Paper Submitted to CDC 2017.

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]]>Communication is not only essential for sharing sensor measurements and performing diagnostics, it is often an integral part of the closed loop control mechanism. In fact, in many applications and scenarios such as extra-terrestrial exploration, high precision manufacturing, and multi-robot testbeds, the robots frequently rely on communicating with a centralized decision maker for their velocity or position commands. In such situations, a failure in the communication network can severely hinder the motion of the robots and performance of the algorithm. This is the premise behind this project, whereby the adverse effects caused by intermittently failing communication networks are mitigated.

In order to calculate the safe time horizon, we first compute the set of all possible locations that can be reached by a robot within a given time (i.e., the reachable set). This is followed by computing the time horizon for which each robot lies outside the reachable set of other robots. But, for the differential-drive robots considered here, performing such set-membership tests is computationally expensive owing to the non-convexity of the reachable set. Consequently, the reachable set is over-approximated by enclosing it within an ellipse whose convex structure allows for simpler set-membership tests and finite representation. By minimizing the area of the ellipse enclosing the convex hull of the reachable set, we obtain the best ellipsoidal over-approximation of the reachable set in terms of the accuracy and effectiveness of set-membership tests.

Taking advantage of this ellipsoidal approximation, the safe time horizon is defined as the longest time duration for which the robot lies outside the ellipsoidal reachable sets of other robots. Thus, the robot experiencing communication failure can execute its last received velocity command for the corresponding safe time horizon and remain safe. Beyond this, the robot stops moving.

**Investigators:**

- Siddharth Mayya
- Magnus Egerstedt

- Zak Costello
- Yancy Diaz-Mercado
- Rowland O’Flaherty
- Daniel Pickem
- Thiagarajan Ramachandran

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In general, self-reconfigurable systems are comprised of individual agents which are able to connect to and disconnect from one another to form larger functional structures. A system with agents or modules that have distinct capabilities, shapes, or sizes, is called a heterogeneous system. Alternatively, a system with identical and interchangeable modules is referred to as a homogeneous system. In this project we will present algorithms that reconfigure homogeneous systems and treat self-reconfiguration as a two- and three-dimensional coverage problem. However, minor modifications to the core algorithm could extend its applicability to heterogeneous systems as well.

Self-reconfiguration in this work tries to solve the following problem. Given an initial geometric arrangement of cubes (called a configuration) and a desired target configuration, the solution to the self-reconfiguration problem is a sequence of primitive cube motions that reshapes/reconfigures the initial into the target configuration. A full sequence from an initial to a target configuration is shown below together with convergence times for configurations of various sizes.

Our approach to homogeneous self-reconfiguration is fully decentralized such that no central decision maker is required. In such a decentralized setup, even though each agent acts as a purely self-interested individual decision maker with local information only, our method guarantees convergence to the global target configuration. Decision making requires no (in the two-dimensional case) or limited communication (in the three-dimensional case). Each agent computes transition probabilities for all actions possible at the current time step (according to a fixed motion model). An agent then selects an action based on these probabilities and accepts/rejects it according to an acceptance probability. The key to guarantee global convergence to the target configuration lies in these acceptance probabilities that are computed according to the Metropolis-Hastings algorithm and ensure that the only stochastically stable state is the target configuration.

- Daniel Pickem
- Magnus Egerstedt
- Jeff Shamma

- [1] D. Pickem, M. Egerstedt, J.S. Shamma, “A Game-theoretic Formulation of the Homogeneous Self-Reconfiguration Problem,”
*Decision and Control (CDC), 2015 IEEE 54nd Annual Conference on*. IEEE, 2015.

How one single operator should influence the performance of a large group of robots is not a simple question in that it depends on various factors, including the size and composition of the team as well as the objective task to be performed. This work introduces music theory as a new approach for multi-robot control that gives the human operator the power of specifying goal locations while controlling the spacial arrangement of the team.

In particular, we are interested in applying harmony rules to the user’s control input provided through a piano in order to modify the control parameters of a swarm. The aim is to abstract the harmonizing process: instead of generating sound combinations — namely chords — that serve as an accompaniment of a melody line, we apply harmony rules to an objective path (the melody) to obtain a geometry to be displayed by the group (the chord).

A leader-follower strategy has been chosen to illustrate the musical-based control. The leader is controlled by the piano input, which is directly mapped to the target locations to be visited by it. The followers must adopt a formation geometry according to the characteristics of the chord that harmonizes the melody tone. In general, within a diatonic scale, each tone could be potentially harmonized using three different chords, that is, the three triads containing that chord. This leads to different chord types associated to each note and, therefore, different geometries. However, harmony rules restrict the choice of a chord for a melody tone, depending on the harmonic progression it generates..

The harmonization process used in this project, where we create the chords on top of the bass line, also relates to the leader-follower paradigm. The melody given by the human operator and, thus, the leader robot are identified with the bass line of the composition. The accompaniment chords, subordinate to the melody, control the movement of the followers. Given this hierarchy, we have encoded the harmony rules that generate the followers’ chords as a finite state machine where the states correspond to each one of the chord’s note and the transitions are triggered by the changes in scale degree commanded by the leader, that is, the input melody.

- María Santos
- Magnus Egerstedt

Trust- based coordination occurs when a team of robots and possibly humans have to work together to achieve some goal, but each robot (and/or human) may not trust all of the other robots (and/or humans). If a robot doesn’t trust one of its neighbors, it may not want to heavily weight the information obtained from that neighbor. So it seems that this trust should play a role in the dynamics of the agents. In addition, the trust levels should depend on whether an agent’s neighbor is doing what it is expected to do. For example, when trying to meet at a common location in the 2D plane, an agent would expect its neighboring agents to be moving towards it.

This work introduces a trust model that couples the change in performance in a team of agents to how the

agents perceive (or trust) each other. This combination of social dynamics and physical update laws not only changes the performance of the system, but it has the potential to make the performance deteriorate in a dramatic fashion. In fact, in the two-agent case, it is shown that the system exhibits finite escape time through an invariance result that carries over also to larger systems and more elaborate trust models. The invariance result states that an increase in performance must be accompanied by an increase in the total trust in the network (and vice versa for deteriorating performance).

We can show that, in the 2-agent case, the agents’ states may actually blow up (go to infinity) in finite time, if the sum of the initial trusts of the agents is negative. This is shown in the figure above. The two trajectories that approach zero depict a two agent case in which the sum of the initial trusts is positive, while the two trajectories that go off to infinity depict the case in which the sum of the initial trust values is negative.

This model also has connections to the belief and group polarization phenomena encountered in group processes driven by social interaction dynamics. It also allows us to start thinking about how humans should interact with teams of robots and what role trust plays in this interaction.

- Tina Setter
- Andrea Gasparri
- Magnus Egerstedt