University of Groningen
Constructing tensegrity frameworks and related applications in multi-agent formation control
Yang, Qingkai
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Publication date: 2018
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Yang, Q. (2018). Constructing tensegrity frameworks and related applications in multi-agent formation control. University of Groningen.
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Conclusions and future work
T
his chapter summarizes the main results of this thesis and indicates the possiblefuture research directions.
8.1
Conclusions
This thesis has addressed the problem of constructing tensegrity frameworks and has discussed the related applications in formation control for multi-agent systems. We have discussed how to design tensegrity frameworks in two situations: one is to assign different types of members to chosen pairs of vertices; the other is to grow existing tensegrity frameworks through merging. To fully utilize tensegrity frameworks, we have also explored stress-based formation controls in different scenarios. Now, we provide the specific conclusions for each technical chapter.
In Chapter 3 we have studied the merging of infinitesimally rigid and rigid tensegrity frameworks in the plane, respectively. In the case of merging infinitesi-mally rigid tensegrity frameworks, we have shown that infinitesimal rigidity can be preserved by guaranteeing the existence of the proper self-stress in combination with the infinitesimal rigidity of the corresponding bar framework. In addition, we further discussed the influence on the pre-existing members caused by merging. By looking at the sign of the new stress, it has been verified that the types of the members, viz. cable or strut, can also be preserved. With respect to rigid tensegrity frameworks, we have proved the existence of appropriate linking members assur-ing the rigidity of the combined tensegrity framework. We have also proposed a distance perturbation method to determine the type of the new members. We have presented the explicit expression on assigning the type of the fourth member based on rigidity matrix.
Chapter 4 has extended the results in Chapter 3 from local rigidity to super stability. In Chapter 4, we start with the problem of how to conduct vertex addition and edge splitting operations on super stable tensegrity frameworks in parallel to bar frameworks. By inserting a set of members, it has been proven that the obtained tensegrity framework is super stable as well. We have also clarified the type of the new members depending on the positions of the new vertex with respect to the existing vertices. In addition, a numerical algorithm has been proposed to specify
118 8. Conclusions and future work
the values of the stresses of the new members. Analogously, we have illustrated that super stability is also preserved under edge splitting. We then proceed to another class of approaches used for growing tensegrity frameworks, i.e., merging. When the super stable tensegrity frameworks share at least d + 1 vertices in d-dimensional Euclidean space, we provide a mild sufficient condition under which the merged tensegrity framework is still super stable. We finally studied optimal merging of super stable tensegrity frameworks when d reduces to 2 or 3. The detailed growing procedures have been provided for better illustration.
We have created some connections between tensegrity frameworks and forma-tion control in Chapter 5, where the problem on how to construct universally rigid tensegrity frameworks under given configurations is first raised, followed by the formation stabilization problem for multi-agent systems modeled by single integra-tors. The construction of universally rigid tensegrity frameworks is equivalent to the design of a positive semi-definite stress matrix with rank n − d − 1 since the members can be consequently assigned based on the associated stress. Because of the fact that the elements of the stress matrix lie in the null space of the generalized configuration matrix at equilibrium, we have designed a numerical algorithm to seek a stress matrix that agrees with a universally rigid tensegrity framework. In view of the desirable features that the cables and struts have strict upper and lower bounds on lengths, respectively, we proposed a class of nonlinear distance-based control laws by mapping the multi-agent system to a virtual tensegrity framework. It has been shown that the formation can be stabilized using the proposed con-trol strategies and at the same time the inter-agent distances never exceed their limitations during the evolution.
In Chapter 6 we further promote the use of tensegrity frameworks in formation control. As a representative application, we investigate the formation scaling problem under the virtual tensegrity framework with vertices and their associated members denoting the agents and links between them, respectively. We first explore the conditions under which the formations can be scaled to the desired size using the proposed stress-based control laws. It has been shown that d pairs of agents are sufficient to change the size of the whole formation if their position vectors
linearly span the whole space Rd. By introducing the orthogonal projections to the
controllers of a portion of agents, we have also proved that the size of the formation can be decided by only one pair of agents. In addition, the formation is shown to be uniquely determined up to the translation and scaling of the given configuration among all the possible affine transformations. These results have been further extended to satisfy the requirement that the scaling information is known to only one agent. In this case, we have proposed a new class of estimator-based control laws, such that the whole formation can be driven to the prescribed size.
Chapter 7 has solved the problem of formation tracking to a given reference signal. In order to precisely follow the given trajectory, we have designed a
distributed finite-time estimator with the advantage that the explicit knowledge of the bound of the agents’ speed is not required in contrast to the existing results. Based on rigidity graph theory, we have proposed distance-based control laws by employing the output of the estimator. It has been shown that the shape of the formation converges to the pre-defined one, and its centroid follows the given reference.
8.2
Future work
Although in this thesis we have solved a sequence of problems related to the construction of tensegrity frameworks and the applications in formation control, several problems still need to be considered in future research. In this section, we identify some future topics listed following the order of the chapters.
• Chapter 3: It has been shown that rigidity and infinitesimal rigidity can be preserved in the process of merging given static separate tensegrity frame-works. However, these results might not hold when the tensegrity framework moves due to the change of geometric relationships with respect to each other. Hence it is desirable to analyze the influence imposed by motion and design rigidity (including infinitesimal and universal rigidity) maintenance control laws.
• Chapter 4: In addition to the research on local rigidity (Chapter 3) and super-stability of tensegrity frameworks, it is also important to investigate the strategies of augmenting globally rigid tensegrity frameworks systematically. The procedures therein can give more freedoms when setting stresses for newly added members. In addition, it is of great interest to study splitting tensegrity frameworks in contrast to merging as have been discussed in this chapter.
• Chapter 5: Regarding the construction of tensegrity frameworks, the cost and complexity of the whole framework are important criteria especially in practical applications. Therefore it is appealing to investigate optimization based construction algorithms to further reduce the number of members required for creating a tensegrity framework.
• Chapter 6: The scaling parameter in this chapter is assumed to be constant. However, the size of the formation might be time-varying. This motivate us to extend our current results to the case involving a dynamic signal indicating the size of the formation. In addition, it is of great interest to make use of tensegrity frameworks in more cooperative control tasks for robots, e.g., autonomous formation enclosing and transformation.
120 8. Conclusions and future work
• Chapter 7: One future study is to generalize the results to fully local coordi-nate systems, where the orientations might be inconsistent. Another possible exploration is to develop more powerful control algorithms to cope with system constraints, such as input saturation and transmission delay.
