Extended Goursat normal forms with applications to ... - IEEE Xplore
Procmllng8 d the a n d Conference on D d r l o n "Id Control 8.n Antonlo, Toxrr * Dscember 1993
FPI - 5 5 0 Extended Goursat Normal Forms with Applications to Nonholonomic Motion Planning* L. Bushnell, D. Tilbury, S. Sastry Electronics Research Laboratory Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720
Abstract The theme of this paper is the generalization of Goursat normal forms for Pfaffian systems with co-dimension greater than two. There are necessary and sufficient conditions for the existence of coordinates that transform a Pfaffian system with co-dimension greater than two into an extended Goursat normal form, which is the dual of the multiple-chain, single-generator chained form mentioned in our earlier work. In this paper, we concentrate on how to find such coordinate transformations for multi-steering, multitrailer mobile robot systems so that we can use available steering and stabilization algorithms for nonholonomic motion planning. We present a methodology for constructing a coordinate transformation and apply it to the example of a five-axle, two-steering mobile robot.
1
Introduction
We are interested in motion planning for nonholonomic systems with nonintegrable velocity constraints. The constraints can be written as ai = 0, where the a' are one-forms on the configuration space. Classical motion planning problem solutions assume that the system can move in all directions at a point and are inadequate for systems with velocity constraints. In previous work [2, 6, 91, nonholonomic systems were considered not from the point of view of their constraints, but rather from the point of view of a control system with the allowable motions in the span of the input vector fields. In these works we gave sufficient conditions in terms of the vector fields for converting multi-input control systems to a chained form. Once the system was in chained form, the states were easily steered using sinusoidal, piecewise constant or polynomial control inputs. The dual to the chained form, in the sense of one-forms, is the Goursat normal form. Although the mathematics literature abounds with the theory of exterior differential systems (we recommend [l, 4, 131 as a survey), only recently have there been attempts to apply this machinery to solve general control problems in steering nonholonomic systems. In [lo], we used techniques from exterior differential algebra to find a coordinate transformation for the N-trailer system that put the cc-dimension two Pfaffian system into Goursat normal form. In this paper we use exterior differential algebra to convert PfafEan systems of co-dimension greater than two into extended Goursat normal form. We refer to a theorem that states the necessary and sufficient conditions for the existence of a coordinate transformation to extended Goursat normal form. If we cannot constructively find these coordinates, however, it is not useful for nonholonomic motion planning since the steering algorithms use the transformed coordinates. In this instance, we consider the 'This research waa supported in part by NSF under grant IRI-9014490. D.Tilbury would like to acknowledge an AT&T Ph.D. Fellowship for partial support of this work.
0191-2216/93/$3.00 0 1993 IEEE
use of partial prolongation to put the system into Goursat normal form. The outline of the paper is as follows: in Section 2, we present the results from exterior differential algebra that we will need. In Section 3, we present an algorithm for transforming a Pfaffian system with co-dimension greater than two into extended Goursat normal form. In Section 4,we give an example of a five-axle, twosteering mobile robot system. We show a problem that arises with the algorithm for multi-steering, multi-trailer systems and propose a new procedure as a solution. In Section 5, conclusions are made.
2
Background
We now present some of the mathematical tools from exterior differential algebra that are used in this paper. A more thorough presentation can be found in [l]. An n-dimensional Pfaffian system, denoted by I , is the differential system =
=
a1
... -
-0
(1)
7
where the constraints, or one-forms, are the directions in which the system cannot move instantaneously. Our first tool is the derived flag. Define the co-distributions
I @ ) := I = {ao,a', ... ,an-l} I(') := { A E Z(i-1) : dX mod
I(i-1) E
0) ,
for i = 1,. . . , N , where N is the step in which this procedure terminates, i.e., I(N+')= I ( N ) . We now restrict ourselves to the class of systems that have I ( N )= {0}, or systems that are completely controllable. Consider the derived flag with basis { a i }adapted to the derived flag. That is, a basis such that the basis of I(j+l) is a subset of the basis of I ( j ) for j = 0 , . . . ,N - 1. Definition (Towers of a derived flag) Let I be a collection of n = s1 . . . ,s smooth linearly independent one-forms defined on U,
+ +
I = {w{,wi ,... Let the one-form T dui
=
#0
TA
qJ# 0
,4
: j = 1,... , m } .
(2)
mod I be such that for j = 1 , . . . ,m ,
dfl mod I(aJ-k), k = 1, . . . ,s,
modI.
-1
(3)
These congruences imply that the derived flag associated with the system Z has the form 16) = { J
ws,. j
.. , w & ~ : j = 1 , . . . , m }
If i 2 s j , then none of the constraints wJ will appear in the ith derived system. We say that the derived Jag of I has m towers. We will refer to the set of relations ( 3 ) as the eztended Goursat congruences.
3447
We now state the necessary and sufficient conditions for the existence of a coordinate transformation into an extended Goursat normal form for a system with co-dimension greater than two.
Theorem 1 (Extended Goursat Normal Form)[9, 51 Let U be a local subset of W" and Z = { w ~ , w ~. ., .,wiI : j = 1,. . . ,m} be a collectton of smooth lznearly Endependent one-forms defined on U . If there exzsts a szngle one-form a # 0 mod I such that the Goursat congruences (3) are satzsfied, then there exists a set of coordinates on U such that I can be written as:
I = { d t i I + l - z:,dzo,. . . , d z i - t i d z o
: 3 =
1,. . . , m } .
+
i; =
U,
i; =
.;Uo
...
(4) 4,+1
ZZUO
= .:,U0
Sufficient conditions for converting to this form were given in [2];the conditions stated above, however, are both necessary and sufficient.
3
Coordinate Transformation
If the constraints of the system (1) satisfy the Goursat congruences (3), by Theorem 1 we know there exist coordinates for the extended Goursat normal form. We present here a purely algebraic algorithm for finding the coordinates. The algorithm is similar to the one used in [3] for linearization to Brunovsky normal form. For co-dimension two systems, or systems with only one tower, the transformation is straightforward: the generator coordinate, t o and the coordinate for the bottom of the tower, z.,+1 are found from the solution to Pfaff's problem [l]. Then the rest of the coordinates are found through differentiation. This was presented in [lo]. The difficulty with having more than one tower is that we must modify the constraints to decouple the towers so that we can use the solution to Pfaff's problem to find the coordinates for each tower. For the most general case with some of the towers having the same length, the one form a that satisfies the extended Goursat congruences (3) is not necessarily integrable, but can be made so as shown below:
Lemma 2 Assume that there exists a single one-form, a # 0 mod I that satisfies the extended Goursat congruences (9). Then there exists an integrable one-form, denoted a' that also satisfies the extended Goursat congruences. Proof We present the proof for the case when the first tower is the longest for the case of distinct length towers. When some of the towers have the same length, the proof is more involved. Given that there exists a T # 0 which satisfies the extended Goursat congruences (3), we know that the derived flag has the structure: p l - 1 ) p l )
~
=
(41
mod w:,
A w: A w: A
df1 = 0 ,
which gives us that dj1 is linearly dependent on T,w:, and w : , i.e., = ko(€) a
+ h(E)w: + h ( l )w:
,
where E is the total state of the system. We define a' = dj1, and note that any such a' with 4 # 0 will also satisfy the extended Goursat congruences. 0
1. Compute a derived flag for the Pfaffian system (1).
2. Construct m towers (2) using an integrable ?r (use Lemma 2 if needed) such that the Goursat congruences (3) are satisfied. 3. From the Goursat congruence
dw: = a A w:
mod
,
we know the rank of U: is one since dw: A w i # 0 and (dw:)' A w: = 0. Therefore, we can use the solution to Pfaff's problem [l] to compute the coordinates for the first tower of the normal form as follows: (a) Define dj1 := ?r. This satisfies the first Pfaff equation
dwl A w l A dfl = 0 and w: A dfl Find a function w: A
f2
#0.
satisfying the second Pfaff equation
df1 A dfz = 0 and df1 A dj2
#0.
(6)
(b) Define zo := f 1 as the generator coordinate and := fi as the coordinate corresponding to the bottom of the first tower. (c) Since wt satisfies the Pfaff equation (6), we can modify it as = b(E) df2 - a ( ( ) dto
for some smooth functions a ( ( ) and b ( 0 , where ( E W" is the total state of the system. We know b ( [ ) # 0 for if it were, dwi = dzo A d a ( ( ) = 0 mod U ; , which contradicts the Goursat congruences (3). Therefore, we can modify this constraint to give
and define the the coordinate z;, to be a ( < ) / b ( [ ) . (d) Modify w;' for i = 2 , . . ,s1 by using the form constraints in I(al-i) from the Goursat congruence equation (3) to satisfy G!, = dt'sl-i+z - c:,-i+l(€) ~ Z O
.
and define z:l-i+l to be c : , - ~ + ~This . is a purely algebraic step which gives all the coordinates for the first tower. 4. Compute the coordinates for the second tower as follows:
(01 .
We will use the last Goursat congruence:
dw: = a A w:
T
Algorithm 1 (Conversion to Extended Goursat Normal Form)
i;" = um i2" = t;"uo
q"+,=
Substituting for dw: from (5), we have
4 1
The extended Goursat normal form is the dual of the multipleinput, single-generator chained form presented in [2]. For example, the (m 1)-input, single-generator chained form is written as
io = U 0
which implies that w: has rank 1. From the solution to Pfaff's problem, we know there exists a function f 1 satisfying the equation
(5) 3448
(a) The Goursat congruences imply that 91, defined analogously to f l above, may be chosen to be fi since we are using the same generator coordinate zo.
from (b) Modify U : by using the form constraints in the Goursat congruence equation (3) to satisfy the equation Gf = d h ( [ ) - c([) dzo
,
(7)
where h(
FPI - 5 5 0 Extended Goursat Normal Forms with Applications to Nonholonomic Motion Planning* L. Bushnell, D. Tilbury, S. Sastry Electronics Research Laboratory Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720
Abstract The theme of this paper is the generalization of Goursat normal forms for Pfaffian systems with co-dimension greater than two. There are necessary and sufficient conditions for the existence of coordinates that transform a Pfaffian system with co-dimension greater than two into an extended Goursat normal form, which is the dual of the multiple-chain, single-generator chained form mentioned in our earlier work. In this paper, we concentrate on how to find such coordinate transformations for multi-steering, multitrailer mobile robot systems so that we can use available steering and stabilization algorithms for nonholonomic motion planning. We present a methodology for constructing a coordinate transformation and apply it to the example of a five-axle, two-steering mobile robot.
1
Introduction
We are interested in motion planning for nonholonomic systems with nonintegrable velocity constraints. The constraints can be written as ai = 0, where the a' are one-forms on the configuration space. Classical motion planning problem solutions assume that the system can move in all directions at a point and are inadequate for systems with velocity constraints. In previous work [2, 6, 91, nonholonomic systems were considered not from the point of view of their constraints, but rather from the point of view of a control system with the allowable motions in the span of the input vector fields. In these works we gave sufficient conditions in terms of the vector fields for converting multi-input control systems to a chained form. Once the system was in chained form, the states were easily steered using sinusoidal, piecewise constant or polynomial control inputs. The dual to the chained form, in the sense of one-forms, is the Goursat normal form. Although the mathematics literature abounds with the theory of exterior differential systems (we recommend [l, 4, 131 as a survey), only recently have there been attempts to apply this machinery to solve general control problems in steering nonholonomic systems. In [lo], we used techniques from exterior differential algebra to find a coordinate transformation for the N-trailer system that put the cc-dimension two Pfaffian system into Goursat normal form. In this paper we use exterior differential algebra to convert PfafEan systems of co-dimension greater than two into extended Goursat normal form. We refer to a theorem that states the necessary and sufficient conditions for the existence of a coordinate transformation to extended Goursat normal form. If we cannot constructively find these coordinates, however, it is not useful for nonholonomic motion planning since the steering algorithms use the transformed coordinates. In this instance, we consider the 'This research waa supported in part by NSF under grant IRI-9014490. D.Tilbury would like to acknowledge an AT&T Ph.D. Fellowship for partial support of this work.
0191-2216/93/$3.00 0 1993 IEEE
use of partial prolongation to put the system into Goursat normal form. The outline of the paper is as follows: in Section 2, we present the results from exterior differential algebra that we will need. In Section 3, we present an algorithm for transforming a Pfaffian system with co-dimension greater than two into extended Goursat normal form. In Section 4,we give an example of a five-axle, twosteering mobile robot system. We show a problem that arises with the algorithm for multi-steering, multi-trailer systems and propose a new procedure as a solution. In Section 5, conclusions are made.
2
Background
We now present some of the mathematical tools from exterior differential algebra that are used in this paper. A more thorough presentation can be found in [l]. An n-dimensional Pfaffian system, denoted by I , is the differential system =
=
a1
... -
-0
(1)
7
where the constraints, or one-forms, are the directions in which the system cannot move instantaneously. Our first tool is the derived flag. Define the co-distributions
I @ ) := I = {ao,a', ... ,an-l} I(') := { A E Z(i-1) : dX mod
I(i-1) E
0) ,
for i = 1,. . . , N , where N is the step in which this procedure terminates, i.e., I(N+')= I ( N ) . We now restrict ourselves to the class of systems that have I ( N )= {0}, or systems that are completely controllable. Consider the derived flag with basis { a i }adapted to the derived flag. That is, a basis such that the basis of I(j+l) is a subset of the basis of I ( j ) for j = 0 , . . . ,N - 1. Definition (Towers of a derived flag) Let I be a collection of n = s1 . . . ,s smooth linearly independent one-forms defined on U,
+ +
I = {w{,wi ,... Let the one-form T dui
=
#0
TA
qJ# 0
,4
: j = 1,... , m } .
(2)
mod I be such that for j = 1 , . . . ,m ,
dfl mod I(aJ-k), k = 1, . . . ,s,
modI.
-1
(3)
These congruences imply that the derived flag associated with the system Z has the form 16) = { J
ws,. j
.. , w & ~ : j = 1 , . . . , m }
If i 2 s j , then none of the constraints wJ will appear in the ith derived system. We say that the derived Jag of I has m towers. We will refer to the set of relations ( 3 ) as the eztended Goursat congruences.
3447
We now state the necessary and sufficient conditions for the existence of a coordinate transformation into an extended Goursat normal form for a system with co-dimension greater than two.
Theorem 1 (Extended Goursat Normal Form)[9, 51 Let U be a local subset of W" and Z = { w ~ , w ~. ., .,wiI : j = 1,. . . ,m} be a collectton of smooth lznearly Endependent one-forms defined on U . If there exzsts a szngle one-form a # 0 mod I such that the Goursat congruences (3) are satzsfied, then there exists a set of coordinates on U such that I can be written as:
I = { d t i I + l - z:,dzo,. . . , d z i - t i d z o
: 3 =
1,. . . , m } .
+
i; =
U,
i; =
.;Uo
...
(4) 4,+1
ZZUO
= .:,U0
Sufficient conditions for converting to this form were given in [2];the conditions stated above, however, are both necessary and sufficient.
3
Coordinate Transformation
If the constraints of the system (1) satisfy the Goursat congruences (3), by Theorem 1 we know there exist coordinates for the extended Goursat normal form. We present here a purely algebraic algorithm for finding the coordinates. The algorithm is similar to the one used in [3] for linearization to Brunovsky normal form. For co-dimension two systems, or systems with only one tower, the transformation is straightforward: the generator coordinate, t o and the coordinate for the bottom of the tower, z.,+1 are found from the solution to Pfaff's problem [l]. Then the rest of the coordinates are found through differentiation. This was presented in [lo]. The difficulty with having more than one tower is that we must modify the constraints to decouple the towers so that we can use the solution to Pfaff's problem to find the coordinates for each tower. For the most general case with some of the towers having the same length, the one form a that satisfies the extended Goursat congruences (3) is not necessarily integrable, but can be made so as shown below:
Lemma 2 Assume that there exists a single one-form, a # 0 mod I that satisfies the extended Goursat congruences (9). Then there exists an integrable one-form, denoted a' that also satisfies the extended Goursat congruences. Proof We present the proof for the case when the first tower is the longest for the case of distinct length towers. When some of the towers have the same length, the proof is more involved. Given that there exists a T # 0 which satisfies the extended Goursat congruences (3), we know that the derived flag has the structure: p l - 1 ) p l )
~
=
(41
mod w:,
A w: A w: A
df1 = 0 ,
which gives us that dj1 is linearly dependent on T,w:, and w : , i.e., = ko(€) a
+ h(E)w: + h ( l )w:
,
where E is the total state of the system. We define a' = dj1, and note that any such a' with 4 # 0 will also satisfy the extended Goursat congruences. 0
1. Compute a derived flag for the Pfaffian system (1).
2. Construct m towers (2) using an integrable ?r (use Lemma 2 if needed) such that the Goursat congruences (3) are satisfied. 3. From the Goursat congruence
dw: = a A w:
mod
,
we know the rank of U: is one since dw: A w i # 0 and (dw:)' A w: = 0. Therefore, we can use the solution to Pfaff's problem [l] to compute the coordinates for the first tower of the normal form as follows: (a) Define dj1 := ?r. This satisfies the first Pfaff equation
dwl A w l A dfl = 0 and w: A dfl Find a function w: A
f2
#0.
satisfying the second Pfaff equation
df1 A dfz = 0 and df1 A dj2
#0.
(6)
(b) Define zo := f 1 as the generator coordinate and := fi as the coordinate corresponding to the bottom of the first tower. (c) Since wt satisfies the Pfaff equation (6), we can modify it as = b(E) df2 - a ( ( ) dto
for some smooth functions a ( ( ) and b ( 0 , where ( E W" is the total state of the system. We know b ( [ ) # 0 for if it were, dwi = dzo A d a ( ( ) = 0 mod U ; , which contradicts the Goursat congruences (3). Therefore, we can modify this constraint to give
and define the the coordinate z;, to be a ( < ) / b ( [ ) . (d) Modify w;' for i = 2 , . . ,s1 by using the form constraints in I(al-i) from the Goursat congruence equation (3) to satisfy G!, = dt'sl-i+z - c:,-i+l(€) ~ Z O
.
and define z:l-i+l to be c : , - ~ + ~This . is a purely algebraic step which gives all the coordinates for the first tower. 4. Compute the coordinates for the second tower as follows:
(01 .
We will use the last Goursat congruence:
dw: = a A w:
T
Algorithm 1 (Conversion to Extended Goursat Normal Form)
i;" = um i2" = t;"uo
q"+,=
Substituting for dw: from (5), we have
4 1
The extended Goursat normal form is the dual of the multipleinput, single-generator chained form presented in [2]. For example, the (m 1)-input, single-generator chained form is written as
io = U 0
which implies that w: has rank 1. From the solution to Pfaff's problem, we know there exists a function f 1 satisfying the equation
(5) 3448
(a) The Goursat congruences imply that 91, defined analogously to f l above, may be chosen to be fi since we are using the same generator coordinate zo.
from (b) Modify U : by using the form constraints in the Goursat congruence equation (3) to satisfy the equation Gf = d h ( [ ) - c([) dzo
,
(7)
where h(
