Higher Order Sliding Modes in Collaborative Robotics

Problem setupMotion planning

Trajectory tracking

Higher order sliding modes in collaborative

robotics

M. Defoort1, T. Floquet 2,3, A. Kokosy2, W. Perruquetti2,3

1 LAMIH CNRS FRE 3304UVHC, 59313 Valenciennes, FRANCE

2 LAGIS CNRS FRE 3303Ecole Centrale de Lille, 59651 Villeneuve d’Ascq, FRANCE

3 Alien Inria - LNE59651 Villeneuve d’Ascq, FRANCE

27th June 2010

W. Perruquetti Higher order sliding modes in collaborative robotics 27th June 2010 1 / 45


Trajectory tracking

Introduction

+ Robocoop project: http://syner.ec-lille.fr/robocoop

Goals

Coordination of a formation of autonomous robots

To propose and to implement tools for the modeling, theanalysis and the control design in the context of cooperativemobile robots

To get complex behaviors by using simple agent based behaviors

Deployment of large scale networks of cooperative mobilerobots



Trajectory tracking

Introduction

Challenges

local information and decision process,

constrained communication + delays (communicationprotocols, . . . ),

large scale system,

hybrid system aspects

uncertain and hostile dynamic environment,

. . .



Trajectory tracking

Introduction

Applicative fields

health (tele-robotics, . . . )

transportation (plane fleet, drones, mobile robots, planes,underwater robots, . . . )

security (fire, data collection for “spying”, . . . )



Trajectory tracking

Introduction

Goal

Autonomous cooperative navigation of a formation of mobile robots inan uncertain environment with obstacles

Requirements

Static obstacle avoidance and avoidance of collision between robots

Maintaining the communication links (exchanged information:positions, intentions, strategies, . . . ) = needed for local or globalcoordination achievement

High performance trajectory tracking (precision, convergence speed,stability, robustness against parametric uncertainties andperturbations)



Trajectory tracking

Introduction

Goal

Autonomous cooperative navigation of a formation of mobile robots inan uncertain environment with obstacles

Requirements

Static obstacle avoidance and avoidance of collision between robots

Maintaining the communication links (exchanged information:positions, intentions, strategies, . . . ) = needed for local or globalcoordination achievement

High performance trajectory tracking (precision, convergence speed,stability, robustness against parametric uncertainties andperturbations)



Trajectory tracking

Outlines

1 Problem setup

2 Motion planning

3 Trajectory tracking



Trajectory tracking

ObjectiveMobile robot model

Outlines

1 Problem setup

2 Motion planning




Trajectory tracking


Objective

Contributions

4 Design of original methods for motion planning and trajectorytracking

4 Implementation on real mobile robots

Robots Mobiles

PerceptionEnvironnement

Estimationof position

PlanningTrajectories

TrackingTrajectories

of MotorsControls

MotorsSensors

Position

Estimated

Trajectories

Map of obstaclesPreparation

Mission

Objectives

Strategy



Trajectory tracking


Objective

Contributions

4 Design of original methods for motion planning and trajectorytracking

4 Implementation on real mobile robots

Robots Mobiles

PerceptionEnvironnement

Estimationof position

PlanningTrajectories

TrackingTrajectories

of MotorsControls

MotorsSensors

Position

Estimated

Trajectories

Map of obstaclesPreparation

Mission

Objectives

Strategy



Trajectory tracking


Hierarchical structure

To achieve computational tractability:

“Strategic layer” (higher level): goal planning (for examplechoose an appropriate functional cost), task scheduling (forexample use a petri net for description),

“Tactical layer” (mid level): guidance, navigation

“Reflexive layer” (low level): (control) state observation orestimation, trajectory tracking, . . .

Questions

How can we get an “integrated layer” ?+ Solve an optimization problem which integrate some of thesefacts (gives a path) and then use a good “trajectory tracking”



Trajectory tracking







Questions




Trajectory tracking







Questions




Trajectory tracking







Questions

How can we get an “integrated layer” ?

+ Solve an optimization problem which integrate some of thesefacts (gives a path) and then use a good “trajectory tracking”



Trajectory tracking







Questions




Trajectory tracking


Unicycle-type model

Characteristics

2 independently controlled fixed wheels

Nonholonomic constraint

Kinematic model

xi = vi cos θiyi = vi sin θiθi = wi

with

{vi = r

2 (ϕ1,i + ϕ2,i )wi = r

2R (ϕ2,i − ϕ1,i )



Trajectory tracking


Unicycle-type model

Characteristics

2 independently controlled fixed wheels

Nonholonomic constraint

Kinematic model

xi = vi cos θiyi = vi sin θiθi = wi

with

{vi = r

2 (ϕ1,i + ϕ2,i )wi = r

2R (ϕ2,i − ϕ1,i )

yi

xi

θi−→i

−→j

O

ϕ1,i

ϕ2,i

Figure 1:

1

xi , yi : centre of the driving wheelsθi : orientationvi ,wi : linear and angular velocitiesr : radius of wheels2R: distance between driving wheels



Trajectory tracking

Motion planning for a single robotMulti-robots coordination

Outlines

1 Problem setup

2 Motion planning




Trajectory tracking


Motion planning for a single robotMotion planning

Computation of an executable collision-free trajectory for a robotbetween an initial given configuration and a final given configurationINTRODUCTION - Path Planning Review

Trajectory planning

1

Global plannerLocal planner

Potential fields

Dynamic window

Cell decompostion

Latombe 1991 Laumond 1997

Pontryagin et al. 1962Bryson et Ho 1975Bobrow 1988

Optimal control Flatness

Fliess et al. 1995

Visibility graph

Voronoïgraph

Khatib 1986 Borenstein et Koren 1991Barraqunad et al. 1992Rimon et Koditschek 1992

Fox 1997

Chazelle et Guibas 1989 Choset 1996

Agrawal et al. 1996Murray et al. 2001Milam 2003



Trajectory tracking


Single robot: off-line algorithm

Dynamic optimisation based on flatness

Optimal control Flatnessbased on flatness

Dynamic optimization

Resolution of optimal control problems

+ Transformation into a nonlinear programming problem, using B-splinefunctions in order to approximate the trajectory of the flat output+ Computation of optimal control points using an optimisationprocedure (CFSQP)+ Computation of the corresponding control inputs using the flatnessproperties of the system



Trajectory tracking


Single robot: on-line algorithm

Main Principle

+ To relax the constraint that the finalpoint is reached during the planninghorizon, allowing the use of an on-linereceding horizon motion planner

Tp(> 0): planning horizon

Tc(> 0): update period

τk(k ∈ N, τk = tinitial + kTc):updates

τk τk+1

Tp

Tc

Legend:

Computed trajectory

Reference Trajectory



Trajectory tracking


Multi-robots coordination

Objective

+ To generate a (sub) optimal trajectory for each robot which satisfy:

terminal constraints

physical constraints (nonholonomic, maximum velocities, . . . )

obstacle avoidance

minimum distances between robots (collision avoidance)

maximum distances between robots (respect of the broadcastingrange) Communication graph (N ,A,S)

Robots N = {1, . . . ,Na}Edges A ⊂ N ×N � communication links

Constraints of the edgesdi ,com ∈ R+: broadcasting range of robot i



Trajectory tracking



Objective




obstacle avoidance







Trajectory tracking



Objective




obstacle avoidance







Trajectory tracking



Objective




obstacle avoidance







Trajectory tracking



1

INTRODUCTION - Multi-robot Planning and Control Review

Trajectory planning: multi-robot framework

With cooperation

Decentralized approach

Centralized approach

Loizou et Kyriakopoulos, 2002,Olfati-Saber et al., 2003,Tanner et al., 2003Ogren, 2003Dunbar et Murray, 2002

Guo et Parker, 2002Gazi et Passino, 2004Gennaro et Jadbabaie, 2006,Keviczky et all, 2006Kuwata et al. 2006

Without cooperation

Worst case approach

Tomlin et al., 1998



Trajectory tracking


Multi-robots coordination: centralized approach

+ Resolution via a supervisor (independent unit or a single robot ofthe formation)

Limitation 1

Prohibitive computation time

Solution 1

Step of simplification of the initial problem:+ Motion planning of a virtual robot which is located at the centreof gravity of the formation

Limitation 2

Problems due to the supervisor



Trajectory tracking




Limitation 1


Solution 1


Limitation 2




Trajectory tracking




Limitation 1


Solution 1


Limitation 2




Trajectory tracking


Multi-robots coordination: decentralized approach

Desired objectives

low computation time

high performances

use of available local information

no supervisor

Solution

Distributed optimisation based on local information. Each vehicle i only takes into account the intentions of the robotsbelonging to the conflict set Ci (τk) (may produce a collision Ci ,collision(τk)or may lost the communication Ci ,com(τk))



Trajectory tracking



Desired objectives

low computation time

high performances

use of available local information

no supervisor

Solution

Distributed optimisation based on local information. Each vehicle i only takes into account the intentions of the robotsbelonging to the conflict set Ci (τk) (may produce a collision Ci ,collision(τk)or may lost the communication Ci ,com(τk))



Trajectory tracking


Multi-robots coordination: decentralized approachConflicts with robot 1:C1,collision(τk) = {2} C1,com(τk) = {4}

Robot 1

d1,com

R1(τk)

Robot 2

Robot 3

Robot 4

Legende :

Zone d’accessibilitePortee de diffusion des informations

Figure 1:

1



Trajectory tracking



Difficulties

Knowledge of the intentions of robots p ∈ Ci(τk)

uniqueness of the presumed trajectory

coherence between the presumed trajectory and the optimalplanned trajectory

Solution

+ Decomposition of the algorithm into 2 steps:

? determination of the presumed trajectory (which only satisfythe individual constraints)

? determination of the optimal planned trajectory from theexchanged information between robots belonging to the subsetCi(τk)



Trajectory tracking



Difficulties




Solution






Trajectory tracking



Difficulties




Solution






Trajectory tracking



Difficulties




Solution






Trajectory tracking



Difficulties




Solution






Trajectory tracking



Difficulties




Solution






Trajectory tracking


Multi-robots coordination: results

Approach Cent. Leader/Follower Weakly Stronglydecent. decent.

Maxi timeof conflict 2050ms 313ms 703ms 121msresolution

Exchanged global local local localInfo.

−− ++ − +Implem. sequential if conflict with

if Na � 1 resolution a lot of robots

Timereaching 35s 39s 36s 36.5s

goal



Trajectory tracking


Strongly decentralized

Video



Trajectory tracking


Strongly decentralized

Video



Trajectory tracking

IntroductionApproaches with coordinationApproach with coordination

Outlines

1 Problem setup

2 Motion planning




Trajectory tracking


Introduction

Challenges

Nonlinear dynamics,

Presence of perturbations (unmodelled dynamics, sensor noise,external disturbances)

How deal with the stabilization problem at low or zero velocity?

How to integrate cooperation into the control design ?

Leader or not ?

Facts:

90 percent of the job is done by nominal control (path planningfrom which the open loop control is obtained thanks todifferential flatness),

10 by feedback !

Several solutions were proposed, a challenging problem being controldesign taking into account cooperation.



Trajectory tracking


Introduction

1

INTRODUCTION - Multi-robot Planning and Control Review

Trajectory tracking

Continuous non stationary state

feedbackSamson, 1990Coron, 1992Pomet, 1992Jiang et Nijmeijer, 1999Jiang et al, 2001

Desai et al ,2001Das et al, 2002Tanner et al., 2004Orquedo et Fierro, 2006

Without cooperation With cooperation

Discontinuous state feedback

Bloch et al., 1992Hespanha et al., 1999Astolfi, 1996Floquet et al., 2003Drakunov et al, 2005

Dynamic state feedback (ex: quasi static state

feedback)

Rudolph, 1993Delaleau et Rudolph, 1998



Trajectory tracking


Tools:Sliding Mode Control

Objective

To constrain the trajectories of system x = f (x) + g(x)u to reach, in afinite time, and then, to stay onto the sliding surface chosen according tothe control objectives

Variable structure control

u =

{u+(s) if sign(s(x)) > 0u−s) if sign(s(x)) < 0

with u+ 6= u−

Classical control design

u = ueq + udisc

given by s = s = 0, (invariance of the sliding surface)

udisc = −ksign(s), (convergence in finite time onto the surface)



Trajectory tracking



Objective



u =


with u+ 6= u−


u = ueq + udisc





Trajectory tracking



Objective



u =


with u+ 6= u−


u = ueq + udisc





Trajectory tracking



Objective



u =


with u+ 6= u−


u = ueq + udisc





Trajectory tracking



Advantages

Insensibility against perturbations (matching perturbations)

The choice of surface s(x , t) = 0 allow to choose a priori theclosed-loop dynamics

Inconvenients

Chattering

Trajectory

s = 0

Chattering phenomenon

s(x , t) must have a relative degree equalto 1 wrt. u

The trajectories are not robust againstperturbations during the reaching phase



Trajectory tracking



Advantages

Insensibility against perturbations (matching perturbations)

The choice of surface s(x , t) = 0 allow to choose a priori theclosed-loop dynamics

Inconvenients

Chattering

Trajectory

s = 0

Chattering phenomenon

s(x , t) must have a relative degree equalto 1 wrt. u

The trajectories are not robust againstperturbations during the reaching phase



Trajectory tracking


Tools:Higher Order Sliding Mode Control

Objective

To constrain the system trajectories to evolve onto the sliding surface:

Sr ={x ∈ Rn : s = s = . . . = s(r−1) = 0

}

IntroductionCommande par modes glissants d’ordre supérieur optimale

Expérimentation sur deux benchmarks industrielsConclusion et perspectives

Commande par modes glissants : avantages et inconvénientsCommande par modes glissants d’ordre supérieur

Modes glissants d’ordre r = ρ

trajectoire trajectoire

chattering

s = 0 s = 0

mode glissant d’ordre r = 1 mode glissant d’ordre r > 1

Stabilisation en TF⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

z1 = z2z2 = z3

...zρ = ϕ(·) + γ(·)us = z1

16 Commande par modes glissants d’ordre supérieur: théorie et applications



Trajectory tracking


Tools:Integral Sliding Mode Control

Objective

To remove the reaching phase+ To guarantee the robustness properties against perturbations in themodel from the initial time instance

Philosophy

+ To choose the sliding variable such that the system trajectories arealready on the sliding surface at the initial time instance



Trajectory tracking


Tools:Integral Sliding Mode Control

Objective

To remove the reaching phase+ To guarantee the robustness properties against perturbations in themodel from the initial time instance

Philosophy

+ To choose the sliding variable such that the system trajectories arealready on the sliding surface at the initial time instance



Trajectory tracking


Problem setup

Reference trajectory

xrefyrefθref

=

cos θref 0sin θref 0

0 1

[

vrefwref

]

Objective

x(t0)

θ

−→i

−→j

O

θre f

Robot de reference

Robot reel

y(t0)

xre f (t0)

yre f (t0)

t1t2

t3

t1

t2

t3t4

t4

Figure 1:

1

Individual tracking of the optimalplanned trajectory for each robot i+ To stabilize the tracking errors:

exeyeθ

=

x − xrefy − yrefθ − θref

Difficulties

+ Presence of perturbations and parametric uncertainties in themodel:

xy

θ

=

cos θ 0sin θ 0

0 1

[

vw

]+ p(q, t)



Trajectory tracking


Problem setup


xrefyrefθref

=


0 1

[

vrefwref

]

Objective

x(t0)

θ

−→i

−→j

O

θre f

Robot de reference

Robot reel

y(t0)

xre f (t0)

yre f (t0)

t1t2

t3

t1

t2

t3t4

t4

Figure 1:

1


exeyeθ

=


Difficulties


xy

θ

=

cos θ 0sin θ 0

0 1

[

vw

]+ p(q, t)



Trajectory tracking


Problem setup


xrefyrefθref

=


0 1

[

vrefwref

]

Objective

x(t0)

θ

−→i

−→j

O

θre f

Robot de reference

Robot reel

y(t0)

xre f (t0)

yre f (t0)

t1t2

t3

t1

t2

t3t4

t4

Figure 1:

1


exeyeθ

=


Difficulties


xy

θ

=

cos θ 0sin θ 0

0 1

[

vw

]+ p(q, t)



Trajectory tracking


Algo. 1

Assumptions

Perturbations satisfy the matching condition

Perturbations are bounded by known positive functions

Reference velocities are continuous and bounded

No stop point



Trajectory tracking


Algo. 1

The tracking errors asymptotically converge toward zero under:

u = unom + udisc

Continuous term unom [Jiang et al., 2001]

unom stabilize the tracking errors without perturbation

unom =

[vref cos e3 + µ3 tanh e1

wref + µ1vref e2

1+e21 +e2

2

sin e3e3

+ µ2 tanh e3

]

with

e1

e2

e3

=

− cos θ − sin θ 0

sin θ − cos θ 00 0 −1

exeyeθ



Trajectory tracking


Algo. 1


u = unom + udisc

Discontinuous term udisc

udisc reject the effect of the perturbation from the initial time instance

udisc =

[−G1(e)sign(σ1)

−G2(e)sign(−e2σ1 + σ2)

]

with σ = [σ1, σ2]T given by:σ = σ0(e) + eaux

σ0(e) = [−e1,−e3]T : linear combinaison of state

integral part

eaux =

[vref cos e3

wref

]−[

1 −e2

0 1

]unom(e)

eaux = − σ0(e(0))



Trajectory tracking


Algo. 1


u = unom + udisc



udisc =

[−G1(e)sign(σ1)


]



integral part

eaux =

[vref cos e3

wref

]−[

1 −e2

0 1

]unom(e)

eaux = − σ0(e(0))



Trajectory tracking


Algo. 1


u = unom + udisc



udisc =

[−G1(e)sign(σ1)


]



integral part

eaux =

[vref cos e3

wref

]−[

1 −e2

0 1

]unom(e)

eaux = − σ0(e(0))



Trajectory tracking


Experimental results: Algo. 1

Single nominal control

0 1 2 3 4 5 6 7 8 9−1

0

1

2

3

4

5

6

x (m)

y (m

)

traj. référencetraj. réelleobstacleobstacle augmenté

0 5 10 150

0.02

0.04

0.06

0.08

0.1

t(s)

erre

ur (

m)

ISMC

2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

y (m

)

traj. référencetraj. réelleobstacleobstacle augmenté

0 5 10 150

0.01

0.02

0.03

0.04

0.05

0.06

t(s)

erre

ur (

m)



Trajectory tracking


Algo. 1

Limitations

conservative assumptions

discontinuities on velocities

perturbations must satisfy the matching condition

Solution

+ Practical stabilization using second order ISMC

T

2ε

x

t

δ

−δ

x(0)

Figure 1:

1



Trajectory tracking


Algo. 1

Limitations

conservative assumptions

discontinuities on velocities

perturbations must satisfy the matching condition

Solution

+ Practical stabilization using second order ISMC

T

2ε

x

t

δ

−δ

x(0)

Figure 1:

1



Trajectory tracking


Algo. 2Transformation into the Heisenberg system

Transformation:[ZX

]=

eθ cos θ − 2 sin θ eθ sin θ + 2 cos θ 00 0 1

cos θ sin θ 0

exeyeθ

State feedback:

U =

[w − wref

v − w (ex sin θ − ey cos θ)− vref cos eθ

]

+ Perturbed Heisenberg Syst.:

Z = UT JX + δZ

X = U + δX

U = T + δU



Trajectory tracking



Transformation:[ZX

]=


cos θ sin θ 0

exeyeθ

State feedback:

U =

[w − wref


]


Z = UT JX

+ δZ

X = U

+ δX

U = T

+ δU

J =

[0 −11 0

]



Trajectory tracking



Transformation:[ZX

]=


cos θ sin θ 0

exeyeθ

State feedback:

U =

[w − wref


]


Z = UT JX + δZ

X = U + δX

U = T + δU

perturbations



Trajectory tracking



Transformation:[ZX

]=


cos θ sin θ 0

exeyeθ

State feedback:

U =

[w − wref


]


Z = UT JX + δZ

X = U + δX

U = T + δU

Avoidance of discontinuities onvelocities



Trajectory tracking


Algo 2

+ To stabilize the Z−dynamics before the X ’s one

δZ = δX = 0{s1 = Z + m1Z + m2Zaux

s2 = ψ + m3ψ + m4ψaux

with

{Zaux = Z

ψaux = ψ+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε

where 0 < ε and Θ : R→ R+ is a positive definitefunction of class C2 such that ψ(0) > 0 and{

mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ ISMC

i.e. [sign(s1), sign(s2)]T

Arbitrary perturbations{σ1 = Z + m1Zaux,1 + m2Zaux,2σ2 = ψ + m3ψaux,1 + m4ψaux,2

with

Zaux,1 = Z

Zaux,2 = Zaux,1

ψaux,1 = ψ

ψaux,2 = ψaux,1

+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε

where 0 < ε� 1 and Θ : R→ R+ is a positive definitefunction of class C2 such that ψ(0) > 0 and{

mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ 2nd order ISMC

i.e. sampled twisting

Conclusion

∀t ≥ 0, ψ(t) ≥ 0 =⇒ XTX ≥ 2Θ(Z ) + ε =⇒ XTX > 0 (becauseΘ(Z ) is positive definite)+ avoidance of singularity in control

The Z and ψ dynamics are exponentially stable. Moreover, sinceZ = 0 and ψ = 0, ‖X‖2 = ε because Θ(0) = 0



Trajectory tracking


Algo 2


δZ = δX = 0{s1 = Z + m1Z + m2Zaux


with

{Zaux = Z

ψaux = ψ+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε


mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ ISMC



with

Zaux,1 = Z

Zaux,2 = Zaux,1

ψaux,1 = ψ

ψaux,2 = ψaux,1

+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε


mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ 2nd order ISMC


Conclusion





Trajectory tracking


Algo 2


δZ = δX = 0{s1 = Z + m1Z + m2Zaux


with

{Zaux = Z

ψaux = ψ+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε


mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ ISMC



with

Zaux,1 = Z

Zaux,2 = Zaux,1

ψaux,1 = ψ

ψaux,2 = ψaux,1

+ I.C.

ψ =1

2XTX − Θ(Z)−

1

2ε


mi > 0, i = 1, . . . , 4m3 ≥ 2

√m4

+ 2nd order ISMC


Conclusion





Trajectory tracking


Approach with coordination

Objective

+ To propose for the trajectory tracking module, a decentralizedstrategy in order to achieve coordination

Advantages

To reduce the planning task when a fixed geometric shape must bekept

To avoid collisions between robots in spite of the perturbations

Difficulties

Robustness properties of the closed-loop controllers+ Repercussion of errors



Trajectory tracking



Objective


Advantages



Difficulties




Trajectory tracking



Objective


Advantages



Difficulties




Trajectory tracking


Approach with coordinationRelative positions between robots

Expression of the state of robot Rj using its relative position wrt. robotRi in polar coordinates:

relative distance lijrelative bearing ψij

Robot i

Robot k

ψik

lik

θi

θk

dcam

θ j

li j

ψi j

Robot j

−→i

−→j

O xi

yi

xck

yck

Figure 1:

1



Trajectory tracking



Objective+ To stabilize the tracking errors using the relative coordinates of the different robots:{

lik → lik,desψik → ψik,des

Algorithms

ISMCRobot k has:

sensors in order to measure itsrelative configuration wrt. roboti and its time derivative

speed sensors in order tocompute its velocity

a wifi antenna in order to receivefrom robot i , its velocity and itsacceleration

2nd order ISMCRobot k only has sensors in order tomeasure its relative configuration wrt.robot i



Trajectory tracking


Experimental results

Miabot mobile robots

3 Miabot (processor AtmelATMega64 + Bluetoothcommunication)

area of 3m × 2m

a centralized camera

a distant computer whichcontrol the whole system



Trajectory tracking



ISM of Order 1

−0.5 0 0.5 1 1.5 2−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x [m]

y [m

]

traj robot 3traj robot 1traj robot 2

ISM of Order 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x [m]

y [m

]

traj robot 2traj robot 1traj robot 3



Trajectory tracking



ISM of Order 1

0 1 2 3 4 5 6 7 8−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t [s]

[m]

erreur l12

−l12,des

erreur l13

−l13,des

0 1 2 3 4 5 6 7 8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t [s]

[rad

]

erreur ψ12

−ψ12,des

erreur ψ13

−ψ13,des

ISM of Order 2

0 1 2 3 4 5 6 7 8−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t [s]

[m]

erreur l12

−l12,des

erreur l13

−l13,des

0 1 2 3 4 5 6 7 8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t [s]

[rad

]

erreur ψ12

−ψ12,des

erreur ψ13

−ψ13,des



Trajectory tracking


Video

Video



Trajectory tracking


Video with 3 miabot

Video



Trajectory tracking


Video with 7 miabots

Video


Conclusion

Contributions

Theoretical and practical contributions

? Design of motion planning modules based on differentialflatness and optimisation

single robot (off-line and on-line algorithms)multi-robots (centralized and decentralized approaches)

? Design of an arbitrary higher order sliding modecontroller based on ISMC

? Design of trajectory tracking modules based on ISMC

without coordinationwith coordination


Conclusion

Contributions








Conclusion

Contributions








Documents

Higher Order Sliding Modes in Collaborative Robotics