IEEE Robotics & Automation Magazine - December 2015 - 47

Pi (s) = % (s 2 + 2p i, j ~ i, j s + ~ 2i, j)
j =1

(11)

for all i = 1fn. Here, ~ i, 1 and ~ i, 2 represent the first
and the second eigenfrequency of the ith eigenmode. The
corresponding damping ratios p i, 1 and p i, 2 can be specified according to the desired behavior (e.g., underdamped
p i, j < 1, critically damped p i, j = 1, overdamped
p i, j > 1) . By writing the obtained diagonal gain matrices
in state space form, proper stiffness and damping matrices are obtained. See Figure 10 (dotted lines) for a sketch
of the design idea.
5) Recoupling: Finally, the diagonal gain matrices from the last
step can be transformed back into the configuration space,
which leads to fully coupled gain matrices.
An overview of the gain design is given in Figure 11. The
main advantage of the modal decoupling is that the obtained
system of equations is a physically interpretable counterpart of
the controlled system. Therefore, the large nonlinearity of the
robot dynamics can be accounted for, and a smooth gain variation is ensured. The transformation simplifies the robot
model by reducing the system order, but it does not introduce
any structural changes. The principal setup of two mechanical
masses interconnected by an elastic element is still valid, as
shown in Figure 10. Notice that the energy-shaping techniques

=- K QP qu Q - K DQ io Q - K TQ (i Q - q Q)
- K QS (io Q - qo Q),

(12)

Q
with qu Q = qr Q - q d is evaluated. The static equivalent qr from
(4) is used again. Equation (12) constitutes the generic form of
the feedback controller and is applied to the decoupled system
(9). Velocity feedback is
defined by the damping
matrices K QD and K QS .
The capability to estimate
Concerning K QP and K QT ,
the following procedure
the joint torques from the
may be chosen: according
to the VSA design princielongation of the elastic
ples ("Variable Stiffness
Actuators and the DLR
elements enables the
Hand Arm System" section), K P should be chosen
application of advanced
as high as possible to overcome motor friction and to
torque-control-based
obtain a compliant behavior mainly introduced by
methods.
the elastic elements. As the
controller is designed in
the decoupled space, K QP has to be diagonal. To achieve such a
diagonal K QP as close as possible to the desired K P, an optimization problem can be stated as, min K QP || K P - Q -T K PQ Q -1 || F,
which is convex. If the resulting gains are too high, the values of
K QP should be lowered. In practice, a tradeoff between high
gains for positioning accuracy and gain limitations due to the

Modal
Damping
Design
SISO

State Feedback
Controller

due to robustness requirements. Since the measurements
are always noisy and delayed, a basic rule can be formulated. The closer B i and B are, the less feedback action has
to be performed, which increases the robustness of the
closed-loop system. A fast convex optimization step is applied to minimize the Frobenius norm $ F of the deviation between B and B i . The last two steps allow to obtain
n-single-input, single-output (SISO) systems with a similar structure as the design model in one DOF.
4) Modal damping: In the decoupled space, the n fourthorder systems can be obtained, whose characteristic polynomials can be formulated as

Recoupling
Transformation

(10)

State Feedback Control Law
The control law

Decoupling
Transformation

Bi = a M + b K
s.t. min
B - Bi F
a, b

find application in step 3. To demonstrate the performance of
the gain design, we present its application with a state feedback
controller.

Inertia Shaping

where the decoupled variables are denoted by superscript Q, in particular, the new coordinates are
q Q = Q -1 q, i Q = Q -1 i, and x Qm = Q T x m . For a more
precise description of this approach see [47] and for the
generalized eigenvalue problem, see [49]. This system is
decoupled, if B is also diagonalized by Q.
3) Inertia shaping: As the transformation diagonalizes only
the matrices M and K, an additional step is needed.
The torque feedback controller (5) is applied to reshape
the remaining matrix B. The new motor inertia matrix
B i is designed to consist of a linear combination of M
and K. As a result, it is also diagonalized by Q. An optimization problem is formulated to choose B i as close as
possible to B

Eigenmode Analyzable
MIMO Gain Design Model
Figure 11. An overview of the gain design algorithm. First, the
kinetic energy shaping (5) is applied. Then, a mechanically motivated
decoupling transformation follows, which is key to enable the damping
gain design. After independently assigning gains to the n fourth-order
systems, the recoupling step allows to obtain fully coupled matrices,
which can be applied in a state feedback controller structure. Compare
this sketch with the steps of the controller description of the "State
Feedback Damping Control" section.

DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

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