Algorithms For Shaping a Particle Swarm With a Shared Control Input Using Boundary Interaction Shiva Shahrokhi, Arun Mahadev, and Aaron T. Becker Abstract—Consider a swarm of particles controlled by global inputs. This paper presents algorithms for shaping such swarms in 2D using boundary walls. The range of configurations created Kilobots with by conforming a swarm to a boundary wall is limited. We photophile behavior (global control input) describe the set of stable configurations of a swarm in two canonical workspaces, a circle and a square. To increase the Distant light diversity of configurations, we add boundary interaction to sources our model. We provide algorithms using friction with walls to place two robots at arbitrary locations in a rectangular workspace. Next, we extend this algorithm to place n agents at desired locations. We conclude with efficient techniques to 3 cm control the covariance of a swarm not possible without wall- friction. Simulations and hardware implementations with 100 robots validate these results. 3 cm These methods may have particular relevance for current micro- and nano-robots controlled by global inputs. Fig. 1. Swarm of kilobots programmed to move toward the brightest light source as explained in xV. The current covariance ellipse and mean are shown in red, the desired covariance is shown in green. Navigating a swarm using I. INTRODUCTION global inputs is challenging because each member receives the same control Particle swarms steered by a global force are common in inputs. This paper focuses on using boundary walls and wall friction to break the symmetry caused by the global input and control the shape of a swarm. applied mathematics, biology, and computer graphics. [_x ; y_ ]> = [u ; u ]>; i 2 [1; n] (1) i i x y can be generated. To extend the range of possible shapes, xII-B The control problem is to design ux(t); uy(t) to make all introduces wall friction to the system model. We prove that n particles achieve a task. As a current example, micro- two orthogonal boundaries with high friction are sufficient to and nano-robots can be manufactured in large numbers, arbitrarily position two robots in xIII-A, and xIII-B extends this see Chowdhury et al. [6], Martel et al. [16], Kim et al. [12], to prove a rectangular workspace with high-friction boundaries Donald et al. [7], Ghosh and Fischer [9], Ou et al. [18] or can position a swarm of n robots arbitrarily within a subset of Qiu and Nelson [19]. Someday large swarms of robots will the workspace. xIV describes implementations of both position be remotely guided ex vivo to assemble structures in parallel control algorithms in simulation and xV describes experiments and through the human body, to cure disease, heal tissue, and with a hardware setup and up to 100 robots, as shown in prevent infection. For each application, large numbers of micro Fig. 1. After a review of recent related work xVI, we end robots are required to deliver sufficient payloads, but the small with directions for future research xVII. size of these robots makes it difficult to perform onboard II. THEORY computation. Instead, these robots are often controlled by a A. Using Boundaries: Fluid Settling In a Tank arXiv:1609.01830v1 [cs.RO] 7 Sep 2016 global, broadcast signal. These applications require control techniques that can reliably exploit large populations despite One method to control a swarm’s shape in a bounded high under-actuation. workspace is to simply push in a given direction until the Even without obstacles or boundaries, the mean position swarm conforms to the boundary. of the swarm in (1) is controllable. By adding rectangular a) Square workplace: This section examines the mean 2 2 boundary walls, some higher-order moments such as the (¯x; y¯), covariance (σx; σy; σxy), and correlation ρxy of a very swarm’s position variance orthogonal to the boundary walls large swarm of robots as they move inside a square workplace (σx and σy for a workspace with axis-aligned walls) are under the influence of gravity pointing in the direction β. also controllable [23]. A limitation is that global control can The swarm is large, but the robots are small in comparison, only compress a swarm orthogonal to obstacles. However, and together occupy a constant area A. Under a global input navigating through narrow passages often requires control of such as gravity, they flow like water, moving to a side of the the variance and the covariance. workplace and forming a polygonal shape, as shown in Fig. 2. The paper is arranged as follows. xII-A provides analytical The range for the global input angle β is [0,2π). In this position control results in two canonical workspaces with fric- range, the swarm assumes eight different polygonal shapes. tionless walls. These results are limited in the set of shapes that The shapes alternate between triangles and trapezoids when 1.0 0.03 x σ xy 3 π 0.8 π 0.02 5 π 4 4 8 3 π 0.01 0.6 π π 8 5 π 3 π 2 0.2 0.4 20.6 0.8 1.0 0.4 8 4 -0.01 3 π π 0.2 8 -0.02 4 β A -0.03 A 0.2 0.4 0.6 0.8 1.0 π 3 π A=3/4 0.08 2 2 σ x 8 0.4 1/2 ρxy 5 π 3 π 0.06 0.2 1/4 8 4 1/8 π 1/16 0.04 π 0.2 0.4 0.6 2 0.8 1.0 4 -0.2 3 π 0.02 π 8 8 A -0.4 π A 0.00 0 0.0 0.2 0.4 0.6 0.8 1.0 4 Fig. 2. Pushing the swarm against a square boundary wall allows limited control of the shape of the swarm, as a function of swarm area A and the commanded movement direction β. Left plot shows locus of possible mean positions for five values of A. The locus morphs from a square to a circle as A increases. The 2 covariance ellipse for each A is shown with a dashed line. Center shows two corresponding arrangements of kilobots. At right is x¯(A); σxy(A); σx(A); and ρ(A) for a range of β values. See online interactive demonstration at [29]. 0.10 3 π σ xy 5 π 4 0.05 8 h π h 0.5 2 1.0 1.5 2.0 π 3 -0.05 π β 4 8 -0.10 π 2 3 π 1.0 0.25 2 σ x 8 0.20 0.5 π 5 π ρxy 3 π 0.15 4 π 8 h 4 0.5 1.02 1.5 2.0 0.10 π -0.5 3 π 0.05 8 h π 8 0 4 0.5 1.0 1.5 2.0 -1.0 Fig. 3. Pushing the swarm against a circular boundary wall allows limited control of the shape of the swarm, as a function of the fill level h and the commanded movement direction β. Left plot shows locus of possible mean positions for four values of h. The locus of possible mean positions are concentric circles. See online interactive demonstration at [28]. the area A<1/2, and alternate between squares with one corner in the lower-left corner is: removed and trapezoids when A>1/2. p p p p R 2 −A tan(β) R 2 −A cot(β)+x cot(β) Computing means, variances, covariance, and correlation 0 0 x dy dx R x¯(A; β) = requires integrating over the region containing the swarm: A 1p p = 2 A tan(β) (5) RR x dx dy RR y dx dy 3 x¯ = R , y¯ = R (2) p p p p A A R 2 −A tan(β) R 2 −A cot(β)+x cot(β) y dy dx RR 2 RR 2 0 0 2 R (x − x¯) dx dy 2 R (y − y¯) dx dy y¯(A; β) = σx = , σy = (3) A A A p RR 2 1 p (x − x¯x)(y − y¯) dx dy σ = 2 A cot(β) (6) σ = R , ρ = x (4) 3 xy A xy σ σ x y The full equations are included in the appendix, and are The region of integration R is the polygon containing the summarized in Fig. 2. A few highlights are that the correlation swarm. If the force angle is β, the mean when the swarm is is maximized when the swarm is in a triangular shape, and is ±1/2. The covariance of a triangle is always ±(A=18). and N = F cos(θ) Variance is minimized in the direction of β and maximized orthogonal to β when the swarm is in a rectangular shape. Fig. 16 shows the resultant forces on two robots when one The range of mean positions are maximized when A is small. is touching a wall. As illustrated, both experiences different b) Circular workplace: Though rectangular boundaries net forces although each receives the same inputs. For ease of are common in artificial workspaces, biological workspaces analysis, the following algorithms assume µf is infinite and are usually rounded. Similar calculations can be computed for robots touching the wall are prevented from sliding along the a circular workspace. The workspace is a circle centered at wall. This means that if one robot is touching the wall and (0,0) with radius 1 and thus area π. For notational simplicity, another robot is free, if the control input is parallel or into the swarm is parameterized by the global control input signal the wall, the touching robot will not move.