  Underactuated tower crane lifting requires time-energy optimal trajectories
for the trolley/slew operations and reduction of the unactuated swings
resulting from the trolley/jib motion. In scenarios involving non-negligible
hook mass or long rig-cable, the hook-payload unit exhibits double-pendulum
behaviour, making the problem highly challenging. This article introduces an
offline multi-objective anti-swing trajectory planning module for a
Computer-Aided Lift Planning (CALP) system of autonomous double-pendulum tower
cranes, addressing all the transient state constraints. A set of auxiliary
outputs are selected by methodically analyzing the payload swing dynamics and
are used to prove the differential flatness property of the crane operations.
The flat outputs are parameterized via suitable B\'{e}zier curves to formulate
the multi-objective trajectory optimization problems in the flat output space.
A novel multi-objective evolutionary algorithm called Collective Oppositional
Generalized Differential Evolution 3 (CO-GDE3) is employed as the optimizer. To
obtain faster convergence and better consistency in getting a wide range of
good solutions, a new population initialization strategy is integrated into the
conventional GDE3. The computationally efficient initialization method
incorporates various concepts of computational opposition. Statistical
comparisons based on trolley and slew operations verify the superiority of
convergence and reliability of CO-GDE3 over the standard GDE3. Trolley and slew
operations of a collision-free lifting path computed via the path planner of
the CALP system are selected for a simulation study. The simulated trajectories
demonstrate that the proposed planner can produce time-energy optimal
solutions, keeping all the state variables within their respective limits and
restricting the hook and payload swings.
