IEEE Robotics & Automation Magazine - December 2015 - 41

collocated if they constitute a power port of a subsystem. A
power port is given if the product of these two variables represents the instantaneous power flow; see [38]. This is a force
and a velocity (or a linear operator of velocity, such as position). This stems from the fact that the motor and the link
dynamics are interconnected by an elastic element (1).
Therefore, the link-side coordinate does not have a direct
input. Controllers that neglect the coupling dynamics are
exposed to stability issues. And due to the introduced
mechanical stiffness of VSA, stable controllers, which only
regulate i and neglect the compliance, are prone to large
position errors. A method to both ensure stability and accurately execute the task has been proposed in [37]. We adapt
that approach to the DLR HASy.
The main idea is to formulate a collocated variable qr (i),
that is statically equivalent to the link-side position q but only
depends on i. Therefore, a static version of (2) is required, i.e.,
io = qo = 0, which can be derived using the potential energy
V (i, q) of the robot. The static equilibrium is achieved if
2V (i, q)
=-f ({, v) + g (q) = 0
2q

(3)

holds. This equation can also be interpreted as the evaluation of only potential energy terms in the Lagrangian
L (i, q) = T (i, io , q, qo ) - V (i, q), where T (i, io , q, qo ) is the
kinetic system energy, for the derivation of the equations of
motion. The special properties of (3) are the key elements
for finding qr (i) . First, the potential function V (i, q) is
positive definite. Second, the joint stiffness and the Hessian
of the gravity potential are bounded. The joint stiffness is
the instantaneous stiffness of the VSA, and it is upper and
lower bounded by the mechanical design of the joint. These
conditions cause the convex nature of (3) and ensure a single solution of q for each i. Hence, the mapping qr (i) is a
diffeomorphism, i.e., each desired position q d can be realized by an appropriate i in a static sense. Nevertheless,
finding the solution of that implicit function requires a
numerical algorithm in case of complex nonlinear systems,
such as VSA robots. Due to these convexity properties, the
simplified, iterative gradient method in Figure 4 rapidly
converges to qr (i) . The algorithm stops after a specified
number of iterations. Although a condition on q i +1 - q i
would indicate the convergence more accurately, specifying
i end ensures determinism, which is an essential requirement
for the real-time implementation. A passive controller can
then be formulated with proportional-derivative (PD) components for minimizing the position error and selective dissipation of kinetic energy:
u = g (qr (i)) - J qr (i) T K p (qr (i) - q d) - K d io .

(4)

The controller torque is denoted by u. The matrices K p and
K d are the positive definite, symmetric P- and D-gains. The
Jacobian matrix J qr (i) T is a mapping between i and qr (i),
the static equivalent of q. A gravity compensation term

g (qr (i)) also based on the static equivalent completes the
control law. To compensate for external loads such as pickedup objects, a standard load estimation algorithm can be used.
This controller has
several major advantages.
The effects of the passive
First, the use of qr (i) enables stable and statically
joint compliance and the
correct link position control. The dynamics origilow-damped oscillatory
nated from the elastic
element do not have to
system are not fully
be considered explicitly.
Second, arbitrarily low
accounted for by the
controller gains can be
adjusted even for large
energy-shaping techniques.
displacements from the
equilibrium, since gravity
is always compensated
accurately. This property shows the flexibility of the controller, as low gains normally let the robot collapse under the
gravitational loads, what is not desired. Third, the approach
provides great versatility for the choice of task coordinates.
A task can be generally formulated to control k independent output variables y = h (i, q) to desired constant values
y d ! 0 k . We control the steady state of the link coordinates
q. An alternative, popular choice is the control of Cartesian
coordinates as carried in the "Active/Passive Impedance
Control" section; see [39].
To demonstrate this approach, experiments on the
shoulder joint have been performed and plotted in Figure  5. In the top plots, a direct noncollocated feedback of
the link-side position q has been used in the controller.

i := i0

qi + 1 := qi -

i := i + 1

No

1 ^V (i, q)
l
^q

q = qi

i = iend?

Yes
q := qi + 1
Figure 4. A numerical iteration scheme is used to calculate the static
link-side equivalent variable qr . Fast convergence can be guaranteed by
the convex nature of the problem and the depicted simplified gradient
method. In terms of the implementation on the real system, one has to
keep in mind that the numerical costs can be greatly reduced by using
the solution from the last time step as starting condition for the current
one. The rate of convergence can be controlled via the parameter k.

DECEMBER 2015

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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41
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