# Multi-Objective Control of an Active Tensegrity Structure

Multi-Criteria Control

Many combinations of contractions and elongations of active struts can satisfy a single serviceability objective to an acceptable degree. This presents an opportunity to enhance control command search through use of additional objectives. Additional objectives should not significantly decrease control command quality with respect to the slope objective. Goals are to increase robustness of both the structure and the active control system in order to carry out multiple control events over service lives. The following four conflicting objectives are used to guide search:

• Slope: maintain top surface slope of the structure constant when subjected to loading,

• Stroke: maintain actuator jacks as close as possible to their midpoint,

• Stress: minimize stress of the most stressed element,

• Stiffness: maximize the stiffness of the structure.

The general form of a multi-objective optimization problem can be expressed as follows:

Minimize objective functions f (x)

subject to inequality constraints g(x) < 0

and equality constraints h (x) = 0

where x є R", f (x) є ^k, g(x) є Щ. m, and h(x) є ^p. Here, n represents the number of variables, k the number of objective functions, m the number of inequality constraints and p the number of equality constraints.

A Pareto filtering approach is employed in order to avoid the use of weight factors. In case of a multi-objective minimization task, a solution x* is said to be Pareto optimal if there exists no feasible vector of decision variables x which would decrease some objective without causing a simultaneous increase in at least one other objective. This concept results in a set of solutions called the Pareto optimal set. The vectors x* corresponding to the solutions included in the Pareto optimal set are called non-dominated (Pareto, 1896).

The multi-objective search method adapted to our tensegrity structure serviceability control task involves building a Pareto optimal solution set and selecting one solution (see Figure 4). The Pareto optimal solution set is identified according to the four objectives and the five constraints described above. Solution generation and Pareto filtering is carried out using the ParetoPGSL algorithm. Solutions are generated in order to minimize all objectives. Dominated solutions are rejected. ParetoPGSL stops after 1500 generated solutions since solution quality does not improve any further.

The selection strategy that is adopted hierarchically reduces the solution space until identification of a control command. It is developed in four steps and reflects the importance of the objectives. Control commands for which slope compensation is less than 95% are first rejected. In practical situations, slope compensation would be acceptable if its value was above this threshold. To keep objectivity with respect to the three remaining objectives, the remaining solutions are divided into thirds according to solution quality. The worst third of the solutions with respect to the stroke objective is rejected. The worst half of the remaining solutions with respect to the stress objective is then rejected. Finally, the best solution with respect to the stiffness objective is identified among solutions that are left. This becomes the control command that is applied to the structure. Therefore, each of the three objectives in the last three steps leads to rejection of the same number of solutions.

Control solutions describe the structural configuration when slopes are compensated. Sequences of application of control commands that transform the altered slope state to the compensated slope state involve verifying that no failure would happen during intermediate steps. The control command is divided into 1 mm steps. Strut contractions are placed at the beginning of the sequence and elongations at the end. In this way, energy is generally first taken out of the structure before it is added. Calculations are made using the dynamic relaxation method. The position of the structure is evaluated for each 0.1 mm of actuator travel. The sequence is then applied to the physical structure for experimental validation.

 Figure 4: Multi-objective methodology: Hierarchical selection of Pareto optimal solutions Results of Multi-Criteria Control

This methodology is tested for 24 load cases involving up to two vertical point loads from 391 N to 1209 N in magnitude (see Table 2). Loaded nodes are numbered according to Figure 2. One and two point load cases numbered from 1 to 24 are presented in Table 2.

Table 2. Load cases applied to the structure

 Load case Loaded Load magnitude 1 26 625 2 26 900 3 26 1209 4 32 625 5 32 859 6 32 1092 7 37 391 8 37 550 9 37 700 10 48 391 11 48 550 12 48 700 13 6 1092 14 37 and 45 391 15 37 and 45 624 16 37 and 45 742 17 39 and 48 157 18 39 and 48 215 19 39 and 48 274 20 41 and 50 391 21 41 and 50 624 22 45 and 48 391 23 45 and 48 624 24 45 and 48 742

Examine load case 5: 859 N point load at node 32. Pareto optimal solutions are generated using the ParetoPGSL algorithm (see Figure 5). Solutions are presented in four dimensions with respect to the four objectives. The slope objective is shown on the vertical axis. Stroke and stress objectives are represented with the horizontal axis. The gray bar evaluates the stiffness objective. Values close to zero are considered best for all objectives.

Pareto-optimal solutions

The first step of the hierarchical selection strategy consists of rejecting all solutions for which slope compensation is less than 95% (see Figure 6).

The second step of the selection strategy involves dividing the remaining solution set into three parts according to stroke objective. The worst third is rejected (see Figure 7).

The third step of the selection strategy results in dividing the remaining solution set into two parts according to stress objective quality. The worst half is rejected (see Figure 8).

Figure: 8. The worst half of the previous set with respect to stress has now been rejected

The last step of the selection strategy consists of identifying the control command as the best solution with respect to stiffness objective. (see Figure 8). This solution represents the configuration of the structure when the slope is compensated. Sequence of application of the control command is then calculated to verify that no failure would happen and to observe slope evolution. The control command is applied to the loaded physical structure for experimental validation (see Figure 9). Slope evolution is plotted versus steps of 1mm of actuator travel.

Figure 9: Experimental and numerical slope compensation sequence

Numerical simulation gives an altered slope of -147mm/100m and a compensated slope of 1mm/100m (99% compensated). Experimental testing gives an altered slope of 138mm/100m and a compensated slope of -4mm/100m (97% compensated). The average actuator travel is 1.5 mm. Stress

Table 3. Multiple load application scenario

 Load event Loaded Load Magnitude 1 32 391 2 50 391 3 37 391 4 48 391 5 26 391 6 6 150

values are numerical only because the structure is not equipped with force sensors that would provide experimental data. Simulation and laboratory test results for slope are generally in good agreement.

Structural control for multiple load application events (Table 3) is presented in Figure 10. Slope evolution is plotted versus steps of 1mm of actuator travel. Structural behavior when control commands are identified using multi-objective search and single objective search are evaluated. Control commands are more rapidly effective when they are identified with multi-objective search. Single objective control command exhibit a more pronounced zig-zag profile that requires more steps to correct the slope. Multi-objective commands are useful to maintain robustness of both the structure and the control system whereas in single objective sequence no such maintenance can be assured. At the sixth control command multi-objective method makes it possible to compensate the slope whereas a single objective method leads to buckling of a strut.

 Figure 10: Multiple loads applied sequentially: multi-objective and slope-objective control commands

Conclusions

This paper presents a study of the application active control to a tensegrity structure so that opportunities for innovative applications can be identified. The following conclusions come out this research: •

These results lead toward more autonomous and self-adaptive structures that evolve in changing environments.

Acknowledgments

The authors would like to thank the Swiss National Foundation for supporting this work. Dr. E. Fest built the structure and the control system. Dr. B. Domer improved control and implemented the case – based reasoning study. Prof. B. Raphael provided support during programming of the control system. We are also grateful to Dr. K. Shea (University of Cambridge, UK), Passera & Pedretti SA (Lugano, Switzerland) and P. Gallay (EPFL, Switzerland) for their contributions.

References

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