IEEE Robotics & Automation Magazine - March 2016 - 25

This desired wrench Fd is then distributed to the supporting contacts at the feet. Therefore, we consider the mapping
between the stacked vector W = (Fi) of contact wrenches Fi
and the total wrench F via the grasp matrix G, i.e., F = GW.
The desired contact wrenches Fi are then computed based on
a constrained minimization of the error
W ) = min ^|| W - W def || + c || Fd - GW ||h
s.t. AW # a,

(3)

where the desired wrench is implemented by means of a soft
constraint using a large weighting parameter c & 1. Furthermore, additional constraints AW # a can be imposed on the
optimization problem to consider unilaterality, friction, and
location of the center of pressure for each contact.
To realize the desired contact wrenches Fi, we map these
contact wrenches to the corresponding joint torques x d . For
an exact implementation of the contact forces, the whole multibody robot dynamics should be considered. As a simpler solution, we can also consider a kinetostatic mapping based on
the relevant Jacobian matrices, which can be derived by considering the embedding of the COM dynamics (1) within the
complete multibody dynamics [12], [19]. This leads to
xd

= J rT Fr + J Tl Fl,

(4)

To illustrate the features of this controller, two experiments were conducted with the TORO [8]. In the ﬁrst experiment, the robot balances on its legs while the arms are not in
contact. In this conﬁguration, a number of arbitrary disturbances are applied to the robot, demonstrating the capability
of this controller to handle 3-D balancing [Figure  1(c)].
The full experiment can be seen on the video accompanying
this publication in IEEE
Xplore. The second exThe bipedal postural
periment illustrates the
ability to exploit multiple
balance has been
contacts for balancing,
including the resulting restudied from a different
dundancy, by solving the
optimization problem
perspective.
mentioned previously. In
this scenario, the robot is
supporting itself with
all four end effectors [see Figure 1(b)]. In the default
pose, the controller distributes the weight of the robot to all
four end effectors by generating vertical contact forces of approximately ffoot, r = 372 N and ffoot, l = 340 N for the feet,
while the arms carry a weight of fhand, r = 35 N and
fhand, l = 39 N (Figure 3). As soon as the robot is manually
pushed down (t = 3-7 s), it tries to counteract the disturbance by increasing the contact forces. When the robot is
pulled up (t = 8-14 s), the optimization in (3) prevents

where J r and J l denote the Jacobian matrices for the right
and left foot with respect
to the COM location x.
In this way, the compliance (2) is implemented
for the whole body dynamics without requiring
an explicit control or
measurement of the conRd
tact forces. More details
xd
Fd
on the derivation of the
mg
controller are presented
in [12].
The resulting control
scheme does not require
Z
Fl
Y
any measurement of conX
tact forces (neither at the
Fr
feet nor at other locations
on the robot), and can be
made robust against unxd, Rd
certainties in the ground
Torque-Controlled Robot [8]
geometry by utilizing a
q
xm
xd
state measurement based
Fi
Robot
Object Force Fd
Torque
Force
Force
on the IMU data. It can
Dynamics
Generation
Control
Mapping
Distribution
readily be extended to a
x
x, R
larger set of contact points
Kinematics
involving the arms and
IMU
legs to realize more complex MC interaction tasks, Figure 2. The notation and block diagram for the model-based balancing controller with an example
as shown in Figure 2.
of a multicontact situation.
march 2016

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2016
