**6. Numerical results**

348 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

It follows that�� = � at the beginning of the optimization process. At the end, �����. By using this scaling procedure, the final objective function value provides a non-dimensional index that describes the percentage of improvement. As an example, the final value *J*=0.3

Along the investigation of the optimization problem, there are two kinds of solution points (Luenberger, 1984): *local minimum points*, and *global minimum points*. A point **u***\** is said to be a *global minimum point* of *f* over if *f*(**u**) *≥ f*(**u***\**) for all **u**. If *f*(**u**)*> f*(**u***\**) for all **u**, **u** *≠*

There are several search methods devoted to find the global minimum of a nonlinear objective function, since it is not an easy task. Well known methods such as genetic algorithms (Vose, 1999), differential evolution algorithm (Price *et al.*, 2005) and simulated annealing (Kirkpatrick, 1983) could be used in this case. The main characteristic of these methods is that the global (or near global) optimum is obtained through a high number of

As proposed by Santos *et al.* (2005), to use the best feature of local optimization method (low computational cost) and global optimization method (global minimum), it is considered using the so-called tunneling strategy (Levy and Gomez, 1985) (Levy and Montalvo, 1985), a methodology designed to find the global minimum of a function. It is composed of a sequence of cycles, each cycle consisting of two phases: a minimization phase having the purpose of lowering the current function value, and a tunneling phase that is devoted to find a new initial point (other than the last minimum found) for the next minimization phase. This algorithm was first introduced in Levy and Gomez (1985), and the name derives

The first phase of the tunneling algorithm (minimization phase) is focused on finding a local minimum **u***\** of Equation (21), while the second phase (tunneling phase) generates a new initial point **u***0***u***\** where *f*(**u***0*) *f*(**u***\**). In summary, the computation evolves through the

a. Minimization phase: Given an initial point **u***0*, the optimization procedure computes a local minimum **u***\** of *f*(**u**). At the end, it is considered that a local minimum is found. b. Tunneling phase: Given **u***\** found above, since it is a local minimum, there exists at least

In other words, there exists **u***0 Z* = {**u**  – {**u***\**} | *f*(**u**) *f*(**u***\**) }. To move from **u***\** to **u***<sup>0</sup>* along the tunneling phase a new initial point **u** = **u***\** + , **u** is defined and used in the

[(�−�∗)�(�−�∗)]�

that has a pole in **u***\** for a sufficient large value of �. By computing both phases iteratively, the sequence of local minima leads to the global minimum. Different values for � are

(29)

�(�) <sup>=</sup> �(�) <sup>−</sup> �(�∗)

means that the overall objective was reduced to 30% of this initial value.

**u***\**, then **u***\** is said to be a *strict global minimum point* of *f* over .

3. Set ��

4. Update �� <sup>=</sup> ��

� = ��(��� ��� �).

5. Set �[�(�)]=���� � ����.

**5. Global optimization** 

functional evaluations.

from its graphic interpretation.

one **u***0* , so that *f*(**u***0*) *f*(**u***\**), **u***0* **u***\**.

following phases:

auxiliary function

�̅ � � � = ���.

Computational evaluation was performed aiming at evaluating the effectiveness of the proposed methodology, as outlined in Figure 2.

Fig. 2. Computational workflow.

Multi-Criteria Optimal Path Planning of Flexible Robots 351

Fig. 4. Contribution of the global strategy.

Fig. 5. Range of improvement obtained by the global strategy.

process. This effect is graphically presented by Figure 6.

discussed in the following.

Performance indexes are affected differently by the presence of other objectives and the corresponding priority. The influence of the weight factor on individual objectives is

The sum of torque is a performance index to be minimized. The higher the importance of the torque in the objective formulation better the improvement achieved by the optimization

An important point when using multi-objective formulation is the choice of performance indexes. A discussion about the influence of torque, manipulability and end-effector positioning error is presented in the following.

Figure 3 presents the Pareto frontier for the case in which torque and manipulability are considered in the same objective function.

Fig. 3. Minimum value of the performance index.

The abscissa contains several values of the weight factorߙ א ሺͲǡͳሻ. If the factor is zero, only manipulability is taken into account. If the factor is one, only the torque is considered. The factor 0.5 sets the same importance to both objectives. The ordinate is the percentage related to the initial value of the objective function. At the beginning, the objective function value is one. The value 0.10 in the improvement scale means that the final value is 10% of the initial value.

Results presented in the figure confirm that the improvement of manipulability is easier to be achieved by the numerical procedure than the reduction of torque. Those are the final results of the global optimization.

Another important point is the contribution of the global optimization strategy. Figure 4 shows the corresponding improvement achieved.

The abscissa contains several values of the weight factor ߙ א ሺͲǡͳሻ. The ordinate represents a range of values where 1.0 means no improvement, that is, no further improvement was obtained with respect to the local minimum. The value 0.75 means that the new (possibly global) minimum has a value corresponding to 75% of the local minimum.

A general view about the contribution of the tunneling strategy to the local minimum is presented in Figure 5. The reduction varies between 0.7 and 1.0, that is, there was a reduction to 70% of the value of the local minimum in the best scenario and no reduction (the corresponding value is 1.0) in the worst case.

An important point when using multi-objective formulation is the choice of performance indexes. A discussion about the influence of torque, manipulability and end-effector

Figure 3 presents the Pareto frontier for the case in which torque and manipulability are

The abscissa contains several values of the weight factorߙ א ሺͲǡͳሻ. If the factor is zero, only manipulability is taken into account. If the factor is one, only the torque is considered. The factor 0.5 sets the same importance to both objectives. The ordinate is the percentage related to the initial value of the objective function. At the beginning, the objective function value is one. The value 0.10 in the improvement scale means that the final value is 10% of the initial

Results presented in the figure confirm that the improvement of manipulability is easier to be achieved by the numerical procedure than the reduction of torque. Those are the final

Another important point is the contribution of the global optimization strategy. Figure 4

The abscissa contains several values of the weight factor ߙ א ሺͲǡͳሻ. The ordinate represents a range of values where 1.0 means no improvement, that is, no further improvement was obtained with respect to the local minimum. The value 0.75 means that the new (possibly

A general view about the contribution of the tunneling strategy to the local minimum is presented in Figure 5. The reduction varies between 0.7 and 1.0, that is, there was a reduction to 70% of the value of the local minimum in the best scenario and no reduction

global) minimum has a value corresponding to 75% of the local minimum.

positioning error is presented in the following.

Fig. 3. Minimum value of the performance index.

shows the corresponding improvement achieved.

(the corresponding value is 1.0) in the worst case.

results of the global optimization.

value.

considered in the same objective function.

Fig. 4. Contribution of the global strategy.

Fig. 5. Range of improvement obtained by the global strategy.

Performance indexes are affected differently by the presence of other objectives and the corresponding priority. The influence of the weight factor on individual objectives is discussed in the following.

The sum of torque is a performance index to be minimized. The higher the importance of the torque in the objective formulation better the improvement achieved by the optimization process. This effect is graphically presented by Figure 6.

Multi-Criteria Optimal Path Planning of Flexible Robots 353

Results comparing the effects of the end-effector positioning error and torque are presented next.The Pareto frontier for torque and end-effector displacement is shown in Figure 8.

Despite the small difference in the results, for all cases studied the optimum is between 31% and 32% of the initial value of the objective function. The contribution of the global optimization strategy is also similar for all cases and the global minimum was found

Fig. 8. Minimum value of the performance index.

Fig. 9. Contribution of the global strategy.

between 80.4% and 81.4% of the value of the local minimum.

Fig. 6. Influence of weight on the torque.

Manipulability is a performance index to be maximized. It follows that larger the performance index value, better the performance. The influence of the weight coefficient over the manipulability is presented in Figure 7.

Fig. 7. Influence of weight coefficients on the manipulability.

According to Figure 7, the lower the value of the weight coefficient, better the manipulability. Note that manipulability and torque are conflicting objectives that are addressed by the present formulation.

Manipulability is a performance index to be maximized. It follows that larger the performance index value, better the performance. The influence of the weight coefficient

According to Figure 7, the lower the value of the weight coefficient, better the manipulability. Note that manipulability and torque are conflicting objectives that are

Fig. 6. Influence of weight on the torque.

over the manipulability is presented in Figure 7.

Fig. 7. Influence of weight coefficients on the manipulability.

addressed by the present formulation.

Results comparing the effects of the end-effector positioning error and torque are presented next.The Pareto frontier for torque and end-effector displacement is shown in Figure 8.

Fig. 8. Minimum value of the performance index.

Despite the small difference in the results, for all cases studied the optimum is between 31% and 32% of the initial value of the objective function. The contribution of the global optimization strategy is also similar for all cases and the global minimum was found between 80.4% and 81.4% of the value of the local minimum.

Fig. 9. Contribution of the global strategy.

Multi-Criteria Optimal Path Planning of Flexible Robots 355

Results presented in Figures 6 and 7 appear frequently when conflicting objectives are taken into account in the optimization procedure. On the other hand, correlated indexes usually

This analysis suggest that torque and manipulability are better choices to compose a multicriteria analysis as compared with results given by torque and end-effector positioning error. This is justified by the fact that end-effector disturbance is affected by the torque profile. As a result, if the torque is reduced, then the effect of flexibility at the end-effector is

This work was dedicated to multi-criteria optimization problems applied to flexible robot manipulators. Initially, the model of deflection, torque and manipulability were presented for the system analyzed. Next, optimal control formalism and multi-criteria strategy were outlined. It was concluded that the effectiveness of the numerical procedure depends on the

The first numerical evaluation considered torque and manipulability as performance indexes. The effect of changing the weighting coefficients was presented in Figure 3. This is a typical trend of concurrent objectives, i.e., when one performance index is improved, the other degenerates. In this context it is important to identify the contribution of each index, as

The effect of the global optimization procedure was also discussed. This point was shown to

The second numerical evaluation considered the end-effector error positioning and the torque as performance indexes. It was shown that there are no significant changes in the performance index while the weighting factor values are changed. It is explained by the correlation between the objectives, i.e., both indexes are directly affected by the joint torque.

have small deviation in the values obtained, as presented in Figures 11 and 12.

Fig. 12. Influence of weight on the end-effector positioning error.

choice of the objective functions and weighting factors.

also reduced.

**7. Conclusion** 

shown in Figures 6 and 7.

be effective to obtain the global minimum.

The average contribution of the tunneling process to the global minimum is shown in Figure 10.

Fig. 10. Range of improvement obtained by the global strategy.

The optimal value of torque index is between 31.6% and 31.9% of the initial value, as presented in Figure 11.

Fig. 11. Influence of weight on the torque.

The end-effector positioning error index was increased about 36 times, as presented in Figure 12.

The average contribution of the tunneling process to the global minimum is shown in Figure 10.

The optimal value of torque index is between 31.6% and 31.9% of the initial value, as

The end-effector positioning error index was increased about 36 times, as presented in

Fig. 10. Range of improvement obtained by the global strategy.

presented in Figure 11.

Fig. 11. Influence of weight on the torque.

Figure 12.

Fig. 12. Influence of weight on the end-effector positioning error.

Results presented in Figures 6 and 7 appear frequently when conflicting objectives are taken into account in the optimization procedure. On the other hand, correlated indexes usually have small deviation in the values obtained, as presented in Figures 11 and 12.

This analysis suggest that torque and manipulability are better choices to compose a multicriteria analysis as compared with results given by torque and end-effector positioning error. This is justified by the fact that end-effector disturbance is affected by the torque profile. As a result, if the torque is reduced, then the effect of flexibility at the end-effector is also reduced.
