FORMATION FLYING International Graduate School of Dynamic Intelligent Systems LEADER FOLLOWER MODEL...... Reading Class. 04 -12-03 -ARVIND IGS .
International Graduate School of Dynamic Intelligent Systems ‘‘FORMATION FLYING CONTROL OF MULTIPLE-SPACECRAFT VIA GRAPHS ,MATRIX INEQUALITIES, AND SWITCHING.´‘ -Mehran Masbahi. -Fred.Y.Hadaegh.
Scope.... • Introduction to the Model. • Preliminaries in the model. • Concepts in the model. • Applications of the model. International Graduate School of Dynamic Intelligent Systems
Leader-Follower Model... • Designate a particular frame (or multiple frames) in the formation as the reference frame(s)(RFs). • Measure and Control rest of the formation w.r.t RFs. International Graduate School of Dynamic Intelligent Systems
LF DIAGRAM....... International Graduate School of Dynamic Intelligent Systems L Fi Fol-1 Inertial Reference Frame-FIN Fol-2 Fj
Terms.... • ri -position of origin of Fi w.r.t FIN . • rdi -the actual desired position . • vi -velocity of origin of Fi w.r.t FIN . • vdi-the actual desired velocity. • ui -Control law of the ith spacecraft. International Graduate School of Dynamic Intelligent Systems
Definitions... • Formation pattern vector :- P(t)=[ r1(t) ,r2(t) ,.... rn(t) ]T € R3n . • Formation Configuration vector:- C(t)=[r1(t),v1(t),r2(t),v2(t),.... rn(t),vn(t) ]T € R6n . International Graduate School of Dynamic Intelligent Systems
Definitions contd... • Desired Formation pattern vector :- Pd(t)=[ rd1(t) ,rd2(t) ,.... rdn(t) ]T € R3n . • Desired Formation Configuration vector:- Cd(t)=[rd1(t),vd1(t),rd2(t),vd2(t),.... rdn(t),vdn(t) ]T € R6n . International Graduate School of Dynamic Intelligent Systems
LF..... • Formation Flying -Relative Spacecraft positions & velocities. • LF -A spacecraft position and velocity is expressed w.r.t to a leader. International Graduate School of Dynamic Intelligent Systems
LF..... • Affine Map:- If V is a vector space ,then a translation t:VV is an affine map that is given by t(x)=x+a for some a in V and for all x in V. International Graduate School of Dynamic Intelligent Systems
LF..... • Define Affine Map h: R3 R3 for kj {1,...,n} k l rdk(t) = Hkl . rl + h kl (t) hkltranslation ..from the leader ´l´. rl Inertial measurement of the ´l´. Hkl matrix € R3*3. International Graduate School of Dynamic Intelligent Systems
LF Graph..... • LF Graph- Digraph . • GLF=(VLF,ELF) where [i,j] € ELF iff i is the leader of j. • GC =(VLF,EC) where [i,j] € EC iff i can send real valued messages to j. International Graduate School of Dynamic Intelligent Systems
LF Graph..... International Graduate School of Dynamic Intelligent Systems L1 F L2 F IN.FR F F
LF Graph..... • Acyclic GLF Devoted follower • Cyclic GLF International Graduate School of Dynamic Intelligent Systems
Complete Graph... • Complete Graph :-If a graph is weakly connected. International Graduate School of Dynamic Intelligent Systems
Formation..... • Qualified Formation A Formation is said to be qualified at time t, if GC(t) is weakly complete. International Graduate School of Dynamic Intelligent Systems
Qualified Formations... Communication graphs..... International Graduate School of Dynamic Intelligent Systems
LF Graph..... • Valid LF An LF is valid at time t, if GLF(t) is complete,acyclic and the indegree of all its vertices is at most one. • The Graph of a valid LF is a directed tree. International Graduate School of Dynamic Intelligent Systems
A valid Formation..... But an invalid LF. International Graduate School of Dynamic Intelligent Systems L1 L2 L5 F L3 L4
A Valid Formation... and a valid LF. International Graduate School of Dynamic Intelligent Systems F1 F2 F5 L F4 F3
Controlling issues..... • Dependency Control action of the follower space craft depends on the states(inertial positions and velocities) of another space craft(leader). International Graduate School of Dynamic Intelligent Systems
Dependency.... • Dependent:- j (follower) is dependent on i(leader) if uj is a function of ri . • Strongly Dependent:- j(follower) is strongly dependent on i(leader) if uj is a function of both ri and ui . International Graduate School of Dynamic Intelligent Systems
Control Objective... • Design a controller for each space-craft such that the origin is globally asymptotically stable equilibrium point of kCd(t) - C(t) k1 International Graduate School of Dynamic Intelligent Systems
Control Laws.. • Control laws for the followers are derived solving LMI‘s such that in the error function for a follower j of leader i given as ej(t)=rdj(t) – rj(t), rj(t) rjd(t) as t1 . International Graduate School of Dynamic Intelligent Systems . .
Control law.... • The control law derived for a follower j of a leader i solving LMI .... uj(t)=ui(t)+ d2hij + YQ-1zj(t) dt2 zj(t) -includes the error function. YQ-1 -Matrices which include the l position . International Graduate School of Dynamic Intelligent Systems
The error dynamics... • The error dynamics of the rotating frame of the follower with respect to a leader is given in the form of ellipsoid. • The ellipsoid describes the time evolution of position error,position rate error and angular velocity of the follower in its rotating reference frame. International Graduate School of Dynamic Intelligent Systems
Control Switching... • Leadership Re-assignment:- International Graduate School of Dynamic Intelligent Systems L2 L1 Rotating frame of The follower 0 0,0,0 0 0
LF Capturing..... International Graduate School of Dynamic Intelligent Systems Existing LF New LF Participants..
Applications..... • Maneuvering swarms through highly obstructed paths.(unmanned flights,auv‘s...etc). • Leader Re-assignment in this LF can be used for avoiding incoming dangers. International Graduate School of Dynamic Intelligent Systems
