IEEE Robotics & Automation Magazine - March 2016 - 76
Stable periodic walking can be found under different u e0
and u s0 with appropriate mechanical parameters. All masses
and lengths of the model in this article are normalized by
the total mass and the leg length, respectively. The dimensionless leg length and foot length are 1 and 0.25, while the
dimensionless leg mass and foot mass are 0.45 and 0.05, respectively. The parameter values are set based on a rough
Hip
Equilibrium Position (Rad)
0.5
30
25
0
-0.5
Stiffness (Nm/Rad)
20
0
1
2
3
25
0
20
Ankle 1
0.5
-0.5
0
1
2
3
25
0
20
Ankle 2
1
2
3
0
1
2
3
0
1
2
3
15
0.5
-0.5
0
15
0
1
2
3
Time (s)
Time (s)
(a)
(b)
0.5
Leg Angle (Rad)
Leg Angle (Rad)
Figure 4. The CPG state variables of the locomotion with u e0 =
0.22 and u 0s = 25 for two gait cycles. (a) state variables u ei ,
namely, the equilibrium position of each joint. (b) state variables
u is, namely, the stiffness of each joint.
0
-0.5
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
Leg Angle (Rad)
Leg Angle (Rad)
0 2 4 6 8 10 12 14
Time (s)
(a)
0.5
0
-0.5
0
2
4
6
5
10
Time (s)
(b)
15
u 0 (n) + G p (Vdes - V (n)), n = 1
u 0 (n + 1) = * u 0 (n) + G p (Vdes - V (n)) +
G d (V (n - 1) - V (n)),
n $ 2,
0.5
0
-0.5
0
Time (s)
(c)
2
4
6
Time (s)
(d)
Figure 5. The leg angle trajectories during the walking pattern
transitions of the simulated model and the prototype. (a) The
varying u e0 of the model. u e0 is changed from 0.1 to 0.25, while
u 0s maintains 22. (b) The varying u 0s of the model. u 0s is changed
from 16 to 31, while u e0 maintains 0.22. (c) The varying u e0 of the
prototype. u e0 is changed from 0.1 to 0.3, while u 0s maintains 15.
(d) The varying u 0s of the prototype. u 0s is changed from 15 to 27,
while u e0 maintains 0.1.
76
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
march 2016
estimation of mass distribution and morphology of humans
[23], [24]. For this model, the ranges of u e0 and u s0 with obtainable periodic gaits are from 0.1 to 0.31 and from 16 to
34, respectively. The CPG state variables u ei and u si in (9)
and (10) (i.e., the equilibrium position and joint stiffness)
of a representative motion cycle with u e0 = 0.22 and
u s0 = 25 are shown in Figure 4. The equilibrium positions
change their directions based on the phase information, obtained by the feedback from foot contact, to form the periodic locomotion. The hip stiffness stays at about the same
level during one gait cycle, while the ankle stiffness has a
relatively large cyclic variation, which shows humanlike
ankle behaviors and is produced primarily by the local
ankle feedback.
To study the respective effects of u e0 and u s0 , we fix one
while changing the other and observe the performance of the
biped. The results indicate that increasing only u e0 leads to an
increase in the magnitude of leg angles, and thus an increase
in step length, while the step frequency is almost unchanged
[see Figure 5(a)]. However, both the magnitude of the leg
angle and the step frequency have apparent increments with
increasing u s0 [see Figure 5(b)]. Thus, we can conclude that
changing joint torque causes variations in the step length,
while it has only a minimal influence on step frequency,
which is consistent with the previous studies. Kuo et al. [25]
reported that the walking period is almost invariant with respect to the energy input and the increase in velocity results
mainly from increasing step length. According to our results,
step frequency is mainly influenced by adjusting joint stiffness. In the following paragraphs, we analyze the respective
effects of changing joint torque and joint stiffness on closedloop speed control.
For closed-loop speed control, we may change each of u e0
and u s0 or both of them according to the variance of the desired speed. These three control methods are denoted as
Method-I (controls only u e0 with fixed u s0 ), Method-II (controls only u s0 with fixed u e0 ), and Method-III (controls both
u e0 and u s0 ), respectively. The control rule is designed based
on a discrete proportion-differentiation method:
(11)
where Vdes is the desired velocity, V (n) is the velocity of the
nth step, G p and G d are the gain coefficients, and u 0 (n) denotes u e0 (Method-I), or u s0 (Method-II), or both u e0 and u s0
(Method-III) of the nth step.
In Method-III, we empirically choose a combination that
the two parameters increase or decrease proportionally [11],
satisfying the following constraint:
u e0 (n + 1) - u e0 (n) = c $ (u s0 (n + 1) - u s0 (n)), n $ 1, (12)
where c is a constant coefficient to adjust the portion of equilibrium position control and stiffness control in controlling
speed. In Method-I, G p = 0.32 and G d = 0.02, in Method-II,
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