The aim of this research is the design of a nonlinear dynamic
system able to generate smooth reference signals for servo systems.
The need for smooth trajectories can be justified by the physical
limitations of the real plants or by task dependent constraints. For
example, in robotic applications the generation of trajectories with
bounded velocity and acceleration is a typical problem. In the case
of motor control, constraints on the maximum torque and on the
maximum rotor velocity must be expected. There are two possible ways
to handle such limitations: a) the structure of the controller can be
modified to take into account these limitations or, better, b) the
external set-point can be ``filtered'' to provide a new internal
reference signal which satisfies the dynamic limitations of the
controlled system. In our research the second way has been considered
since it leads to a simpler controller structure.
In robotics, the
problem of trajectory generation is usually solved by calculating
optimal profiles. This is done off-line by using optimization
routines that usually require some a priori knowledge of the desired
trajectory. Kinematic saturations such as bounded velocities or
bounded accelerations are handled by imposing some constraints to the
optimization problem. Then, if the desired trajectory changes, also
the tracking profile needs to be changed. We investigate the problem
of the trajectory generation from a different point of view. The
proposed trajectory generator does not require any a priori
knowledge: by using a nonlinear dynamic feedback, it calculates the
trajectory which tracks ``at best'' the external set-point while
satisfying imposed constraints. Moreover, the final set-point is
reached (almost) in the ``minimum-time'' and ``without
overshoot''.
The design of the trajectory generator is mainly
based on Variable Structure (VS) control techniques. With a proper
choice of the sliding surface it is possible to guarantee good
performances both in transient and in final staty-state conditions.
Different versions of the filter have been developed. At the moment,
the two and three dimensional cases has been treated, corresponding
to dynamic limitations on velocity, acceleration, but also on jerk.
Since the kernel of the generator is based on a state variable
filter, the first, the second [1]
[2]
[3]
[4]
and the third [5]
time derivatives of the output signal are available. The advantage
deriving from the knowledge of such signals is well known: they can
be used in the controller to generate feedforward actions. A demo
version of the three level filter is available at the following
page.
The
same approach can also be used to take into account the existence of
dynamic constraints in robotic applications. For example, a modified
version of the dynamic filter is used to bound joints' torques and
torque derivatives within assigned limits. [6][7]
