rigidity theory in multi-agent coordination a fundamental system architecture D aniel Zelazo Geometric Constraint Systems: Rigidity, Flexibility and Applications Lancaster University CoNeCt Cooperative Networks and Controls Lab
what is control theory? 1
a classic control system Disturbances y System r e u C ontroller − x = f ( x, u, d ) ˙ y m Measurements A control systems engineer aims to design a controller that en- sures the closed-loop system ◮ is stable ◮ satisfies some performance criteria 2
what are multi-agent systems? 3
what are multi-agent systems? What is the right control architecture? ◮ of each agent ◮ of the information exchange layer 3
control architectures 4
control architectures 5
coordination objectives rendezvous formation control localization ◮ Does the control strategy need to change with different sensing/communication? ◮ Are there common architectural requirements that do not depend on the choice of sensing? 6
formation control Formation Control Objective Given a team of robots endowed with the ability to sense/communicate with neighboring robots, design a control for each robot using only local information that moves the team into a desired spatial configuration - the formation 7
agent dynamics Control Theory provides us with an analytical justification for using simple models! INTEGRATOR DYNAMICS x = u x ˙ y = u y ˙ z = u z ˙ UNICYCLE DYNAMICS x = v lin cos( ψ ) ˙ y = v lin sin( ψ ) ˙ ˙ ψ = v ang 8
agent configurations ◮ we consider a team of n agents in a d -dimensional Euclidean space ◮ agents interact p i ( t ) ∈ R d according to a sensing graph ◮ the configuration of the agents at time t is the vector G = ( V , E ) p 1 ( t ) . . ∈ R nd p ( t ) = . p n ( t ) ◮ agents modelled by single integrator dynamics p i ( t ) = u i ( t ) , i = 1 , . . . , n ˙ 9
a formation potential THE ”FORMATION” POTENTIAL Φ( p ) = 1 � ( � p i − p j � 2 − d 2 ij ) 2 4 i ∼ j A GRADIENT FLOW p = −∇ p Φ( p ) ˙ 10
a formation potential THE ”FORMATION” POTENTIAL Φ( p ) = 1 � ( � p i − p j � 2 − d 2 ij ) 2 4 i ∼ j A GRADIENT FLOW p = −∇ p Φ( p ) ˙ Theorem The gradient dynamical system asymptotically converges to the critical points of the formation potential. 10
a distributed implementation Distributed Control � ( � p i − p j � 2 − d 2 p i = ˙ ij )( p j − p i ) i ∼ j 11
a distributed implementation Distributed Control � ( � p i − p j � 2 − d 2 p i = ˙ ij )( p j − p i ) i ∼ j ◮ Does this strategy solve the formation control problem? ◮ Does it reveal a necessary control architecture for the multi-agent system? 11
rigidity meets formation control For a framework ( G , p ) , we have Edge Function Rigidity Matrix R D ( p ) = ∂f D ( p ) . . . f D ( p ) = 1 ∂p � p i − p j � 2 2 . . . 12
rigidity meets formation control For a framework ( G , p ) , we have Edge Function Rigidity Matrix R D ( p ) = ∂f D ( p ) . . . f D ( p ) = 1 ∂p � p i − p j � 2 2 . . . p = ∇ Φ( p ) = ∂ ∂p � f D ( p ) − 1 2 d 2 � 2 ˙ = − R D ( p ) T R D ( p ) p − R D ( p ) T d 2 12
rigidity meets formation control Theorem (Stability and Rigidity) If the target formation is infinitesimally rigid, then the dynamics are (locally) asymptotically stable and satisfy t →∞ p ( t ) = p ⋆ lim j � 2 = d 2 where � p ⋆ ij for all { i, j } ∈ E . i − p ⋆ L. Krick, M. E. Broucke & B. A. Francis, Stabilisation of infinitesimally rigid formations of multi-robot networks , International Journal of Control, 82(3):423-439, 2009. 13
a real robot 14
bearing sensing Bearing Sensing The bearing between two agents is defined as the unit vector p j ( t ) − p i ( t ) g ij ( t ) = � p j ( t ) − p i ( t ) � , where p i ( t ) is the position of agent i . ◮ NOTE: g ij can be expressed in a common frame or local frame 15
bearing-only formation control target formation specified by desired bearings Formation Control Objective Design u i for each agent using only bearing measurements such that t →∞ g ij ( t ) = g ∗ lim ij for all pairs ( i, j ) in the sensing graph. 16
what is bearing rigidity? Bearing Rigidity ◮ If we fix the bearing of each edge in a network, can the geometric pattern of the network be uniquely determined? ◮ Intuitive definition: a network is bearing rigid if its bearings can uniquely determine its geometric pattern. 17
bearing-edge function ⋄ How can one determine if a given network is bearing rigid? (a) (b) (c) (d) 18
bearing-edge function ⋄ How can one determine if a given network is bearing rigid? (a) (b) (c) (d) The Bearing-Edge Function For a network with |E| = m edges, the bearing-edge function is defined as g 1 . . ∈ R dm . f B ( p ) � . g m 18
bearing-preserving motions Bearing Trivial Motions Trivial motions preserve the bearing between all pairs of agents for any framework ◮ (rigid body) translations ◮ scaling 19
infinitesimal motion Consider the Taylor-series expansion of the bearing-edge function: f B ( p + δ p ) = f B ( p ) + ∂f B ( p ) δ p + h.o.t. ∂p Infinitesimal Motions An infinitesimal motion, δ p , of a network satisfies ∂f B ( p ) δ p = 0 . ∂p ◮ first order ”bearing-preserving” motions ◮ trivial motions are always infinitesimal motions 20
a rank test The Rigidity Matrix R B ( p ) � ∂f B ( p ) ∂p Rank-Test for Bearing Rigidity A network is infinitesimally bearing rigid if and only if rank( R B ( p )) = dn − d − 1 . Examples: 21 (a) (b) (c) (d)
a rank test The Rigidity Matrix R B ( p ) � ∂f B ( p ) ∂p Rank-Test for Bearing Rigidity A network is infinitesimally bearing rigid if and only if rank( R B ( p )) = dn − d − 1 . Examples: 22 (a) (b) (c) (d)
some features of bearing rigidity Bearing Rigidity Distance Rigidity infinitesimal infinitesimal distance rigidity bearing rigidity global global distance rigidity bearing rigidity distance rigidity bearing rigidity ◮ in R 2 , infinitesimal distance rigidity and infinitesimal bearing rigidity are equivalent ◮ infinitesimal bearing rigidity is preserved in lifted spaces ◮ Laman graphs are generically bearing rigid in arbitrary dimension ◮ at most 2 n − 3 edges are sufficient to ensure bearing rigidity in arbitrary dimension ◮ infinitesimal bearing rigid frameworks uniquely define a shape (modulo scale and translation) 23
directed bearing rigidity (a) R 2 (b) R 2 × S 1 (c) R 3 × S 1 (d) SE (3) Bearing Rigidity Function Given a n -agent formation modeled as a framework ( G , χ ) in ¯ D , the bearing rigidity function is the map b G : ¯ D → ¯ M , χ �→ b G ( χ ) = [ b T · · · b T m ] T 1 24
se (2) framework example Trivial Motions Trivial motions in SE (2) are translations, scaling, and coordinated rotations • • • 3 1 2 3 • • • 1 2 In directed bearing rigidity, local rigidity does not imply global rigidity 25
examples (a) n = 3 (b) n = 4 (c) n = 6 (d) n = 8 (e) n = 3 (f) n = 4 (g) n = 6 (h) n = 8 IBF frameworks in ( R 2 × S 1 ) n ((a),(b)), in ( R 3 × S 1 ) n with n = e 3 ((c),(d)). Examples of IBR frameworks in ( R 2 × S 1 ) n ((e),(f)) and in ( R 3 × S 1 ) n with n = e 3 ((g),(h)). 26
a general bearing rigidity matrix For a framework ( G , χ ) , the bearing rigidity matrix takes the form B G ( χ ) = [ B p B o ] ∈ R 3 m × 6 n , with E ⊤ ∈ R 3 m × 3 n B p = D p ¯ and B o = D o ¯ E ⊤ o ∈ R 3 m × 3 n (1) D p i R i D p D o � > � p x p y p z � α i , β i , γ i , { e h } 3 � diag( d ij R > diag( R > SE (3) R i P (¯ p ij )) i [¯ p ij ] × I 3 ) i i i h =1 � > R 3 × S 1 p y R ( α i , n ) , n = P 3 diag( d ij R > diag( R > � p x p z h =1 n h e h i P (¯ p ij )) i [¯ p ij ] × [ 0 3 × 2 n ]) i i i 0 � > R 2 × S 1 � p x p y diag( d ij R > diag( R > R ( α i , e 3 ) i P (¯ p ij )) i [¯ p ij ] × [ 0 3 × 2 e 3 ]) i i � > R 3 � p x p y p z diag( d ij I > diag( I > R ( α i , 0 3 × 1 ) = I 3 3 P (¯ p ij )) 3 [¯ p ij ] × 0 3 × 3 ) i i i p y 0 � > R 2 � p x diag( d ij I > 3 P (¯ diag( I > 3 [¯ R ( α i , 0 3 × 1 ) = I 3 p ij )) p ij ] × 0 3 × 3 ) i i 27
bearing rigidity ...back to formation control 28
the bearing potential Consider the potential function of bearing errors: Φ( t ) = 1 � � g ij ( t ) − g ∗ ij � 2 2 A Gradient-descent control p = −∇ p Φ( t ) ˙ 1 � � e ij ( t ) � P gij ( t ) g ∗ p i ( t ) = − ˙ ij j ∈N i ◮ e ij ( t ) = p j ( t ) − p i ( t ) ◮ implementation requires distance and bearing measurements! 29 ◮ P gij ( t ) is an orthogonal projection matrix
bearing-only strategy Proposed Control Law � P gij ( t ) g ∗ p i ( t ) = − ˙ ij j ∈N i B ( p )diag {� e ij �} g ∗ p ( t ) = R T ˙ p j g ij − P g ij g ∗ ij g ∗ ij p i P g ij g ∗ ij 30 Figure 1: Geometric interpretation
examples 31
examples 32
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