Edinburgh Napier University

  The balancing of an inverted pendulum is a classic control problem of some thirty years or so standing, which
has been revisited of late in the light of recent developments in symbolic computation. The controllability of the
damped double inverted pendulum has previously been investigated with the aid of algebraic computation, and
some interesting results have been obtained. In particular, the transfer functions with respect to the possible
control inputs, for three balancing states, have been calculated symbolically, and the cancelling poles have been
extracted in full algebraic form.
The more complex problem of the triple inverted pendulum, which has, not a curve of uncontrollability (with
respect to the force on the trolley), but a surface of uncontrollability, is considered in this paper. Numerical
and symbolic investigations have been undertaken for the cases when only one of the three damping coefficients
is non-zero, for the quasi-uniform system - i.e. when the masses of the arms and the trolley are set equal, and
the lengths of the arms are set equal. This simpli�cation is necessary to enable the symbolic investigation to be
undertaken. The increased complexity of this problem has necessitated an exploratory numerical investigation
of the transfer functions. The numerical results for the cancelling pole, along with some deductions concerning
its symbolic form, have facilitated a symbolic investigation.