35

II
0.
00
2
0.
00
6
0.
01
0.
01
4
0.
01
8
0.
02
5
0
M .03
et
ho
dI

Method-I
Method-II
Method-III

ho
d-

0.5

0.6

M
et

Normalized Step Length (-)

(a)

Final Step Length of
Walking Pattern Transition (m)

Froude Number (-)

0.21

three control methods. Method-I has the fastest responses to
the varying desired speed but it also has the longest rise time.
The motion does not enter the relative steady state until the
14th step. In addition, the speed shows oscillatory characteristics during the transition. The transition motion with
Method-II is quite insensitive to the change in the desired
speed. Steady state is achieved after the 12th step with satisfactory control precision. Method-III combines the advantages of the two former methods. The speed rises rapidly at
the beginning of the transition and converges to the desired
value quickly. The transition is basically completed at the
ninth step. All the three methods have acceptable control
precision, although the accuracy of Method-I is slightly
worse than those of the other two methods.
The step lengths after the transition by applying each one
of the three control methods are different [see Figure 6(b)]. In
Method-I with invariant stiffness, the step frequency almost
maintains unchanged and the speed variation results mainly
from the change of step length. Thus, the step length of Method-I has the largest change during the speed transition. Method-II has the largest portion of step frequency variation in
speed control. Therefore, the change in the step length of
Method-II is the smallest. The results of step length variation
imply that the proposed walking model can realize different
walking patterns with the same desired speed. Tuning the coefficient c in (12) can adjust the portions of torque control
and stiffness control, which influence the final step length and
step frequency in walking pattern transitions. c = 0 is equivalent to applying Method-II, and Method-I means c has an infinite value. Figure 7 shows the final step lengths of walking
pattern transitions studied in Figure 6 with different values
of c. The results indicated that the final step length could increase with increasing c. Thus, the choice of c depends on the
desired step length variation during speed control. In this article, the value of c is empirically chosen. In the future, we may

The Coeffiicient c

Figure 7. The final step length of walking pattern transitions with
different values of the coefficient c in (12). The desired Fr is
changed from 0.157 to 0.207. The case of Method-II is equivalent
to c = 0 and the case of Method-I is equivalent to c = 3 .

march 2016

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