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Dipartimento di ingegneria dell’Informazione – Università di ParmaDottorato di Ricerca in Tecnologie dell’Informazione
a.a. 2005/2006
Introduction to Dynamic Path Inversion
Aurelio PIAZZIDII, Università di Parma
25 January 2006
A. Piazzi Introduction to Dynamic Path Inversion
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Outline• Introduction• The problem and differential flatness• A selection of solved problems• Geometric continuity of Cartesian paths
A. Piazzi Introduction to Dynamic Path Inversion
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Introduction
In the previuos lesson we have posed and solved (forlinear and scalar systems) the
stable dynamic input-output signal inversion problem
Σ( )y t( )u t
For multivariable systems the signal inversion problem is:
Given a desired bounded ( ) pdy t ∈ find a
bounded ( ) pdu t ∈ such that ( )( ), ( )d du t y t ∈ B .
A. Piazzi Introduction to Dynamic Path Inversion
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Introduction
For multivariable systems the inversion problem can alsobe posed as a
stable dynamic input-output path inversion problem
The idea is to consider the output signal y(t) a function(curve) parameterization of a path Γ in the output space Rp. For a given time interval [0, t1] Γ = y([0, t1]).
1y2y
3yΓ
A. Piazzi Introduction to Dynamic Path Inversion
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The problem and differential flatness
Dynamic Input-Output Path Inversion Problem:
Given a path pΓ ⊂ and a traveling time 1 0t > find initial conditions and input ( )u t for which the system output ( )y t safisfies ( )1[0, ]y t = Γ
This problem is quite general and especially relevantfor the motion control of nonholonomic wheleedvehicles.
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The problem and differential flatness
The path inversion problem has a strong connection with differential flatness (Fliess et al. 1993).
A system with m (scalar) inputs is said to be (differentially) flat if there exist m outputs yF1 , . . . , yFmfor which the system variables (the states and the inputs) can be algebraically expressed as functions of the yFi’s and their derivatives (till a finite order).
1The vector ( , , ) is called fthe lat outp . utF F Fmy y y= …
A. Piazzi Introduction to Dynamic Path Inversion
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The problem and differential flatness
1
Consider the nonlinear system in state-space form:( , )
, ,( , )
is differentially flat if there exists a vector-valued function ( )for which defining
[ , , ] ,
m n p
TF F Fm
x f x uu x y
y g x uh
y y y
y
Σ
=⎧∈ ∈ ∈⎨ =⎩
Σ ⋅
= …
( )1
1
1
1 1
1 1
111 1
, , , , , , ,
there exist functions ( ) and ( ) satisfying:( , , , , , , ),
( , , , , , , ).
m
m
m
F m m
F F Fm Fm
F F Fm Fm
h x Du D u Du D u
A Bx A y D y y D y
u B y D y y D y
ββ
αα
αα ++
=
⋅ ⋅
=
=
… … …
… … …… … …
A. Piazzi Introduction to Dynamic Path Inversion
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The problem and differential flatness
The dynamic path inversion problem is (relatively) easy to solve when the system is differentially flat and the actual output is flat (y = yF ).
( )1
:Given the path choose a velocity planning on itto find the trajectory ( ) for which [0, ] .Then, determine the initial conditions (0) and theinput signal ( ) by applying
y t y tx
u t
Γ
= Γ
Conceptual solution
the functions ( ) and ( ).A B⋅ ⋅
Proving that y is flat may be not trivial…
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A selection of solved problems
Solved path inversion problems:
1) car-like vehicle (Nelson 1989, Rouchon et al. 1993, Reuter1998, ARGO Project: Guarino, Piazzi, Bertozzi, Broggi, Fascioli, 1999, 2002 )
cossin
tan
x vy v
vl
θθ
θ δ
⎧⎪ =⎪
=⎨⎪⎪ =⎩
±
µ
x
y
l
A. Piazzi Introduction to Dynamic Path Inversion
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A selection of solved problems
δ
θ
P
Q
d
Γ
l
wz
τν
Consolini, Piazzi, Tosques 2001, 2003
A. Piazzi Introduction to Dynamic Path Inversion
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A selection of solved problems
2) Unicycle mobile robot (solution with smooth velocities, Guarino, Piazzi, Romano 2004 TR)
cossin
x vy v
θθ
θ ω
=⎧⎪ =⎨⎪ =⎩
x
yθ
A. Piazzi Introduction to Dynamic Path Inversion
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A selection of solved problems
3) Wheeled omnidirectional robot (Guarino, Piazzi, Romano 2002): an holonomous model.
x
yθ1
3
2
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A selection of solved problems
4) General n-trailer system (Rouchon et al. 1993, Altafini 2002, …)
5) VTOL model (Consolini, Tosques 2004 CDC): a nonminimum-phase system.
6) Chaplygin-like nonholonomic systems (Tosques, Consolini 2003 ECC).
7)….
The general dynamic path inversion is an open research problem
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A selection of solved problems
The solution to the path inversion problem is an input signal thatcan be used as a feedforward control.
In preview of a practical application how to complement thisfeedforward with a feedback action?
1. Path-error feedback correction (classic approach to pathfollowing: various schemes can be devised)
2. Iterative steering (Lucibello, Oriolo 1996 CDC, Automatica 2001). Originally it was proposed as a novel approach tostabilization of nonlinear systems.
3. Path-error feedback correction plus Iterative steering.
A. Piazzi Introduction to Dynamic Path Inversion
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A selection of solved problems
Iterative Iterative steeringsteering conceptconcept appliedapplied toto the the pathpath followingfollowing problemproblem
idial desired path
replanned path
actual path
Iterative steering requires a supervisor architecture…
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
Relevant issues for the path inversion problemRelevant issues for the path inversion problem
Apart differential flatness other issues are:• Nonholonomy• Minimum-phase/Nonminimum-phase• Geometric continuity of paths
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Geometric continuity of Cartesian paths
A curve on the Cartesian plane can be described by
the map p(u), u ∈ [u0, u1] :
y
x
1u
0u
20 1: [ , ]
(u) ( )
(u)
u u
u uαβ
→
⎡ ⎤→ = ⎢ ⎥
⎣ ⎦
p
p
( )0 1
0 1
The associated to the curve ( ) is the image of [ , ] according to ( ) :
[ , ] .
uu u u
u u
path pp
p
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Geometric continuity of Cartesian paths
def.
( )0 1
0 1
0 1
A curve ( ), [ , ] is if1. ( ) [ , ]2. ( ) 0 [ , ]
u u u uP u u
u u u
∈
⋅ ∈
⋅ ≠ ∀ ∈
p regularpp
( )0 1
0 1
[ , ] is the set of piecewise-continuous functions over the domain [ , ].P u u
u u
A regular curve has a well-defined( ) ( )( )
uτ p uunit tangent vectorp u
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
The arc length function is
0
0 1 1
1 1
:[ , ] [0, ]
( ) ( )
( ) is the total curve length
u
u
f u u s
u f u s d
s f u
ξ ξ
→
→ ≡ = ∫ p
( )0 1
1
Given a regular curve ( ), ( ) [ , ] and it is bijective.
Hence, there exists the inverse :
u f C u u
f −
⋅ ∈p
11 0 1
1
:[0, ] [ , ]
( )
f s u u
s u f s
−
−
→
→ =
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
Attached to every point of a regular curve p(u) there is the orthonormal moving frame {τ(u), ν(u) } congruent to the axes of the {x, y }-plane.
x
yτν
osculating circle
( )0 1If ( ) [ , ] thenthe of ( ) is well-definedaccording to the Frenet
curvature
(
formula
( ) ( ), ( )
1 is the radius of the osculating circle( )
) cc
c
P u uu
d u u k u
k
k uds
u
τ ν
⋅ ∈
= ∈
pp
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
( )
0 1
3/ 22 2
The curvature function is :[ , ] , ( )
( )( ( ) ( ) (
( )) )
( )
c c
c
k u u u k u
u u u u
uk
uu α β α β
α β
→ →
−=
+
( )1
1
The curvature as a function of the arc length is:[0, ] ,
(
)
( ))
(
cs
s s s
k f s
κ
κ
κ−
→ →
=
( )0
( ) ( ) ( )u
cu
k u f u dκ κ ξ ξ⎛ ⎞
= = ⎜ ⎟⎜ ⎟⎝ ⎠∫ p
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
( )
( )
1
00 1
1. A curve ( ) has
and we say ( ) is a -curve if1. ( ) is regular;2. the unit tangent vector
-curves first order
is continuous alon
geometric continuity
g the curve: ( ) [ , ]
u
u Gu
G
C u uτ ⋅ ∈
def p
pp
( )
( )
2
1
0 1
2. A curve ( ) has
and we say ( ) is a -curve if1. ( ) is a -curve;2. ( ) [ , ] ;
3. the curvature is continuous along the cur
-curves second order g
ve
eometric continuity
: (c
u
u Gu G
P u
G
u
k
⋅ ∈
⋅
def p
ppp
( )( )
00 1
01
) [ , ]
or ( ) [0, ] .
C u u
C sκ
∈
⋅ ∈
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
( ) 0
1 2
00 1
- and -curves were introduced in computer graphics by (1983).
A curve ( ) [ , ] can be defined as a .-curve
G G
u C u u G∈
Barsky and Beattyp
Generalization to Gk-curves (Piazzi, Romano, Guarino 2003 ECC)
( )
( )
1
0 1
. A curve ( ) has
and we say ( ) is a -curve if1. ( ) is a -curve;2. ( ) [ , ] ;3. the ( 2)-nd order derivative wit
-curves; 2 -th order geometric
continuity
h respect
k
k
k
k
G u
u Gu G
D P u u
k
k
k
−
⋅ ∈
−
≥def p
pp
p
( )2 01
to the arc length of the curvature is continuous along the curve: ( ) [0, ] .kD C sκ− ⋅ ∈
A. Piazzi Introduction to Dynamic Path Inversion
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Geometric continuity of Cartesian paths
( ).
A set of points of a Cartesian plane is a -path, i.e., a path with , if
-paths;
there e
0
- xith o sts rder geo a -curvewhose image is the given path
metric cont.
inuity
k
k
k
G k
kG
G
≥def
( )
2
0 1 0 1
Formally: is a -path if there existsa -curve ( ), [ , ] such that [ , ] .
k
k
GG u u u u u u
Γ ⊂
∈ = Γp p
A. Piazzi Introduction to Dynamic Path Inversion
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References
• M. Fliess, J. Levine, Ph. Martin, P. Rouchon, “Flatness and defect of nonlinear systems: introductory theory and examples”, Int. J. Control, Vol. 61, No. 6, pp. 1327-1361, 1995.
• P. Rouchon, M. Fliess, J. Levine, Ph. Martin, , “Flatness, motion planning and trailer systems”, Proc. Conf. Decision and Control, pp. 2700-2705, 1993.
• P. Lucibello, G. Oriolo, “Stabilization via iterative state steering with application to chained-form systems”, Proc. Decision and Control, Vol. 3, pp. 2614-2619, 1996.
• P. Lucibello, G. Oriolo, “Robust stabilization via iterative state steering with an application to chained-form systems”, Automatica, Vol. 37, pp. 71-79, 2001.
• W.L. Nelson, “Continuous Steering-Function Control of robot carts”, Transactions on Industrial Electronics, Vol. 36, No. 3, pp. 330-337, 1989.
• J. Reuter, “Mobile robot trajectories with continuously diffirentiable curvature: an optimal control approach”, Proc.Int. Conf. Intelligent Robots and Systems, Victoria B.C. (Canada), October 1998.
• C. Altafini, “Following a Path of Varying Curvature as an Output Regulation Problem”, Transactions on Automatic Control, Vol. 47, No. 9, pp. 1551-1556, September 2002.
• A. Broggi, M. Bertozzi, A. Fascioli, C. Guarino Lo Bianco, and A. Piazzi, “The ARGO autonomous vehicle’s vision and control systems”, Int. J. of Intelligent Control and Systems, Vol. 3, No. 4, pp. 409-441, 1999.
• A. Piazzi, C. Guarino Lo Bianco, M. Bertozzi, A. Fascioli, and A. Broggi, “Quintic G^2-splines for the iterative steering of vision-based autonomous vehicles”, IEEE Transactions on Intelligent Transportation Systems, Vol. 3, No. 1, pp. 27-36, March 2002.
• L. Consolini, A. Piazzi, M. Tosques, “Path following of car-like vehicles using dynamic inversion”, Int. J. Control, Vol. 76, No. 17, pp. 1724–1738, November 2003.
• C. Guarino Lo Bianco, A. Piazzi, M. Romano, “Smooth motion generation for unicycle mobile robots via dynamic path inversion”, IEEE Transactions on Robotics, Vol. 20, No. 5, pp. 884—891, October 2004.
• C. Guarino Lo Bianco, A. Piazzi, M. Romano, “Smooth control of a wheeled omnidirectional robot”, Proc .IFAC 2004 Intelligent Autonomous Vehicles Conference, Lisboa, Portogal, 5-7 July 2004.
• L. Consolini, M. Tosques, “A controlled invariance problem for the VTOL aircraft with bounded internal dynamics”, Proc. Conf. Decision Control, December 2004.
• M. Tosques, L. Consolini, “A path-following problem for a class of non-linear uncertain system”, Proc. European Control Conf., September 2003.
• B.A. Barsky, J.C. Beatty, “Local control of bias and tension in beta-spline”, Computer Graphics, Vol. 17, No. 3, pp. 193–218, 1983.• A. Piazzi, M. Romano, C. Guarino Lo Bianco, “G3- splines for the path planning of wheeled mobile robots”, Proc. European Control Conf.,
September 2003.