The cables pass through the robot along a precise path and their interaction with the structure generates loads that affect the robot itself. These loads vary based on the robot’s payload and the articulation angles during movement. Different cable routes change the loads on the cables significantly, so it is possible to investigate the behavior of the forces by varying the path followed by each cable.
Each joint is driven by two antagonist cables anchored at points A and B, as shown in Fig. 2. There are 20 ports on the joint, arranged in a circumference, through which the cables can pass. The cable routing matrix T describes the ports through which the cables pass. T is an upper triangular matrix and any element t_i,j represents the port on the joint i through which the cable that guides joint j passes. The number of free variables can be reduced from 55 to 36 by considering the following design choices:

• t_i,i =5: the anchor points on the joint are aligned with port 5;
• t_i,i+1 =0 or 10: the cables operating joint i+1 do not change direction in module i.

In addition, the elements of the routing matrix are subject to some constraints:

• t_i,j ≠t_i,k, where ijoint and j≠k. This means that within joint i, cable j cannot pass through the same port as cable k.
• t_i,j )≠t_i,k +10, where ijoint and j≠k. Cable k has an antagonist cable that passes in the opposite direction inside the joint, so that port is already assigned.

Furthermore, the routings that satisfy these constraints are not always feasible because cables may intersect along the structure of the robot.
The goal of this study is to find the routing matrix that minimizes the maximum forces of the actuation cables.

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.