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From: "Victor B. Naumov" <nau@iias.spb.su>
Subject: Russian Robotics 3/8: Automatic Path Generation
Message-ID: <MBOYER.95Jul5164415@pellan.ireq-robot.hydro.qc.ca>
Lines: 102
Sender: news@ireq.hydro.qc.ca (Netnews Admin)
Organization: St.-Petersburg Institute for Informatics and Automation, RAN
Date: Wed, 5 Jul 1995 20:44:15 GMT
Approved: mboyer@ireq-robot.hydro.qc.ca, crr@ireq-robot.hydro.qc.ca

                 ACADEMY OF SCIENCES OF RUSSIA
   ST.-PETERSBURG INSTITUTE FOR INFORMATICS AND AUTOMATION
                         ROBOTICS LAB.
                     of Prof.F.M.Kulakov


                  METHOD OF AUTOMATIC FORMATION 
                  -----------------------------
                   OF THE DESIRED TRAJECTORIES
                   ---------------------------

Fields of Application
----------------------
- Robotics

Description of Idea
-------------------
Some research being conducted here is targeted at perfecting the
tactical level of robot control; this level forms the goals (desired
program trajectories) for the lowest level.  These problems have been
investigated in our country for a long time.  One of the first methods
of automatic formation of the desired trajectories was developed
here [Kulakov,1980; Kulakov, 1982].

This method is based on the methods of nonlinear programming. The
desired trajectory is formed in the space G of the joint coordinates g as the
discrete sequence of the arguments g1,g2,..., gn of some function, which
provides the global minimum of this functional; in the case where this
minimum is found, the task is executed. 

This function can be defined, for instance as a squared distance
between vectors, one of which formulizes task, the second - the
current state of the environment model.  In the simplest case these
vectors can be represented by the desired finite Xd and current X= (g)
positions of the tool, mounted at the end of manipulator.

              2               2
F(x) = X - X  = X - X(g)
         d          d

Obviously, this function has a global minimum if the current and the
desired positions coincide, i.e. task will execute.

The minimization of the function is carried out, taking into
consideration the limitation on its argument (joint coordinates)
defining the area of allowed values of the argument.  These
limitations can be:  - nonequations, separating space occupied by
obstacles from free space


Y (x) = Y [X(g)]  0, i=1,...,m
 i       i

- nonequations, representing the space of change of joint coordinates,
allowed by the constructions of the manipulator gmin<g<gmax
and other limitations.

Therefore algorithms of minimization in such an approach must provide
the formed sequence of the argument on the way to the minimum belong
to the allowed area of the argument defined by the above mentioned
limitations.  In addition, the length of the formed trajectory must be
a minimum.  For this goal another algorithm of nonlinear programming
was used based on a modified barriers.

To form the continuous geometrical trajectory in joint coordinates
space which passes through the points of the formed sequence
g1,g2,...,gn interpolation is used.  A method was developed for
forming desired trajectories as a function of time.  It provides the
possibility to form a trajectory on basis of an obtained geometrical
trajectory and allows one to reach a minimum of the same criteria:
time T or expenditure of energy

        g
         1           T
                      T
 e =     Qg dg +     Q RQdt
        g            0
         n

Q - vector of active forces/torques formed by drives, 
R - matrix of energy loss gains. The limitation of the press applied 
  to links manipulator, taken into consideration.

This method is based on the use of the variation approach for task 
decomposition to find the optimal trajectory, which has the fixed
initial and end points g1 and gn; g1 = gn = 0. It presupposes to solve
the Puasson equations.

This work has been conducted with the International Association for
the promotion of cooperation with Scientists - from Independent States
of Former Soviet Union (INTAS).

The title of this project is uTask-level optimal programming of
industrial robot (TaPro).  The coordinator of the project is
Fraunhofer Gesellschaft zur Forderung der angewandten Forschung e.v.,
Institute for Production Systems and Design Technology, Germany.


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