IEEE Robotics & Automation Magazine - March 2016 - 24
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1. The setup of the experiments for the comparison of the model-based and the bioinspired controllers. (a) Transient behavior.
(b) Redundancy of the MC controller. (c) 3-D balancing. (d) Tilt disturbance in the support surface. (e) Translational acceleration
disturbance in the support surface. (f) Compliant support surface.
the balancing approach is to control the robot's total COM
x ! R 3 in such a way that it shows a compliant response
with respect to external disturbances. It is well known that
for every mechanical multibody system, the COM dynamics
fulfills the equation
mxp = mg + fc + fext,
(1)
where m is the total mass, g is the gravity vector, fc is the
sum of all external forces at the robot's end effectors in contact, and fext is an external (disturbance) force. If fc is
considered as a control input and the virtual equilibrium position of the COM is denoted by x d, such a compliant behavior is easily achieved via the control law fc = mg + K t
(x d - x) - D t xo for some positive definite stiffness and
damping matrices K t and D t . The question is then how to
realize the total force fc via the robot's actuators. For that, we
have to take the force distribution to the end effectors in contact into account. In this section, this conceptually simple
24
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IEEE ROBOTICS & AUTOMATION MAGAZINE
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march 2016
idea is generalized to attain balance and posture control for a
complete humanoid.
Given a desired equilibrium x d ! R 3 for the COM position and a desired orientation R d ! SO (3) for the trunk, a
desired control wrench Fd acting at the COM is computed via
the compliance control law
Fd = c
mg
K t (x - x d )
xo
m - Dc m - c
m.
0
x k (R , R d )
~
(2)
Here, R and ~ denote the orientation and the angular velocity of the trunk, respectively, both measured by the onboard
inertial measurement unit (IMU). Damping is implemented
by the matrix D. K t and x k denote a translational stiffness and
the torque from a virtual rotational spring between R and R d.
The torque is computed according to x k = - 2 (dK r + e #
K r) e, with the scalar part d and the vector part e of the quaternion describing the error in the rotation (R Td R) [18]. A formation similar to (2) can also be used for controlling the
angular momentum instead of the trunk orientation [19].
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2016
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