Leader-to-formation stability
1 Leader-to-Formation Stability
Herbert G.Tanner George J.Pappas Vijay Kumar
Abstract—We investigate the stability for robot formations from a different perspective,focusing on the dependence of group con-?guration to incoming input singals or disturbances.Our idea builds on the notion of input-to-state stability to de?ne leader-to-formation stability(LFS),as a means to analyze error propagation and performance characterization.Contrary to other notions of stability for interconnected systems,in leader-to-formation stabil-ity the focus is shifted from disturbance rejection to quantifying transient and steady state behaviour,thus being able to relax con-ditions,address a larger class of systems and provide insight to issues of performance improvement in relation to interconnection topology.The new concepts are implemented numerically in the case of a formation of mobile robots where(LFS)also indicates the formation structures that ensure the smallest errors during maneuvering.
I.I NTRODUCTION
Interconnected systems have lately received considerable at-tention,motivated by recent advances in computation and com-munication,which provide the enabling technology for appli-cations such as automated highway systems[1],cooperative robot reconnaissance[2],[3]and manipulation[4],[5],forma-tion?ight control[6],[7],satellite clustering[8]and control of groups of unmanned vehicles[9],[10],[6].Formation control is one aspect of the study of a patricular class of interconnected systems.
One research thrust aims at network architectures and coor-dination methods as a means of generating a group behavior in a formation of vehicles.In behavior-based approaches[2], [11],[12]the group behavior emerges as a combination of group member behaviors,selected among a set of primitive ac-tions.Agent behavior has alternatively been designed so that the group members move as being particles in a rigid virtual structure[13].Similar ideas have been combined with potential ?eld-like controllers for multi agent control[14],[4],[15],and decentralized formation forming[16].The leader-follower ap-proach[17],[18],[19]distinguishes a designated group leader which the other agents follow either directly or indirectly. Another research direction focuses on the stability of the in-terconnected system.In[20]a distributed control scheme is designed for spatially interconnected systems that is shown to inherit the same topological structure with the target system. String stability[21],[22],[1]and mesh stability[23],the latter Herbert Tanner is with the Department of Electrical and Systems Engineer-ing,3401Walnut Street,Suite301C,University of Pennsylvania,Philadelphia, PA19104-6228(e-mail:tanner@https://www.360docs.net/doc/8d6503420.html,)
George Pappas is with the Department of Electrical and Systems Engineering, 200South33rd Street,University of Pennsylvania,Philadelphia,PA19104(e-mail:pappasg@https://www.360docs.net/doc/8d6503420.html,)
Vijay Kumar is with the Department of Mechanical Engineering and Ap-plied Mechanics,3401Walnut Street,Suite301C,University of Pennsylvania, Philadelphia,PA19104-6228,(e-mail:kumar@https://www.360docs.net/doc/8d6503420.html,)being the generalization of the former in more than two dimen-
sions,express the property of the system to attenuate distur-bances as they propagate through the interconnections.While
earlier works[24],[21],[22],[25]have used the notion of string stability in the frequency domain,string stability of in-terconnected systems has recently been studied in a state space
framework[1].In[1],suf?cient conditions for string stability were derived,requiring global Lipschitz continuity of vector ?elds and exponential stability of the unconnected subsystems.
Weaker notions of string stability include string stability [1],which is the only type of stability that can be guaranteed for an interconnected system without a special structure.It is
also known that in certain cases string stability is impossible [26],[25],[22].
The class of formations discussed in this paper generally do
not fall under the class of string stable interconnected systems. On the other hand,it is generally the case that formations are stable at least locally,in the sense that local controllers can en-sure some boundedness of errors.A question then arises as to whether further stability analysis can be performed to address issues such as boundedness of propagating errors and perfor-mance characterization.This question motivates the introduc-tion of a different framework,that brings forward the depen-dence of the internal state of an interconnected system to in-coming input signals or disturbances,rather than ensuring con-vergence of signals to zero.
In this paper we de?ne Leader-to-Formation Stability(LFS), a different notion of stability for leader-following formations, and we use it to characterize how the motion of the group lead-ers can affect the motion of the group.This notion is based on input-to-state stability[27]and its propagation properties through certain interconnections[28],[29].In previous work [30],[31],[32]we have been able to characterize the effect of the behavior of the group leaders to the rest of the group for cer-tain types of formation interconnections and obtain quantitative measures of the stability of the formation with respect to the motion of the leaders.In this paper we generalize these results and exploit LFS for analysis and design of robot formations.
II.D EFINITIONS AND P RELIMINARY R EMARKS
We consider formations that are based on leader-follower ar-chitectures.In that framework,a formation will be broadly de-?ned as a network of vehicles interconnected through their con-trollers,with the latter being designed so that the motion of the vehicles meet certain speci?cations.In the formations consid-ered,a number of vehicles are identi?ed as designated leaders in the sense that their motion is not constrained by the forma-tion speci?cations.Related work[6],[17],[18],[33],[34]has motivated the use of graphs to represent agent interaction.We
2
therefore brie?y introduce some notions from graph theory[35]
that will facilitate our analysis.
A.Graph Theory Preliminaries
A directed graph consists of a vertex set and an di-rected edge set,where a directed edge is an ordered pair
of distinct vertices.A vertex is incident with an edge if it is one of the two vertices of the edge.An edge in a directed graph is said to be incoming with respect to and outcoming
with respect to.Such an edge has vertex as a tail and vertex as a head.The indegree of a vertex in a directed graph is de-?ned as the number of edges that have this vertex as a head.A
subgraph of a graph is a graph such that
and.A subgraph of is an induced sub-
graph if two vertices of are adjacent in if and only they are adjacent in.A path of length in a directed graph is a sequence of distinct vertices such that for every ,.A weak path is a sequence
of distinct vertices such that for each either or is an edge in.A directed graph is weakly connected or simply connected if any two vertices can be joined with a weak path.The distance between two vertices and in a graph is the length of the shortest path from to .The diameter of a graph is the maximum distance between two distinct vertices.A(directed)cycle is a connected graph where every vertex is incident with one incoming and one out-coming edge.An acyclic graph is a graph with no cycles.The incidence matrix of an oriented graph with vertices and edges(assuming some enumeration on the edge set)is an matrix,the of which is if is the head of edge,if it is the tail and otherwise.
B.Formation Graphs
A formation is generally described in terms of its shape. We have seen,for example,‘line’,‘column’,‘diamond’and ‘wedge’formations[2],[9].This is a global geometric descrip-tion of a formation as opposed to the particular interconnection structure that the control designer has chosen to implement this geometry.We will refer to this geometric formation description using the term shape.In this paper we will consider formations that are based on leader-following.In this case shape can be de?ned as follows:
De?nition II.1.The shape of a formation of robots with leaders moving in is a point in a-dimensional submanifold of.
Broadly speaking,the shape will be a point in a hyper-surface of the total dimensional state space,de?ned after imposing constraints related to the position of the formation leaders.The desired shape,,is a particular region in that hypersurface.In the case where can be locally parame-terized as a point then the shape error is simply given by the Euclidean distance1:
The choice of an appropriate metric for describing shape in SE(3)is a dif-ferent important issue[36],and will not be discussed in this paper.The agent interconnections can then be designed to implement
the desired shape.The type of agent interconnections consid-ered are leader-following relationships,which we chose to rep-resent by means of a graph.The leader-follower architecture
dictates an orientation on this graph.
The formation will be associated with and identi?ed by a di-
rected graph that represents both its shape and the control spec-i?cations that realize it.
De?nition II.2(Formation Control Graph).A formation control graph is a directed graph with: A?nite set of vertices and a map as-signing to each vertex a control system
where and.
An edge set encoding leader-follower rela-tionships between agents.The ordered pair
belongs to if depends on the state of agent,.
A collection of edge speci?cations,de?ning
control objectives(setpoints)for each
for some.The speci?cations,implement a certain desired shape for the formation.
According to De?nition II.2,each vertex represents the dy-
namics of a particular robot.For robot,the tails of all incom-ing edges to vertex represent leaders of,their set denoted by .Robot is controlled so as to meet the speci?cations for all.Similarly,we can de?ne the set of im-mediate followers of agent as the heads of all edges having as tail.Vertices of indegree zero,represent formation leaders, denoted by.For the formation leaders in no speci?cation is prescribed with respect to other agents.In-stead,formation leaders aim at achieving group objectives such as following a reference trajectory or navigating within an ob-stacle populated environment.
In the present analysis,the formations control graphs consid-ered are acyclic.The case of cycles within a formation graph is considered elsewhere[30].The primary reason for this assump-tion is that a cycle may not always be stable[37]and stability analysis will involve the use of versions of the small gain theo-rem[30].
Formation speci?cations capture shape information.The for-mation shape can speci?ed with respect to some common ref-erence frames,which can be assumed to be the local frames of the formation leaders:
Formation speci?cations are de?ned in terms of a desired shape vector.If is the desired shape component corresponding to agent,then the desired state for will be expressed as,where can be the state of any formation leader.
Edges,on the other hand,express possible ways to realize a desired shape.The set of possible edges can be restricted due to sensing and/or communication constraints.Given a set of possible edges,one can decompose the formation speci?cations to individual edge speci?cations as follows:
(1)
3 where is the incidence matrix of the formation control graph
and denotes the Kronecker matrix product[38].If the set
of edges can realize the desired shape,then the left hand side
of(1)will be independent of the formation leaders states.For
agents with multiple leaders,on the other hand,there will be a
degree of redundancy in their incoming edges speci?cations.
With the speci?cation for edge being cal-
culated using(1),a setpoint for agent can be expressed as
follows:
For agents with multiple leaders,the speci?cation redundancy
can be resolved by projecting the incoming edges speci?cations
into orthogonal components:
(2)
where are projection matrices with.
Then the error for the closed loop system of robot can be
de?ned as the deviation from the prescribed setpoint:
Since there are no edge speci?cations for the formation leaders,
errors for the leaders are de?ned with respect to some overall
‘high level’formation objective.Such an objective could be,
for example,tracking of a reference trajectory.In general,the
closed loop system for any leader would take the form:
where is(asymtptotically)stable and is an ex-
ogenous input signal related to this formation objective.Alter-
natively,can be viewed as a disturbance affecting the closed
loop leader dynamics.The formation error is de?ned as the
deviation of the formation shape from the desired:
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to which the formation is viewed as a nonlinear operator from the space of leader input/disturbances to the space of the for-
mation internal state.LFS expresses the dependence of the for-
mation shape on the input signals given to the leaders.Inequal-ity(3)provides a‘nonlinear gain estimate’through functions and,taking into account the initial conditions.In-tuitively,a formation control scheme based on local controllers
would be able to stabilize interconnection errors if the reference
signals sent to the leaders are set to zero.On the other hand,a high speed maneuver from the part of the leaders in response to a reference input signal is expected to have adverse effects on the stability of the followers.
It is known[41],[28]that certain interconnections of ISS
systems preserve the ISS property.As indicated by our pre-vious work[32],[30],certain formation interconnections pre-serve ISS and therefore it would be straightforward to estab-lish Leader-to-Formation Stability.In this paper we general-ized these results to arbitrary interconnections of leaders and followers conforming with the conditions of De?nition II.2. Based on alternative characterizations of input-to-state sta-bility,De?nition II.3implies the following:
Corollary II.4.If a formation is LFS,in the sense of De?nition II.3,then the formation error satis?es:
Corollary II.4reveals that the steady state LFS gain serves as an ultimate bound for the formation error.This motivates the de?nition of the following LFS measure:
De?nition II.5.Consider a formation that is LFS.Then the scalar quantity:
is called the LFS performance measure for the formation.
The LFS measure can be thought of as a radius of the ball in which the steady state formation error will remain,when the inputs to the formation leaders are bounded inside unit balls.
III.LFS P ROPAGATION
Fig.2.A generic formation control graph structure.
Any induced formation control subgraph of depth two has the structure depicted in Figure2.For simplicity,assume an enumeration of the induced formation control graph where the leaders(depth zero)are assigned the numbers,agents at depth one are assigned the numbers and the followers at depth two are assigned the numbers. Let the dynamics of the agents be expressed as follows:
(4a)
(4b)
(4c)
The agents are driven by control laws of the form:
(5a)
(5b)
(5c)
resulting in closed loop error dynamics which can be written as:
(6a)
(6b)
(6c)
The main result of the paper is based on the invariance of the LFS property under cascading and draws from well known results on ISS:
Lemma III.1([28]).Suppose that in the system:
the subsystem is ISS with respect to and,and the-subsystem is ISS with respect to,that is,
where and are class functions and and are class functions.Then the complete-system is ISS with:
where
5 and(6c)is LFS with respect to:
then the induced formation control graph is LFS with respect to :
with and
(7a)
(7b) Proof.See Appendix.
In the case where the agent dynamics are linear,then LFS does not require speci?c assumptions on the agent dynamics. Moreover,one can obtain much less conservative results than those derived by direct application of(7),by exploiting the lin-ear structure.Application of the feedback control laws
(8a)
(8b)
(8c) where,,and are such that,
are Hurwitz,and,and satisfy:
results in closed loop error dynamics that can be written as:
(9a)
(9b)
(9c) This model is equivalent to the one used for a string of LTI systems in[42].In this case,the LFS gains are given as follows: Proposition III.3.Consider the formation of Figure2where the closed loop error dynamics of the agents are given by(9). Then,(9)is LFS with respect to:
where a parameter,
,and each satisfying:
Proof.See Appendix.
We conclude this section with an analytical justi?cation of the intuitive fact that the larger the diameter of a formation control graph,the larger the ampli?cation of leader perturba-tions,unless the formation is string stable[1].Using(10)we can calculate the input LFS gain of a string of vehicles,which for simplicity it is assumed to be described by linear dynamics. Figures3-4depict the change in the LFS performance measure of the formation as more vehicles are considered in the string. In Figure3,the LFS input gain of the leader-follower pair is small enough to make errors attenuate and maintain stability. The performance measure approaches a steady state value.On the contrary,if the LFS input gain of the leader-follower pair is such that allows signi?cant error propagation the formation performance measure deteriorates rapidly(note the logarithmic
allows the string to be augmented without performance degradation.
IV.T HE E FFECT OF F EEDFORWARD I NFORMATION The results of Section III assumed for each agent feedback information from its leaders.The use of local information al-lows the construction of decentralized control schemes which are generally scalable and robust.As it could be expected, however,utilization of additional information could possibly enhance the performance of the formation[32],although this is not necessarily the case[33].This section presents ways in which such additional information can be used to improve the LFS performance characteristics of a formation. Information obtained by some agents may have more signi?-cant effect to the stability of a particular leader-follower dynam-ics than from others.One way to improve the stability charac-teristics of some leader-follower dynamics is to use feedforward
6
measure implies error ampli?cation.
information from the pair leader,denoted by the head of a par-
ticular edge:
Corollary IV.1.Assume that in system(4),for some
control law a control law can
render(6c)asymptotically stable.Then,for the leader-
follower pair,.
Proof.The result is a consequence of the fact that if(6c)is
asymptotically stable,then there will be a class-function
such that for the edge error,.
It follows that the pair is LFS with.
For the linear case,the result requires only a rank condition:
Corollary IV.2.Assume that in(9)for some,
is full column rank.Then the control law
where yields.
Proof.Since is full column rank,it has a left inverse and
therefore one can?nd a satisfying
.Then for the closed loop error dynamics of:
and since is stable,
Let.Then for(12)with
,the Lyapunov function will satisfy:
7
where
where is a modeling parameter https://www.360docs.net/doc/8d6503420.html,ing input-output feedback linearization:the interconnection error dynamics can be brought to the form:
(13) In the case where separation and bearing are de?ned with re-spect to different leading robots,say and,respectively,the control inputs for have to be selected as follows:
The internal dynamics of can be shown to be stable[37]. Then,using as a Lyapunov func-tion for(13),and denoting by,we can arrive at:
which yields for:
(14b) establishing the LFS property of the leader-follower pair.
VII.S IMULATIONS
In this section we will turn our attention to a simple formation of three mobile robots(Figure6).We will use LFS to assess and numerically verify the stability properties of three different formation structures,based on
(13).
Fig.6.Formation of three mobile robots using separation-bearing controllers.
8
A.Line Formation
In the line formation,the three robots are linked in cascade.Robot is the formation leader,robot follows robot try-ing to maintain constant separation
and constant bearing
with respect to .Similarly,robot fol-lows robot aiming at a constant separation
and constant bearing
with respect to robot .The desired shape for the formation is the one in which the robots should move while remaining on a straight line.The controller
gains for all robots were set to
,and the value selected for the model parameter was
.The leader of the formation is required to track a target that moves on a circle.The control interconnection is depicted in the graph of Figure 7.Figure 8shows snapshots of the robots motion.Figure 9gives the evolution of the formation errors where for reasons of clar-ity and consistency with respect to units,the formation errors associated with separation and bearing are provided
separately.
Fig.7.Cascade interconnec-
Fig.8.Robot paths for the
From Figure 9it is evident that after an initial transient,the formation errors converge to a steady state.Equation (7)predict a very conservative LFS steady state gain function:
and a corresponding LFS measure:.The conservativeness of the calculated bounds is primarily due to the propagation of the overapproximations made during the estimation of the leader-follower gains,which are more severe in the nonlinear case.It is actually the price paid for decom-posing the calculation of the gains of the complete https://www.360docs.net/doc/8d6503420.html,-ing linear expressions for the leader-follower gain estimates can yield a less conservative measure:.
B.Wedge Formation
In wedge formation both and follow the formation leader ,with the desired separation for robot now being
so that the shape of the formation remains unchanged.For that con?guration,equations (7)predict an LFS steady state gain
function
and a LFS performance measure of
.A linear over-approximation of (14),gives through (10):
and .The results suggest that the wedge formation is clearly more stable than the line formation.In Figures 10-12it is clear that the desired shape of the formation (the robots in straight line)is maintained much more accurately than in the case of the line formation and errors are signi?cantly smaller.
Fig.11.Robot paths in wedge for-
C.Triangle Formation
In the triangular formation,robot has two leaders;is trying
to keep a distance
from robot and a bearing of with respect to the leader.The LFS input gain is:
and .The linear over-approximation
using (10)results in
and .The simulation results for the triangle formation are shown in Figure 13-15.
9
We close this section by simulating a string of ten mobile robots.In this case,the formation leader is assigned to follow a target moving in sinusoidal path while the string of vehicles following the leader is trying to keep a straight line.Figure 17shows the path of the?rst and last robot in the string and outlines the formation shape.The formation errors are given in Figure18where it is seen that after the initial transient,the
cap-
per-
in-
con-This research is partially supported under DARPA MICA Contract Number N66001-01-C-8076,and by DARPA/AFRL Software-Enabled Control Grant F33615-01-C-1848.
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A PPENDIX
Proof of Proposition III.2
For the generic formation of Figure2,note that LFS of each follower with respect to,,for the time inter-val and yields:
(15)
(16)
In case agent does not follow agent,the corresponding term is zero.Similarly,the LFS property of with respect to is equivalent to:
(17) and implies:
(18a)
11
Substituting(15),(18)into(16)yields a new bound:
(19) Observe now how the shape information of agent can be re-constructed from the agent errors:
Combining the above with(19)-(17)and recalling that for any class-function,: Summing over all and including the shape deviation of every,we obtain for:
Proof of Proposition III.3
The proof follows the same lines as that of Proposition III.2. Note that every follower is a perturbed system with expo-nentially stable nominal error dynamics having as a Lyapunov function:.This implies that:
(21) The LFS gains(10)are derived in the same way as Proposition III.2,with the sole difference being that due to linear class-functions,we can have: