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. At the moment only the two dimensional case has
been treated, corresponding to dynamic limitation on velocity and acceleration. The three dimensional case is currently being developed, with the aim of also bounding trajectories jerks.
Since the kernel of the generator is based on a state variable filter,
both the first and the second 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
[1]
[2]
[3]
[4].
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.
[5]
