The swarm paradigm of multirobot cooperation relies on a distributed architecture, where each robot makes its own decisions based on locally available knowledge. But occasionally the swarm members may need to share information about their environment or actions through some type of ad hoc communication channel, such as a radio modem, infrared communication, or an optical connection. In all of these cases robust operation is best attained when the transmitter/receiver robot pair is (1) separated by less than some maximum distance (range constraint); and (2) not obstructed by large dense objects (line-of-sight constraint). Therefore to maintain a wireless link between two robots, it is desirable to simultaneously comply with these two spatial constraints. Given a swarm of point robots with specified initial and final configurations and a set of desired communication links consistent with the above criteria, we explore the problem of designing inputs to achieve the final configuration while preserving the desired links for the duration of the motion. Some interesting conclusions about the feasibility of the problem are offered. A potential field-based optimization algorithm is provided, along with a novel composition scheme, and its operation is demonstrated through both simulation and experimentation on a group of small robots.
Large teams of mobile robots, referred to as
However, in many applications, it is impossible or inefficient to employ truly independent control algorithms on each agent. In order to complete their task the swarm members may need to share information about their intentions or their environment. Indeed, many proposed control laws in the literature require that each member of the swarm is connected to the group through some type of ad hoc, low-power, wireless communication channel, such as a radio or optical links. Power limitations and phenomena such as secondary reflections and shadow effects create a variety constraints on the relative positions of the transmitter and receiver. We abstract these more complex electromagnetic phenomena and work with a simpler two-component communication constraint. The first is a limitation on the maximum distance between the transmitter and receiver, referred to here, and in other works, as the
In this paper we address the problem of navigating the swarm, in the plane, from an initial configuration to a specified final configuration, while maintaining a pre-specified list of wireless links between certain robots (range plus line-of-sight). After a review of related work below, we provide a formal problem statement in Section
In this section we review some common notions of robot swarm connectivity and their role in flocking behavior and formation control. We then narrow the scope of the discussion to related work on explicit control of swarm connectivity, and the relationship to the approach presented here. We end with a discussion of the line-of-sight constraint.
Most discussions on connectivity within robot swarms employ a neighbor or proximity graph modeling paradigm [
All works on flocking and formation control rely on some underlying notion of swarm connectivity to prove stability and convergence. In general the primary objective of
There are several works that explicitly treat connectivity as the primary control objective. They can be loosely divided into
An example of a feedback-based method is in [
Like the works discussed above, the work presented here and our earlier work [
Several authors in the wireless networking community have studied the adverse impact of transmitter/receiver occlusions from an electromagnetic field strength perspective and the modeling implications of this for mobile robotics have been discussed in detail [
In some of these cases, loss of line of sight is considered an unfortunate “fact of life” [
The principle difference between the connectivity work discussed in the previous section and the work presented in this paper, is that we modify the local connection/edge formation criteria to explicitly maintain line-of-sight constraints as well as the widely used single-hop, range constraint. To our knowledge this issue has only been considered in a few previous works. A behavior-based approach is described in [
Given
Any given swarm state
Note that
We are concerned with the following problem (see Figure
The basic problem considered in this paper. Design a control law to guide the swarm from the initial position,
Given an initial connected configuration
Clearly there are certain combinations of the free space
If
Of particular interest in this paper is the
Obviously, when
Furthermore, if
A path
For a given cycle
Geometry of the line-of-sight problem, required to compute
Frames depict an example of a combination of
To ensure the existence of solutions in this paper we only consider path-connected free spaces; and
As remarked earlier, in order to maintain an edge
This notion is difficult to formalize however. The traditional path-homotopy equivalence relation does not apply because the endpoints of
Given two paths
Later we will explore some connections between the definitions in this section and our control laws. Namely, our range potential produces motions equivilent to the straight line homotopy; the line-of-sight potential preserves path-loop homotopic equivalence; the pathloop homotopic equivalence condition suggests that a truly distributed controller is incapable of solving the problem for arbitrary initial conditions—either some “lead" robot must select a path class for the entire swarm or some type of bidirectional messaging must be used to reach a dynamic consensus on path-class selection.
The range constraint dictates that if The potential only possesses minima at configurations where the range constraint is satisfied. The partial derivatives are antisymmetric: The motion induced by
If two robots
The left frame illustrates a situation where line of sight is not maintained (no straight-line homotopy between paths; loop is not homotopic to constant loop). The right frame shows two paths that maintain line of sight (straight-line homotopy between paths; loop is homotopic to the constant loop).
We point out several observations. The potential only possesses minima at configurations where the line-of-sight constraint is satisfied. The partial derivatives are symmetric The motion induced by The set of measure zero, along which
To see why the third property holds, consider two paths that possess path-loop equivalence, such as in Figure
In this paper we use Navigation Functions as the basis for ensuring the goal completion portion of the problem (
For robot polar at admissible on a Morse function (i.e., its Hessian is nonsingular at the critical points); smooth on
As an example consider that in the simplest case of circular obstacles in a circular workspace, a navigation function for robot
A fundamental result from topology (the Morse Index Theorem [
At any time, robot
Consider a single robot
Given a Navigation function
The proof [
Observe that
Next we show the obstacle avoidance property is preserved.
For any point on the boundary of the free space
This proof suggests that we are able to select values of
This fact that many vector fields can decrease the value of the potential was observed in [
We now turn our attention to
Assume at time
Thus
Assume robot
We call such a
Consider Figure
An illustration of the velocity selection problem for robot
If the intersection of the two cones and half plane is empty, no solution exists; and one of the constraints must be relaxed. If the intersection is not empty the formulation of the objective function reflects a preference toward rapid goal attainment (large
Computationally, this is a essentially a semidefinite programming problem, which can be solved in polynomial-time [
One attractive feature of the semidefinite programming approach used in
In cases where the set of feasible directions is empty, simultaneously satisfying the three objectives is impossible.
This approach is summarized in Algorithm
Let Determine Compute Compute Compute Move. Infeasible. Drop a constraint or terminate.
So far we have show that all
A generic pair of robots, with active line of sight and range constraints. Figure indicates possible directions of
The method is tested both in simulation and on a group of mobile robots.
First, the method was simulated using Matlab (version 2010A) on a desktop PC. We created a 300 by 300 unit workspace, populated with random triangular obstacles, as seen in Figure
For the first set of experiments 5 robots are required to maintain a “P-" shaped connectivity graph. For example see the configuration labeled “frame 1" in the lower left quadrant of Figure
The robots are placed in randomly generated initial scenarios using a uniform distribution. In order to generate acceptable scenarios, each initial condition was tested for two criteria. Is the desired P-shaped connectivity graph a subset of the connectivity graph induced by the range and line of sight conditions? Are all of the robots inside the free space?
If the answer to either of these questions was “no", the condition was rejected and a new set of random positions was generated. This methodology is identical to the one used to generate samples for Monte-Carlo integration over nonrectangular domains.
Even though the simulation is done on a single PC, we try to replicate the typical information dependencies that would arise on a distributed multirobot system. Specifically, robot its own position the geometry of the obstacles and the workspace; the position and velocity,
Regarding the first two assumptions, they are difficult to guarantee for physical implementation, but are none the less standard simulation assumptions in the literature. Regarding the third, recall that the motivation for designing this motion controller is to maintain communication links. Therefore, it is reasonable that those links could be used for peer-to-peer communication that is helpful for that purpose. Finally, we point out that the robots do
In the context of the simulation, the robots exchange the required information as permitted and recompute their velocities via numerical optimization every time step (0.1 second). Their top speed is 10 units/sec. Because there are no higher order dynamics, we use a forward Euler numerical integration method. To numerically solve the velocity selection problem at each time step, we used
We selected one typical example scenario for illustration shown in Figure
A second scenario illustrates this with a larger group of 12 robots and 15 desired links and is shown in Figure
The table in Figure
Simulation results for connectivity algorithm.
A swarm of 5 robots maintaining a specified communication graph, beginning in the lower left corner and moving toward the upper right corner.
A swarm of 12 robots maintaining a specified communication graph. Beginning in the upper right corner and moving toward the lower left corner.
The next column lists how many scenarios resulted in a minor constraint violation. We define a minor violation as a short duration (2 or fewer time steps), small magnitude < (0.1 sec × 10 m/s)2 violations of (
An illustration of a minor constraint violation.
The final column lists the number of scenarios deemed infeasible by
An illustration of an infeasible problem.
The algorithm was implemented on 6 iRobot Creates, shown in Figure
Three sequential snapshots of the 6 iRobot Creates executing the motion planner presented in this paper. The first frame show the connectivity ring and the goal positions.
Three sequential snapshots of the 6 iRobot Creates executing the motion planner presented in this paper. Frame 6 is the goal position.
Motivated by the use of wireless communication among swarm members, in this paper we consider the problem of steering
The author was supported by ONR Grant N0001405WRY20391. He wishes to thank Thomas Dunbar for help with the experimental results and numerous discussions.