Problem formulation
A multi-objective optimization problem was defined to minimize
the maximum forces of the actuation cables. The model was
developed in modeFRONTIER, which is an integration platform
for process automation, optimization, and data mining. The
optimization variables are a subset of the 36 elements of T. This
subset is the result of a sensitivity analysis performed to discard
the less discriminating parameters.
Both the sensitivity analysis and the optimization problem
must compute the maximum cable forces for a given routing
configuration.
Maximum cable forces of the robot
The maximum cable forces are the solution of the static equilibrium
of the system formulated as an optimization problem.
For each joint, the cable
forces that ensure the
equilibrium depend on the
actuation forces affecting
the subsequent joints,
and the configuration
of the joint, so starting
from the last module we
were able to derive the
actuation forces by solving
an optimization problem
in which the optimization
variables were the angles
of the joint, and the
objective function was the
actuation force.
The flowchart of this optimization problem is shown in Fig. 3 and
the parameters used for the optimization are provided in the table
below.
Sensitivity analysis
Once the maximum forces on the actuation
cables for a given matrix T were obtained, a
sensitivity analysis was performed to identify
the most significant parameters of the routing
matrix. To meet all the above constraints,
the DOE generation was implemented in
MATLAB, and the scatter matrix tool was
used to verify that this DOE was able to cover
the entire domain of variables.
The main and interaction effects of the 36 T parameters on the
cable forces were analyzed using SS-ANOVA. In particular, the
forces of the first and second joints, which are the most stressed,
and the norm of the cable force vector were selected as the
performance indicators. Considering all the independent terms
of T, this method requires a DOE table of 703 elements. This
analysis made it possible to reduce the number of the optimization
variables from 36 to 17.
Optimization problem
This optimization problem
aims to find the optimal
routing matrix to minimize
the cable forces on the first
and second joints and on
the norm of the cable force
vector.
As shown in Fig. 4,
modeFRONTIER generates
the set of 17 variables
that satisfy the routing
constraints, then MATLAB
takes these parameters as
input and tries to generate a
feasible T matrix by verifying the cable intersection constraint. If
this constraint is satisfied, the objective functions are evaluated,
otherwise modeFRONTIER generates a new set of input variables.
Since the optimization variables are integers and the problem is
multi-objective, the Multi-objective Genetic Algorithm II (MOGAII)
algorithm has been chosen for the optimization strategy using
manual settings. The table below lists the solver parameters.
