IEEE Robotics & Automation Magazine - December 2015 - 43

the VSA elements can also influence the achievable Cartesian
stiffness. For example, the elastic elements are mounted separately in each joint, whereas the human arm has biarticular
muscles, which allow for coupling stiffnesses. An analysis of
Cartesian stiffness matrices that can be achieved by a sevenDOF VSA arm with only diagonal joint stiffness revealed that
errors up to 25-55%, depending on the pose, may be encountered [41]. An example of mechanical stiffness that is impossible to reach by a VSA, is zero stiffness (gravity compensation).
Moreover, the technical realization of VSA joints leads to several further limitations. In particular, the mechanical construction limits the feasible stiffness range.
To extend the capabilities of the passive stiffness elements, we
present the approach of combining it with an active impedance
controller. This helps to overcome many of the aforementioned
limitations. Active control adapts the stiffness in a wide range,
and the elastic elements can absorb impacts and increase the
energy efficiency. By combining the two concepts, one can benefit from their individual advantages. This is achieved by the
serial interconnection of the active stiffness K active and the passive one K passive . The overall stiffness K res results in
-1
1
K -res1 = K active
+ K -passive
,

(6)

as shown in Figure 6. The stiffness matrix K passive results from
the diagonal and bounded passive joint stiffness, while K active
can be adjusted almost arbitrarily, however, it is limited by
conditions for the controller stability and practical gain limitations. For the realization of the active impedance, the same
approach as presented in the "Potential Energy Shaping" section with Cartesian coordinates as output variables
y = h (i, q) = h x (q) is used. The controller is then
u = g (qr ) - J Tqr (i) J Tx (qr ) K active (yr - y d) - K d io

(7)

with J x (q) = 2h x (q) /2q (4). The passive Cartesian stiffness
results from the joint elasticities and the robot configuration.
Controlling a VSA robot with (7) leads to a coupled active/
passive Cartesian impedance in all the cases. A proper active
and passive stiffness design is required for the desired Cartesian compliance. To achieve the desired TCP stiffness K des,
we implement an optimization algorithm with the two following steps.
● The passive stiffness is designed to approach the desired
stiffness. Therefore, compliance matrices are used, which
are the inverses of the respective stiffness matrices:
C = K -1 . A leastsquares optimization problem with inequality constraints is solved in the compliance space
such that the compliance matrix C passive is chosen "as
close as possible'' to C des . A residual term arises in most
cases due to the diagonal and bounded joint stiffness
matrix. The problem is formulated in the compliance
space to simplify mathematical computation. By specifying a suitable weighting matrix, the results are the same.
● The active impedance is constructed to account for the
residual stiffness. A matrix nearness problem is solved so

that C active is chosen in a way such that the desired compliance C des is approached under the constraint that the stability of the Cartesian impedance controller is preserved.
The nearness of these matrices is defined by the weighted
Frobenius norm. The optimization algorithm is computationally very efficient and it is computed at a rate of 366 Hz on the
real-time operating system. A detailed description of the
approach can be found in [42].
The algorithms are implemented and evaluated on the
arm of the DLR HASy. The following experiments demonstrate the stiffness behavior for different parameterizations.
All the measurements are performed with the robot in the
nonsingular configuration, as shown in Figure 3. The user
chooses a desired Cartesian stiffness and triggers the parameterization. Then, the joint stiffness variation parameter v is
changed and held constant by a PD position controller for the
adjuster motor. In addition, the active Cartesian stiffness
matrix K active is commanded and held constant. The robot
stiffness behavior can be
analyzed by deflecting the
robot from its equilibrium
Popular control approaches
pose.
Figure 7 shows meainclude energy shaping
surements of the TCP
position coordinates x q,
techniques, which
and z q and the motorbased TCP position x i
constitute a solid basis for
and z i, where the Cartesian coordinates of the
VSA robot control.
T C P a re g i v e n by
x i = (x i, y i, z i) T. In the
latter ones, the motor positions instead of the link positions
are used to calculate the forward kinematics (see Figure 8).
The corresponding stiffness values for the time instants t 1, t 2,
and t 3 are given in Table 1. The desired Cartesian stiffness
K des remains constant throughout the trajectory. K passive
shows the locally valid Cartesian stiffness matrix as it is generjoint
ated by the passive stiffnesses k passive of the joints
joint

k passive = J Tx (q) K passive J x (q) .

(8)

At t 1, the optimization is triggered, and the necessary values for v are set (optimization step 1). Therefore, K passive

KActive

KPassive

Figure 6. A sketch of an active impedance controller with passively
compliant joints. The active impedance controller is combined with the
diagonal and bounded passive joint elasticity. Thereby, a wide stiffness
variation range is achieved together with mechanical robustness and
energy-efficiency properties.

DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

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