IEEE Robotics & Automation Magazine - March 2016 - 75

uo ei = 1e (a ei $ u e0 - u ei - b e v ei + / w eij iu j + Feed ei (q, qo ))
xi

vo ei = 1 e (- v ei + iu i)
xl i
Z
r
, ue > r
]
2 i 2
]
ui = [ u ei , - r # u ei # r ,
i
2
2
]
] - r , u ei < - r
2
2
\

i = 1, 2, ..., n - 1
(9)

uo si = 1s (a si $ u s0 - u si - b s v si + / w sij k j + Feed si (q, qo ))
xi

vo = 1 s (- v si + k i)
xl i
k i = max (0, u si ), i = 1, 2, ..., n - 1,
s
i

(10)

where the superscripts e and s indicate that the oscillator is
for the control of the equilibrium position and for the control of the stiffness, respectively. The subscript i is the joint
index. The state variables of CPG are u ei and u si and v ei and
v si represent the degree of the self-inhibition effect of the
oscillator. The outputs of the oscillator, i.e., the equilibrium
position and the stiffness of the ith joint, are iu i and k i , furthermore, u e0 and u s0 are input signals of CPG and a ei and
a si are signal coefﬁcients. The constants of rising time are
e
s
e
s
x i and x i , while xl i and xl i are the constants of adaption
e
s
time and b and b are the coefficients of the adaptation effect. The connection weights are w eij and w sij and
Feed ei (q, qo ) and Feed si (q, qo ) are the feedback from the walker. The schematic diagram of the CPG-based control system is shown in Figure 2. The two driven signals u e0 and u s0
are the inputs of the control system. Changing u e0 can adjust the equilibrium position of each joint, while changing
u s0 can adjust the stiffness. Thus, u e0 and u s0 control the
level of joint torque and joint stiffness, respectively.
The proposed torque-stiffness-controlled dynamic walking provides a general paradigm of bipedal walkers with both
controllable joint torque and adaptable joint stiffness. For the
walkers of this kind, the external actuation and the intrinsic
dynamics can be controlled independently and simultaneously, which offers an alternative method of motion control other
than adjusting only joint torques. Therefore, in the experiments of this study, we will focus on the respective effects of
joint torque and joint stiffness on walking pattern transitions.

biped travels on ground level. Since the introduction of flat
foot and compliant ankle, the actuated ankle also provides
energy during push-off phases. In each step, there are two
impulses, heel-strike and foot-strike, representative of the
initial impact of the heel and the following impact as the
whole foot contacts the ground. Foot scuffing of the swing
leg is ignored, since the biped has no knee. The bipedal
model is shown in Figure 3.
The control inputs of the CPG-based control system are
the two driven signals u e0 and u s0 in (9) and (10). Increasing
u e0 can increase the difference between the actual joint angle
and the joint angle of equilibrium position, thus generating
larger joint torque. For the four-link model, u e0 mainly influences the hip torque during swing phases and the ankle
torque at push off. Increasing u s0 can make the hip joint and
the two ankle joints stiffer. Here, each step of the locomotion
is segmented into different phases: 1) push off, 2) swing, and
3) foot rotation. The coefficients and feedback functions in
(9) and (10) are phase-dependent piecewise constants.
The CPG-based control system of the four-link biped includes six oscillators for equilibrium position and stiffness
control of the three joints. Connections among the equilibrium positions and stiffnesses are established for the limb coordination (see Figure 3). Derivative feedback of hip and ankle
angles are added to the oscillators for decreasing time-delay effects and preventing the limb moving too fast to maintain stable walking. Biological studies indicated that hip stiffness has a
larger value around double-support phases in normal human
walking [13]. Thus, we add feedback that increase hip stiffness
in double-support phases. The unit oscillator for the ankle
stiffness of the stance leg receives sensory feedback from the
ankle angle and the angular velocity. The stiffness increases
adaptively in dorsiflexion, which is consistent with the general
tendency of normal human walking [13]. The parameter values and the expression of feedback functions of the CPG
model in simulations can be found in the supplemental material for this article, available on IEEE Xplore.

Hip
Leg 2
Leg 1

A Four-Link Model Case
Previous studies [22] have reviewed some cases where simple
models can also give insights into human motion. Consequently, in this article, a simple four-link model is taken as an
example to study the performance of torque-stiffness-controlled dynamic walking. We employ a simple model also for
the decreasing uncertainties generated from complex mechanical mechanisms. The proposed model consists of two
rigid legs interconnected individually through a hinge at the
hip. Either leg includes a flat foot mounted on the ankle.
Each joint is actuated by a torsional-spring-like actuator, with
adjustable equilibrium position and spring constant. The

Foot 2
Foot 1
Ground
Figure 3. A four-link torque-stiffness-controlled walking model
and the coupled oscillators. The dashed-pink lines indicate
the equilibrium positions of leg 1, leg 2, and foot 2 relative to
foot 1, leg 1, and leg 2, respectively. The pink oscillators are for
equilibrium position control, while the blue oscillators are for
stiffness control. The red lines represent the interactions among
the oscillators. The green regions indicate the sensory feedback.

march 2016

IEEE ROBOTICS & AUTOMATION MAGAZINE

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