3.2. Controller design and optimization
The introduced control strategy consists of a feed-forward
control unit and a feedback control unit, and the
two units are designed separately. As shown in
Figure 6, the feed-forward control unit involves an
optimal multi-input shaping technique. It is implemented
outside of the feedback loop and is used to
suppress induced vibration of the reference model at
the starting and the stop stages of the robot motion.
The motion between the two stages is controlled by a
feedback control unit making error compensation. This
subsection focuses on the starting maneuvers from
some prescribed input state to the achievement of uniform
rectilinear motion, where a dynamic balance has
been reached between a specific coulomb friction and
input torques. The feed-forward controller is developed
to reduce the vibration of the suspension system using
optimal ZVD shapers which are optimized by employing
the PSO with chaos method.
Each of the ZVD shapers functions with three sets of
impulses (i.e. six variables including three timings and
three amplitudes). Then, as the first amplitude timing is
at zero and the three amplitudes are constrained to add
up to unity, the number of variables for each shaper is
reduced to four. Two shapers act on the left and right
input torques for the robot considered here, and the
optimization of multi-input shapers is transformed
into an optimization problem of eight variables which
are expressed as