This chapter presented a method for constructing the configuration spaces for a group of robots. The task configuration space we construct is a space composed of
polytopes, where the agents cannot collide, lose communication, or violate any of the set constraints. Any state that is within the task configuration is considered a safe state.
We choose to model the configuration space as the Cartesian product of the configuration spaces of the individual robots in order to preserve as many solutions as possible. Since the task configuration space is a space composed of polytopes with matching facets, we can automatically synthesize controllers on this space which drive the group to the goal configuration without exiting the space in Chapters 5 and 6. This means the group will avoid violating any of the set constraints, inter-robot collisions, and collisions with the environment.
One limitation of this algorithm is that the complexity is exponential in the
number of agents. Because of the size of Pmax, constructing CT is the most time
consuming portion of preprocessing. By using a heuristic-based graph search such as that presented in Section 3.2, we can significantly decrease precomputation time.
Furthermore, although Pmax is exponential in the number of agentsn, the proximity
constraint dependency combined with connectivity constraints significantly decrease the number of polytopes in the space, as we will show in Chapter 5.
In the following chapter, we construct a different configuration space specifically for a group of robots without unique goal positions. Then, we will translate the paths on the polytope and simplex graphs into feedback controllers to solve Problems 3.1.7, 3.1.8, and 3.1.9.
Chapter 4
Group Configuration Space
Modeling
In this chapter we construct configuration spaces specifically for groups of robots by using an abstraction. The abstraction defines a virtual boundary for the group of robots which is used to navigate the group through the space. By this construc- tion, we are naturally able to establish bounds on the positions of the robots. This allows us to guarantee safety–the controllers are designed so that the virtual bound- ary associated with the abstraction does not pass through the obstacles. Thus the
complexity of the problem of synthesizing controllers is independent of the number
of robots, which promises scalability to large groups.
4.1
Problem formulation
Consider again a group G, of n kinematic agents with dynamics (3.1). The group
must together navigate an obstacle-filled environment to a specified task location. Since the group is working closely together, we assume that communication occurs
very rapidly within the group, so that control can be centralized over the group. We use an abstraction on the group of robots to reduce the computational com-
plexity of the problem of navigating n robots. Our abstraction defines a virtual
adaptiveboundary for the group of robots, while controllers at the robot level ensure
the robots stay within the boundary. The abstraction boundary is determined by automatically synthesized controllers, and the robots react to the changing size and shape of the abstraction boundary. Thus, a group of robots can navigate a space knowing only the boundary and the local (within the boundary) state of the group, decoupling planning and control of the agents from the physical environment. The
dimension of the abstraction is independent of the number of robots n. We discuss
in detail the specific abstraction we use in Section 4.2.
Figure 4.1 is a graphical representation of the hierarchical structure of the ap- proach. At the bottom level, individual robots execute the continuous controllers that are designed to satisfy specifications and desired formation properties. At the middle level, individual robots interact with each other to maintain constraints (this level may not be necessary for all problems). At the top level, the abstracted group navigates the space while maintaining constraints designed to ensure there is enough room for the robots. We sum controllers at these levels to generate the input for the individual robots. We assume for now that there are no obstacles within the group boundary.
Our hierarchical control system enables a multiple time scale approach, as shown in Fig. 4.1, to eliminate the possibility of local minima which may occur when sum- ming two controllers. Group dynamics (motion within a group) are assumed to evolve on a much faster time-scale than abstraction dynamics (overall motion of an entire group); sufficient time scale separation between group and abstraction dynamics ensures the two controllers can be designed independently.
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Figure 4.1: Hierarchical structure. At the bottom level, robots implement the con- tinuous controller. At the top level, the abstracted group navigates the space.
Since the abstraction is independent of the robots’ configuration, synthesis of and planning in the abstraction workspace occurs in advance. At least one robot must have knowledge of the evolution of the abstraction over time. This information, as well as individual robot state information, propagates through the group rapidly via explicit communication.