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Morse Theory and Formation Control formulas for computing bias in localization problem Brian D O Anderson

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What is Morse Theory and our Motivation? Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 2 Outline 13 May 2011 Morse Theory and Formation Control

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Morse theory is a set of results regarding the number of minima, maxima and saddle points of a scalar real function of a number of real variables By extension, it will often give information about the number of stable, unstable and saddle point equilibria for an autonomous system or closed-loop control system Researchers, including ourselves, have found that many formation shape control laws can have multiple equilibria. It is of great interest then to know what restrictions this will impose on use of the system. You don’t want to end up at the wrong equilibrium point Hence we study Morse Theory. 3 What is Morse Theory and our Motivation? 13 May 2011 Morse Theory and Formation Control

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What is Morse Theory and our Motivation? Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 4 Outline 13 May 2011 Morse Theory and Formation Control

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Very simple motivating example Consider a function f: R 1 R 1 What can we say about the number of minima and maxima? Impose restrictions: –Function is smooth –At every point where f’(x 0 ) = 0, the second derivative f’’(x 0 ) ≠0 –f goes to +∞ when x goes to ±∞. –Extreme points are isolated. 13 May 2011 5 There are 4 minima and 3 maxima (discounting x =±∞.)

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General Rule Consider a smooth function f: R 1 R 1 such that –At every point where f’(x 0 ) = 0, the second derivative f’’(x 0 ) ≠0 –f goes to +∞ when x goes to ±∞. –Extreme points are isolated. Let m 0, m 1 denote the number of minima and maxima. Then 13 May 2011 6Morse Theory and Formation Control m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 1 (Morse equality) There are 4 minima and 3 maxima (discounting x =±∞.) If f: (a,b) R 1 and f’(a) 0, same result holds.

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General Rule Consider a smooth function f: R 1 R 1 such that –At every point where f’(x 0 ) = 0, the second derivative f’’(x 0 ) ≠0 –f goes to +∞ when x goes to ±∞. –Extreme points are isolated. Let m 0, m 1 denote the number of minima and maxima. Then What happens if f: R 2 R 1 or f: R n R 1 where n > 2? –Morse theory provides the answer, and numbers of saddle points (as well as minima, maxima) enter the inequality/equality set. 13 May 2011 7Morse Theory and Formation Control m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 1 (Morse equality)

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Another Example In R 2 the unit circle is known as S 1. In R 3, the unit sphere is known as S 2. In R n, the sphere (defined by x 1 2 +x 2 2 +…x n 2 =1) is known as S n-1. Consider a function f: S 1 R 1. How many minima and maxima does it have? Assume: –f is smooth –At any point where first derivative is zero, second derivative is nonzero –Extreme points are isolated. Then there holds 13 May 2011 8Morse Theory and Formation Control m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 0 (Morse equality) But what if f: S n R 1 where n > 1?

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Comparing R 1, S 1 For functions on R 1 : For functions on S 1 : 13 May 2011 9Morse Theory and Formation Control m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 1 (Morse equality) m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 0 (Morse equality)

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Looking ahead Associated with a space are a number of nonnegative integers –Betti numbers β i –Euler characteristic χ (related to Betti) Consider a function f defined on some space and taking real values. –Suppose at critical points of f, i.e. points x 0 where f’(x 0 )=0, the second derivative Hessian f”(x 0 ) is nonsingular. – Let j be the number of negative eigenvalues of f”(x 0 ). Let m j be the number of critical points with index j. Then the set of m j can be related to the set of β i by a collection of inequalities and one equality. –The equality can also be expressed with the Euler characteristic. 13 May 2011 10Morse Theory and Formation Control Characteristics of the space determine inequalities and equalities for the number of critical points of different degrees of any function defined on the space.

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Comparing R 1, S 1 For functions on R 1 : For functions on S 1 : 13 May 2011 11Morse Theory and Formation Control m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 1 (Morse equality) m 0 ≥ 1 (Morse inequality) m 0 —m 1 = 0 (Morse equality) Right side determined by Betti numbers of R 1 or S 1 = Euler characteristic of R 1 or S 1 Euler characteristic can be found from Betti numbers but is independent concept

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What is Morse Theory and our Motivation? Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 12 Outline 13 May 2011 Morse Theory and Formation Control

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A control problem (Koditschek 1989) This work involves the notion of a potential function, and a steepest descent control law. Consider a particle moving in R n under the influence of a potential field. There is an underlying scalar potential function V(x) and the particle moves according to Where does the particle end up? –At an equilbrium point, i.e. where V(x) has zero derivative or f(x) has a zero. The navigation problem: given a region where a robot moves, specify a V(x) guaranteeing convergence to a prescribed x 0. 13 May 2011 13Morse Theory and Formation Control Equilibrium points of differential equation are same as critical points of V(x)

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Some relevant questions What is the region in which a robot moves? –Is it R 2, R 3 or something more complicated? –Is it R 2, R 3 perhaps with obstacles? If the robot is to go to x 0 always, –Do we not want any other equilibria? –Or do we not want any other stable equilibria? There have been lots or examples in the robot literature showing that when you introduce obstacles, the robot can hang up at an incorrect point. Can Morse theory, distinguishing as it does the stability/instability of equilibria, help answer the above questions? 13 May 2011 14Morse Theory and Formation Control

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Simple answer for R 2 Suppose the robot is moving in R 2 and there are no obstacles. Then one could choose Notice that V is smooth, has a unique isolated minimum, its Hessian is nonzero at the minimum and it goes to infinity when x goes to ±∞. The trajectory will then be a steepest descent from initial state to x 0 and convergence is exponentially fast: 13 May 2011 15Morse Theory and Formation Control Same idea also works for R 3

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Sphere worlds A sphere world is a compact connected subset of R n whose boundary is formed from the disjoint union of a finite number, say M+1 of (n-1)-spheres One large sphere bounds the workspace: 13 May 2011 16Morse Theory and Formation Control

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Sphere worlds A sphere world is a compact connected subset of R n whose boundary is formed from the disjoint union of a finite number, say M+1 of (n-1)-spheres One large sphere bounds the workspace: M smaller spheres bound the obstacles within the workspace: 13 May 2011 17Morse Theory and Formation Control

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Sphere worlds A sphere world is a compact connected subset of R n whose boundary is formed from the disjoint union of a finite number, say M+1 of (n-1)-spheres One large sphere bounds the workspace: M smaller spheres bound the obstacles within the workspace: The robot operates in the free space To ensure nonintersecting obstacles with closures in interior of workspace: 13 May 2011 18Morse Theory and Formation Control

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More on Sphere World Sphere World can be distorted so that obstacles are not circular and results are still valid Koditschek works with sphere world Answer to the navigation problem is provided using Euler Characteristic –Now look at Euler characteristic –Then return to sphere world navigation problem 13 May 2011 19Morse Theory and Formation Control

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What is Morse Theory and our Motivation? Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 20 Outline 13 May 2011 Morse Theory and Formation Control

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Euler Characteristic—oversimplified view Consider a cube. It is said to have a cellular decomposition with 6 (open) 2-cells, 12 (open) 1-cells and 8 0-cells –2-cell is an open rectangle, or something distortable (homotopically equivalent) to this, e.g. an open disk—face of cube. –A 1-cell is an open interval or something distortable (homotopically equivalent) to an open interval: e.g. the curved part of the boundary of a semicircle, without the end points– edge of cube –A 0-cell is point—corner of cube. Consider a compact topological space/manifold/cell complex X (includes cube, sphere world, sphere) with a (not necessarily unique) cell-decomposition K. Suppose there are α q q-cells in K. The Euler characteristic of X is 13 May 2011 21Morse Theory and Formation Control Note the word compact: it means closed and bounded. R 2 does not qualify. Unit disk minus origin does not qualify. It is possible to fiddle with the definition to handle these issues.

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Connection with minima/maxima/saddle points X is a compact topological space/manifold/cell complex with cell decomposition K. f is a smooth function on X; critical points isolated with Hessians nonsingular. m j is the number of critical points of f of index j. Then the Morse equality holds: Sometimes one must cheat a bit with f. –For example, f may have to be increasing all along the boundary of X, if there is a boundary. (This is a bit like requiring when discussing R n that f goes to infinity as x goes to infinity). 13 May 2011 22Morse Theory and Formation Control

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Euler characteristic for closed disk in R 2 Consider the closed unit disk in R 2. It is the union of: –The open unit disk in R 2 (α 2 =1) –The circumference of the unit disk, less an identified single point— which is homotopic to an open interval (α 1 =1) –The identified single point on the circumference (α 0 =1) 13 May 2011 23Morse Theory and Formation Control =++

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Euler characteristic for closed disk in R 2 It is also the union of: –Two open regions bounded by semicircles (α 2 =2) –The diameter lying between the semicircles, less its endpoints, plus two circular arcs, open at each end, associated with the two semicircles (α 1 =3) –Two points (α 0 =2) 13 May 2011 24Morse Theory and Formation Control =++

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Euler characteristic for closed disk in R 2 Consider the closed unit disk in R 2. It is the union of: –The open unit disk in R 2 (α 2 =1) –The circumference of the unit disk, less an identified single point— which is homotopic to an open interval (α 1 =1) –The identified single point on the circumference (α 0 =1) It is also the union of: –Two open regions bounded by semicircles (α 2 =2) –The diameter lying between the semicircles, less its endpoints, plus two circular arcs, open at each end, associated with the two semicircles (α 1 =3) –Two points (α 0 =2) 13 May 2011 25Morse Theory and Formation Control

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Euler characteristic for closed disk in R n Consider the closed unit disk in R n [x 1 2 +x 2 2 +…x n 2 ≤1]. It is the union of: –The open unit disk in R n : [x 1 2 +x 2 2 +…x n 2 <1] (α n =1) –The unit disk boundary [x 1 2 +x 2 2 +…x n 2 =1], (which is the sphere S n-1 ) less an identified single point (e.g. [0,0…,0,1]) This boundary less the single point is the same as points satisfying x 1 2 +x 2 2 +…x n 2 =1 and x n ≠ 1, or equivalently x 1 2 +x 2 2 +…x n-1 2 <1, which is the unit disk D n-1 (α n-1 =1) –The identified single point on the sphere S n-1 (α 0 =1) Therefore 13 May 2011 26Morse Theory and Formation Control This argument generalizes to arbitrary n:

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Euler characteristic for closed disk in R n Consider the closed unit disk in R n [x 1 2 +x 2 2 +…x n 2 ≤1]. It is the union of: –The open unit disk in R n : [x 1 2 +x 2 2 +…x n 2 <1] (α n =1) –The unit disk boundary [x 1 2 +x 2 2 +…x n 2 =1], (which is the sphere S n-1 ) less an identified single point (e.g. [0,0…,0,1]) This boundary less the single point is the same as points satisfying x 1 2 +x 2 2 +…x n 2 =1 and x n ≠ 1, or equivalently x 1 2 +x 2 2 +…x n-1 2 <1, which is the unit disk D n-1 (α n-1 =1) –The identified single point on the sphere S n-1 (α 0 =1) Therefore 13 May 2011 27Morse Theory and Formation Control

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Euler characteristic of S 2, S n It is also easy to get the Euler characteristic of S 2 or S n. S 2 is the union of – its surface x 1 2 + x 2 2 +x 3 2 =1 minus an identified point, say [0,0,1], –And the identified point. The surface x 1 2 + x 2 2 +x 3 2 =1 minus the point [0,0,1] is the same as x 1 2 + x 2 2 <1, which is the unit disk. Thus α 2 =1, α 1 =0, α 0 =1. So More generally In particular, 13 May 2011 28Morse Theory and Formation Control

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Euler characteristic of S 2, S n It is also easy to get the Euler characteristic of S 2 or S n. S 2 is Also 13 May 2011 29Morse Theory and Formation Control Sphere = 2 hemispheres + equator 1 hemisphere is like 1 open disk (α 2 =2) 1 equator is like open interval line plus single point (α 1 =1, α 0 =1) α 2- -- α 1 + α 0 = 2

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What is Morse Theory and our Motivation? Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 30 Outline 13 May 2011 Morse Theory and Formation Control

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Euler Characteristic of sphere world One large sphere bounds the workspace (closed disk): M smaller spheres bound the obstacles: The robot operates in the free space A finite cellular decomposition of workspace is provided by a finite cellular decomposition of free space plus obstacles Hence : 13 May 2011 31Morse Theory and Formation Control

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Key conclusions: With one or more obstacles, the Euler characteristic cannot be 1. Suppose one designs a potential function such that its gradient points outwards on the boundaries of the free space The Morse equality then holds: m 0 – m 1 + m 2 - …= 1 - M( -1) n Suppose there is a unique attracting equilibrium point. With M>0, there are necessarily saddles and or maxima for V(x) in the free space. This arises no matter what V(x) is used. We may be able to determine more if we use Morse inequalities as well! 13 May 2011 32Morse Theory and Formation Control

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What is Morse Theory and our Motivation Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 33 Outline 13 May 2011 Morse Theory and Formation Control

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Formation Shape Control Consider a set of point agents, and a sufficient number of specified distance pairs d * ij between agents i,j that the shape of the formation is thereby specified. –Rigidity theory sorts out sets of allowed distance pairs. –Translation and rotation are permitted with the specification. Suppose the actual formation shape at time t is defined by distances d ij and a control law is adopted to move these values towards the specified values d * ij. This is a formation shape control law. Typically such laws are distributed, rely on sensing certain data etc. Also they are of the gradient descent form Behavior of laws for certain formations are well understood. 13 May 2011 34Morse Theory and Formation Control

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Typical Shape Control Results A triangular formation always converges to a correctly dimensioned triangle, unless the initial formation is collinear. A formation with 4 agents and 6 distances has saddle points but otherwise appears to always converge correctly. Any rigid formation can be locally controlled (at least). 13 May 2011 Morse Theory and Formation Control 35

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Example of a typical law. 36 13 May 2011 Morse Theory and Formation Control

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A more detailed tool—Betti numbers Betti numbers—all nonnegative integers-- can be defined for various topological spaces, including cell complexes, spheres, disks, R n, C n, and real projective spaces. If a manifold is n-dimensional (example: S 1, the circle is 1- dimensional, the equator of a sphere in R 3 is also 1-dimensional) the Betti numbers beyond β n are all zero. For entities living in R 3, (torus, sphere, cube, etc), there are rules of thumb: –β 0 is number of connected components (pretzel has one) –β 1 is number of holes or tunnels (donut has one) –β 2 is number of voids (sphere has one) 13 May 2011 37Morse Theory and Formation Control

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A more detailed tool—Betti numbers Betti numbers—all nonnegative integers-- can be defined for various topological spaces, including cell complexes, spheres, disks, R n, C n, and real projective spaces. If a manifold is n-dimensional (example: S 1, the circle is 1- dimensional, the equator of a sphere in R 3 is also 1-dimensional) the Betti numbers beyond β n are all zero. No explanation here of how to find them. They are recorded in various text books. Examples: 13 May 2011 38Morse Theory and Formation Control

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Betti numbers—one or two facts Let A be a topological space with Betti numbers. Then Suppose Betti numbers β iA, β iB, correspond to topological spaces A,B. Define C = A× B. Then 13 May 2011 39Morse Theory and Formation Control

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Morse Inequalities and Equalities Suppose f is a smooth function on the space A with –Isolated critical points –Nonsingular second derivative at the critical points –Maybe having some constraint at the boundary –m j critical points with index j 13 May 2011 40Morse Theory and Formation Control

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The space of formations To apply this to the formation shape control problem, we need to know what is the topological space A, and its associated Betti numbers The space has been worked out by Kendall. Neglecting formations in which all points are collocated, 13 May 2011 41Morse Theory and Formation Control In an ambient two-dimensional space, the space of formations of n point agents can be represented as Some intuition: R + takes care of a scaling. CP n-2 is meant to take care of all formations neglecting scaling—paying no attention to translation or rotation. Park agent 1 at origin. Plot other n-2 agents in complex plane. After scaling and allowing for rotations, a lengthy argument gives CP n-2. R + accounts for one degree of freedom, and CP n-2 for another 2n-4.

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Morse inequalities and equality for formation The Betti numbers for the space of formations can be computed from the product formula and knowledge of Betti numbers for R + and CP n-2. –An adjustment can be made for the all-collocated formation, which always corresponds to a maximum of the function V(x). For distance-based control laws, the equations are: 13 May 2011 42Morse Theory and Formation Control Triangle Case

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A general conclusion Look at the first three inequalities. They show you cannot have m 0 =1 or 2, and m 1, m 2 both zero. So there will always be unwanted equilibrium points, not matter what form of gradient descent law is used. 13 May 2011 43Morse Theory and Formation Control

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Triangular Formations A collection of control laws have been found, which are similar. Following remarks apply to almost all laws. There are two correct equilibria, which are mirror images of one another There is one maximum, with all points collocated There are three saddle points with one unstable mode with the three agents collinear; they differ according to which agent is between other two There are three saddle points with two unstable modes with two agents collocated and the third placed distinctly (this is new) Summing up: m 0 = 2, m 1 = 3, m 2 = 3, m 3 = 1 One strict equality and three inequalities follow. There is nothing about the Morse inequalities/equality that forces m 0 to be 1 or 2; thus incorrect stable equilibria are not automatically precluded. 13 May 2011 44Morse Theory and Formation Control

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Formations with four agents Over the past several years, formations with four agents have been intensively studied. –They may have five distance links (rigidity) –They may have six distance links (global rigidity/complete graph) At this stage, there is no complete analysis of all the equilibria. It is known for the five link case there are stable equilibria which are not congruent (flip ambiguity) but have correct distances For the six distance link case, it is not known if all stable equilibria are correct. However, it may be possible to identify m j for j>0, and then m 0 (=2?) would follow. 13 May 2011 45Morse Theory and Formation Control

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What is Morse Theory and our Motivation Simple Example of Morse Theory Koditschek’s Navigation Function Problem Euler Characteristic Koditschek’s problem and Euler Characteristic Formation Shape Control Conclusions 46 Outline 13 May 2011 Morse Theory and Formation Control

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Conclusions Morse theory can be used to obtain relations among the number of equilibrium points of different type for a control system using a steepest descent control law. Work must be done to define the topological space in which the underlying system lies Betti numbers must be obtained for the topological space in order that the Morse inequalities/equality can be recorded For the navigation problem and the formation shape stabilization problem based on distances, there are always incorrect equilibria, but they may be saddle points or unstable. 13 May 2011 47Morse Theory and Formation Control

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Some open issues Classify equilibria for four agent formations –Maybe show that with a complete graph, all stable equilibria have correct distances Identify particular results for n>4 agent formations Handling formations in R 3 : issue is what space to use to describe them (space is needed for Betti numbers) Analyse bearing-based control laws. Consider the possibility of dealing with control laws which are not gradient descent 13 May 2011 48Morse Theory and Formation Control

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References Y. Matsumoto, An Introduction to Morse Theory, American Mathematical Society, Providence, RI, 1997. J. Milnor, Morse Theory, Princeton University Press, Princeton, NJ, 1969 D.E. Koditschek and E. Rimon, Robot Navigation Functions on Manifolds with Boundary, Advances in Applied Math., vol. 11, 1990, pp. 412-442. B.D.O. Anderson, Morse Theory and Formation Control, 19th Mediterranean Conference on Control and Automation, to appear. D.G. Kendall, A Survey of the Statistical Theory of Shape, Statistical Science, vol. 4, 1989, pp. 87-99. 13 May 2011 49Morse Theory and Formation Control

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