On Goursat normal forms, prolongations, and control ... - CiteSeerX
Proceedings of the 33rd Conference on Decision and Control Lake Buena Vita, FL December 1994
-
TA-11 11140
On Goursat Normal Forms, Prolongations, and Control Systems D. Tilbury and S. S. Sastry Electronics Research Laboratory Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720 dawnt@eecs. berkeley.edu
I n this paper, the method of exterior diflerential systems for analyzing nonlinear systems is presented. Conditions are given for converting Pfafian systems into Goursat normal f o r m , a n d for converting control systems into Brunovsky form. All of the existing results on feedback linearization for control systems can be restated in the language of Pfafian systems, a n d in addition, new conditions for linearizing control systems using dynamic extension are given.
1.
Introduction
There has been a great deal of interest in the use of exterior differential systems for analyzing nonlinear control systems. This paper brings together some of the results which have been recently published and also adds some new contributions to this area. For mechanical systems with linear velocity constraints, such as mobile robots with wheels that roll without slipping, the problem that is considered is that of finding a feasible path between two given points. Using the method of exterior differential systems, necessary and sufficient conditions can be given for transforming the Pfaftian system defined by the rolling constraints into a normal form for which feasible paths are easily found. The outline of this paper is as follows. First, the definitions for Goursat normal form and extended Goursat normal form are given. Necessary and sufficient conditions for converting PfafFian systems into these normal forms are stated, and several examples of mobile robot systems are studied. The concept of prolongation of a Pfaftian system is presented next, and sufficient conditions are given for converting a PfafFian system to extended Goursat normal form using a specific type of prolongation. Since the dual of Brunovsky linear form is a special case of extended Goursat normal form, all of the results for converting Pfaffian systems to extended Goursat form can be specialized to give conditions for converting control systems to Brunovsky linear form. Necessary and sufficient conditions for linearizing control systems by dynamic extension are finally given.
2.
Pfaffian systems
The interested reader is encouraged to consult [2] and [6], from which most of this introductory material was taken, for more details. A real vector space V or its dual (covector) space V' can be expanded into an exterior algebra with the exterior or wedge product defined by
+(ac)aAy
f o r a , P , y E V (orV*),anda,b,cEB.Thewedgeproductoftwovectors is called a two-vector. Define A2(V) as the space of two-vectors. Similarly, higher vectors can be built up, and A k (V) is defined to be the space of all k-vectors. For completeness, define Ao(V) = P and A'(V) = V. The dimension of A k ( V ) is (;). From the axioms, it follows that A k ( V )is empty for k > n. The exterior algebra over V is a graded algebra,
A(V) = A o ( V ) @ A ' ( V ) @ ~ . ~ @ A " ( V ) Any element X E h ( V )can be written uniquely as X = A0 XI .. A, where A, E A'(V) for i = 1,. . . ,n. Now, consider a differentiable manifold M of dimension n and its cotangent bundle T * M . Construct the bundle A ( T * M ) whose fibers are the exterior algebra of T:M, that is:
+ +. +
A(T:M) = Ao(T:) @ A'(T:) @ A2(T:)
U Ap(T:M)
AP(T'M)=
=EM over M is called an ezterior differential f o r m of degree p or simply a p-form. For local coordinates on M denoted by I = ( X I , . . . ,zn), a local basis for T,M is: { L, .. , Its dual basis on T:M is denoted 811 by { d q , . . . , dx,} where the d x , ' s are defined by the relationships
.
e}.
In terms of these local coordinates, a p-form w can be written as
where the coefficient functions w t l . . . , p ( x ) are smooth functions on M. The notation W ( M ) will be used to mean the module (over the ring of smooth functions) of all smooth sections of A p ( T ' M ) , and O ( M )= @ O p ( M ) as the module of forms on M. Consider a codistribution I on M , spanned by s one-forms,
I={a', ... ,a"} where a' is in fl' (M) for i = 1 , . . . , s. Definition 1 (Pfaffian S y s t e m s ) On a manifold of dimension n, a Pfaftian system is the smallest ideal 1 C R(M), generated b y a codidribation I of one-forms spanned b y { a ' , ,aS}which is closed under wedge products.
. ..
Throughout the c o m e of this paper, by deliberately confusing the notation, the codistribution I will be referred to as the Pfaftian system. The dimension of a PfafFian system is defined to be s, the number of independent one-forms which generate it. Any n - s linearly independent one-forms which are independent of I form a complement to I. The codimension of I is defined to be n - s. An integral curve for a PfafFian system is a curve c(t) : ( - e , E ) --t M which satisfies the constraints, that is, c * ( a ' ) = 0 for all a' E I. Here the notation c * ( a ' ) is taken to mean a'( A (local) independence condition for a Pfaftian system is a oneform T which does not vanish on integral curves, that is c * ( T ) # 0. If T is integrable, the Pfaftian system will correspond t o a system of firstorder ordinary differential equations. Throughout this paper, it will be assumed that the independence condition is integrable. Indeed, for control systems, the independence condition will often be d t , where t represents time. The notion of congruence modulo a Pfaffian system will be needed. For a system I = {a', . ,a'}, it is said that 1) ( mod I if
g).
~ A D = - O A ~ aAol=O
a a A ( b p + q )= ( a b ) a A D
The bundle A ( T ' M ) has AP(T'M) as sub-bundles. A section of the bundle
@.
. . @ A"(T:)
This research was supported in part by the NSF under grant IRI-9014490 and by the ARO under grant DAAL03-91-6-0171. D. Tilbury would also like to acknowledge an AT&T Ph.D. Fellowship for financial support of this work
..
s 1)
= .f -I-
A ai
,=I
for some forms 8' in fl(M). Definition 2 (Exterior Derivative) The exteriorderivative is d e fined as the unique map d : Q k ( M )+ R k + ' ( M ) which satisfies the following properties: I. FOTf E Q o ( M ) ,df = z d x l ... Zdz,,, relative to a local coordinate chart, O T the usual gradient. 2. F ~ ~ a E f l ' , f l E dn (~a ,A p ) = d a A P + ( - l ) r a A d p . 3. dZ = 0 .
+
0-7803-1968-0/94$4.0001994 IEEE 1797
+
Definition 3 (Derived Flag) Let I bystem. Define I(') = I and
I(~+') =
{w
= { a 1 , . . , C Y " ) ,be a P f a f i a n
E I ( ~ :' dw E o
mod
dk)}
The construction is assumed to t e r m i n a t e at s o m e N , when I ( N )= T h e derived Rag is defined to br the sequence o f nested codistributions, I = I(') 3 I(') 3 . . . 3 I ( N )
11 zs assumed that t h r dzmension of
1s
well-defined f o r all k.
Remark 1 . M a x i m a l l y N o n h o l o n o m i c . The last member of the chain of codistributions, is called the bottom derived s y s t e m . Sincr I(") = by definition dw 5 0 mod I ( N ) for all w E I ( N ) . and by the Frobenius theorem, the bottom derived system is integrable. That is, there there exist functions h i , . . . , h , such that I ( N ) = { d h l , . . . , dh,}. Solution trajectories of I are then constrained to lie on level surfaces of h. A Pfaffian system I is said to be n o n h o l o n o m i c if I ( ' ) is a proper subset of I. This paper will be concerned with systems which are maximally n o n h o l o n o m i c , that is, with bottomderivedsystem I ( N ) =
{OhA
variant of the familiar Frobenius theorem is stated here:
Theorem 1 (Frobenius [2]) Let { w ' , . . . , w p j be a set of linearly independent o n e - f o r m s , and f l , . . . , f , a set of f u n c t i o n s whose differentials are linearly independent of each other and of the w"s. If dw'Aw1A...AwPAdf1 A...Adfq=O f o r i = 1 , . . . , p , then there ezist coordinate J u n c t i o n s 2 1 , . . . , z p and coeficient f u n c t i o n s a,,, b,, such that the o n e - f o r m s w' can be written as:
,=I
,=I
The proof follows from the standard Frobenius theorem and the fact chat the codistribution { w ' , , , . , w P , d f l , . . . , d f q } is integrable.
3.
Definition 5 (Extended Goursat Normal Form) A P f a f i a n sysof codimension m 1 is i n extended Goursat t e m I on normal form i f it is generated b y n constraints of the f o r m :
+
I = {dz: -
z:tl dzo : a
Definition 4 (Goursat Normal Form) A codimension two Pfaff i a n s y s t e m I o n E n w i t h generators of the f o r m I = {dzy, - zn-l d z l , . . . ,d:g - z z d z l j
(2)
There are conditions due to Murray [12] for converting a PfafXan system to extended Goursat normal form. This theorem is restated and proven here with the additional condition (correction) that the one-form R must be integrable: Theorem 3 (Extended Goursat Normal Form [12]) Let I be a P f a f i a n s y s t e m of codimension m 1 . If (and only i f ) there e z i s f s a set of generators { m 3 : i = 1,. . . , s ; j - 1 , .. . , m } f o r I and an integrable o n e - f o r m T h c h that f o r a h j , -
+
t h e n there exists a set of coordinates z such that I is i n Goursat normal form. Proof. If the Pfaffian system is already in extended Goursat normal form, the congruences are satisfied with T = dzo and the basis of constraints a: = dz: - ~ i - ~ d z ~ . Now assume that a basis of constraints for I has been found which satisfies the congruences (3). It is easily checked that this basis is adapted to the derived flag, that is:
I ( k ) = {a: : j = 1 , . . . , m ; i = 1 , . . . ,s, - k } Since x is integrable, any first integral of T can be used for the coordinate .'2 If necessary, the constraints CY: can be rescaled so that the congruences (3) are satisfied with d z o . Now consider the last nontrivialderived system, I("1-l). The one-forms { a i , . . . form a basis for this codistribution, where SI = s2 = . . . = s r l . From the fact that d a : -ai Adz' mod I ( ' ' - ' ) , it follows that the one-forms e:,. . . satisfy the Frobenius condition:
,ai'
dei
Goursat Normal Forms
A great deal of work has been done on transforming Pfaffian systems into normal forms. The Goursat normal form will be of special interest in this paper. It w a s originally proposed by Goursat for Pfaffian systems of codimension two, and has the property that all its integral curves can be expressed in terms of two arbitrary functions. An extended Goursat normal form for Pfaffian systems of codimension k > 2 will also be examined. For the extended Goursat normal form, solution trajectories can be expressed in terms of k arbitrary functions.
= 1 , . . . , s , ; j = 1 , . . . ,m } ,
A
cri A
. . . A 0;' Adz'
=0
and thus, by the Frobenius theorem, coordinates z i , . . . ,z;' found such that
[ ":] [ ] = A
a;'
d;
can be
+Bdzo
dz;'
The matrix A must be nonsingular, since the a ; ' s are a basis for 1('1-'j and they are independent of dz'. Therefore, a new basis &{ can be defined as:
is said to be i n Goursat Normal Form.
From the form of the Pfaffian system in the z coordinates, it follows that integral curves of the system are unconstrained in their zl, 2, coordinates alone. Once 21 ( t ) ,z,,(t) are specified as functions of some parameter t , the other coordinates are determined as some functions of t . The following classical theorem gives necessary and sufficient conditions for converting a Pfaffian system into Goursat normal form: Theorem 2 (Goursat Normal Form [Z]) A P f a f i a n s y s t e m I of codimension two on I nhas a set of generators which are i n G O U T sat n o r m a l f o r m if and only iJ there ezists U basis set of f o r m s { c ~ ' , . . . ,an-') J O T I und a o n e - f o r m T satisfying the congruences: de' E -e'+' A R mod a ' , , , , ,m' i = 1 , . . . , n - 3 &n-2
so
mod I
Define the coordinates z: := - ( A - ' B ) , , so that the one-forms E: have the form = dz: - z;dz' f o r j = 1, ... , T I . By the proof of the standard Goursat theorem, all of the coordinates in the j t h tower can be found from and z o , thus by the above procedure, all the coordinates in the first T I towers have effectively been found. To find the coordinates for the other towers, look a t the lowest derived systems in which they appear. Consider the smallest integer k such that dim I($ITkj> k r l ; more towers will appear at this level. It can be shown from the congruences ( 3 ) and the calculations up to this point that a basis for I("1-k) is
{ s i , . . , s i , . . . , & ; I , . . . ,E;'
(1)
where 6; = d z f In [17] it w a s shown how the Pfaffian system associated with the system of a car towing n trailers, generated by the constraints that each axle of wheels roll without slipping, satisfied the conditions for conversion to Goursat normal form. This w a s a system of codimension two, corresponding to the fact that the linear and angular velocities of the front car are the inputs, and thus fredy specifiable. In order to consider Pfaffian systems associated with mobile robots such as the firetruck [4, 161 or the multi-sterring multi-trailer system [18], PfafXan systems of codimensiongreater than two must be studied as well. Consider the following definition:
,a;1+1,.
.. ,a;l+r2}
dz' for j = 1,. . . , T I , as found in the first step,
-
+ +
and a' for j = r1 1 , . . . , r2 are the original one-forms, which are adaptdd to the derived flag. The lengths of these towers: s r l + l = ' . s T 1t T 2= SI - k 1, are now known. For notational convenience, for = 1 , . . . , T I . define z i k ) := ( z ; , . . . );:, By the Goursat congruences, d a 3 = -a; A d z o mod 1(31-k) for 1 , . . . ,r1 r 2 , thus the FrobGius condition
j = r1
+
da;Aal'"A.
1798
+
. .A a r 1 + ' 2 A d z ~ A . ~ . A d z ~ A . . . A d z ; ' A . . . A d z ~ ' A d = z' 0
+
"::*'
is satisfied for j = r1 1 , . . . ,rl , z;'"' new coordinates 2;' ",
...
+ r2. Using the Frobenius theorem,
[ ] [ ] =A
ar1+r2
+
BdzO
+C
t'2
&'l
[ "';*I ]
dai G O
dz;;)
[ ] [ ]
+Bdzo
dz;i +ra
o',
[ ] [ ] [ ] :;
=
'r~ t r z
tr2
+
.
The coordinates := ( A - I B ) ' for j = rl 1 , . . ,r1 -tr2 are then defined so that a: = dz' - z ~ d z ' . Again, by the standard Goursat theorem, all of the coordinates in the towers rl t 1, , rl rz are now defined. The coordinates for the rest of the towers are defined in a manner exactly analogous to those of the second-longest tower. If x is not integrable, then the Frobenius theorem cannot be used to find the coordinates. In the special case where si > s ~ that , is, there is only one tower which is longest, it can be shown that if there exists any P which satisfies the congruences, then there also exists an integrable P' which also satisfies the congruences (with a rescaling of the basis forms), see [3, 121. However, if si = s2, or there are at least two towers which are longest, this is no longer true. Thus, the assumption that r is integrable must be stated in the assumptions. 0
...
+
Another version of the theorem for converting systems to extended Goursat normal form will also be stated here. This version is slightly easier to check, since it does not require finding a basis which satisfies the congruences but only one which is adapted to the derived flag. The proof for this theorem is also given, since only a special case is proved in [14]. T h e o r e m 4 ( E x t e n d e d G o u r s a t N o r m a l Form [14]) A Pfafian system I of codimension m + l on Rn+m+l can be converted to Goursat normal f o r m if and only if The system is mazimally nonholonomic, that is I ( N ) = { 0 } for some N , and There ezists a one-form A such that { I ( k ) , r }is integrable for k = 0 , ... , N - l . Proof. The only if part is easily shown by taking A = dzo and noting that
+ 1 , . .. ,sJ} j = 1 , . .. ,m ; i = k + 1 , . . . , s J } ,
I ( k ) = { d z i - z:-ldzO : j = 1 , . . . , m ; i = k
{ f i k ) , r=} { d z i , d z o : which is integrable. Now, assume that such a P exists. Find the derived flag of the system, I =: I(') 3 I ( ' ) 3 . 3 I('1) = ( 0 ) . The lengths of each tower are determined from the dimensions of the derived flag. Indeed, the longest tower of forms has length 8 1 . If the dimension of I('1-l) is r 1 , then there are rl towers which each have length SI; and thus s1 = s2 = . . = srl . Now, if the dimensionof I(s1-2) is 2rl +r2, then there are r2 towers with length s1 - 1, and sr1+l = .. = sr,tr:,= s1 - 1 . Each sJ is found similarly. Note that a A which satisfies the conditions must be in the complement to I, for if P were in I, then {I,P} integrable means that I is integrable, which contradicts the assumption that I is maximally nonholonomic, that is I ( N )= ( 0 ) for some N. Consider the last nontrivial derived system, I ( ' l - ' ) , with a basis given by {a:,.. . , a;'}. The definition of the derived flag, specifically I(sl) = {0}, implies that
..
.
J=1
30
mod
fi'l-')
j = 1, ... , r 1
(4)
Also, the assumption that { I ( k ) , r }is integrable gives da' 1 -= 0 mod
{d'l-l), T }
=
j = 1 , . .. , r1
(5)
,=l
'=I
T
( c b J P J ) mod
A
E 0 mod
I('1-I)
I('1-I)
J=1
which implies that a is in I('1). However, this contradicts the assumption that 1 ( 3 1 ) = ( 0 ) . Thus the PJ'S are linearly independent mod I('l-'), and the claim is proven. Define a; := PJ for j = 1 , . ,'1. Note these basis elements satisfy the first level of Goursat congruences, that is:
..
j = l ,...,rl
doli ~ - a ; A ? r modI('1-')
If the dimensionof I(sl-2)is greater than 2r1, then choose one-forms * r11 t 1 ,... , a;, tra such that that is a basis for I ( '1 -'I. For the inductionstep, assume that a basis for I ( ' ) has been found
. ai1,a:,. . . ,a ; 2 , .. . ,a;,. . . ,a i c }
{a:, . . ,
which satisfies the Goursat congruences up to this level: dai
= -ai+l
A
T
= 1 , . . . ,c;
mod
k = 1 , . . . , kJ - 1
It is assumed that c towers of forms have appeared in I ( ' ) . Consider only the last form in each tower that appears in I ( ' ) , that is , J = 1 , . . , c. The construction of this basis (or from the Gour-
.
aiJ
sat congruences) implies that
ai,
is in I ( ' ) but is not in I('+'), thus
d e i J $ 0 mod I ( ' )
J
= 1 , . .. , c
The assumption that { I ( t ) ,r} is integrable gives
daiJ G O
mod{I('),r}
j=l,
...,c
thus d a i , must be a multiple of r mod I ( ' ) , daJk3 = r A P '
modi(')
~ = 1 . ., . , c
for some PJ $ 0 mod I ( ' ) . Also,from the fact that a:, is in I ( ' ) and the definition of the derived flag,
dai, f O
modi('-') j = 1 , ...
,c
which implies that PJ 6 I('-'). By a similar argument to the claim above, it can be shown that the PJ'S are independent mod I ( ' ) . Define = P J , and thus
aiJtl
.
.
... , Q E , + I )
{a;,. . , a i l t l , a ; , .. . 'a;atl,. . ,a;,
.
dai
... ,
,orl
+(A-'B)dzo
&'1+'2
1
j=l,
+ .+
Again, A must be nonsingular because the a i ' s are linearly independent mod I('l-') and also independent of dz', and thus new one-forms E r 1 + l , . . . , 6 r l + r a can be defined as: $ltl t1 dzrl t :=A-'
modI("1-2)
which combined with (6) implies that PJ is in I('l-'). Claim. . .. are linearly independent mod I ( ' 1 - l ) . Proof of Claim. The proof is by contradiction. Suppose there exists some combination of the ~ J ' S ,say = blP1 . . brlp'l 0 mod I('1-') with not all of the b,'s equal to zero. Consider a = bla: b,, ay'. The one-form a # 0 because the a: are a basis for I(31-1).The exterior derivative of a can be found by the product rule,
+ .. . +
dzrl t1
= A
j = l , ... , r l
d a i E r A P J modI('1-'1
0 mod I('1-l). results for some 0' Now, from the definition of the derived flag,
Since the congruences are only defined up to mod I ( ' l P k ) , the last group of terms (those multiplied by the matrix C) can be eliminated by adding in the appropriate multiples of 6: = dz: dz'. This will change the B matrix, and what remains has the form:
01r1tr2
combining equations (4) and ( 5 ) , the congruence
can be found such that
forms part of a basis of I('-'). If the dimension of I('-') is greater k2 k, c. then complete the basis of I('-') with than kl such that any linearly independent one-forms aEtl,. . . ,
+ + . .. + +
ailtl ,a;,... ,a;2tl,... ,a;,. . . ,a i c t 1 ,a y , . .. 1 a=+'c} 1
{ai,. .. ,
is a basis for I('-'). Repeated application of this procedure will construct a basis for I which is not only adapted to the derived flag, but also satisfies the Goursat congruences. 0 Remark 2. Dtflerential Flatness. There are connections between Goursat forms and the concept of differential flatness. The precise definition of differential flatness is given in the language of differential
1799
algebra [8,7, 101 and is beyond the scope of this paper. Informally, however, a system is said to be d i g e r e n t i a l l y f l a t if a set of flat outputs (equal t.o the number of inputs) can be found such that the entire state and input trajectories can be recovered from the flat outputs as functions of time and finitely many of their derivatives. Although there exists no general technique to determine whether or not a given system is flat (other than guessing flat outputs), it is easy to see that any system which can be converted into Goursat normal form is differentially flat. The flat outputs can be chosen as !.he states z o , z : , . . . ,z;".
are satisfied instead of the original congruences (3). The Gounat coordinates for this system can be found as
= x3
= Y3
.:
=
I
- 84
tan
if L4 = L5
. , / T G t a n ( v )
$yarctan' + e
+
a t . . ( w ) + i + e
- 7 7 1% (7 I-! tan(-)-]-!
Mobile Robot Examples
4.
20 2:
In this section, some multi-steering mobile robot systems will be considered and it will be shown that the PfafFian systems generated by the constraints that the wheels roll without slipping satisfy the extmded Goursat conditions. A multi-steering trailer system was examined in [18],and it was shown how to transform such a system into chained form (which is the dual of Goursat normal form) using dynamic state feedback. That. is, states were added to the system and this augmented system was transformed into the dual of Goursat normal form. In this section, it is shown that this augmentation is not always necessary; some arrangements of the multi-steering trailer system can be transformed into Goursat form using only static state feedback. The 5-axle system with two steering wheels is the first example in which interesting things begin to happen. Assuming that the first axle is steerable, there are four possible positions for the second steering wheel. Three of these four cases satisfy the conditions for converting t,o extended Goursat normal form; the fourth does not. Two of the examples will be presented in some detail to give the reader a flavor for the type of calculations which are required. Example 1. 5-axle, 1-4 steering trailer system. Consider the :,-axle system with the first and fourth axles steerable, as sketched in Figure 1. The configuration space can be parameterized by the x, y position of the third axle, the hitch angles e,, and the steering angle of the third axle 6. Let q = {x3, y3,05,91,03,92,81, d} represent the state.
1
if L4
> Ls
if L4
< L5
where e = L4 l L 5 . Remark 3. 5 - a x l e , 1 - 2 s t e e r i n g a n d 1 - 5 s t e e r i n g . Without presenting the calculations, which are similar than those above, it is noted here that the five-axle system with either the first and second axles steerable or the first and last axles steerable satisfies the conditions for conversion into extended Goursat normal form. The constraints have the same form. For the 1-2 steering system, T can he chosen as dx5 and the Goursat coordinates are found as zo = x5,z: = y5, and 2: = 4. For the 1-5 steering system, x can be chosen as dxl, and the Goursat coordinates are defined as'2 = x(, 2: = y4, and 2: = 4. Example 2. 5-axle, 1-3 steering. The final instance of the 5-axle trailer system has the first and third axles steerable, as sketched in Figure 2. The vector 9 = ( 2 5 , y 5 , 8 5 , 8 4 , 9 3 , 8 ~ , 8 1 , ~represents } the state.
Figure 2: A 5-axle trailer system with the f i s t and third axles steerable. This is the only configuration of the 5-axle system with two steering wheels which does not satisfy the conditions for converting to extended Goursat normal form. The constraints are that each axle roll without slipping:
Figure 1: A 5-axle trailer system with the first and fourth axles steerable. The constraints are that each axle rolls without slipping: sinBidx, - cosO,dy, i = 1,2,3,5 sinddxq - cosddyr a 5 }a compleThe Pfaffian system is thus I = { a 1 , a z , a ~ , a 4 , and ment to this system is: (dd,d&,dx3}. This basis of constraints is adapted to the the derived flag, I = t a l , az, a3, a 4 , 0 5 ) I(') = {aZ,a3, a"} I(2) = {a3}
(01
1(3) =
and satisfies the congruences: do' E q ( 9 ) d81
A
A
dx3 mod I
dx3
da4 E c * ( q ) dd
mod I ( ' )
do5
A
dx3 mod I
~ ( 9 a)4A dx3 mod I ( ' )
du3 E c3(q) u 2 A dx3 mod I ( 2 ) By a simple rescaling of the basis, the functions c , ( q ) can be eliminated to get the Goursat congruences (3) exactly. This is done as follows. First defiles2 := -c3(y) a 2 toget da3 -C2Adx3 mod I ( 2 ) . 'The exterior derivative of a2 can be written as d?i2 = - c 2 ( q ) =;do' - dc2(g) A a' E - - c z ( q ) c l ( q )a' r,dx3 mod I ( ' ) since a' is in I ( ' ) . The other constraints are scaled similarly. For the rest of this paper, it will asserted that the Goursat congruences arc satisfied if the modified congruences d a ~ ~ c ~ + l ( q ) u ~ +modI('I-') lA?r da;] Z 0
-
- cosddy3
mod I
i = l , ... , s - 1 (7)
cos9,dy,
i = 1,2,4,5
ThePfaftiansystemis I = {a',a2,a3,a4,a5},andacomplementto the system is given by {dd, d81, dx3 This basis is adapted to the derived flag, I = {a', a2, a3, a4, a5}
1.
= =
a'
a4
da2 E c 2 ( 9 ) a'
a' = sinO,dx, a3 = sinbdx3
I(') =
{a2,
a4, a5}
= 1(3) =
{a5}
1(2)
io}
however, the congruences are nof satisfied: da' E CI
dB1 A
da2 I c2 a'
A
dx5 dx5
da3 E c3 d$
+ k2 a'
A a3
A
dx5 mod I
da4 E c4 a3 A dxs mod I ( ' ) de5 E cg a4 A dx5 mod
d2)
In order to have { l ( 2 ) , rintegrable, } x must be chosen as x = dx5 This will also give {IO,.} integrable, but { I ' , x } is (mod {a4,a5}). n o t integrable. Thus, this system does not satisfy the conditions for conversion to extended Goursat normal form. This example will be considered again in the next section.
5.
Prolongations
If a Pfaffian system I of codimension k satisfies the necessary and sufficient conditions for converting into extended Goursat form, then its solution trajectories are determined by k arbitrary functions. However, even if a system cannot be transformed into Goursat form, its solution trajectories may still have this property. If so, it is said that I is a b s o l z l t e l y e q u i v a l e n f (in the sense of Cartan) to the trivial sysAlthough the concept tem (the system with no constraints) on Ik. of absolute equivalence will not be examined in its full generality, some sufficient conditions will be given for a PfaEan system to have a p r o l o n g a t i o n by d z f f e r e n t i a t r o n which can be converted to Goursat form, and thus the solution trajectories of I are determined by k independent functions. A general type of prolongation which preserves a one-one correspondence between solution trajectories of the original and prolonged system is a Cartan prolongation.
1800
Definition 6 ( C a r t a n Prolongation) L e t I be a P f a f i a n s y s t e m o n a manifold M . A s y s t e m J o n M x P P i s a Cartan prolongation of I i f : 1. T * ( Ic)J 2. For every s o l u t i o n curve c : ( - € , E ) + M o f I there ezists a u n i q u e s o l u t i o n curve 2 : ( - E , E ) -+ M x BP of J w i t h T O i. = c. If I i s equipped w i t h a g i v e n independence c o n d i t i o n T , t h e n T ' T m u s t be t h e independence c o n d i t i o n f o r J .
A canonical way to prolong a system with independence condition d t is to take an integrable one-form dq in the complement of I, and augment I with the additional form dq - y d t , where y is a new coordinate on R . In effect, this adds the derivative of q (with respect to the independence condition) as a state variable. As long as all solution trajectories are "smooth enough" (assume C"), there will be a one-one correspondence between solution trajectories of the original and the prolonged system. Consider a special type of Cartan prolongation which consists of many of these canonical prolongations. Definition 7 (Prolongation by differentiation) L e t I be a P f a f f i a n s y s t e m of codimension m+l o n I"+"'+'w i t h coordinates (2,U ,t )
f o r which d t i s a n independence c o n d i t i o n and { d u i , ... , d v m , d t } f o r m s a complement. L e t 6 1 , . . , bm be nonnegative integers and let b d e n o t e t h e i r s u m . T h e s y s t e m I augmented by
.
dvl - v i d t ,
... ,
+.
Since control systems are a special type of Pfaffian system, all of the results presented thus far can be specialized to control systems. Definition 8 (Control S y s t e m ) A controlsystemi = f ( x , u ) with t h e state x E B", t h e i n p u t U E B'", and t h e derivative of t h e state t a k e n w i t h respect t o t i m e t E 1,generates a P f a f i a n s y s t e m I o n Rntmt1
I = { d x , - f ' ( x , u ) d t : i = 1 , . . . , n} (8) w i t h complement { d u l , . . . , d u m , d t } . T h e n a t u r a l independence cond i t i o n t o choose i s d t , since d t # 0 along all s o l u t i o n trajectories of the system. A n y P f a f i a n s y s t e m I of codimension m 1 o n R n t m t l w i t h coordinates ( x , u , t ) can be called a control system if i t h a s a s e t of generators of f o r m ( 8 ) .
+
Brunovsky showed that any controllable linear system j. = Ax Bu with x E R", U E Rm can be converted to a canonical form given by
.
dv, - v A d t , . , , ... dv&"-' - vLmdt, i s called a prolon ationby differentiation of I. T h e augmented s y s t e m i s defined o n Rnfmtbtl.
Since the original system and independence condition corresponded to a set of first order ordinary differential equations, the prolonged system has the same independence condition and also corresponds to a set of first order ordinary differential equations. Sufficient conditions can now be given for a Pfaffian system to have a prolongation which is equivalent to extended Goursat form. T h e o r e m 5 ( G o u r s a t f o r m via prolongation) Consider a P f a f f i a n s y s t e m I = { a ' , . . ,a"} o n w i t h independence condit i o n d z o and complement { d v l , .. , d v m r d z o } . If there ezists a list of integers b l , . . ,bm s u c h t h a t t h e prolonged s y s t e m
.
.
.
Control Systems
6.
+
- vi'dt,
dv;'-'
and the systems {J(')), d x 5 } are integrable for all k. R e m a r k 4. 5 - a z / e , 1-4 steering, revisited. In [3],it was noted that the 5-axle system with steering on the first and fourth axles could be converted into Goursat form after an order two prolongation (U' = d 4 - v 1 d x 5 , w 2 = d v l - v z d x ~ )and , that the Goursat coordinates for the system after prolongation were much simpler than those given in example 4. The one-form x which satisfies the congruences for the extended system is d x 5 , and the other two coordinates which define the Goursat form are y5 and
J = {a',. . . ,a", d v l - v i ds', .. . ,dv;'-'
v1 bi
dzo '
. . . ,dum - v L d z O , . . . ,dv&m-lv!"dto}
x1
;;
...
- U1
e;. = U, xy = X;.
...
(9)
"'
qm =xrm4 = Xi1-'
+ .+
with n = kl . . k,. Therefore, a control system is said to be lrnearitable if and only if it can be converted to Brunovsky form. Brunovsky linear form for a control system is a special case of extended Goursat normal form (2) with dro = d t and t1 = U, ; thus, the theorems for transforming to Goursat form can be specialized to give conditions for exact linearization. T h e o r e m 6 (Exact Linearization [ l l ] ) If a control s y s t e m I defined o n In+"'+'h a s a s e t of generators {a: : j = l , ,m;i = 1 , . . . , s,} s u c h t h a t f o r all j ,
4,
...
satisfies t h e c o n d i t i o n t h a t {J(k),dzo} i s integrable f o r all k, t h e n I can be transformed t o extended G o u r s a t n o r m a l f o r m u s i n g a prolong a t i o n by differentiation.
Proof. The proof is by application of Theorem 4 to the prolonged system J . 0
Although this is a very specific form of prolongation of a P f d a n system, and the conditions of the theorem must be checked in a specific coordinate system with a given independence condition, there do exist practical systems which can be converted into extended Goursat form using this type of prolongation. E x a m p l e 3. 5-axle, 1-3 steering, revisited. Consider again the 5-axle trailer system with the first and third axles steerable, which did not satisfy the conditions for conversion to extended Goursat form. The equations for the exterior derivativesof the constraintscan be examined to see if after prolongation, the augmented P f d a n system will satisfy the conditions for conversion to extended Goursat form. It was seen that ?r = d x 5 will give { I ( 2 ) , x } integrable. However, { I ( ' ) , d x s } is not integrable since d o 2 has a term k 2 ( q ) a ' A a3. If either a' or a3 could be added to I('), that term would no longer ' were added to I ( ' ) , then d e 2 0 mod I ( ' ) cause a problem. If a and the same problem would recur, except now with I ( 2 ) .If a3 could somehow be added to I ( ' ) , it appears that the conditionsof Theorem 4 will be satisfied (the only - thing- remaining- to be checked is that da3 f 0 moda2,a3,a4,a5,dx5.) If I is prolonged by differentiation, and augmented by the additional form w = d+ - v d x 5 , then the derived flag of the augmented system is: J = { a', a 2 , w , a 3 , cr4, a5}
J(') = { J(2) = { J(3) =
{
J(') = {
a2,
t h e n there exists a s e t of coordinates z s u c h t h a t I is in B r u n o v s k y normal fo rm, I = { d r ; ) - ~ : + ~ d t : j = l,... , m ; i = l , . . . ,sJ}.
The control systems version of Theorem 4 is given by T h e o r e m 7 (Exact Linearization [15]) A control s y s t e m I can be converted t o linear f o r m i f and only if { d k ) , d t } i s integrable f o r every k. Another version of this theorem is given in [ l ]with slightly different notation. The problem of feedback linearization by time-scaling was studied in [ 1 3 ] ; the Goursat formulation gives a simple set of conditions to check if a time-scaled version of a system can be linearized, as is shown by the following simple example. Time-transformations are also studied in 191. Example 4. G o u r s a t n o r m a l f o r m for a control system. Consider the single-input control system [ 5 ] , 2' = 1 2 1 3 2
+
= x3
i 3
=U
This control system generates a P f a a n system, I = { d x l - ( ~ 2 ~ 3 ' )d t , d ~ z- x 3 d t , d x 3 of codimension two on R5 with derived flag
+
I=
(al,ff2,ff3)
dl) =
a3, a4, a5} cr4,
x2
a"} a51
I(2)
0)
1(3)
1801
= {al} = {O}
- udt}
(11)
n o t e t h a t { I ( ' ) , d t ) is n o t i n t e g r a b l e , t h u s t h e s y s t e m i s n o t linearizable b y s t a t i c s t a t e feedback. C o n s i d e r a p r o l o n g a t i o n b y d i f f e r e n t i a t i o n of I d e f i n e d b y
where t h e one-forms a d a p t e d t o t h e d e r i v e d flag are given b y
- 2x3dzz a' = d x z - x 3 d t a3 = dx3 - u d t a' = d x l
+ (x3'
- xz)dt
J = { I , w = dui - ~ l d t }
N o t e t h a t t h i s is n o t t h e b a s i s of (11) which g e n e r a t e d I . S i n c e { I ( 2 ) ,d t } is not. i n t e g r a b l e , t h e s y s t e m i s n o t f e e d b a c k linearizable b y T h e o r e m 7. The G o u r s a t c o n g r u e n c e s ( l ) ,however, are satisfied, for ?r = d i = d t - 2 d r 3 . T h u s , t h e r e d o e s e x i s t a t r a n s f o r m a t i o n @ ( z , U. t ) = ( 2 , v, T ) t o G o u r s a t n o r m a l f o r m , which is given b y
22
= 23 = 2 2 - 232
23
= 2' - 2x223
21
r=t
-
2x3
U
=-
2'
1 - 2u
+ -322 3 3
and i t is easily c h e c k e d t h a t
dz' =
d ~ 3 = 22
dz 2 = z,
dT
dr
dr
T h e p r o l o n g a t i o n b y d i f f e r e n t i a t i o n which was defined i n S e c t i o n 5 is t h e dual of d y n a m i c e x t e n s i o n ( a d d i n g i n t e g r a t o r s t o t h e inputs) i n t h e l a n g u a g e of f o r m s . Thus, t h e c o n t r o l s y s t e m s version of T h e o r e m 5 c a n b e s t a t e d as:
Theorem 8 ( L i n e a r i z a t i o n by d y n a m i c e x t e n s i o n ) Consider a control s y s t e m I on iKn+m+' with coordinates ( x , u , t ) , independence condition d t , and complement { d u l , . . . ,du,,dt}. I f (and only i f ) there erists Q prolongation by differentiation of d i m e n s i o n b = bl . . . b , such that t h e augmented s y s t e m
+
+
J={a'=dx,--
f'(x.v)dt:
j = 1 ,... , m ; k= 0 ,... , b , }
- u:dt:
= duk-'
i = l , . . . ,n;
o n R n t m t b + l satisfies the condition ( J ( ' ) ) , d t } i s integrable f o r every k, t h e n the original s y s t e m I is linearizable by d y n a m i c extension. Proof. A p p l y T h e o r e m 7 t o t h e e x t e n d e d s y s t e m J .
0
T h i s t h e o r e m i s s i m i l a r t o t h e o n e s t a t e d b y C h a r l e t , Levine, and M a r i n o [ 5 ] which g a v e sufficient c o n d i t i o n s f o r linearizing s y s t e m s b y d y n a m i c e x t e n s i o n . T h e i r c o n d i t i o n s also relied o n t h e existence of s o m e i n t e g e r s b , w h i c h a r e t h e number of i n t e g r a t o r s added t o t h e z r h i n p u t c h a n n e l . However, t h e e x i s t e n c e of a d y n a m i c e x t e n s i o n of which is l i n e a r i z a b l e d o e s n o t i m p l y t h a t t h e o r d e r b = ( b l , . . . , b,) c o n d i t i o n s of t h e i r t h e o r e m are satisfied f o r t h a t b; w h e r e a s if t h e r e e x i s t s a d y n a m i c e x t e n s i o n of o r d e r b = ( 6 1 . . . . , bm) which can be linearized, t h e c o n d i t i o n s of 'Theorem 8 will a l w a y s b e satisfied for t h a t b. The p r o b l e m of l i n e a r i z a t i o n b y d y n a m i c e x t e n s i o n has also b e e n s t u d i e d b y A r a n d a - B r i c a i r e , M o o g , and P o m e t [l] u s i n g t o o l s f r o m t h e t h e o r y of differential forms as well as differential a l g e b r a . T h e i r c o n d i t i o n s a r e necessary and sufficient, b u t m a y be difficult t o check in p r a c t i c e . A s i m p l e e x a m p l e will b e p r e s e n t e d t o s h o w h o w t h e t h e o r e m c a n b e a p p l i e d t o linearize c o n t r o l s y s t e m s u s i n g d y n a m i c extension. E x a m p l e 5 . [5] Consider a k t a t e , 2-input control system:
x, =xz i-2
= U'
x3 = U2 x 4 = x3 - X3Ul T h e c o r r e s p o n d i n g Pfaffian s y s t e m o n
iK7
is
u l d t , d x 3 - uzdt,dx4 - ( x 3 - X3Ul)dt) w i t h i n d e p e n d e n c e c o n d i t i o n dt and c o m p l e m e n t { d u l , d u z , d t } . T h e
I={dq
q d t , dx2
-
-
derived flag h a s t h e f o r m :
I=
{a',
11') = 1'2)
2, 0 3 , a4} {el,
=
04}
to}
t h e o n e - f o r m s a' which are a d a p t e d t o t h e d e r i v e d flag a r e not t h e same as t h o s e which g e n e r a t e d I . o1 = d x l - x 2 d t
a3 = dx3
-q d t
= d x r - U ] dt
a4 = dxq
+ x3dx2
(1'
dol2 =
-dui
A
dt
A
dt
d a 3 = -du2
A
-
x3dt
+ ( u l - l ) a 3A dt -
J(2)
= { a ] ,a 2 , = {a', = {a'}
J(3)
= {O}
J(')
w,
a3, & 4 } a3, & 4 }
w h e r e the b a s i s a d a p t e d to t h e d e r i v e d flag has c h a n g e d s t i g h t l y , w i t h EU4 = dx1 ( ~ 1 x3 2 3 ) d t . I t i s n o w e a s y t o see j u s t f r o m the expressions of t h e f o r m s a d a p t e d t o t h e d e r i v e d flag t h a t { I ( ' ) , & } i s i n t e g r a b l e f o r i = 0,1,2,a n d thus t h e extended s y s t e m can b e converted t o Brunovsky form. Remark 5. D y n a m i c State Feedback. I t has been s h o w n t h a t a p r o l o n g a t i o n b y differentiation of a c o n t r o l s y s t e m c o r r e s p o n d s t o a dynamic extension. A general dynamic s t a t e feedback corresponds t o a d d i n g some s t a t e s t o t h e s y s t e m and p u t t i n g f e e d b a c k around them, and i n g e n e r a l d o e s n o t c o r r e s p o n d t o a Cartan p r o l o n g a t i o n , since t h e r e m a y not be a one-one c o r r e s p o n d e n c e b e t w e e n t r a j e c t o r i e s of t h e e x t e n d e d s y s t e m and t r a j e c t o r i e s of t h e o r i g i n a l s y s t e m . This i s especially o b v i o u s if t h e a d d e d z s t a t e s h a v e t h e i r o w n d y n a m i c s , s e e f o r i n s t a n c e [19].
+
Acknowledgements The a u t h o r s would like t o t h a n k the reviewers f o r p r o v i d i n g e x t e n s i v e and insightful c o m m e n t s .
7. References E. Aranda-Bricaire, C . H . Moog, and J-B. Pomet. Feedback linearization: A linear algebraic approach. In Proc. I E E E Conf. Decisaon and Contr., pp 3441-3446, 1993. R. Bryant, S. Chern, R.G a r d n e r , H . Goldschmidt, and P Griffiths. Exterior Daflerential Systems Springer-Verlag, 1991 L. Bushnell, D Tilbury, and S. S Sastry. Extended Goursat normal forms with applications t o nonholonomic motion planning In Proc. I E E E Conf. Decrston and Contr., pp. 3447-3452, 1993. L. Bushnell, D. Tilbury, and S . S. Sastry. Steering three-input chained form nonholonomic systems using sinusoids: T h e fire truck example. In Proc. E u r . Contr. Conf., p p . 1432-1437, 1993 To Appear in rntl J . Rob. Res.. B. Charlet, J . Levine, and R. Marino. Sufficient conditions for dynamic s t a t e feedback linearization. S I A M J . Contr. and O p t . , 29(1):38-57, 1991. H . Flanders. Diflerentaal Forms, wrth Appltcations t o the Phystcal Sczences Dover Publications, Mineola, N Y , 1989 M Fliess, J Levine, P Martin, and P. Rouchon. O n differentially flat nonlinear systems. In Proc. IFAC Nonlan. C o n t r . Syst. Desrgn S y m p . ( N O L C O S ) , pp. 408-412, 1992. M Fliess, J Lgivine, P Martin, and P. Rouchon. Sur les s y s t h e s nonlineaires difrientiellenient plats C. R . Acad. Sci. Parts, 315.619624, 1992 M. Fliess, J . IAvine, P. Martin, a n d P. Rouchon Linearisation par bouclage dynamique e t transformations d e Lie-Backlund. C. R . Acad. Scr. Parrs. 317:981-986. 1993. M. Fliess, J Levine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples Intl. J. C o n t r . , 1994 To Appear. R . B . Gardner and'W. F. Shadwick. T h e GS algorithm for exact linearization t o Brunovsky normal form. I E E E Trans. A u t o . C o n t r . , 37(2):224-230, 1992. R 'M. Murray. Applicat~onsand extensions of goursat normal form t o control of nonlinear systems. In Proc. I E E E Conf. Decrsron and Contr , pp 3425-3430, 1993 M. Sampei and K . F u r u t a . On time scaling for nonlinear systems: Application t o linearization. I E E E Trans. A u t o . C o n t r . , AC31(5) 459-462, May 1986. W. F . Shadwick and W M. Sluis. Dynamic feedback for classical geometries. Tech Report FI93-CT23, T h e Fields Inst., Ontario, C a n a d a , 1993. W . M. Sluis. Absolute Equrualence and r t s Appltcations t o Control Theory. P h D thesis, Univ. of Waterloo, 1992. D Tilbury and A. Chelouah. Steering a three-input nonholonomic system using multlrate controls In P r o c . E u r . C o n t r . Conf., pp. 1428-1431, 1993. D . Tilbury, R.Murray, and S . Sastry. Trajectory generation for t h e N-trailer problem using Goursat normal form. In Proc. I E E E Conf. Dectszon and ('ontr., pp. 971-977, 1993. T o a p p e a r in I E E E Trans. Auto.
Contr.
I E E E Trans. Rob. and Auto..
dt
d a 4 = -a2 A a3
J
D Tilbury, 0 Sgrdalen, L. Bushnell, and S. Sastry. A multi-steering trailer system Conversion into chained form using dynamic feedback In Proc IFAC S y m p . Robot C o n t r . , 1994. To A p p e a r in
The s t r u c t u r e e q u a t i o n s a r e fairly s i m p l e t o find, da' = - a 2
c o r r e s p o n d i n g t o a d d i n g an i n t e g r a t o r t o the first input channel. T h e d e r i v e d flag i s n o w
21201'
A
dt
M. van Nieuwstadt, M R a t h i n a m , and R. M. Murray. Differential flatness and absolute equivalence. Tech. Report CIT-CDS 94-006, Caltech, 1994.
-
TA-11 11140
On Goursat Normal Forms, Prolongations, and Control Systems D. Tilbury and S. S. Sastry Electronics Research Laboratory Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720 dawnt@eecs. berkeley.edu
I n this paper, the method of exterior diflerential systems for analyzing nonlinear systems is presented. Conditions are given for converting Pfafian systems into Goursat normal f o r m , a n d for converting control systems into Brunovsky form. All of the existing results on feedback linearization for control systems can be restated in the language of Pfafian systems, a n d in addition, new conditions for linearizing control systems using dynamic extension are given.
1.
Introduction
There has been a great deal of interest in the use of exterior differential systems for analyzing nonlinear control systems. This paper brings together some of the results which have been recently published and also adds some new contributions to this area. For mechanical systems with linear velocity constraints, such as mobile robots with wheels that roll without slipping, the problem that is considered is that of finding a feasible path between two given points. Using the method of exterior differential systems, necessary and sufficient conditions can be given for transforming the Pfaftian system defined by the rolling constraints into a normal form for which feasible paths are easily found. The outline of this paper is as follows. First, the definitions for Goursat normal form and extended Goursat normal form are given. Necessary and sufficient conditions for converting PfafFian systems into these normal forms are stated, and several examples of mobile robot systems are studied. The concept of prolongation of a Pfaftian system is presented next, and sufficient conditions are given for converting a PfafFian system to extended Goursat normal form using a specific type of prolongation. Since the dual of Brunovsky linear form is a special case of extended Goursat normal form, all of the results for converting Pfaffian systems to extended Goursat form can be specialized to give conditions for converting control systems to Brunovsky linear form. Necessary and sufficient conditions for linearizing control systems by dynamic extension are finally given.
2.
Pfaffian systems
The interested reader is encouraged to consult [2] and [6], from which most of this introductory material was taken, for more details. A real vector space V or its dual (covector) space V' can be expanded into an exterior algebra with the exterior or wedge product defined by
+(ac)aAy
f o r a , P , y E V (orV*),anda,b,cEB.Thewedgeproductoftwovectors is called a two-vector. Define A2(V) as the space of two-vectors. Similarly, higher vectors can be built up, and A k (V) is defined to be the space of all k-vectors. For completeness, define Ao(V) = P and A'(V) = V. The dimension of A k ( V ) is (;). From the axioms, it follows that A k ( V )is empty for k > n. The exterior algebra over V is a graded algebra,
A(V) = A o ( V ) @ A ' ( V ) @ ~ . ~ @ A " ( V ) Any element X E h ( V )can be written uniquely as X = A0 XI .. A, where A, E A'(V) for i = 1,. . . ,n. Now, consider a differentiable manifold M of dimension n and its cotangent bundle T * M . Construct the bundle A ( T * M ) whose fibers are the exterior algebra of T:M, that is:
+ +. +
A(T:M) = Ao(T:) @ A'(T:) @ A2(T:)
U Ap(T:M)
AP(T'M)=
=EM over M is called an ezterior differential f o r m of degree p or simply a p-form. For local coordinates on M denoted by I = ( X I , . . . ,zn), a local basis for T,M is: { L, .. , Its dual basis on T:M is denoted 811 by { d q , . . . , dx,} where the d x , ' s are defined by the relationships
.
e}.
In terms of these local coordinates, a p-form w can be written as
where the coefficient functions w t l . . . , p ( x ) are smooth functions on M. The notation W ( M ) will be used to mean the module (over the ring of smooth functions) of all smooth sections of A p ( T ' M ) , and O ( M )= @ O p ( M ) as the module of forms on M. Consider a codistribution I on M , spanned by s one-forms,
I={a', ... ,a"} where a' is in fl' (M) for i = 1 , . . . , s. Definition 1 (Pfaffian S y s t e m s ) On a manifold of dimension n, a Pfaftian system is the smallest ideal 1 C R(M), generated b y a codidribation I of one-forms spanned b y { a ' , ,aS}which is closed under wedge products.
. ..
Throughout the c o m e of this paper, by deliberately confusing the notation, the codistribution I will be referred to as the Pfaftian system. The dimension of a PfafFian system is defined to be s, the number of independent one-forms which generate it. Any n - s linearly independent one-forms which are independent of I form a complement to I. The codimension of I is defined to be n - s. An integral curve for a PfafFian system is a curve c(t) : ( - e , E ) --t M which satisfies the constraints, that is, c * ( a ' ) = 0 for all a' E I. Here the notation c * ( a ' ) is taken to mean a'( A (local) independence condition for a Pfaftian system is a oneform T which does not vanish on integral curves, that is c * ( T ) # 0. If T is integrable, the Pfaftian system will correspond t o a system of firstorder ordinary differential equations. Throughout this paper, it will be assumed that the independence condition is integrable. Indeed, for control systems, the independence condition will often be d t , where t represents time. The notion of congruence modulo a Pfaffian system will be needed. For a system I = {a', . ,a'}, it is said that 1) ( mod I if
g).
~ A D = - O A ~ aAol=O
a a A ( b p + q )= ( a b ) a A D
The bundle A ( T ' M ) has AP(T'M) as sub-bundles. A section of the bundle
@.
. . @ A"(T:)
This research was supported in part by the NSF under grant IRI-9014490 and by the ARO under grant DAAL03-91-6-0171. D. Tilbury would also like to acknowledge an AT&T Ph.D. Fellowship for financial support of this work
..
s 1)
= .f -I-
A ai
,=I
for some forms 8' in fl(M). Definition 2 (Exterior Derivative) The exteriorderivative is d e fined as the unique map d : Q k ( M )+ R k + ' ( M ) which satisfies the following properties: I. FOTf E Q o ( M ) ,df = z d x l ... Zdz,,, relative to a local coordinate chart, O T the usual gradient. 2. F ~ ~ a E f l ' , f l E dn (~a ,A p ) = d a A P + ( - l ) r a A d p . 3. dZ = 0 .
+
0-7803-1968-0/94$4.0001994 IEEE 1797
+
Definition 3 (Derived Flag) Let I bystem. Define I(') = I and
I(~+') =
{w
= { a 1 , . . , C Y " ) ,be a P f a f i a n
E I ( ~ :' dw E o
mod
dk)}
The construction is assumed to t e r m i n a t e at s o m e N , when I ( N )= T h e derived Rag is defined to br the sequence o f nested codistributions, I = I(') 3 I(') 3 . . . 3 I ( N )
11 zs assumed that t h r dzmension of
1s
well-defined f o r all k.
Remark 1 . M a x i m a l l y N o n h o l o n o m i c . The last member of the chain of codistributions, is called the bottom derived s y s t e m . Sincr I(") = by definition dw 5 0 mod I ( N ) for all w E I ( N ) . and by the Frobenius theorem, the bottom derived system is integrable. That is, there there exist functions h i , . . . , h , such that I ( N ) = { d h l , . . . , dh,}. Solution trajectories of I are then constrained to lie on level surfaces of h. A Pfaffian system I is said to be n o n h o l o n o m i c if I ( ' ) is a proper subset of I. This paper will be concerned with systems which are maximally n o n h o l o n o m i c , that is, with bottomderivedsystem I ( N ) =
{OhA
variant of the familiar Frobenius theorem is stated here:
Theorem 1 (Frobenius [2]) Let { w ' , . . . , w p j be a set of linearly independent o n e - f o r m s , and f l , . . . , f , a set of f u n c t i o n s whose differentials are linearly independent of each other and of the w"s. If dw'Aw1A...AwPAdf1 A...Adfq=O f o r i = 1 , . . . , p , then there ezist coordinate J u n c t i o n s 2 1 , . . . , z p and coeficient f u n c t i o n s a,,, b,, such that the o n e - f o r m s w' can be written as:
,=I
,=I
The proof follows from the standard Frobenius theorem and the fact chat the codistribution { w ' , , , . , w P , d f l , . . . , d f q } is integrable.
3.
Definition 5 (Extended Goursat Normal Form) A P f a f i a n sysof codimension m 1 is i n extended Goursat t e m I on normal form i f it is generated b y n constraints of the f o r m :
+
I = {dz: -
z:tl dzo : a
Definition 4 (Goursat Normal Form) A codimension two Pfaff i a n s y s t e m I o n E n w i t h generators of the f o r m I = {dzy, - zn-l d z l , . . . ,d:g - z z d z l j
(2)
There are conditions due to Murray [12] for converting a PfafXan system to extended Goursat normal form. This theorem is restated and proven here with the additional condition (correction) that the one-form R must be integrable: Theorem 3 (Extended Goursat Normal Form [12]) Let I be a P f a f i a n s y s t e m of codimension m 1 . If (and only i f ) there e z i s f s a set of generators { m 3 : i = 1,. . . , s ; j - 1 , .. . , m } f o r I and an integrable o n e - f o r m T h c h that f o r a h j , -
+
t h e n there exists a set of coordinates z such that I is i n Goursat normal form. Proof. If the Pfaffian system is already in extended Goursat normal form, the congruences are satisfied with T = dzo and the basis of constraints a: = dz: - ~ i - ~ d z ~ . Now assume that a basis of constraints for I has been found which satisfies the congruences (3). It is easily checked that this basis is adapted to the derived flag, that is:
I ( k ) = {a: : j = 1 , . . . , m ; i = 1 , . . . ,s, - k } Since x is integrable, any first integral of T can be used for the coordinate .'2 If necessary, the constraints CY: can be rescaled so that the congruences (3) are satisfied with d z o . Now consider the last nontrivialderived system, I("1-l). The one-forms { a i , . . . form a basis for this codistribution, where SI = s2 = . . . = s r l . From the fact that d a : -ai Adz' mod I ( ' ' - ' ) , it follows that the one-forms e:,. . . satisfy the Frobenius condition:
,ai'
dei
Goursat Normal Forms
A great deal of work has been done on transforming Pfaffian systems into normal forms. The Goursat normal form will be of special interest in this paper. It w a s originally proposed by Goursat for Pfaffian systems of codimension two, and has the property that all its integral curves can be expressed in terms of two arbitrary functions. An extended Goursat normal form for Pfaffian systems of codimension k > 2 will also be examined. For the extended Goursat normal form, solution trajectories can be expressed in terms of k arbitrary functions.
= 1 , . . . , s , ; j = 1 , . . . ,m } ,
A
cri A
. . . A 0;' Adz'
=0
and thus, by the Frobenius theorem, coordinates z i , . . . ,z;' found such that
[ ":] [ ] = A
a;'
d;
can be
+Bdzo
dz;'
The matrix A must be nonsingular, since the a ; ' s are a basis for 1('1-'j and they are independent of dz'. Therefore, a new basis &{ can be defined as:
is said to be i n Goursat Normal Form.
From the form of the Pfaffian system in the z coordinates, it follows that integral curves of the system are unconstrained in their zl, 2, coordinates alone. Once 21 ( t ) ,z,,(t) are specified as functions of some parameter t , the other coordinates are determined as some functions of t . The following classical theorem gives necessary and sufficient conditions for converting a Pfaffian system into Goursat normal form: Theorem 2 (Goursat Normal Form [Z]) A P f a f i a n s y s t e m I of codimension two on I nhas a set of generators which are i n G O U T sat n o r m a l f o r m if and only iJ there ezists U basis set of f o r m s { c ~ ' , . . . ,an-') J O T I und a o n e - f o r m T satisfying the congruences: de' E -e'+' A R mod a ' , , , , ,m' i = 1 , . . . , n - 3 &n-2
so
mod I
Define the coordinates z: := - ( A - ' B ) , , so that the one-forms E: have the form = dz: - z;dz' f o r j = 1, ... , T I . By the proof of the standard Goursat theorem, all of the coordinates in the j t h tower can be found from and z o , thus by the above procedure, all the coordinates in the first T I towers have effectively been found. To find the coordinates for the other towers, look a t the lowest derived systems in which they appear. Consider the smallest integer k such that dim I($ITkj> k r l ; more towers will appear at this level. It can be shown from the congruences ( 3 ) and the calculations up to this point that a basis for I("1-k) is
{ s i , . . , s i , . . . , & ; I , . . . ,E;'
(1)
where 6; = d z f In [17] it w a s shown how the Pfaffian system associated with the system of a car towing n trailers, generated by the constraints that each axle of wheels roll without slipping, satisfied the conditions for conversion to Goursat normal form. This w a s a system of codimension two, corresponding to the fact that the linear and angular velocities of the front car are the inputs, and thus fredy specifiable. In order to consider Pfaffian systems associated with mobile robots such as the firetruck [4, 161 or the multi-sterring multi-trailer system [18], PfafXan systems of codimensiongreater than two must be studied as well. Consider the following definition:
,a;1+1,.
.. ,a;l+r2}
dz' for j = 1,. . . , T I , as found in the first step,
-
+ +
and a' for j = r1 1 , . . . , r2 are the original one-forms, which are adaptdd to the derived flag. The lengths of these towers: s r l + l = ' . s T 1t T 2= SI - k 1, are now known. For notational convenience, for = 1 , . . . , T I . define z i k ) := ( z ; , . . . );:, By the Goursat congruences, d a 3 = -a; A d z o mod 1(31-k) for 1 , . . . ,r1 r 2 , thus the FrobGius condition
j = r1
+
da;Aal'"A.
1798
+
. .A a r 1 + ' 2 A d z ~ A . ~ . A d z ~ A . . . A d z ; ' A . . . A d z ~ ' A d = z' 0
+
"::*'
is satisfied for j = r1 1 , . . . ,rl , z;'"' new coordinates 2;' ",
...
+ r2. Using the Frobenius theorem,
[ ] [ ] =A
ar1+r2
+
BdzO
+C
t'2
&'l
[ "';*I ]
dai G O
dz;;)
[ ] [ ]
+Bdzo
dz;i +ra
o',
[ ] [ ] [ ] :;
=
'r~ t r z
tr2
+
.
The coordinates := ( A - I B ) ' for j = rl 1 , . . ,r1 -tr2 are then defined so that a: = dz' - z ~ d z ' . Again, by the standard Goursat theorem, all of the coordinates in the towers rl t 1, , rl rz are now defined. The coordinates for the rest of the towers are defined in a manner exactly analogous to those of the second-longest tower. If x is not integrable, then the Frobenius theorem cannot be used to find the coordinates. In the special case where si > s ~ that , is, there is only one tower which is longest, it can be shown that if there exists any P which satisfies the congruences, then there also exists an integrable P' which also satisfies the congruences (with a rescaling of the basis forms), see [3, 121. However, if si = s2, or there are at least two towers which are longest, this is no longer true. Thus, the assumption that r is integrable must be stated in the assumptions. 0
...
+
Another version of the theorem for converting systems to extended Goursat normal form will also be stated here. This version is slightly easier to check, since it does not require finding a basis which satisfies the congruences but only one which is adapted to the derived flag. The proof for this theorem is also given, since only a special case is proved in [14]. T h e o r e m 4 ( E x t e n d e d G o u r s a t N o r m a l Form [14]) A Pfafian system I of codimension m + l on Rn+m+l can be converted to Goursat normal f o r m if and only if The system is mazimally nonholonomic, that is I ( N ) = { 0 } for some N , and There ezists a one-form A such that { I ( k ) , r }is integrable for k = 0 , ... , N - l . Proof. The only if part is easily shown by taking A = dzo and noting that
+ 1 , . .. ,sJ} j = 1 , . .. ,m ; i = k + 1 , . . . , s J } ,
I ( k ) = { d z i - z:-ldzO : j = 1 , . . . , m ; i = k
{ f i k ) , r=} { d z i , d z o : which is integrable. Now, assume that such a P exists. Find the derived flag of the system, I =: I(') 3 I ( ' ) 3 . 3 I('1) = ( 0 ) . The lengths of each tower are determined from the dimensions of the derived flag. Indeed, the longest tower of forms has length 8 1 . If the dimension of I('1-l) is r 1 , then there are rl towers which each have length SI; and thus s1 = s2 = . . = srl . Now, if the dimensionof I(s1-2) is 2rl +r2, then there are r2 towers with length s1 - 1, and sr1+l = .. = sr,tr:,= s1 - 1 . Each sJ is found similarly. Note that a A which satisfies the conditions must be in the complement to I, for if P were in I, then {I,P} integrable means that I is integrable, which contradicts the assumption that I is maximally nonholonomic, that is I ( N )= ( 0 ) for some N. Consider the last nontrivial derived system, I ( ' l - ' ) , with a basis given by {a:,.. . , a;'}. The definition of the derived flag, specifically I(sl) = {0}, implies that
..
.
J=1
30
mod
fi'l-')
j = 1, ... , r 1
(4)
Also, the assumption that { I ( k ) , r }is integrable gives da' 1 -= 0 mod
{d'l-l), T }
=
j = 1 , . .. , r1
(5)
,=l
'=I
T
( c b J P J ) mod
A
E 0 mod
I('1-I)
I('1-I)
J=1
which implies that a is in I('1). However, this contradicts the assumption that 1 ( 3 1 ) = ( 0 ) . Thus the PJ'S are linearly independent mod I('l-'), and the claim is proven. Define a; := PJ for j = 1 , . ,'1. Note these basis elements satisfy the first level of Goursat congruences, that is:
..
j = l ,...,rl
doli ~ - a ; A ? r modI('1-')
If the dimensionof I(sl-2)is greater than 2r1, then choose one-forms * r11 t 1 ,... , a;, tra such that that is a basis for I ( '1 -'I. For the inductionstep, assume that a basis for I ( ' ) has been found
. ai1,a:,. . . ,a ; 2 , .. . ,a;,. . . ,a i c }
{a:, . . ,
which satisfies the Goursat congruences up to this level: dai
= -ai+l
A
T
= 1 , . . . ,c;
mod
k = 1 , . . . , kJ - 1
It is assumed that c towers of forms have appeared in I ( ' ) . Consider only the last form in each tower that appears in I ( ' ) , that is , J = 1 , . . , c. The construction of this basis (or from the Gour-
.
aiJ
sat congruences) implies that
ai,
is in I ( ' ) but is not in I('+'), thus
d e i J $ 0 mod I ( ' )
J
= 1 , . .. , c
The assumption that { I ( t ) ,r} is integrable gives
daiJ G O
mod{I('),r}
j=l,
...,c
thus d a i , must be a multiple of r mod I ( ' ) , daJk3 = r A P '
modi(')
~ = 1 . ., . , c
for some PJ $ 0 mod I ( ' ) . Also,from the fact that a:, is in I ( ' ) and the definition of the derived flag,
dai, f O
modi('-') j = 1 , ...
,c
which implies that PJ 6 I('-'). By a similar argument to the claim above, it can be shown that the PJ'S are independent mod I ( ' ) . Define = P J , and thus
aiJtl
.
.
... , Q E , + I )
{a;,. . , a i l t l , a ; , .. . 'a;atl,. . ,a;,
.
dai
... ,
,orl
+(A-'B)dzo
&'1+'2
1
j=l,
+ .+
Again, A must be nonsingular because the a i ' s are linearly independent mod I('l-') and also independent of dz', and thus new one-forms E r 1 + l , . . . , 6 r l + r a can be defined as: $ltl t1 dzrl t :=A-'
modI("1-2)
which combined with (6) implies that PJ is in I('l-'). Claim. . .. are linearly independent mod I ( ' 1 - l ) . Proof of Claim. The proof is by contradiction. Suppose there exists some combination of the ~ J ' S ,say = blP1 . . brlp'l 0 mod I('1-') with not all of the b,'s equal to zero. Consider a = bla: b,, ay'. The one-form a # 0 because the a: are a basis for I(31-1).The exterior derivative of a can be found by the product rule,
+ .. . +
dzrl t1
= A
j = l , ... , r l
d a i E r A P J modI('1-'1
0 mod I('1-l). results for some 0' Now, from the definition of the derived flag,
Since the congruences are only defined up to mod I ( ' l P k ) , the last group of terms (those multiplied by the matrix C) can be eliminated by adding in the appropriate multiples of 6: = dz: dz'. This will change the B matrix, and what remains has the form:
01r1tr2
combining equations (4) and ( 5 ) , the congruence
can be found such that
forms part of a basis of I('-'). If the dimension of I('-') is greater k2 k, c. then complete the basis of I('-') with than kl such that any linearly independent one-forms aEtl,. . . ,
+ + . .. + +
ailtl ,a;,... ,a;2tl,... ,a;,. . . ,a i c t 1 ,a y , . .. 1 a=+'c} 1
{ai,. .. ,
is a basis for I('-'). Repeated application of this procedure will construct a basis for I which is not only adapted to the derived flag, but also satisfies the Goursat congruences. 0 Remark 2. Dtflerential Flatness. There are connections between Goursat forms and the concept of differential flatness. The precise definition of differential flatness is given in the language of differential
1799
algebra [8,7, 101 and is beyond the scope of this paper. Informally, however, a system is said to be d i g e r e n t i a l l y f l a t if a set of flat outputs (equal t.o the number of inputs) can be found such that the entire state and input trajectories can be recovered from the flat outputs as functions of time and finitely many of their derivatives. Although there exists no general technique to determine whether or not a given system is flat (other than guessing flat outputs), it is easy to see that any system which can be converted into Goursat normal form is differentially flat. The flat outputs can be chosen as !.he states z o , z : , . . . ,z;".
are satisfied instead of the original congruences (3). The Gounat coordinates for this system can be found as
= x3
= Y3
.:
=
I
- 84
tan
if L4 = L5
. , / T G t a n ( v )
$yarctan' + e
+
a t . . ( w ) + i + e
- 7 7 1% (7 I-! tan(-)-]-!
Mobile Robot Examples
4.
20 2:
In this section, some multi-steering mobile robot systems will be considered and it will be shown that the PfafFian systems generated by the constraints that the wheels roll without slipping satisfy the extmded Goursat conditions. A multi-steering trailer system was examined in [18],and it was shown how to transform such a system into chained form (which is the dual of Goursat normal form) using dynamic state feedback. That. is, states were added to the system and this augmented system was transformed into the dual of Goursat normal form. In this section, it is shown that this augmentation is not always necessary; some arrangements of the multi-steering trailer system can be transformed into Goursat form using only static state feedback. The 5-axle system with two steering wheels is the first example in which interesting things begin to happen. Assuming that the first axle is steerable, there are four possible positions for the second steering wheel. Three of these four cases satisfy the conditions for converting t,o extended Goursat normal form; the fourth does not. Two of the examples will be presented in some detail to give the reader a flavor for the type of calculations which are required. Example 1. 5-axle, 1-4 steering trailer system. Consider the :,-axle system with the first and fourth axles steerable, as sketched in Figure 1. The configuration space can be parameterized by the x, y position of the third axle, the hitch angles e,, and the steering angle of the third axle 6. Let q = {x3, y3,05,91,03,92,81, d} represent the state.
1
if L4
> Ls
if L4
< L5
where e = L4 l L 5 . Remark 3. 5 - a x l e , 1 - 2 s t e e r i n g a n d 1 - 5 s t e e r i n g . Without presenting the calculations, which are similar than those above, it is noted here that the five-axle system with either the first and second axles steerable or the first and last axles steerable satisfies the conditions for conversion into extended Goursat normal form. The constraints have the same form. For the 1-2 steering system, T can he chosen as dx5 and the Goursat coordinates are found as zo = x5,z: = y5, and 2: = 4. For the 1-5 steering system, x can be chosen as dxl, and the Goursat coordinates are defined as'2 = x(, 2: = y4, and 2: = 4. Example 2. 5-axle, 1-3 steering. The final instance of the 5-axle trailer system has the first and third axles steerable, as sketched in Figure 2. The vector 9 = ( 2 5 , y 5 , 8 5 , 8 4 , 9 3 , 8 ~ , 8 1 , ~represents } the state.
Figure 2: A 5-axle trailer system with the f i s t and third axles steerable. This is the only configuration of the 5-axle system with two steering wheels which does not satisfy the conditions for converting to extended Goursat normal form. The constraints are that each axle roll without slipping:
Figure 1: A 5-axle trailer system with the first and fourth axles steerable. The constraints are that each axle rolls without slipping: sinBidx, - cosO,dy, i = 1,2,3,5 sinddxq - cosddyr a 5 }a compleThe Pfaffian system is thus I = { a 1 , a z , a ~ , a 4 , and ment to this system is: (dd,d&,dx3}. This basis of constraints is adapted to the the derived flag, I = t a l , az, a3, a 4 , 0 5 ) I(') = {aZ,a3, a"} I(2) = {a3}
(01
1(3) =
and satisfies the congruences: do' E q ( 9 ) d81
A
A
dx3 mod I
dx3
da4 E c * ( q ) dd
mod I ( ' )
do5
A
dx3 mod I
~ ( 9 a)4A dx3 mod I ( ' )
du3 E c3(q) u 2 A dx3 mod I ( 2 ) By a simple rescaling of the basis, the functions c , ( q ) can be eliminated to get the Goursat congruences (3) exactly. This is done as follows. First defiles2 := -c3(y) a 2 toget da3 -C2Adx3 mod I ( 2 ) . 'The exterior derivative of a2 can be written as d?i2 = - c 2 ( q ) =;do' - dc2(g) A a' E - - c z ( q ) c l ( q )a' r,dx3 mod I ( ' ) since a' is in I ( ' ) . The other constraints are scaled similarly. For the rest of this paper, it will asserted that the Goursat congruences arc satisfied if the modified congruences d a ~ ~ c ~ + l ( q ) u ~ +modI('I-') lA?r da;] Z 0
-
- cosddy3
mod I
i = l , ... , s - 1 (7)
cos9,dy,
i = 1,2,4,5
ThePfaftiansystemis I = {a',a2,a3,a4,a5},andacomplementto the system is given by {dd, d81, dx3 This basis is adapted to the derived flag, I = {a', a2, a3, a4, a5}
1.
= =
a'
a4
da2 E c 2 ( 9 ) a'
a' = sinO,dx, a3 = sinbdx3
I(') =
{a2,
a4, a5}
= 1(3) =
{a5}
1(2)
io}
however, the congruences are nof satisfied: da' E CI
dB1 A
da2 I c2 a'
A
dx5 dx5
da3 E c3 d$
+ k2 a'
A a3
A
dx5 mod I
da4 E c4 a3 A dxs mod I ( ' ) de5 E cg a4 A dx5 mod
d2)
In order to have { l ( 2 ) , rintegrable, } x must be chosen as x = dx5 This will also give {IO,.} integrable, but { I ' , x } is (mod {a4,a5}). n o t integrable. Thus, this system does not satisfy the conditions for conversion to extended Goursat normal form. This example will be considered again in the next section.
5.
Prolongations
If a Pfaffian system I of codimension k satisfies the necessary and sufficient conditions for converting into extended Goursat form, then its solution trajectories are determined by k arbitrary functions. However, even if a system cannot be transformed into Goursat form, its solution trajectories may still have this property. If so, it is said that I is a b s o l z l t e l y e q u i v a l e n f (in the sense of Cartan) to the trivial sysAlthough the concept tem (the system with no constraints) on Ik. of absolute equivalence will not be examined in its full generality, some sufficient conditions will be given for a PfaEan system to have a p r o l o n g a t i o n by d z f f e r e n t i a t r o n which can be converted to Goursat form, and thus the solution trajectories of I are determined by k independent functions. A general type of prolongation which preserves a one-one correspondence between solution trajectories of the original and prolonged system is a Cartan prolongation.
1800
Definition 6 ( C a r t a n Prolongation) L e t I be a P f a f i a n s y s t e m o n a manifold M . A s y s t e m J o n M x P P i s a Cartan prolongation of I i f : 1. T * ( Ic)J 2. For every s o l u t i o n curve c : ( - € , E ) + M o f I there ezists a u n i q u e s o l u t i o n curve 2 : ( - E , E ) -+ M x BP of J w i t h T O i. = c. If I i s equipped w i t h a g i v e n independence c o n d i t i o n T , t h e n T ' T m u s t be t h e independence c o n d i t i o n f o r J .
A canonical way to prolong a system with independence condition d t is to take an integrable one-form dq in the complement of I, and augment I with the additional form dq - y d t , where y is a new coordinate on R . In effect, this adds the derivative of q (with respect to the independence condition) as a state variable. As long as all solution trajectories are "smooth enough" (assume C"), there will be a one-one correspondence between solution trajectories of the original and the prolonged system. Consider a special type of Cartan prolongation which consists of many of these canonical prolongations. Definition 7 (Prolongation by differentiation) L e t I be a P f a f f i a n s y s t e m of codimension m+l o n I"+"'+'w i t h coordinates (2,U ,t )
f o r which d t i s a n independence c o n d i t i o n and { d u i , ... , d v m , d t } f o r m s a complement. L e t 6 1 , . . , bm be nonnegative integers and let b d e n o t e t h e i r s u m . T h e s y s t e m I augmented by
.
dvl - v i d t ,
... ,
+.
Since control systems are a special type of Pfaffian system, all of the results presented thus far can be specialized to control systems. Definition 8 (Control S y s t e m ) A controlsystemi = f ( x , u ) with t h e state x E B", t h e i n p u t U E B'", and t h e derivative of t h e state t a k e n w i t h respect t o t i m e t E 1,generates a P f a f i a n s y s t e m I o n Rntmt1
I = { d x , - f ' ( x , u ) d t : i = 1 , . . . , n} (8) w i t h complement { d u l , . . . , d u m , d t } . T h e n a t u r a l independence cond i t i o n t o choose i s d t , since d t # 0 along all s o l u t i o n trajectories of the system. A n y P f a f i a n s y s t e m I of codimension m 1 o n R n t m t l w i t h coordinates ( x , u , t ) can be called a control system if i t h a s a s e t of generators of f o r m ( 8 ) .
+
Brunovsky showed that any controllable linear system j. = Ax Bu with x E R", U E Rm can be converted to a canonical form given by
.
dv, - v A d t , . , , ... dv&"-' - vLmdt, i s called a prolon ationby differentiation of I. T h e augmented s y s t e m i s defined o n Rnfmtbtl.
Since the original system and independence condition corresponded to a set of first order ordinary differential equations, the prolonged system has the same independence condition and also corresponds to a set of first order ordinary differential equations. Sufficient conditions can now be given for a Pfaffian system to have a prolongation which is equivalent to extended Goursat form. T h e o r e m 5 ( G o u r s a t f o r m via prolongation) Consider a P f a f f i a n s y s t e m I = { a ' , . . ,a"} o n w i t h independence condit i o n d z o and complement { d v l , .. , d v m r d z o } . If there ezists a list of integers b l , . . ,bm s u c h t h a t t h e prolonged s y s t e m
.
.
.
Control Systems
6.
+
- vi'dt,
dv;'-'
and the systems {J(')), d x 5 } are integrable for all k. R e m a r k 4. 5 - a z / e , 1-4 steering, revisited. In [3],it was noted that the 5-axle system with steering on the first and fourth axles could be converted into Goursat form after an order two prolongation (U' = d 4 - v 1 d x 5 , w 2 = d v l - v z d x ~ )and , that the Goursat coordinates for the system after prolongation were much simpler than those given in example 4. The one-form x which satisfies the congruences for the extended system is d x 5 , and the other two coordinates which define the Goursat form are y5 and
J = {a',. . . ,a", d v l - v i ds', .. . ,dv;'-'
v1 bi
dzo '
. . . ,dum - v L d z O , . . . ,dv&m-lv!"dto}
x1
;;
...
- U1
e;. = U, xy = X;.
...
(9)
"'
qm =xrm4 = Xi1-'
+ .+
with n = kl . . k,. Therefore, a control system is said to be lrnearitable if and only if it can be converted to Brunovsky form. Brunovsky linear form for a control system is a special case of extended Goursat normal form (2) with dro = d t and t1 = U, ; thus, the theorems for transforming to Goursat form can be specialized to give conditions for exact linearization. T h e o r e m 6 (Exact Linearization [ l l ] ) If a control s y s t e m I defined o n In+"'+'h a s a s e t of generators {a: : j = l , ,m;i = 1 , . . . , s,} s u c h t h a t f o r all j ,
4,
...
satisfies t h e c o n d i t i o n t h a t {J(k),dzo} i s integrable f o r all k, t h e n I can be transformed t o extended G o u r s a t n o r m a l f o r m u s i n g a prolong a t i o n by differentiation.
Proof. The proof is by application of Theorem 4 to the prolonged system J . 0
Although this is a very specific form of prolongation of a P f d a n system, and the conditions of the theorem must be checked in a specific coordinate system with a given independence condition, there do exist practical systems which can be converted into extended Goursat form using this type of prolongation. E x a m p l e 3. 5-axle, 1-3 steering, revisited. Consider again the 5-axle trailer system with the first and third axles steerable, which did not satisfy the conditions for conversion to extended Goursat form. The equations for the exterior derivativesof the constraintscan be examined to see if after prolongation, the augmented P f d a n system will satisfy the conditions for conversion to extended Goursat form. It was seen that ?r = d x 5 will give { I ( 2 ) , x } integrable. However, { I ( ' ) , d x s } is not integrable since d o 2 has a term k 2 ( q ) a ' A a3. If either a' or a3 could be added to I('), that term would no longer ' were added to I ( ' ) , then d e 2 0 mod I ( ' ) cause a problem. If a and the same problem would recur, except now with I ( 2 ) .If a3 could somehow be added to I ( ' ) , it appears that the conditionsof Theorem 4 will be satisfied (the only - thing- remaining- to be checked is that da3 f 0 moda2,a3,a4,a5,dx5.) If I is prolonged by differentiation, and augmented by the additional form w = d+ - v d x 5 , then the derived flag of the augmented system is: J = { a', a 2 , w , a 3 , cr4, a5}
J(') = { J(2) = { J(3) =
{
J(') = {
a2,
t h e n there exists a s e t of coordinates z s u c h t h a t I is in B r u n o v s k y normal fo rm, I = { d r ; ) - ~ : + ~ d t : j = l,... , m ; i = l , . . . ,sJ}.
The control systems version of Theorem 4 is given by T h e o r e m 7 (Exact Linearization [15]) A control s y s t e m I can be converted t o linear f o r m i f and only if { d k ) , d t } i s integrable f o r every k. Another version of this theorem is given in [ l ]with slightly different notation. The problem of feedback linearization by time-scaling was studied in [ 1 3 ] ; the Goursat formulation gives a simple set of conditions to check if a time-scaled version of a system can be linearized, as is shown by the following simple example. Time-transformations are also studied in 191. Example 4. G o u r s a t n o r m a l f o r m for a control system. Consider the single-input control system [ 5 ] , 2' = 1 2 1 3 2
+
= x3
i 3
=U
This control system generates a P f a a n system, I = { d x l - ( ~ 2 ~ 3 ' )d t , d ~ z- x 3 d t , d x 3 of codimension two on R5 with derived flag
+
I=
(al,ff2,ff3)
dl) =
a3, a4, a5} cr4,
x2
a"} a51
I(2)
0)
1(3)
1801
= {al} = {O}
- udt}
(11)
n o t e t h a t { I ( ' ) , d t ) is n o t i n t e g r a b l e , t h u s t h e s y s t e m i s n o t linearizable b y s t a t i c s t a t e feedback. C o n s i d e r a p r o l o n g a t i o n b y d i f f e r e n t i a t i o n of I d e f i n e d b y
where t h e one-forms a d a p t e d t o t h e d e r i v e d flag are given b y
- 2x3dzz a' = d x z - x 3 d t a3 = dx3 - u d t a' = d x l
+ (x3'
- xz)dt
J = { I , w = dui - ~ l d t }
N o t e t h a t t h i s is n o t t h e b a s i s of (11) which g e n e r a t e d I . S i n c e { I ( 2 ) ,d t } is not. i n t e g r a b l e , t h e s y s t e m i s n o t f e e d b a c k linearizable b y T h e o r e m 7. The G o u r s a t c o n g r u e n c e s ( l ) ,however, are satisfied, for ?r = d i = d t - 2 d r 3 . T h u s , t h e r e d o e s e x i s t a t r a n s f o r m a t i o n @ ( z , U. t ) = ( 2 , v, T ) t o G o u r s a t n o r m a l f o r m , which is given b y
22
= 23 = 2 2 - 232
23
= 2' - 2x223
21
r=t
-
2x3
U
=-
2'
1 - 2u
+ -322 3 3
and i t is easily c h e c k e d t h a t
dz' =
d ~ 3 = 22
dz 2 = z,
dT
dr
dr
T h e p r o l o n g a t i o n b y d i f f e r e n t i a t i o n which was defined i n S e c t i o n 5 is t h e dual of d y n a m i c e x t e n s i o n ( a d d i n g i n t e g r a t o r s t o t h e inputs) i n t h e l a n g u a g e of f o r m s . Thus, t h e c o n t r o l s y s t e m s version of T h e o r e m 5 c a n b e s t a t e d as:
Theorem 8 ( L i n e a r i z a t i o n by d y n a m i c e x t e n s i o n ) Consider a control s y s t e m I on iKn+m+' with coordinates ( x , u , t ) , independence condition d t , and complement { d u l , . . . ,du,,dt}. I f (and only i f ) there erists Q prolongation by differentiation of d i m e n s i o n b = bl . . . b , such that t h e augmented s y s t e m
+
+
J={a'=dx,--
f'(x.v)dt:
j = 1 ,... , m ; k= 0 ,... , b , }
- u:dt:
= duk-'
i = l , . . . ,n;
o n R n t m t b + l satisfies the condition ( J ( ' ) ) , d t } i s integrable f o r every k, t h e n the original s y s t e m I is linearizable by d y n a m i c extension. Proof. A p p l y T h e o r e m 7 t o t h e e x t e n d e d s y s t e m J .
0
T h i s t h e o r e m i s s i m i l a r t o t h e o n e s t a t e d b y C h a r l e t , Levine, and M a r i n o [ 5 ] which g a v e sufficient c o n d i t i o n s f o r linearizing s y s t e m s b y d y n a m i c e x t e n s i o n . T h e i r c o n d i t i o n s also relied o n t h e existence of s o m e i n t e g e r s b , w h i c h a r e t h e number of i n t e g r a t o r s added t o t h e z r h i n p u t c h a n n e l . However, t h e e x i s t e n c e of a d y n a m i c e x t e n s i o n of which is l i n e a r i z a b l e d o e s n o t i m p l y t h a t t h e o r d e r b = ( b l , . . . , b,) c o n d i t i o n s of t h e i r t h e o r e m are satisfied f o r t h a t b; w h e r e a s if t h e r e e x i s t s a d y n a m i c e x t e n s i o n of o r d e r b = ( 6 1 . . . . , bm) which can be linearized, t h e c o n d i t i o n s of 'Theorem 8 will a l w a y s b e satisfied for t h a t b. The p r o b l e m of l i n e a r i z a t i o n b y d y n a m i c e x t e n s i o n has also b e e n s t u d i e d b y A r a n d a - B r i c a i r e , M o o g , and P o m e t [l] u s i n g t o o l s f r o m t h e t h e o r y of differential forms as well as differential a l g e b r a . T h e i r c o n d i t i o n s a r e necessary and sufficient, b u t m a y be difficult t o check in p r a c t i c e . A s i m p l e e x a m p l e will b e p r e s e n t e d t o s h o w h o w t h e t h e o r e m c a n b e a p p l i e d t o linearize c o n t r o l s y s t e m s u s i n g d y n a m i c extension. E x a m p l e 5 . [5] Consider a k t a t e , 2-input control system:
x, =xz i-2
= U'
x3 = U2 x 4 = x3 - X3Ul T h e c o r r e s p o n d i n g Pfaffian s y s t e m o n
iK7
is
u l d t , d x 3 - uzdt,dx4 - ( x 3 - X3Ul)dt) w i t h i n d e p e n d e n c e c o n d i t i o n dt and c o m p l e m e n t { d u l , d u z , d t } . T h e
I={dq
q d t , dx2
-
-
derived flag h a s t h e f o r m :
I=
{a',
11') = 1'2)
2, 0 3 , a4} {el,
=
04}
to}
t h e o n e - f o r m s a' which are a d a p t e d t o t h e d e r i v e d flag a r e not t h e same as t h o s e which g e n e r a t e d I . o1 = d x l - x 2 d t
a3 = dx3
-q d t
= d x r - U ] dt
a4 = dxq
+ x3dx2
(1'
dol2 =
-dui
A
dt
A
dt
d a 3 = -du2
A
-
x3dt
+ ( u l - l ) a 3A dt -
J(2)
= { a ] ,a 2 , = {a', = {a'}
J(3)
= {O}
J(')
w,
a3, & 4 } a3, & 4 }
w h e r e the b a s i s a d a p t e d to t h e d e r i v e d flag has c h a n g e d s t i g h t l y , w i t h EU4 = dx1 ( ~ 1 x3 2 3 ) d t . I t i s n o w e a s y t o see j u s t f r o m the expressions of t h e f o r m s a d a p t e d t o t h e d e r i v e d flag t h a t { I ( ' ) , & } i s i n t e g r a b l e f o r i = 0,1,2,a n d thus t h e extended s y s t e m can b e converted t o Brunovsky form. Remark 5. D y n a m i c State Feedback. I t has been s h o w n t h a t a p r o l o n g a t i o n b y differentiation of a c o n t r o l s y s t e m c o r r e s p o n d s t o a dynamic extension. A general dynamic s t a t e feedback corresponds t o a d d i n g some s t a t e s t o t h e s y s t e m and p u t t i n g f e e d b a c k around them, and i n g e n e r a l d o e s n o t c o r r e s p o n d t o a Cartan p r o l o n g a t i o n , since t h e r e m a y not be a one-one c o r r e s p o n d e n c e b e t w e e n t r a j e c t o r i e s of t h e e x t e n d e d s y s t e m and t r a j e c t o r i e s of t h e o r i g i n a l s y s t e m . This i s especially o b v i o u s if t h e a d d e d z s t a t e s h a v e t h e i r o w n d y n a m i c s , s e e f o r i n s t a n c e [19].
+
Acknowledgements The a u t h o r s would like t o t h a n k the reviewers f o r p r o v i d i n g e x t e n s i v e and insightful c o m m e n t s .
7. References E. Aranda-Bricaire, C . H . Moog, and J-B. Pomet. Feedback linearization: A linear algebraic approach. In Proc. I E E E Conf. Decisaon and Contr., pp 3441-3446, 1993. R. Bryant, S. Chern, R.G a r d n e r , H . Goldschmidt, and P Griffiths. Exterior Daflerential Systems Springer-Verlag, 1991 L. Bushnell, D Tilbury, and S. S Sastry. Extended Goursat normal forms with applications t o nonholonomic motion planning In Proc. I E E E Conf. Decrston and Contr., pp. 3447-3452, 1993. L. Bushnell, D. Tilbury, and S . S. Sastry. Steering three-input chained form nonholonomic systems using sinusoids: T h e fire truck example. In Proc. E u r . Contr. Conf., p p . 1432-1437, 1993 To Appear in rntl J . Rob. Res.. B. Charlet, J . Levine, and R. Marino. Sufficient conditions for dynamic s t a t e feedback linearization. S I A M J . Contr. and O p t . , 29(1):38-57, 1991. H . Flanders. Diflerentaal Forms, wrth Appltcations t o the Phystcal Sczences Dover Publications, Mineola, N Y , 1989 M Fliess, J Levine, P Martin, and P. Rouchon. O n differentially flat nonlinear systems. In Proc. IFAC Nonlan. C o n t r . Syst. Desrgn S y m p . ( N O L C O S ) , pp. 408-412, 1992. M Fliess, J Lgivine, P Martin, and P. Rouchon. Sur les s y s t h e s nonlineaires difrientiellenient plats C. R . Acad. Sci. Parts, 315.619624, 1992 M. Fliess, J . IAvine, P. Martin, a n d P. Rouchon Linearisation par bouclage dynamique e t transformations d e Lie-Backlund. C. R . Acad. Scr. Parrs. 317:981-986. 1993. M. Fliess, J Levine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples Intl. J. C o n t r . , 1994 To Appear. R . B . Gardner and'W. F. Shadwick. T h e GS algorithm for exact linearization t o Brunovsky normal form. I E E E Trans. A u t o . C o n t r . , 37(2):224-230, 1992. R 'M. Murray. Applicat~onsand extensions of goursat normal form t o control of nonlinear systems. In Proc. I E E E Conf. Decrsron and Contr , pp 3425-3430, 1993 M. Sampei and K . F u r u t a . On time scaling for nonlinear systems: Application t o linearization. I E E E Trans. A u t o . C o n t r . , AC31(5) 459-462, May 1986. W. F . Shadwick and W M. Sluis. Dynamic feedback for classical geometries. Tech Report FI93-CT23, T h e Fields Inst., Ontario, C a n a d a , 1993. W . M. Sluis. Absolute Equrualence and r t s Appltcations t o Control Theory. P h D thesis, Univ. of Waterloo, 1992. D Tilbury and A. Chelouah. Steering a three-input nonholonomic system using multlrate controls In P r o c . E u r . C o n t r . Conf., pp. 1428-1431, 1993. D . Tilbury, R.Murray, and S . Sastry. Trajectory generation for t h e N-trailer problem using Goursat normal form. In Proc. I E E E Conf. Dectszon and ('ontr., pp. 971-977, 1993. T o a p p e a r in I E E E Trans. Auto.
Contr.
I E E E Trans. Rob. and Auto..
dt
d a 4 = -a2 A a3
J
D Tilbury, 0 Sgrdalen, L. Bushnell, and S. Sastry. A multi-steering trailer system Conversion into chained form using dynamic feedback In Proc IFAC S y m p . Robot C o n t r . , 1994. To A p p e a r in
The s t r u c t u r e e q u a t i o n s a r e fairly s i m p l e t o find, da' = - a 2
c o r r e s p o n d i n g t o a d d i n g an i n t e g r a t o r t o the first input channel. T h e d e r i v e d flag i s n o w
21201'
A
dt
M. van Nieuwstadt, M R a t h i n a m , and R. M. Murray. Differential flatness and absolute equivalence. Tech. Report CIT-CDS 94-006, Caltech, 1994.
