IEEE Robotics & Automation Magazine - March 2016 - 73
The n-link biped can be characterized by the Euclidean coordinates, which include the position of the center of mass
(CoM) and the direction of each link:
x = (x 1, y 1, i 1, x 2, y 2, i 2, ..., x n, y n, i n)l.
(2)
The superscriptil denotes the transposed matrix (the same
in the following paragraphs). (x n, y n) and i n are the position of CoM and the direction of the nth link, respectively. The walker can also be described by the generalized
coordinates q:
q = (x 0, y 0, i 1, i 2, ..., i n)l,
(3)
where (x 0, y 0) is a point on the biped to characterize the position of the walker. The n angles depict the posture. We denote
j as the Jacobian:
J = dx .
dq
(4)
We define constraint function p (q) to represent constraint conditions, which include external constraint, e.g.,
the contact between a foot and the ground, and internal
constraint, e.g., knee locking. Each component of p should
keep zero to satisfy the constraint conditions. For example,
the full contact between a flat foot and the ground can be
modeled as H heel = 0, H toe = 0, and x ankle - x c = 0. H heel
and H toe are the heights of the heel and the toe above the
ground, respectively. x ankle denotes the horizontal position
of ankle and x c means the contact point between ankle and
ground. When a contact is detected, an impact event occurs
and leads to a change in the generalized velocity, then, a
new constraint condition is added to the motion. Since the
contact between the biped and the ground is a unilateral
constraint, the directions of ground reaction forces are used
to determine the contact release. Thus, the dimension of
p (q) may change during the gait cycle.
Under the constraint mentioned above, the Euler-Lagrange equations in the matrix form can be obtained as
follows:
Fq
M q - Ul qp
E; E = > 2 (Uqo ) o H,
;
q
U 0 Fc
2q
2J o o
qq,
2q
M q - Ul qo +
M q qo E; E = ;
;
E,
0 mc
U
0
(7)
where qo + and qo - are the generalized velocities just after and
just prior to the impact, respectively, and m c is the impulse
acted on the walker.
Equations (5) and (7) form a hybrid dynamical system.
Note that the dimensions of p, Fc , and m c may vary when
constraint conditions change.
The control inputs of the proposed torque-stiffness-controlled dynamic walking are the equilibrium position and stiffness of all the n - 1 joints, i.e., the rest angle and spring
constant of each spring. Joint angle, angular velocity, and contact information are important feedback for control. The general form can be expressed as follows:
= iui (q, qo , C inf , t)
k i = k i (q, qo , C inf , t),
iui
i = 1, 2, ..., n - 1,
(8)
where iui and k i represent the equilibrium position and the
stiffness of joint i. C inf is the contact information and t represents the time. We can further develop control methods based
on (8) for torque-stiffness-controlled dynamic walking.
`
ยท
Ti = -Ki : (i
( i - ii) -d : ii
Link i
Joint i
ii
g
Equilibr
Equilibrium
Positio
Position
Link i + 1
`
ii
Spring
C4
Link 3
g
g
Link n
Link 2
(5)
Link
Lin
L
i k1
C1
C3
where M q is the mass matrix in the generalized coordinates.
U is defined as ^2p/2qh $ Fc represents the constraint force
vector. Fq is the active external force in the generalized coordinates, which can be expressed as follows:
Fq = J l F - J l M
trolled biped is concentrated on the adjustment of equilibrium
positions and spring constants [ Ji and k in (1), respectively] to
change F and, thus, modulate the motion behaviors.
The equations at impacts can be represented as follows:
(6)
where M and F are the mass matrix and the active external force
vector, respectively, in Euclidean coordinates. F includes gravitation, the damping torques, and the joint torques generated by the
torsional springs. The control problem of torque-stiffness-con-
Ground
C2
Figure 1. A schematic diagram of an n- link torque-stiffnesscontrolled dynamic walking biped. The walker travels on level
ground. g stands for gravitational acceleration. Each link is
hinged with adjacent ones at the joints. Each joint is equipped
with a torsional spring. For clarity, only the spring at joint i is
plotted. i i represents the relative angle between link i and link
i + 1. iu i denotes the equilibrium position of link i + 1 relative to
link i , where the spring achieves its rest angle. Ti is the torque
acted on link i + 1 generated by the torsional spring. C 1 and
C 2 are the vertical constraints at the two end points of link 1
(the heel and the toe of the stance foot in common cases). C 3
represents the horizontal constraint of link 1. C 4 denotes an
internal constraint between link 2 and link 3 (commonly occurs in
knee locking), which keeps the two links in the same direction.
march 2016
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
73
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