IEEE Robotics & Automation Magazine - December 2019 - 100

system that computes the joint actuation torques x, given a
preferred joint-space motion to track. To achieve this, a computed torque-control law in the quasi-velocity vector was
developed, starting from the underactuated and kinematically
constrained model of ALTER-EGO. Note that the model discussed in [14] does not take into account the variable stiffness
of the robot's upper body explicitly, which remains an open
subject of research.
The idea of [14] follows in brief. Let q be the generalized coordinates of the robot, n the number of DOF, n fb the number of
independent variables needed to describe the floating base
motion, and n c the number of constraints acting on the robot,
e.g., due to base kinematics. Then, let o ! R n + nfb - nc be the quasivelocity vector so that qo = S (q) o. Consider the error dynamics

Control
In a simplified model, the state-space of the lower body subsystem has three generalized coordinates (i, }, and z),
which describe the semisum of the wheel angles and the
robot's yaw and tilt angles, respectively. Figure 3(a) expresses
the balance control scheme, with u = [u 1 u 2] representing
the torque control for the two wheels, and e = r - y being
the error between the current y and required r robot states.
The anticipated state can also be modified by the operator
using the available teleoperation devices (see the "Operating
Modes" section). A classical LQR method can be applied to
stabilize the wheel base (as in [16] and [17]), which can be
simply designed but does not take advantage of the arms'
fast-balancing motions. This method is suitable when arms
are not available to balance (e.g., because they are used in
other tasks) and provides good balancing performance, as
presented in the "Experiments and Discussion" section.
To improve independent LQR control performance, in
[14], we developed a new whole-body dynamic control

oo

- oo + K d (o d - o) + K p

δ = f (G, θpre)
+

J+

.
qdes

1

qdes

s

-

δ

G (qdes)

+

+

qcmd

UpperBody
System

u

MobileBody
System

(ii)
FK

(i)

r

#0 t (o d - o) = 0,

+

e

LQR

-
(iii)

y
(a)

.
vref, vref, 3vref
τ

Dynamic
Control

v,3v

.
qmsr, qmsr

Compute
Quasi-Velocities

(b)

Figure 3. The ALTER-EGO control schema. (a)The full-state feedback control system obtained with LQR. (b) The whole-body control
schema. FK: forward kinematics.

100

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

DECEMBER 2019

(5)

where K p and K d are positive definite gain matrices
and o d are the desired quasi-velocities. The resulting

θ pre
xdes

d



IEEE Robotics & Automation Magazine - December 2019

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2019

Contents
IEEE Robotics & Automation Magazine - December 2019 - Cover1
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