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Configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom
Systems & Control Letters 61 (2012) 334–342
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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom Kazuhiro Sato ∗ , Toshihiro Iwai Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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Article history: Received 29 November 2010 Received in revised form 27 July 2011 Accepted 14 November 2011 Available online 5 January 2012 Keywords: Configuration flatness Lagrangian control systems Riemannian geometry
abstract A Lagrangian control system is said to be configuration flat if its states are determined by configuration flat outputs depending on configuration variables only. This paper gives a necessary condition and two types of sufficient conditions for Lagrangian control systems with n degrees of freedom and m controls with m < n to be configuration flat, and further makes a remark on the choice of output variables for configuration flatness. Two examples are given in which the sufficient conditions are utilized for finding configuration flat outputs. © 2012 Published by Elsevier B.V.
1. Introduction Fliess et al. were the first who defined the concept of differentially flat systems [1,2]. If a system is differentially flat, the states are completely determined by specific functions called flat outputs. In addition, if a system is differentially flat, it is linearizable [3,4]. From these points of view, a number of studies have been made on tracking control of differentially flat systems [3,5–11]. Some necessary conditions and sufficient conditions for flatness have been given in previous studies [12,6,13,14,9,15,16]. In particular, Rathinam and Murray [16] defined configuration flatness for Lagrangian control systems and gave a necessary and sufficient condition for a Lagrangian control system with n degrees of freedom and n − 1 controls to be configuration flat. In other words, they provided a complete characterization for configuration flatness in the case of n − 1 controls. Roughly speaking, if a system is configuration flat, there exists as many functions on the configuration space, called configuration flat outputs, as controls such that the system’s behavior is determined by them. The aim of this paper is to extend their result [16]. This paper studies Lagrangian control systems with a Lagrangian given by L(q, q˙ ) =
∗
1 2
g (˙q, q˙ ) − V (q),
Corresponding author. E-mail address: [email protected] (K. Sato).
0167-6911/$ – see front matter © 2012 Published by Elsevier B.V. doi:10.1016/j.sysconle.2011.11.007
where q denote a point of the configuration space, g is a Riemannian metric on the configuration space, V is a potential energy function. As long as mechanical systems are concerned, the Lagrangian control systems of this type cover many of mechanical systems with control, and some of them are known to be configuration flat [9,17,10,16]. Further, for the mechanical systems of this type, Riemannian geometry is known to be effectively used in the study of control [18,19,16]. In fact, Rathinam and Murray [16] have shown how to find configuration flat outputs using concepts in Riemannian geometry such as Riemannian metric and Levi-Civitá connection. However, their results leave the door open for extension. The purpose of this paper is to put forward conditions for a Lagrangian control system with fewer controls than the degrees of freedom to be configuration flat. Though Lévine gave a necessary and sufficient condition for an implicit system to be differentially flat [6,13], the condition is too general to be applied for finding flat outputs in mechanical systems with external force as control. Though this paper’s results can apply to mechanical control systems with fewer controls than degrees of freedom, unfortunately the characterization of configuration flatness is still incomplete. Nevertheless, the sufficient conditions obtained in this paper provide a way to find configuration flat outputs. This paper is organized as follows. Section 2 is a brief review of concepts from Lagrangian control systems theory and also provides the definition of configuration flatness. Section 3 starts with the introduction of Levi-Civitá connections from Riemannian geometry, and proceeds to give a necessary condition and a sufficient condition for configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom. In addition, this paper gives a sufficient condition for output variables not to be configuration flat. Section 4 gives another sufficient condition
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
for configuration flatness of Lagrangian control systems with n degrees of freedom and n − 2 controls. Finally, in Section 5, the present results are applied to a quadrotor unmanned aerial vehicle and a mass spring system to find configuration flat outputs. In addition, a simple example is given which satisfies the necessary condition but does not satisfy the sufficient conditions for configuration flatness.
Definition 2. Suppose y1 , . . . , ym are differentially independent around q ∈ Q . Then y1 , . . . , ym are said to be configuration flat outputs around q ∈ Q if there exist functions z 1 , . . . , z n−m such that ( y1 , . . . , ym , z 1 , . . . , z n−m ) is a local coordinate system of Q , and further if there exist functions fk and positive integers rk such that
d drk (z k ◦ c )(t ) = fk ( y ◦ c )(t ), ( y ◦ c )(t ), . . . , r ( y ◦ c )(t ) , dt
2. Lagrangian control systems and configuration flatness
dt
∂L ∂ q˙ i
∂L − i = ua Fia , ∂q
i = 1, . . . , n,
(1)
where Fia is a tensor field and the control ua Fia dqi is viewed as an element in the cotangent bundle T ∗ Q . This is because forces are naturally paired with velocities to give instantaneous power. Here and in the sequel, we adopt the Einstein summation convention: An index occurring twice in a product is to be summed from 1 up to the last number. We assume that the external control forces determine a codistribution
F := span Fi1 dqi , . . . , Fim dqi ⊂ T ∗ Q
of rank m. We call a triple (TQ , L, F) a Lagrangian control system. The annihilator of F is expressed as
F⊥ = span aik
∂ , k = 1, . . . , n − m , ∂ qi
(2)
where rank(aik ) = n − m. From the definition of the annihilator, all feasible solutions of (1) have to satisfy the following equations: aik
d dt
∂L ∂ q˙ i
−
∂L ∂ qi
= 0,
k = 1, . . . , n − m.
(3)
We introduce the jet space J 2 (R , Q ) of the second-order to describe the notion of configuration flatness [20]. In view of (3), we define a submanifold E ⊂ J 2 (R , Q ), in terms of local coordinates (t , q1 , . . . , qn , q˙ 1 , . . . , q˙ n , q¨ 1 , . . . , q¨ n ) of J 2 (R , Q ), to be
(t , q1 , . . . , qn , q˙ 1 , . . . , q˙ n , q¨ 1 , . . . , q¨ n ) ∈ J 2 (R , Q ) 2 ∂ 2L j ∂L ∂ L j ¨ ˙ = 0, aik q + q − ∂ q˙ i ∂ q˙ j ∂ q˙ i ∂ qj ∂ qi k = 1, . . . , n − m .
k
(5)
where c : R → Q is a generic solution of (1). Definition 3. A Lagrangian control system (TQ , L, F) is said to be configuration flat around q ∈ Q if it admits configuration flat outputs around q ∈ Q . We now make a remark on the relation of configuration flatness to differential flatness. We assume that the Lagrangian control system (1) has a nonsingular matrix
∂2L ∂ q˙ i ∂ q˙ j
1≤i,j≤n
. If
this control system is configuration flat, then it can be described as a differential flat system. We recall that a system x˙ = f (x, u) with states x and control inputs u = (u1 , . . . , um ) is differentially flat if andonly if there exist functions α, β and (r )
(rm ) ,u = γ such that x = α y˜ 1 , . . . , y˜ 1 1 , . . . , y˜ m , . . . , y˜ m (r1 +1) (rm +1) , . . . , y˜ m , . . . , y˜ m and y˜ = γ x, u1 , . . . , β y˜ 1 , . . . , y˜ 1 (l ) (l ) u1 1 , . . . , um , . . . , umm , where y˜ := ( y˜ 1 , . . . , y˜ m ) are called flat
outputs [1,3,6,2,21]. We now put the Lagrangian system (1) in the form of first order differential equations. Let (Aij )1≤i,j≤n :=
∂2L ∂ q˙ i ∂ q˙ j
−1
1≤i,j≤n
. Then, introducing velocity variables v i , we can bring
Eq. (1) into dqi dt dv i dt
= vi , = Aij
i = 1, . . . , n,
Since (Fia )
∂ 2L k ∂L a − v + u F . a j ∂ qj ∂v j ∂ qk
1≤i≤n 1≤a≤m
of (1) is of rank m at each point on Q , there exists
the Moore–Penrose pseudoinverse (Fa+ i )
1≤i≤n 1≤a≤m
of (Fia )
1≤i≤n 1≤a≤m
at each
point on Q [22]. Then, from (1) the control inputs ua , 1 ≤ a ≤ m, can be expressed as
ua =
E :=
dt
k = 1, . . . , n − m,
This section is a brief review from [16]. Let Q and TQ denote a configuration space of dimension n and the tangent bundle over Q , respectively. Let (q1 , . . . , qn , q˙ 1 , . . . , q˙ n ) be local coordinates of TQ and L : TQ → R be a Lagrangian. When external control forces u1 , . . . , um are applied to the Lagrangian system, the equations of motion are described as d
335
∂ 2 L dv j ∂ 2L k ∂L + v − i ∂v i ∂v j dt ∂v i ∂ qk ∂q
Fa+ i .
Hence, if the Lagrangian control system (1) with
∂2L ∂ q˙ i ∂ q˙ j
1≤i,j≤n
dy1 , . . . , dym , dy˙ 1 , . . . , dy˙ m , dy¨ 1 , . . . , dy¨ m
being nonsingular is configuration flat, it has configuration flat outputs, and thereby the state variables (q1 , . . . , qn , v 1 , . . . , v n ) along with the control variables (ua ) can take the form referred to in the definition of differential flatness. If a system is differentially flat, then the system is endogenous dynamic feedback linearizable [3,6,9,21] and locally reachable [6]. Therefore, if a Lagrangian control system is configuration flat, the system has these properties. Nevertheless, a question as to whether a Lagrangian control system (TQ , L, F) is configuration flat or not is non-trivial. Hence, our interest will center on how to find configuration flat outputs. The following lemma on configuration flatness [16] will be of use later.
are linearly independent on π2−1 (U ) ∩ E, where U is an open neighborhood of q ∈ Q and where π2 : J 2 (R , Q ) → Q is the standard projection.
Lemma 1. Let y1 , . . . , ym be configuration flat outputs for (1). Then the set of solutions c : R → Q to (1) that project down to the same curve y ◦ c are all isolated.
(4)
Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion locally defined around q ∈ Q . Then, the configuration flatness is defined as follows [16]. Definition 1. Functions y1 , . . . , ym are called differentially independent around q ∈ Q if
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3. Lagrangian control systems with n degrees of freedom and m controls Suppose that m < n. Let Q be a Riemannian manifold endowed with a Riemannian metric g. Consider a mechanical system (TQ , L) with the Lagrangian given by L(q, q˙ ) =
1
g (˙q, q˙ ) − V (q), 2 where V : Q → R is a potential energy function on Q .
(6)
3.2. Conditions for configuration flatness So far we have introduced necessary concepts from Riemannian geometry. We now proceed to work with configuration flatness of our Lagrangian control system. Given a smooth submersion y : Q → R m , by adding complementary functions z r , we can choose local coordinates (q1 , . . . , qn ) on Q such that
3.1. A brief review of Riemannian geometry Before proceeding further, we make a brief review of connections in Riemannian geometry [23,18].
qa = ya , a = 1, . . . , m, qr +m = z r , r = 1, . . . , n − m.
From now on, given a smooth submersion y : Q → R m , we choose local coordinates like (13). Now, let y : Q → R m be a smooth submersion locally defined around q ∈ Q . Then we can put F⊥ and Ker y∗ in the form
∂ ∂ i , . . . , ξ := a , n − m n −m ∂ qi ∂ qi
Definition 4. Let X(Q ) denote the set of all smooth vector fields on Q and C ∞ (Q ) the set of all smooth functions on Q . A linear connection ∇ on Q is a map
F⊥ = span ξ1 := ai1
∇ : (X , Y ) → ∇X Y ∈ X(Q ),
Ker y∗ = span η1 := br1
X , Y ∈ X(Q ),
which satisfies the following properties:
∇X Y is R-bilinear in X and Y , ∇fX Y = f ∇X Y ,
(7)
f ∈ C ∞ (Q ),
∇X ( fY ) = (Xf )Y + f ∇X Y ,
(8)
f ∈ C (Q ). ∞
(9)
The ∇X Y is called the covariant derivative of Y along X . 3
If ∇ is given on Q , the n smooth functions
Γjki
are determined
∇
∂ ∂ qj
∂ ∂ = Γjki i . ∂ qk ∂q
The Γjki are called the Christoffel symbols of the linear connection. Claim 1. Let Q be a Riemannian manifold endowed with a Riemannian metric g. Then there is a unique linear connection ∇ on Q such that (10) (11)
where X , Y , Z ∈ X(Q ), and [X , Y ] := XY − YX . The ∇ is called the Levi-Civitá connection. In the following, we denote by ∇ the Levi-Civitá connection. Then, from the properties (10), Γjki is symmetric; Γjki = Γkji . In addition, owing to (11), Γjki are expressed in terms of the Riemannian metric g as
Γjkl =
1 2
∂ gik ∂ gij ∂ gjk + k − i ∂ qj ∂q ∂q
g il ,
j, l = 1, . . . , n,
(12)
where g ik gkj = δji with (g ik ) = (gik )−1 . Further, we introduce the wedge product of vector fields and the inner product of them [24], which will make it compact to describe a necessary condition for configuration flatness. Definition 5. For vector fields Xa and Ya , a = 1, . . . , n − m, the inner product of X1 ∧ · · · ∧ Xn−m and Y1 ∧ · · · ∧ Yn−m is defined to be G(X1 ∧ · · · ∧ Xn−m , Y1 ∧ · · · ∧ Yn−m )
:= det
g (X1 , Y1 )
···
g (Xn−m , Y1 )
···
.. .
g (X1 , Yn−m )
.. . . g (Xn−m , Yn−m )
∂ ∂ , . . . , ηn−m := brn−m r ∂ zr ∂z
(14)
,
(15)
where y∗ : TQ → T R m is the differential map of y, and where rank(brs ) = n − m. For basis vectors of F⊥ and Ker y∗ , Definition 5 provides the following. Lemma 2. As for (14) and (15), the following two conditions are equivalent;
n
∇X Y − ∇Y X = [X , Y ], X (g (Y , Z )) = g (∇X Y , Z ) + g (Y , ∇X Z ),
∂ ∂ = 0, G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ ∂z ∂ z n −m G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m ) = 0.
in terms of local coordinates (q , . . . , q ) through 1
(13)
Proof. For the vector fields ξ1 , . . . , ξn−m , and η1 , . . . , ηn−m given in (14) and (15), respectively, we have G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m )
∂ ∂ = det brs 1≤r ,s≤n−m G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ n−m . ∂z ∂z Since η1 , . . . , ηn−m are basis vectors in Ker y∗ , one has det brs 1≤r ,s≤n−m ̸= 0. This ends the proof. We return to our key equation (3). In terms of local coordinates
(qi , q˙ i ) of TQ , the Lagrangian (6) is expressed as L(q, q˙ ) := 12 gij q˙ i q˙ j − V (q). On account of (13), Eq. (3) is rewritten as ∂V l + gi,m+r z¨ r Wk := aik gia y¨ a + Γab gli y˙ a y˙ b + ∂ qi l a r l r1 r2 + 2Γa,m+r gli y˙ z˙ + Γm+r1 ,m+r2 gli z˙ z˙ = 0, r1 , r2 , k = 1, . . . , n − m.
(16)
We are now in a position to describe a necessary condition for Lagrangian control systems to be configuration flat. Theorem 1. Let (TQ , L, F) be a Lagrangian control system, where L is a Lagrangian defined on the tangent bundle TQ over a Riemannian manifold Q , and where F is the space of control forces of rank m. Suppose that y1 , . . . , ym are configuration flat outputs around q ∈ Q . Then, for F⊥ and Ker y∗ given in (14) and (15), respectively, one obtains G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m ) = 0.
(17)
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
Proof. We put (16) in the form
gi,m+1 ai1
···
gin ai1
1
z¨ . .. i i z¨ n−m gi,m+1 an−m · · · gin an−m f1 ( y, y˙ , y¨ , z , z˙ ) .. = (18) , . fn−m ( y, y˙ , y¨ , z , z˙ ) l gli y˙ a y˙ b + ∂∂qVi + where fk ( y, y˙ , y¨ , z , z˙ ) := −aik gia y¨ a + Γab 2Γal,m+r gli y˙ a z˙ r + Γml +r1 ,m+r2 gli z˙ r1 z˙ r2 . If G(ξ1 ∧ · · · ∧ ξn−m , ∂∂z 1
.. .
∧ ··· ∧
.. .
∂ ) ∂ z n−m
̸= 0, i.e., if the determinant of the coefficient matrix of the left-hand side of (18) does not vanish, then (18) can be brought into z1 d2 . .. dt 2 z n −m
gi,m+1 ai1
=
gin ai1
···
.. .
gin ain−m
···
−1
f1 ( y, y˙ , y¨ , z , z˙ )
.. . . fn−m ( y, y˙ , y¨ , z , z˙ )
This equation implies that given any curve y(t ) we get a 2(n − m)-parameter family of solutions q(t ) = ( y(t ), z (t )) that project to y(t ). Therefore, Lemma 1 shows that y1 , . . . , ym are not configuration flat outputs. This contradicts the assumption. Hence
∂ ∂ G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ = 0. ∂z ∂ z n −m This and Lemma 2 result in (17).
+r ∂ , k = 1, . . . , n − m, so that the form ξk = am k ∂ zr
∂ ∂ ∧ · · · ∧ n −m 1 ∂z ∂z
∆n−m
g
ξ1 ,
∂ ∂ z1
···
.. := . ∂ g ξn−m , 1 ∂z
···
g
ξ1 ,
∂
∂ z n −m .. . ∂ g ξn−m , n−m ∂z
(19)
Assumption 1. Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion and y1 , . . . , ym be differentially independent around q ∈ Q . For ξ1 , . . . , ξn−m ∈ F⊥ , the following conditions hold,
∆n−m = 0, ∂ = 0, g ξk1 , ∇X k ∂z 2
(20)
∀X ∈ X(Q ), k1 , k2 = 1, . . . , n − m. (21)
As is easily verified, (20) and (21) are equivalent to
= 0,
k1 , k2 = 1, . . . , n − m, j = 1, . . . , n,
(22) (23)
Since ξ1 , . . . , ξn−m are basis vectors in F⊥ , one has det
+r am ̸= 0, the factor G ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m , ∂∂z 1 ∧ · · · ∧ k 1≤k, r ≤n−m
in the right-hand side of the above equation does not van-
ish, either. As a matter of fact, since (gij ) is positive definite matrix, we have ζ i ζ j gij > 0 for any non-zero ζ = ζ i ∂∂qi ∈ X(Q ). In
particular, for any ζ with ζ = 0, a = 1, . . . , m, and ζ ̸= 0, r = 1, . . . , n − m, we obtain ζ r1 +m ζ r2 +m gr1 +m,r2 +m > 0, so r +m
a
that G ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m , ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m
Wk = aik
l gia y¨ a + Γab gli y˙ a y˙ b +
∂V ∂ qi
= 0,
k = 1, . . . , n − m.
(24)
Let
Jn−m
∂W 1 ∂ z1 . := .. ∂ Wn−m ∂ z1
···
···
∂ W1 ∂ z n−m .. . . ∂ Wn−m ∂ z n−m
(25)
We are now in a position to describe a sufficient condition for Lagrangian control systems to be configuration flat.
+r = det am k 1≤k, r ≤n−m ∂ ∂ ∂ ∂ G ∧ · · · ∧ , ∧ · · · ∧ . ∂ z1 ∂ z n −m ∂ z 1 ∂ z n−m
respectively. Then, Eq. (16) is brought into
Proof. Since F = span dy1 , . . . , dym , the ξk given in (14) take
∂ ∂ z n−m
We note that if the codistribution F spanned by the control 1 m forces is integrable, then there exist functions y , . . . , y such that F is expressed as F = span dy1 , . . . , dym . Lemma 2 and Theorem 1 mean that the matrix
aik1 Γj,l m+k2 gli
Theorem 2. Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion and y1 , . . . , ym be differentially independent around q ∈ Q . If F = span dy1 , . . . , dym , then y1 , . . . , ym are not configuration flat outputs around q ∈ Q .
G ξ1 ∧ · · · ∧ ξn−m ,
It follows from Lemma 2 and Theorem 1 that y1 , . . . , ym are not configuration flat outputs around q ∈ Q .
aik1 gi,m+k2 = 0,
As an immediate consequence of Theorem 1, the choice of output variables for which (17) does not hold is inadequate for configuration flatness. In fact, we obtain the following theorem.
̸= 0 with r1 , r2 = 1, . . . , n − m. Hence, ∂ ∂ G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ ̸= 0. ∂z ∂ z n−m
is of rank less than n − m. To treat sufficient conditions for configuration flatness, we start by setting a simple assumption on ∆n−m .
.. .
gi,m+1 ain−m
337
= det gr1 +m,r2 +m
Theorem 3. Suppose that Assumption 1 holds and rank Jn−m = n − m on π2−1 (U )∩ E, where U is a neighborhood of q ∈ Q and E is defined by (4). Then, y1 , . . . , ym are configuration flat outputs around q ∈ Q , so that the Lagrangian control system (TQ , L, F) is configuration flat. Proof. Note that Assumption 1 implies (24). If rank Jn−m = n − m on π2−1 (U ) ∩ E, the implicit function theorem implies that there exist functions h1 , . . . , hn−m such that z 1 = h1 ( y, y˙ , y¨ ), . . . , z n−m = hn−m ( y, y˙ , y¨ ). This means that y1 , . . . , ym are configuration flat outputs around q ∈ Q. We have to note that if m = n − 1, a necessary and sufficient condition is already known for configuration flatness:
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K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
Proposition 1 ([16]). Let y = ( y1 , . . . , yn−1 ) : Q → R n−1 be a smooth submersion and y1 , . . . , yn−1 be differentially independent around q ∈ Q . If y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q , then
g
ξ1 ,
∂ ∂ z1
= g ∇X ξ1 ,
∂ ∂ z1
= 0,
(26)
where for F⊥ and Ker y∗ given in (14) and (15), respectively, ξ1 ∈ F⊥ and ∂∂z 1 ∈ Ker y∗ .
Conversely, if (26) is satisfied, y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q .
We now remark how the theorems in the present article are related to this proposition. If m equals n − 1, and if y1 , . . . , yn−1 are differentially independent around q ∈ Q , then Jn−m becomes J1 and is necessarily of rank 1 on π2−1 (U ) ∩ E, where U is a neighborhood of q ∈ Q and E is defined by (4). Hence, Assumption 1 reduces to g ξ1 , ∂∂z 1
= g ξ1 , ∇X ∂∂z 1 = 0. ∂ However, the latter condition can be expressed as g ξ1 , ∇X ∂ z 1 = −g ∇X ξ1 , ∂∂z 1 = 0. Thus, we find that Assumption 1 reduces
to (26), and that Theorem 3 reduces to the ‘‘converse’’ part of Proposition 1. Conversely, if y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q , then Theorem 1 provides the condition g ξ1 , ∂∂z 1
= 0, one of equations in (26), but does not refer to the condition g ∇X ξ1 , ∂∂z 1 = 0. However, we note that if
y1, . . . , yn−1 are configuration flat outputs around q ∈ Q , then ∂ ∂ g ∇X ξ1 , ∂ z 1 = 0 can be derived from g ξ1 , ∂ z 1 = 0, as is shown in [16]. Hence, if m = n − 1, the present theorems could reproduce Proposition 1 obtained in [16]. 4. Mechanical systems with n degrees of freedom and n − 2 controls To study further a sufficient condition for configuration flatness, we move a step by relaxing the condition (20) but setting m = n − 2. As is already stated, Lemma 2 and Theorem 1 mean that det ∆2 = 0 for m = n − 2, so that rank ∆2 = 0 or 1
on π2−1 (U ) ∩ E .
(27)
Suppose that rank ∆2 = 0 on π2 (U ) ∩ E and that Eq. (21) holds. If rank J2 = 2 on π2−1 (U ) ∩ E, then Assumption 1 holds with m = n − 2, so that Theorem 3 implies that y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q . We turn to the case of rank ∆2 = 1 on π2−1 (U ) ∩ E. For the sake of convenience, we treat a simple case such that ∆2 is of the form
Assumption 2. Let y = ( y1 , . . . , yn−2 ) : Q → R n−2 be a smooth submersion and y1 , . . . , yn−2 be differentially independent around q ∈ Q . For ξ1 , ξ2 ∈ F⊥ , the following condition holds,
∂ ∂ ∂ g ξ1 , 1 g ξ1 , 2 g ξ1 , 1 0 ∂ z ∂ z ̸= 0, (31) = ∂z ∂ ∂ 0 0 g ξ2 , 2 g ξ2 , 1 ∂z ∂z ∂ g ξk1 , ∇X k = 0, ∀X ∈ X(Q ), k1 , k2 = 1, 2. (32) ∂z 2 We are now in a position to describe another sufficient condition for Lagrangian control systems to be configuration flat. Theorem 4. Suppose that Assumption 2 holds and that
∂ W1 ∂ W2 , ̸= 0, ∂ z2 ∂ z1
∂ W2 = 0 on π2−1 (U ) ∩ E . ∂ z2
Then y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q , so that the Lagrangian control system (TQ , L, F) is configuration flat. Proof. From Assumption 2, W1 and W2 are expressed as (29) and ∂W (30), respectively. If ∂ z 22 = 0 on π2−1 (U ) ∩ E , W2 is independent ∂W
of z 2 . Further, if ∂ z 12 ̸= 0 on π2−1 (U ) ∩ E, by the implicit function theorem there exists a function h1 such that z 1 = h1 ( y, y˙ , y¨ ).
(33)
Substituting this equation into (29), we obtain the equation W1 = ∂W 0 with z 1 replaced by (33). From ∂ z 21 ̸= 0 on π2−1 (U ) ∩ E, it follows that there exists a function h2 such that z 2 = h2 ( y, y˙ , y¨ , y(3) , y(4) ).
(34)
Eqs. (33) and (34) mean that y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q . Though we have confined ourselves to m = n − 2, a similar argument would be possible for m < n − 2 to obtain further sufficient conditions. 5. Examples
−1
∂ g ξ1 , 1 ∆2 = ∂z 0
0
̸= 0.
(28)
0
Needless to say, a similar discussion will run in parallel in the case where only one of entries of ∆2 does not vanish and the others vanish. From (21) or (23) along with (28), Eq. (16) with m = n − 2 proves to be expressed as
∂V + gi,n−1 z¨ 1 ∂ qi ∂V l W2 = ai2 gia y¨ a + Γab gli y˙ a y˙ b + = 0. ∂ qi W1 = ai1
l gia y¨ a + Γab gli y˙ a y˙ b +
= 0,
We now replace Assumption 1 by the following.
(29)
(30)
5.1. Quadrotor UAV In this subsection, we apply Theorem 3 to a quadrotor unmanned aerial vehicle (UAV) [25–28], which is a Lagrangian control system with six degrees of freedom and four controls, as is shown in Fig. 1. Previous works [26,27] claimed that the system treated in [25] is (configuration) flat under certain conditions. However, Kim et al. have pointed out that the setting up of the system lacks in mathematical rigorousness, and given an accurate Lagrangian control system [28]. Following their setting up, we show that Theorem 3 provides a way to find configuration flat outputs. We regard the quadrotor UAV as a rigid body, whose configuration space is R 3 × SO(3) [29]. Let (X , Y , Z , φ, θ , ψ) be local coordinates of R 3 × SO(3), where (X , Y , Z ) denotes the position of the center of gravity of the quadrotor UAV, and φ, θ and ψ denote the roll, pitch and yaw angles of UAV in an inertial frame, respectively [28]. The Lagrangian of this system L : T (R 3 × SO(3)) → R is given by L :=
1 2
m(X˙ 2 + Y˙ 2 + Z˙ 2 ) +
1 2
ω T I ω − mγ Z ,
(35)
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
By definition, the coefficients aik should satisfy the following equations,
f2 M2
f3
a1k (cos φ sin θ cos ψ + sin φ sin ψ) f1
+ a2k (cos φ sin θ sin ψ − sin φ cos ψ) + a3k cos φ cos θ = 0, (40) 4 ak 1 0 − sin θ 0 cos φ sin φ cos θ a5k = 0. (41) 0 − sin φ cos φ cos θ a6k
M3 Z
f4
M1 mγ
ψ
M4 φ
θ
We first consider Eq. (41). If the pitch angle θ does not equal (2n + 1) π2 with n being an arbitrary integer, the coefficient matrix of (41) is nonsingular, so that we obtain
Y
X
a4k = a5k = a6k = 0,
Fig. 1. Quadrotor UAV.
I :=
I1 0 0
0 I2 0
0 0 I3
a11 = cos θ cos ψ,
a12 = 0,
is the inertia matrix of UAV in the body frame, and γ is the gravitational acceleration. Further, ω in (35) denotes the angular velocity of the vehicle in the body frame [29], and is expressed as 1
0 cos φ − sin φ
ω= 0
0
1
= span ξ1 := cos θ cos ψ
(36)
2
g (˙q, q˙ ) − mγ Z ,
where the Riemannian metric g is given in Box I. The Lagrangian control system of the quadrotor UAV [28] is then subject to the equations of motion d dt
∂L ∂ q˙ i
j Fi
where
4 ∂L j − i = uj Fi (q), ∂q j=1
i = 1, . . . , 6,
(37)
(q) is given in Box II.
Here, u1 is the total thrust produced by the four rotors Mi , i = 1, . . . , 4, that is, it is given by u1 := f1 + f2 + f3 + f4 , where fi := ki wi2 is the thrust generated by Mi and ki > 0 is a constant, and wi is the angular speed of Mi . The control inputs u2 , u3 , and u4 are the generalized moments; they are given by u2 := ( f3 − f1 )d, u3 := ( f2 − f4 )d and u4 := ( f2 + f4 − f1 − f3 )κ , where d represents the distance from each Mi to the center of gravity of the quadrotor UAV and κ is a constant. Then, the control inputs determine the following codistribution:
F := span (cos φ sin θ cos ψ + sin φ sin ψ)dX
+ (cos φ sin θ sin ψ − sin φ cos ψ)dY + (cos φ cos θ )dZ , dφ − sin θ dψ, cos φ dθ + sin φ cos θ dψ, − sin φ dθ + cos φ cos θ dψ . (38) To find the annihilator of F, we put it in the form
ξ2 := − cos φ cos θ
∂ ∂ ∂ + cos θ sin ψ − sin θ , ∂X ∂Y ∂Z
∂ ∂Y
+ (cos φ sin θ sin ψ − sin φ cos ψ)
∂ . ∂Z
(42)
To find configuration flat outputs y1 , y2 , y3 , and y4 with the help of Theorem 3, we need to calculate the coefficients of the LeviCivitá connection. A straightforward calculation shows that the covariant derivatives of basis vectors ∂∂qi take the form
∇
∂
∂ qs1
∂ ∂
qs2
∂ ∂ + Γs51 s2 5 4 ∂q ∂q ∂ + Γs61 s2 6 , s1 , s2 = 1, . . . , 6, ∂q
= Γs41 s2
(43)
where (q1 , q2 , q3 , q4 , q5 , q6 ) = (X , Y , Z , φ, θ , ψ). In view of (13), we wish to find local coordinates y1 , . . . , y4 and z 1 , z 2 for which Assumption 1 holds. If we can perform a coordinate transformation of the form z 1 = z 1 (φ, ψ), z 2 = z 2 (φ, ψ), y1 = y1 (q), . . . , y4 = y4 (q), we see from the Riemannian metric g that ∆2 given in (19) with n = 6 and m = 4 vanishes, and further from (42) and (43) we verify that Eq. (21) holds. In addition, if there are differentially independent functions ya = ya (X , Y , Z , θ ), a = 1, . . . , 4, then ∂ , ∂ are shown to be in Ker y∗ , and ( y1 , y2 , y3 , y4 , z 1 , z 2 ) are ∂ z1 ∂ z2 local coordinates in the configuration space R 3 × SO(3). The easiest choice of ( y1 , y2 , y3 , y4 , z 1 , z 2 ) may be ( y1 , y2 , y3 , y4 , z 1 , z 2 ) := (X , Y , Z , θ , φ, ψ). The last task is to determine whether J2 defined by (25) with n = 6 and m = 4 is of rank 2 or not. To this end, we write out the translation motion part of (37) as follows: my¨ 1 = (cos z 1 sin y4 cos z 2 + sin z 1 sin z 2 )u1 , my¨ 2 = (cos z 1 sin y4 sin z 2 − sin z 1 cos z 2 )u1 ,
(44)
my¨ 3 = −mγ + cos z 1 cos y4 · u1 . From (37), (42) and (44), W1 and W2 of (24) are shown to be expressed as
∂ ∂ ∂ F⊥ = span a1k + a2k + a3k ∂X ∂Y ∂Z
+ a4k
a22 = − cos φ cos θ ,
In terms of the local coordinates q := (X , Y , Z , φ, θ , ψ), the Lagrangian (35) is put in the form L=
a31 = − sin θ ,
Defining basis vectors ξ1 and ξ2 with these two sets of coefficients, as is shown in the following, we can express F⊥ as
F
˙ φ − sin θ sin φ cos θ θ˙ . cos φ cos θ ψ˙
a21 = cos θ sin ψ,
a32 = cos φ sin θ sin ψ − sin φ cos ψ.
⊥
k = 1, 2.
We turn to Eq. (40), which is satisfied by non-zero (a1k , a2k , a3k ). For example, we can choose solutions to (40) as follows:
where m denotes the mass of the vehicle and
339
W1 := cos z 2 cos y4 · y¨ 1 + sin z 2 cos y4 · y¨ 2 − sin y4 · ( y¨ 3 + γ ),
∂ ∂ ∂ + a5k + a6k , k = 1, 2 . ∂φ ∂θ ∂ψ
(39)
W2 := − cos z 1 cos y4 · y¨ 2
+ (cos z 1 sin y4 sin z 2 − sin z 1 cos z 2 ) · ( y¨ 3 + γ ).
340
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
m 0 0 (gij ) := 0 0 0
0 m 0 0 0 0
0 0 m 0 0 0
0 0 0 I1 0 −I1 sin θ
0 0 0 0 I2 cos2 φ + I3 sin2 φ (I2 − I3 ) sin φ cos φ cos θ
0 0 0 −I1 sin θ (I2 − I3 ) sin φ cos φ cos θ 2 2 2 2 2 I1 sin θ + I2 sin φ cos θ + I3 cos φ cos θ
Box I.
cos φ sin θ cos ψ + sin φ sin ψ 0 j Fi (q) := 0 0
cos φ sin θ sin ψ − sin φ cos ψ 0 0 0
cos φ cos θ 0 0 0
0 1 0 0
0 0 cos φ − sin φ
0 − sin θ sin φ cos θ cos φ cos θ
Box II.
J2 =
0 sin z 1 cos y4 y¨ 2 − (sin z 1 sin y4 sin z 2 + cos z 1 cos z 2 )( y¨ 3 + γ )
− cos y4 · (sin z 2 y¨ 1 − cos z 2 y¨ 2 ) 1 (cos z sin y4 cos z 2 + sin z 1 sin z 2 )( y¨ 3 + γ )
Box III.
u1
Hence, J2 is put in the form as given in Box III. This implies that J2 is of rank 2 if and only if
u2 k1
k2
m1
cos y4 ̸= 0,
x1
sin z 2 · y¨ 1 − cos z 2 · y¨ 2 ̸= 0,
(45)
sin z 1 cos y4 · y¨ 2
− (sin z sin y sin z + cos z cos z )( y¨ + γ ) ̸= 0. 1
4
2
m2
1
2
k3 m3
x2
m4 x3
x4
Fig. 2. Mass–spring system.
3
To investigate these conditions, we use (44). By straightforward calculation along with (44), we obtain
We assume that control inputs u1 and u2 are applied to particles of mass m1 and m2 , respectively. Then, the equations of motion of this system are given by
m(sin z 2 · y¨ 1 − cos z 2 · y¨ 2 ) = sin z 1 · u1 ,
m1 x¨ 1 − k1 (x2 − x1 ) = u1 ,
1
m2 x¨ 2 + k1 (x2 − x1 ) − k2 (x3 − x2 ) = u2 ,
m sin z cos y · y¨ 2
4
− (sin z sin y sin z + cos z cos z )( y¨ + γ ) 1
4
2
1
2
3
(46)
= − cos z 2 cos y4 · u1 .
m3 x¨ 3 + k2 (x3 − x2 ) − k3 (x4 − x3 ) = 0, m3 x¨ 4 + k3 (x4 − x3 ) = 0.
(47)
The codistribution F spanned by the control inputs is expressed as F = span dx1 , dx2 . Clearly, F is integrable, so that x1 and x2 are not configuration flat outputs on account of Theorem 2. ⊥ The annihilator of F is easily found to be expressed as F =
If u1 equals zero, then Eq. (44) implies that ya , a = 1, . . . , 3 are not differentially independent. Hence, we can assume that u1 ̸= 0. With this assumption, Eqs. (45) and (46) are put together to imply that J2 is of rank 2 if and only if y4 ̸= (2n + 1) π2 , z 1 ̸= nπ , and z 2 ̸= (2n + 1) π2 , where n is an arbitrary integer. Therefore, under the assumption u1 ̸= 0, the variables X , Y , Z , and θ are configuration flat outputs on R 3 × SO(3) for φ ̸= nπ , θ ̸= (2n + 1) π2 , and ψ ̸= (2n + 1) π2 , where n is an arbitrary integer. We can verify in a straightforward manner that φ and ψ are configuration flat outputs. In fact, by (44) we can put φ and ψ in the form of (5) as given in Box IV. We note that under the assumption u1 ̸= 0, configuration flat outputs can be defined on R 3 × SO(3) with some exceptional sets of measure zero.
span ξk := a1k ∂∂x + a2k ∂∂x , k = 1, 2 , where det(ak12 )1≤k1 ,k2 ≤2 ̸= 0. 3 4 We wish to find new coordinates ( y1 , y2 , z 1 , z 2 ) such that y1 , y2 are configuration flat outputs. If such coordinates exist, xi , i = 1, . . . , 4, are functions of ( y1 , y2 , z 1 , z 2 ). Then, Eq. (19) with n = 4 and m = 2 is put in the form
5.2. Mass–spring system
Since det(ak12 )1≤k1 ,k2 ≤2 ̸= 0, each element of ∆2 equals zero if and
4 1
2 i =1
mi x˙ 2i −
3 ki i=1
2
∆2 =
a11 a12
a21 a22
m3 ∂ x3 ∂ z1 ∂x m4
4
∂ z1
∂ x3 ∂ z2 . ∂ x4 m4 2 ∂z m3
k
In this subsection, we give a system for which Assumption 1 does not hold. Although we cannot apply Theorem 3 to it, we can apply Theorem 4 to show that the system is configuration flat. Consider a mass–spring system of four degrees of freedom with two control inputs, which is shown in Fig. 2. Let xi , i = 1, . . . , 4, denote displacements of particles of mass mi from their equilibrium positions, respectively. The Lagrangian of this system is given by L :=
k
(xi+1 − xi )2 .
only if x3 and x4 are independent of z 1 , z 2 . It turns out that ∆2 = 0 if and only if x3 = x3 ( y1 , y2 ), x4 = x4 ( y1 , y2 ). Then, it follows from (47) that y1 and y2 are not differentially independent, which implies that y1 and y2 are not configuration flat outputs. Thus, we observe that Assumption 1 does not hold in this case. Hence, Theorem 3 does not apply to this system. However, this system is shown to be configuration flat as follows: Suppose that there are differentially independent functions y1 = y1 (x1 , x4 ), y2 = y2 (x1 , x4 ), and that there are functionally independent functions z 1 = z 1 (x3 ), z 2 = z 2 (x2 ), and that a11 = a22 = 1, a12 = a21 = 0. In addition, if this
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
φ = arccos ±
Z¨ + γ cos θ ·
X¨ 2 + Y¨ 2 + (Z¨ + γ )2
341
,
√
sin arccos ± cos θ · ψ = arcsin ±
Z¨ +γ
X¨ 2 +Y¨ 2 +(Z¨ +γ )2
X¨ 2 + Y¨ 2
¨X 2 + Y¨ 2 + (Z¨ + γ )2 ¨ + arctan Y X¨ Box IV.
transformation is invertible, that is, x1 , . . . , x4 are put in the form x1 = x1 ( y1 , y2 ), x2 = x2 (z 2 ), x3 = x3 (z 1 ), x4 = x4 ( y1 , y2 ), then W1 and W2 of (29) and (30) are expressed as
∂ x3 1 ∂ 2 x3 1 2 ¨ z + (˙ z ) + k2 x3 (z 1 ) − x2 (z 2 ) 2 1 1 ∂z ∂ z − k3 x4 ( y1 , y2 ) − x3 (z 1 ) = 0, ∂ x4 1 ∂ 2 x4 1 2 ∂ x4 2 ∂ 2 x4 2 2 ¨ ˙ ¨ ˙ + ) + + ) W2 = m4 y ( y y ( y ∂ y1 ∂ y2 ∂ y1 2 ∂ y2 2 1 2 1 + k3 x4 ( y , y ) − x3 (z ) = 0.
W1 = m3
Now, we perform a coordinate transformation of the form y = y(q1 , q2 , q3 ) and z i = z i (q1 , q2 , q3 ), i = 1, 2, where y is a i candidate of configuration flat output and z the other coordinates.
ηk := b1k ∂∂z 1 + b2k ∂∂z 2 , k = 1, 2 . By a straightforward calculation, the quantity G(ξ1 ∧ ξ2 , η1 ∧ η2 ) is put Then, one has Ker y∗ =
in the form G(ξ1 ∧ ξ2 , η1 ∧ η2 ) = q42 det
b11 b21
b12 b22
a11 det 1
Since ξ1 = ∂∂x and ξ2 = ∂∂x , and z 1 = z 1 (x3 ), z 2 = z 2 (x2 ) 3 4 and since the coefficients of the Levi-Civitá connection all vanish, Assumption 2 holds. Further, the following conditions hold
×
∂ W1 ∂ W2 , ̸= 0, ∂ z2 ∂ z1
a1 + det 11 a2
m3 m4 (4) m3 + m4 m4 x4 + + k2 k3 k2 k3 m4 (2) x3 = x + x4 . k3 4
x2 =
(2)
x4 + x4 ,
5.3. A gap between the necessary condition and the sufficient conditions In this subsection, we give a simple example which satisfies the necessary condition for a Lagrangian control system to be configuration flat, but does not satisfy the sufficient conditions. Given a Lagrangian with only a kinetic energy, we caneasily find such examples. Let Q := (q1 , q2 , q3 ) ∈ R 3 |q2 ̸= 0 be a configuration space and L : TQ → R a Lagrangian. Suppose that the Lagrangian L given by L := 12 g (˙q, q˙ ) has the Riemannian metric
g of the form (gij ) :=
q22
0
0
q22
0
0
0 0
. Consider a Lagrangian control
q22
system with one control input, d dt
∂L ∂ q˙ i
−
∂L = uFi , ∂ qi
i = 1, 2, 3,
where u is a control input and F1 = 0, F2 = F3 = 1. The codistribution F spanned by the control input is expressed as F := span {dq2 + dq3 }. The annihilator of F is easily found to be expressed as
∂ ∂ ∂ 2 + a2k + a3k a ∂ q1 ∂ q2 ∂ q3 k + a3k = 0, k = 1, 2 .
F⊥ = span
ξk := a1k
a2
∂ W2 = 0. ∂ z2
Hence, Theorem 4 is applied to show that y1 and y2 are configuration flat outputs. Note here that x1 and x4 are configuration flat outputs as well. This fact can be verified in a straightforward manner. In fact, we can put x2 and x3 in the form of (5) as follows:
∂q
1
a21 a22
1 · ∂z ∂ q2 ∂ z1 ∂q 1 a31 ∂ z1 · ∂ q3 a32 ∂ z1
∂ q1 ∂ z 2 ∂ q2 ∂ z2 ∂ q1 ∂ z 2 . ∂ q3 ∂ z2
Thus, if we choose y = q1 , z 1 = q2 and z 2 = q3 , we obtain G(ξ1 ∧ ξ2 , η1 ∧ η2 ) = 0, which is the necessary condition of Theorem 1 for a Lagrangian control system to be configuration flat. Hence, we verify that y = q1 serves as a configuration flat output. However, this system does not satisfy the sufficient conditions for any choice of y, z 1 and z 2 . In fact, by computing a part of the left-hand side of (23), we obtain l aik Γ12 gli = a1k q2 ,
k = 1, 2.
(48)
If a11 and a12 are equal to zero, ξ1 and ξ2 in F⊥ are not linearly independent at each point on Q . This is a contradiction to the fact that ξ1 and ξ2 are basis vectors in F⊥ . Thus, either a11 or a12 is not equal to zero. If this is the case, Eq. (48) never vanish, so that Eq. (23) does not hold on Q . Since (21), (23) and (32) are equivalent, this system does not satisfy the sufficient conditions stated in Theorems 3 and 4. Hence, the present system satisfies the necessary condition (17) for a Lagrangian control system to be configuration flat, but does not satisfy the sufficient conditions stated in Theorems 3 and 4. 6. Conclusion We have presented a necessary condition and a sufficient condition for configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom. In addition, we have given another sufficient condition for configuration flatness of Lagrangian control systems with n degrees of freedom and n − 2 controls. Finally, we have shown that the sufficient conditions provide a way to find configuration flat outputs with examples. Acknowledgments The authors would like to thank two anonymous referees for helpful comments, which served to improve this paper.
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Contents lists available at SciVerse ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom Kazuhiro Sato ∗ , Toshihiro Iwai Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
article
info
Article history: Received 29 November 2010 Received in revised form 27 July 2011 Accepted 14 November 2011 Available online 5 January 2012 Keywords: Configuration flatness Lagrangian control systems Riemannian geometry
abstract A Lagrangian control system is said to be configuration flat if its states are determined by configuration flat outputs depending on configuration variables only. This paper gives a necessary condition and two types of sufficient conditions for Lagrangian control systems with n degrees of freedom and m controls with m < n to be configuration flat, and further makes a remark on the choice of output variables for configuration flatness. Two examples are given in which the sufficient conditions are utilized for finding configuration flat outputs. © 2012 Published by Elsevier B.V.
1. Introduction Fliess et al. were the first who defined the concept of differentially flat systems [1,2]. If a system is differentially flat, the states are completely determined by specific functions called flat outputs. In addition, if a system is differentially flat, it is linearizable [3,4]. From these points of view, a number of studies have been made on tracking control of differentially flat systems [3,5–11]. Some necessary conditions and sufficient conditions for flatness have been given in previous studies [12,6,13,14,9,15,16]. In particular, Rathinam and Murray [16] defined configuration flatness for Lagrangian control systems and gave a necessary and sufficient condition for a Lagrangian control system with n degrees of freedom and n − 1 controls to be configuration flat. In other words, they provided a complete characterization for configuration flatness in the case of n − 1 controls. Roughly speaking, if a system is configuration flat, there exists as many functions on the configuration space, called configuration flat outputs, as controls such that the system’s behavior is determined by them. The aim of this paper is to extend their result [16]. This paper studies Lagrangian control systems with a Lagrangian given by L(q, q˙ ) =
∗
1 2
g (˙q, q˙ ) − V (q),
Corresponding author. E-mail address: [email protected] (K. Sato).
0167-6911/$ – see front matter © 2012 Published by Elsevier B.V. doi:10.1016/j.sysconle.2011.11.007
where q denote a point of the configuration space, g is a Riemannian metric on the configuration space, V is a potential energy function. As long as mechanical systems are concerned, the Lagrangian control systems of this type cover many of mechanical systems with control, and some of them are known to be configuration flat [9,17,10,16]. Further, for the mechanical systems of this type, Riemannian geometry is known to be effectively used in the study of control [18,19,16]. In fact, Rathinam and Murray [16] have shown how to find configuration flat outputs using concepts in Riemannian geometry such as Riemannian metric and Levi-Civitá connection. However, their results leave the door open for extension. The purpose of this paper is to put forward conditions for a Lagrangian control system with fewer controls than the degrees of freedom to be configuration flat. Though Lévine gave a necessary and sufficient condition for an implicit system to be differentially flat [6,13], the condition is too general to be applied for finding flat outputs in mechanical systems with external force as control. Though this paper’s results can apply to mechanical control systems with fewer controls than degrees of freedom, unfortunately the characterization of configuration flatness is still incomplete. Nevertheless, the sufficient conditions obtained in this paper provide a way to find configuration flat outputs. This paper is organized as follows. Section 2 is a brief review of concepts from Lagrangian control systems theory and also provides the definition of configuration flatness. Section 3 starts with the introduction of Levi-Civitá connections from Riemannian geometry, and proceeds to give a necessary condition and a sufficient condition for configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom. In addition, this paper gives a sufficient condition for output variables not to be configuration flat. Section 4 gives another sufficient condition
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
for configuration flatness of Lagrangian control systems with n degrees of freedom and n − 2 controls. Finally, in Section 5, the present results are applied to a quadrotor unmanned aerial vehicle and a mass spring system to find configuration flat outputs. In addition, a simple example is given which satisfies the necessary condition but does not satisfy the sufficient conditions for configuration flatness.
Definition 2. Suppose y1 , . . . , ym are differentially independent around q ∈ Q . Then y1 , . . . , ym are said to be configuration flat outputs around q ∈ Q if there exist functions z 1 , . . . , z n−m such that ( y1 , . . . , ym , z 1 , . . . , z n−m ) is a local coordinate system of Q , and further if there exist functions fk and positive integers rk such that
d drk (z k ◦ c )(t ) = fk ( y ◦ c )(t ), ( y ◦ c )(t ), . . . , r ( y ◦ c )(t ) , dt
2. Lagrangian control systems and configuration flatness
dt
∂L ∂ q˙ i
∂L − i = ua Fia , ∂q
i = 1, . . . , n,
(1)
where Fia is a tensor field and the control ua Fia dqi is viewed as an element in the cotangent bundle T ∗ Q . This is because forces are naturally paired with velocities to give instantaneous power. Here and in the sequel, we adopt the Einstein summation convention: An index occurring twice in a product is to be summed from 1 up to the last number. We assume that the external control forces determine a codistribution
F := span Fi1 dqi , . . . , Fim dqi ⊂ T ∗ Q
of rank m. We call a triple (TQ , L, F) a Lagrangian control system. The annihilator of F is expressed as
F⊥ = span aik
∂ , k = 1, . . . , n − m , ∂ qi
(2)
where rank(aik ) = n − m. From the definition of the annihilator, all feasible solutions of (1) have to satisfy the following equations: aik
d dt
∂L ∂ q˙ i
−
∂L ∂ qi
= 0,
k = 1, . . . , n − m.
(3)
We introduce the jet space J 2 (R , Q ) of the second-order to describe the notion of configuration flatness [20]. In view of (3), we define a submanifold E ⊂ J 2 (R , Q ), in terms of local coordinates (t , q1 , . . . , qn , q˙ 1 , . . . , q˙ n , q¨ 1 , . . . , q¨ n ) of J 2 (R , Q ), to be
(t , q1 , . . . , qn , q˙ 1 , . . . , q˙ n , q¨ 1 , . . . , q¨ n ) ∈ J 2 (R , Q ) 2 ∂ 2L j ∂L ∂ L j ¨ ˙ = 0, aik q + q − ∂ q˙ i ∂ q˙ j ∂ q˙ i ∂ qj ∂ qi k = 1, . . . , n − m .
k
(5)
where c : R → Q is a generic solution of (1). Definition 3. A Lagrangian control system (TQ , L, F) is said to be configuration flat around q ∈ Q if it admits configuration flat outputs around q ∈ Q . We now make a remark on the relation of configuration flatness to differential flatness. We assume that the Lagrangian control system (1) has a nonsingular matrix
∂2L ∂ q˙ i ∂ q˙ j
1≤i,j≤n
. If
this control system is configuration flat, then it can be described as a differential flat system. We recall that a system x˙ = f (x, u) with states x and control inputs u = (u1 , . . . , um ) is differentially flat if andonly if there exist functions α, β and (r )
(rm ) ,u = γ such that x = α y˜ 1 , . . . , y˜ 1 1 , . . . , y˜ m , . . . , y˜ m (r1 +1) (rm +1) , . . . , y˜ m , . . . , y˜ m and y˜ = γ x, u1 , . . . , β y˜ 1 , . . . , y˜ 1 (l ) (l ) u1 1 , . . . , um , . . . , umm , where y˜ := ( y˜ 1 , . . . , y˜ m ) are called flat
outputs [1,3,6,2,21]. We now put the Lagrangian system (1) in the form of first order differential equations. Let (Aij )1≤i,j≤n :=
∂2L ∂ q˙ i ∂ q˙ j
−1
1≤i,j≤n
. Then, introducing velocity variables v i , we can bring
Eq. (1) into dqi dt dv i dt
= vi , = Aij
i = 1, . . . , n,
Since (Fia )
∂ 2L k ∂L a − v + u F . a j ∂ qj ∂v j ∂ qk
1≤i≤n 1≤a≤m
of (1) is of rank m at each point on Q , there exists
the Moore–Penrose pseudoinverse (Fa+ i )
1≤i≤n 1≤a≤m
of (Fia )
1≤i≤n 1≤a≤m
at each
point on Q [22]. Then, from (1) the control inputs ua , 1 ≤ a ≤ m, can be expressed as
ua =
E :=
dt
k = 1, . . . , n − m,
This section is a brief review from [16]. Let Q and TQ denote a configuration space of dimension n and the tangent bundle over Q , respectively. Let (q1 , . . . , qn , q˙ 1 , . . . , q˙ n ) be local coordinates of TQ and L : TQ → R be a Lagrangian. When external control forces u1 , . . . , um are applied to the Lagrangian system, the equations of motion are described as d
335
∂ 2 L dv j ∂ 2L k ∂L + v − i ∂v i ∂v j dt ∂v i ∂ qk ∂q
Fa+ i .
Hence, if the Lagrangian control system (1) with
∂2L ∂ q˙ i ∂ q˙ j
1≤i,j≤n
dy1 , . . . , dym , dy˙ 1 , . . . , dy˙ m , dy¨ 1 , . . . , dy¨ m
being nonsingular is configuration flat, it has configuration flat outputs, and thereby the state variables (q1 , . . . , qn , v 1 , . . . , v n ) along with the control variables (ua ) can take the form referred to in the definition of differential flatness. If a system is differentially flat, then the system is endogenous dynamic feedback linearizable [3,6,9,21] and locally reachable [6]. Therefore, if a Lagrangian control system is configuration flat, the system has these properties. Nevertheless, a question as to whether a Lagrangian control system (TQ , L, F) is configuration flat or not is non-trivial. Hence, our interest will center on how to find configuration flat outputs. The following lemma on configuration flatness [16] will be of use later.
are linearly independent on π2−1 (U ) ∩ E, where U is an open neighborhood of q ∈ Q and where π2 : J 2 (R , Q ) → Q is the standard projection.
Lemma 1. Let y1 , . . . , ym be configuration flat outputs for (1). Then the set of solutions c : R → Q to (1) that project down to the same curve y ◦ c are all isolated.
(4)
Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion locally defined around q ∈ Q . Then, the configuration flatness is defined as follows [16]. Definition 1. Functions y1 , . . . , ym are called differentially independent around q ∈ Q if
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K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
3. Lagrangian control systems with n degrees of freedom and m controls Suppose that m < n. Let Q be a Riemannian manifold endowed with a Riemannian metric g. Consider a mechanical system (TQ , L) with the Lagrangian given by L(q, q˙ ) =
1
g (˙q, q˙ ) − V (q), 2 where V : Q → R is a potential energy function on Q .
(6)
3.2. Conditions for configuration flatness So far we have introduced necessary concepts from Riemannian geometry. We now proceed to work with configuration flatness of our Lagrangian control system. Given a smooth submersion y : Q → R m , by adding complementary functions z r , we can choose local coordinates (q1 , . . . , qn ) on Q such that
3.1. A brief review of Riemannian geometry Before proceeding further, we make a brief review of connections in Riemannian geometry [23,18].
qa = ya , a = 1, . . . , m, qr +m = z r , r = 1, . . . , n − m.
From now on, given a smooth submersion y : Q → R m , we choose local coordinates like (13). Now, let y : Q → R m be a smooth submersion locally defined around q ∈ Q . Then we can put F⊥ and Ker y∗ in the form
∂ ∂ i , . . . , ξ := a , n − m n −m ∂ qi ∂ qi
Definition 4. Let X(Q ) denote the set of all smooth vector fields on Q and C ∞ (Q ) the set of all smooth functions on Q . A linear connection ∇ on Q is a map
F⊥ = span ξ1 := ai1
∇ : (X , Y ) → ∇X Y ∈ X(Q ),
Ker y∗ = span η1 := br1
X , Y ∈ X(Q ),
which satisfies the following properties:
∇X Y is R-bilinear in X and Y , ∇fX Y = f ∇X Y ,
(7)
f ∈ C ∞ (Q ),
∇X ( fY ) = (Xf )Y + f ∇X Y ,
(8)
f ∈ C (Q ). ∞
(9)
The ∇X Y is called the covariant derivative of Y along X . 3
If ∇ is given on Q , the n smooth functions
Γjki
are determined
∇
∂ ∂ qj
∂ ∂ = Γjki i . ∂ qk ∂q
The Γjki are called the Christoffel symbols of the linear connection. Claim 1. Let Q be a Riemannian manifold endowed with a Riemannian metric g. Then there is a unique linear connection ∇ on Q such that (10) (11)
where X , Y , Z ∈ X(Q ), and [X , Y ] := XY − YX . The ∇ is called the Levi-Civitá connection. In the following, we denote by ∇ the Levi-Civitá connection. Then, from the properties (10), Γjki is symmetric; Γjki = Γkji . In addition, owing to (11), Γjki are expressed in terms of the Riemannian metric g as
Γjkl =
1 2
∂ gik ∂ gij ∂ gjk + k − i ∂ qj ∂q ∂q
g il ,
j, l = 1, . . . , n,
(12)
where g ik gkj = δji with (g ik ) = (gik )−1 . Further, we introduce the wedge product of vector fields and the inner product of them [24], which will make it compact to describe a necessary condition for configuration flatness. Definition 5. For vector fields Xa and Ya , a = 1, . . . , n − m, the inner product of X1 ∧ · · · ∧ Xn−m and Y1 ∧ · · · ∧ Yn−m is defined to be G(X1 ∧ · · · ∧ Xn−m , Y1 ∧ · · · ∧ Yn−m )
:= det
g (X1 , Y1 )
···
g (Xn−m , Y1 )
···
.. .
g (X1 , Yn−m )
.. . . g (Xn−m , Yn−m )
∂ ∂ , . . . , ηn−m := brn−m r ∂ zr ∂z
(14)
,
(15)
where y∗ : TQ → T R m is the differential map of y, and where rank(brs ) = n − m. For basis vectors of F⊥ and Ker y∗ , Definition 5 provides the following. Lemma 2. As for (14) and (15), the following two conditions are equivalent;
n
∇X Y − ∇Y X = [X , Y ], X (g (Y , Z )) = g (∇X Y , Z ) + g (Y , ∇X Z ),
∂ ∂ = 0, G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ ∂z ∂ z n −m G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m ) = 0.
in terms of local coordinates (q , . . . , q ) through 1
(13)
Proof. For the vector fields ξ1 , . . . , ξn−m , and η1 , . . . , ηn−m given in (14) and (15), respectively, we have G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m )
∂ ∂ = det brs 1≤r ,s≤n−m G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ n−m . ∂z ∂z Since η1 , . . . , ηn−m are basis vectors in Ker y∗ , one has det brs 1≤r ,s≤n−m ̸= 0. This ends the proof. We return to our key equation (3). In terms of local coordinates
(qi , q˙ i ) of TQ , the Lagrangian (6) is expressed as L(q, q˙ ) := 12 gij q˙ i q˙ j − V (q). On account of (13), Eq. (3) is rewritten as ∂V l + gi,m+r z¨ r Wk := aik gia y¨ a + Γab gli y˙ a y˙ b + ∂ qi l a r l r1 r2 + 2Γa,m+r gli y˙ z˙ + Γm+r1 ,m+r2 gli z˙ z˙ = 0, r1 , r2 , k = 1, . . . , n − m.
(16)
We are now in a position to describe a necessary condition for Lagrangian control systems to be configuration flat. Theorem 1. Let (TQ , L, F) be a Lagrangian control system, where L is a Lagrangian defined on the tangent bundle TQ over a Riemannian manifold Q , and where F is the space of control forces of rank m. Suppose that y1 , . . . , ym are configuration flat outputs around q ∈ Q . Then, for F⊥ and Ker y∗ given in (14) and (15), respectively, one obtains G(ξ1 ∧ · · · ∧ ξn−m , η1 ∧ · · · ∧ ηn−m ) = 0.
(17)
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
Proof. We put (16) in the form
gi,m+1 ai1
···
gin ai1
1
z¨ . .. i i z¨ n−m gi,m+1 an−m · · · gin an−m f1 ( y, y˙ , y¨ , z , z˙ ) .. = (18) , . fn−m ( y, y˙ , y¨ , z , z˙ ) l gli y˙ a y˙ b + ∂∂qVi + where fk ( y, y˙ , y¨ , z , z˙ ) := −aik gia y¨ a + Γab 2Γal,m+r gli y˙ a z˙ r + Γml +r1 ,m+r2 gli z˙ r1 z˙ r2 . If G(ξ1 ∧ · · · ∧ ξn−m , ∂∂z 1
.. .
∧ ··· ∧
.. .
∂ ) ∂ z n−m
̸= 0, i.e., if the determinant of the coefficient matrix of the left-hand side of (18) does not vanish, then (18) can be brought into z1 d2 . .. dt 2 z n −m
gi,m+1 ai1
=
gin ai1
···
.. .
gin ain−m
···
−1
f1 ( y, y˙ , y¨ , z , z˙ )
.. . . fn−m ( y, y˙ , y¨ , z , z˙ )
This equation implies that given any curve y(t ) we get a 2(n − m)-parameter family of solutions q(t ) = ( y(t ), z (t )) that project to y(t ). Therefore, Lemma 1 shows that y1 , . . . , ym are not configuration flat outputs. This contradicts the assumption. Hence
∂ ∂ G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ = 0. ∂z ∂ z n −m This and Lemma 2 result in (17).
+r ∂ , k = 1, . . . , n − m, so that the form ξk = am k ∂ zr
∂ ∂ ∧ · · · ∧ n −m 1 ∂z ∂z
∆n−m
g
ξ1 ,
∂ ∂ z1
···
.. := . ∂ g ξn−m , 1 ∂z
···
g
ξ1 ,
∂
∂ z n −m .. . ∂ g ξn−m , n−m ∂z
(19)
Assumption 1. Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion and y1 , . . . , ym be differentially independent around q ∈ Q . For ξ1 , . . . , ξn−m ∈ F⊥ , the following conditions hold,
∆n−m = 0, ∂ = 0, g ξk1 , ∇X k ∂z 2
(20)
∀X ∈ X(Q ), k1 , k2 = 1, . . . , n − m. (21)
As is easily verified, (20) and (21) are equivalent to
= 0,
k1 , k2 = 1, . . . , n − m, j = 1, . . . , n,
(22) (23)
Since ξ1 , . . . , ξn−m are basis vectors in F⊥ , one has det
+r am ̸= 0, the factor G ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m , ∂∂z 1 ∧ · · · ∧ k 1≤k, r ≤n−m
in the right-hand side of the above equation does not van-
ish, either. As a matter of fact, since (gij ) is positive definite matrix, we have ζ i ζ j gij > 0 for any non-zero ζ = ζ i ∂∂qi ∈ X(Q ). In
particular, for any ζ with ζ = 0, a = 1, . . . , m, and ζ ̸= 0, r = 1, . . . , n − m, we obtain ζ r1 +m ζ r2 +m gr1 +m,r2 +m > 0, so r +m
a
that G ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m , ∂∂z 1 ∧ · · · ∧ ∂ z n∂−m
Wk = aik
l gia y¨ a + Γab gli y˙ a y˙ b +
∂V ∂ qi
= 0,
k = 1, . . . , n − m.
(24)
Let
Jn−m
∂W 1 ∂ z1 . := .. ∂ Wn−m ∂ z1
···
···
∂ W1 ∂ z n−m .. . . ∂ Wn−m ∂ z n−m
(25)
We are now in a position to describe a sufficient condition for Lagrangian control systems to be configuration flat.
+r = det am k 1≤k, r ≤n−m ∂ ∂ ∂ ∂ G ∧ · · · ∧ , ∧ · · · ∧ . ∂ z1 ∂ z n −m ∂ z 1 ∂ z n−m
respectively. Then, Eq. (16) is brought into
Proof. Since F = span dy1 , . . . , dym , the ξk given in (14) take
∂ ∂ z n−m
We note that if the codistribution F spanned by the control 1 m forces is integrable, then there exist functions y , . . . , y such that F is expressed as F = span dy1 , . . . , dym . Lemma 2 and Theorem 1 mean that the matrix
aik1 Γj,l m+k2 gli
Theorem 2. Let y = ( y1 , . . . , ym ) : Q → R m be a smooth submersion and y1 , . . . , ym be differentially independent around q ∈ Q . If F = span dy1 , . . . , dym , then y1 , . . . , ym are not configuration flat outputs around q ∈ Q .
G ξ1 ∧ · · · ∧ ξn−m ,
It follows from Lemma 2 and Theorem 1 that y1 , . . . , ym are not configuration flat outputs around q ∈ Q .
aik1 gi,m+k2 = 0,
As an immediate consequence of Theorem 1, the choice of output variables for which (17) does not hold is inadequate for configuration flatness. In fact, we obtain the following theorem.
̸= 0 with r1 , r2 = 1, . . . , n − m. Hence, ∂ ∂ G ξ1 ∧ · · · ∧ ξn−m , 1 ∧ · · · ∧ ̸= 0. ∂z ∂ z n−m
is of rank less than n − m. To treat sufficient conditions for configuration flatness, we start by setting a simple assumption on ∆n−m .
.. .
gi,m+1 ain−m
337
= det gr1 +m,r2 +m
Theorem 3. Suppose that Assumption 1 holds and rank Jn−m = n − m on π2−1 (U )∩ E, where U is a neighborhood of q ∈ Q and E is defined by (4). Then, y1 , . . . , ym are configuration flat outputs around q ∈ Q , so that the Lagrangian control system (TQ , L, F) is configuration flat. Proof. Note that Assumption 1 implies (24). If rank Jn−m = n − m on π2−1 (U ) ∩ E, the implicit function theorem implies that there exist functions h1 , . . . , hn−m such that z 1 = h1 ( y, y˙ , y¨ ), . . . , z n−m = hn−m ( y, y˙ , y¨ ). This means that y1 , . . . , ym are configuration flat outputs around q ∈ Q. We have to note that if m = n − 1, a necessary and sufficient condition is already known for configuration flatness:
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K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
Proposition 1 ([16]). Let y = ( y1 , . . . , yn−1 ) : Q → R n−1 be a smooth submersion and y1 , . . . , yn−1 be differentially independent around q ∈ Q . If y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q , then
g
ξ1 ,
∂ ∂ z1
= g ∇X ξ1 ,
∂ ∂ z1
= 0,
(26)
where for F⊥ and Ker y∗ given in (14) and (15), respectively, ξ1 ∈ F⊥ and ∂∂z 1 ∈ Ker y∗ .
Conversely, if (26) is satisfied, y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q .
We now remark how the theorems in the present article are related to this proposition. If m equals n − 1, and if y1 , . . . , yn−1 are differentially independent around q ∈ Q , then Jn−m becomes J1 and is necessarily of rank 1 on π2−1 (U ) ∩ E, where U is a neighborhood of q ∈ Q and E is defined by (4). Hence, Assumption 1 reduces to g ξ1 , ∂∂z 1
= g ξ1 , ∇X ∂∂z 1 = 0. ∂ However, the latter condition can be expressed as g ξ1 , ∇X ∂ z 1 = −g ∇X ξ1 , ∂∂z 1 = 0. Thus, we find that Assumption 1 reduces
to (26), and that Theorem 3 reduces to the ‘‘converse’’ part of Proposition 1. Conversely, if y1 , . . . , yn−1 are configuration flat outputs around q ∈ Q , then Theorem 1 provides the condition g ξ1 , ∂∂z 1
= 0, one of equations in (26), but does not refer to the condition g ∇X ξ1 , ∂∂z 1 = 0. However, we note that if
y1, . . . , yn−1 are configuration flat outputs around q ∈ Q , then ∂ ∂ g ∇X ξ1 , ∂ z 1 = 0 can be derived from g ξ1 , ∂ z 1 = 0, as is shown in [16]. Hence, if m = n − 1, the present theorems could reproduce Proposition 1 obtained in [16]. 4. Mechanical systems with n degrees of freedom and n − 2 controls To study further a sufficient condition for configuration flatness, we move a step by relaxing the condition (20) but setting m = n − 2. As is already stated, Lemma 2 and Theorem 1 mean that det ∆2 = 0 for m = n − 2, so that rank ∆2 = 0 or 1
on π2−1 (U ) ∩ E .
(27)
Suppose that rank ∆2 = 0 on π2 (U ) ∩ E and that Eq. (21) holds. If rank J2 = 2 on π2−1 (U ) ∩ E, then Assumption 1 holds with m = n − 2, so that Theorem 3 implies that y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q . We turn to the case of rank ∆2 = 1 on π2−1 (U ) ∩ E. For the sake of convenience, we treat a simple case such that ∆2 is of the form
Assumption 2. Let y = ( y1 , . . . , yn−2 ) : Q → R n−2 be a smooth submersion and y1 , . . . , yn−2 be differentially independent around q ∈ Q . For ξ1 , ξ2 ∈ F⊥ , the following condition holds,
∂ ∂ ∂ g ξ1 , 1 g ξ1 , 2 g ξ1 , 1 0 ∂ z ∂ z ̸= 0, (31) = ∂z ∂ ∂ 0 0 g ξ2 , 2 g ξ2 , 1 ∂z ∂z ∂ g ξk1 , ∇X k = 0, ∀X ∈ X(Q ), k1 , k2 = 1, 2. (32) ∂z 2 We are now in a position to describe another sufficient condition for Lagrangian control systems to be configuration flat. Theorem 4. Suppose that Assumption 2 holds and that
∂ W1 ∂ W2 , ̸= 0, ∂ z2 ∂ z1
∂ W2 = 0 on π2−1 (U ) ∩ E . ∂ z2
Then y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q , so that the Lagrangian control system (TQ , L, F) is configuration flat. Proof. From Assumption 2, W1 and W2 are expressed as (29) and ∂W (30), respectively. If ∂ z 22 = 0 on π2−1 (U ) ∩ E , W2 is independent ∂W
of z 2 . Further, if ∂ z 12 ̸= 0 on π2−1 (U ) ∩ E, by the implicit function theorem there exists a function h1 such that z 1 = h1 ( y, y˙ , y¨ ).
(33)
Substituting this equation into (29), we obtain the equation W1 = ∂W 0 with z 1 replaced by (33). From ∂ z 21 ̸= 0 on π2−1 (U ) ∩ E, it follows that there exists a function h2 such that z 2 = h2 ( y, y˙ , y¨ , y(3) , y(4) ).
(34)
Eqs. (33) and (34) mean that y1 , . . . , yn−2 are configuration flat outputs around q ∈ Q . Though we have confined ourselves to m = n − 2, a similar argument would be possible for m < n − 2 to obtain further sufficient conditions. 5. Examples
−1
∂ g ξ1 , 1 ∆2 = ∂z 0
0
̸= 0.
(28)
0
Needless to say, a similar discussion will run in parallel in the case where only one of entries of ∆2 does not vanish and the others vanish. From (21) or (23) along with (28), Eq. (16) with m = n − 2 proves to be expressed as
∂V + gi,n−1 z¨ 1 ∂ qi ∂V l W2 = ai2 gia y¨ a + Γab gli y˙ a y˙ b + = 0. ∂ qi W1 = ai1
l gia y¨ a + Γab gli y˙ a y˙ b +
= 0,
We now replace Assumption 1 by the following.
(29)
(30)
5.1. Quadrotor UAV In this subsection, we apply Theorem 3 to a quadrotor unmanned aerial vehicle (UAV) [25–28], which is a Lagrangian control system with six degrees of freedom and four controls, as is shown in Fig. 1. Previous works [26,27] claimed that the system treated in [25] is (configuration) flat under certain conditions. However, Kim et al. have pointed out that the setting up of the system lacks in mathematical rigorousness, and given an accurate Lagrangian control system [28]. Following their setting up, we show that Theorem 3 provides a way to find configuration flat outputs. We regard the quadrotor UAV as a rigid body, whose configuration space is R 3 × SO(3) [29]. Let (X , Y , Z , φ, θ , ψ) be local coordinates of R 3 × SO(3), where (X , Y , Z ) denotes the position of the center of gravity of the quadrotor UAV, and φ, θ and ψ denote the roll, pitch and yaw angles of UAV in an inertial frame, respectively [28]. The Lagrangian of this system L : T (R 3 × SO(3)) → R is given by L :=
1 2
m(X˙ 2 + Y˙ 2 + Z˙ 2 ) +
1 2
ω T I ω − mγ Z ,
(35)
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By definition, the coefficients aik should satisfy the following equations,
f2 M2
f3
a1k (cos φ sin θ cos ψ + sin φ sin ψ) f1
+ a2k (cos φ sin θ sin ψ − sin φ cos ψ) + a3k cos φ cos θ = 0, (40) 4 ak 1 0 − sin θ 0 cos φ sin φ cos θ a5k = 0. (41) 0 − sin φ cos φ cos θ a6k
M3 Z
f4
M1 mγ
ψ
M4 φ
θ
We first consider Eq. (41). If the pitch angle θ does not equal (2n + 1) π2 with n being an arbitrary integer, the coefficient matrix of (41) is nonsingular, so that we obtain
Y
X
a4k = a5k = a6k = 0,
Fig. 1. Quadrotor UAV.
I :=
I1 0 0
0 I2 0
0 0 I3
a11 = cos θ cos ψ,
a12 = 0,
is the inertia matrix of UAV in the body frame, and γ is the gravitational acceleration. Further, ω in (35) denotes the angular velocity of the vehicle in the body frame [29], and is expressed as 1
0 cos φ − sin φ
ω= 0
0
1
= span ξ1 := cos θ cos ψ
(36)
2
g (˙q, q˙ ) − mγ Z ,
where the Riemannian metric g is given in Box I. The Lagrangian control system of the quadrotor UAV [28] is then subject to the equations of motion d dt
∂L ∂ q˙ i
j Fi
where
4 ∂L j − i = uj Fi (q), ∂q j=1
i = 1, . . . , 6,
(37)
(q) is given in Box II.
Here, u1 is the total thrust produced by the four rotors Mi , i = 1, . . . , 4, that is, it is given by u1 := f1 + f2 + f3 + f4 , where fi := ki wi2 is the thrust generated by Mi and ki > 0 is a constant, and wi is the angular speed of Mi . The control inputs u2 , u3 , and u4 are the generalized moments; they are given by u2 := ( f3 − f1 )d, u3 := ( f2 − f4 )d and u4 := ( f2 + f4 − f1 − f3 )κ , where d represents the distance from each Mi to the center of gravity of the quadrotor UAV and κ is a constant. Then, the control inputs determine the following codistribution:
F := span (cos φ sin θ cos ψ + sin φ sin ψ)dX
+ (cos φ sin θ sin ψ − sin φ cos ψ)dY + (cos φ cos θ )dZ , dφ − sin θ dψ, cos φ dθ + sin φ cos θ dψ, − sin φ dθ + cos φ cos θ dψ . (38) To find the annihilator of F, we put it in the form
ξ2 := − cos φ cos θ
∂ ∂ ∂ + cos θ sin ψ − sin θ , ∂X ∂Y ∂Z
∂ ∂Y
+ (cos φ sin θ sin ψ − sin φ cos ψ)
∂ . ∂Z
(42)
To find configuration flat outputs y1 , y2 , y3 , and y4 with the help of Theorem 3, we need to calculate the coefficients of the LeviCivitá connection. A straightforward calculation shows that the covariant derivatives of basis vectors ∂∂qi take the form
∇
∂
∂ qs1
∂ ∂
qs2
∂ ∂ + Γs51 s2 5 4 ∂q ∂q ∂ + Γs61 s2 6 , s1 , s2 = 1, . . . , 6, ∂q
= Γs41 s2
(43)
where (q1 , q2 , q3 , q4 , q5 , q6 ) = (X , Y , Z , φ, θ , ψ). In view of (13), we wish to find local coordinates y1 , . . . , y4 and z 1 , z 2 for which Assumption 1 holds. If we can perform a coordinate transformation of the form z 1 = z 1 (φ, ψ), z 2 = z 2 (φ, ψ), y1 = y1 (q), . . . , y4 = y4 (q), we see from the Riemannian metric g that ∆2 given in (19) with n = 6 and m = 4 vanishes, and further from (42) and (43) we verify that Eq. (21) holds. In addition, if there are differentially independent functions ya = ya (X , Y , Z , θ ), a = 1, . . . , 4, then ∂ , ∂ are shown to be in Ker y∗ , and ( y1 , y2 , y3 , y4 , z 1 , z 2 ) are ∂ z1 ∂ z2 local coordinates in the configuration space R 3 × SO(3). The easiest choice of ( y1 , y2 , y3 , y4 , z 1 , z 2 ) may be ( y1 , y2 , y3 , y4 , z 1 , z 2 ) := (X , Y , Z , θ , φ, ψ). The last task is to determine whether J2 defined by (25) with n = 6 and m = 4 is of rank 2 or not. To this end, we write out the translation motion part of (37) as follows: my¨ 1 = (cos z 1 sin y4 cos z 2 + sin z 1 sin z 2 )u1 , my¨ 2 = (cos z 1 sin y4 sin z 2 − sin z 1 cos z 2 )u1 ,
(44)
my¨ 3 = −mγ + cos z 1 cos y4 · u1 . From (37), (42) and (44), W1 and W2 of (24) are shown to be expressed as
∂ ∂ ∂ F⊥ = span a1k + a2k + a3k ∂X ∂Y ∂Z
+ a4k
a22 = − cos φ cos θ ,
In terms of the local coordinates q := (X , Y , Z , φ, θ , ψ), the Lagrangian (35) is put in the form L=
a31 = − sin θ ,
Defining basis vectors ξ1 and ξ2 with these two sets of coefficients, as is shown in the following, we can express F⊥ as
F
˙ φ − sin θ sin φ cos θ θ˙ . cos φ cos θ ψ˙
a21 = cos θ sin ψ,
a32 = cos φ sin θ sin ψ − sin φ cos ψ.
⊥
k = 1, 2.
We turn to Eq. (40), which is satisfied by non-zero (a1k , a2k , a3k ). For example, we can choose solutions to (40) as follows:
where m denotes the mass of the vehicle and
339
W1 := cos z 2 cos y4 · y¨ 1 + sin z 2 cos y4 · y¨ 2 − sin y4 · ( y¨ 3 + γ ),
∂ ∂ ∂ + a5k + a6k , k = 1, 2 . ∂φ ∂θ ∂ψ
(39)
W2 := − cos z 1 cos y4 · y¨ 2
+ (cos z 1 sin y4 sin z 2 − sin z 1 cos z 2 ) · ( y¨ 3 + γ ).
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m 0 0 (gij ) := 0 0 0
0 m 0 0 0 0
0 0 m 0 0 0
0 0 0 I1 0 −I1 sin θ
0 0 0 0 I2 cos2 φ + I3 sin2 φ (I2 − I3 ) sin φ cos φ cos θ
0 0 0 −I1 sin θ (I2 − I3 ) sin φ cos φ cos θ 2 2 2 2 2 I1 sin θ + I2 sin φ cos θ + I3 cos φ cos θ
Box I.
cos φ sin θ cos ψ + sin φ sin ψ 0 j Fi (q) := 0 0
cos φ sin θ sin ψ − sin φ cos ψ 0 0 0
cos φ cos θ 0 0 0
0 1 0 0
0 0 cos φ − sin φ
0 − sin θ sin φ cos θ cos φ cos θ
Box II.
J2 =
0 sin z 1 cos y4 y¨ 2 − (sin z 1 sin y4 sin z 2 + cos z 1 cos z 2 )( y¨ 3 + γ )
− cos y4 · (sin z 2 y¨ 1 − cos z 2 y¨ 2 ) 1 (cos z sin y4 cos z 2 + sin z 1 sin z 2 )( y¨ 3 + γ )
Box III.
u1
Hence, J2 is put in the form as given in Box III. This implies that J2 is of rank 2 if and only if
u2 k1
k2
m1
cos y4 ̸= 0,
x1
sin z 2 · y¨ 1 − cos z 2 · y¨ 2 ̸= 0,
(45)
sin z 1 cos y4 · y¨ 2
− (sin z sin y sin z + cos z cos z )( y¨ + γ ) ̸= 0. 1
4
2
m2
1
2
k3 m3
x2
m4 x3
x4
Fig. 2. Mass–spring system.
3
To investigate these conditions, we use (44). By straightforward calculation along with (44), we obtain
We assume that control inputs u1 and u2 are applied to particles of mass m1 and m2 , respectively. Then, the equations of motion of this system are given by
m(sin z 2 · y¨ 1 − cos z 2 · y¨ 2 ) = sin z 1 · u1 ,
m1 x¨ 1 − k1 (x2 − x1 ) = u1 ,
1
m2 x¨ 2 + k1 (x2 − x1 ) − k2 (x3 − x2 ) = u2 ,
m sin z cos y · y¨ 2
4
− (sin z sin y sin z + cos z cos z )( y¨ + γ ) 1
4
2
1
2
3
(46)
= − cos z 2 cos y4 · u1 .
m3 x¨ 3 + k2 (x3 − x2 ) − k3 (x4 − x3 ) = 0, m3 x¨ 4 + k3 (x4 − x3 ) = 0.
(47)
The codistribution F spanned by the control inputs is expressed as F = span dx1 , dx2 . Clearly, F is integrable, so that x1 and x2 are not configuration flat outputs on account of Theorem 2. ⊥ The annihilator of F is easily found to be expressed as F =
If u1 equals zero, then Eq. (44) implies that ya , a = 1, . . . , 3 are not differentially independent. Hence, we can assume that u1 ̸= 0. With this assumption, Eqs. (45) and (46) are put together to imply that J2 is of rank 2 if and only if y4 ̸= (2n + 1) π2 , z 1 ̸= nπ , and z 2 ̸= (2n + 1) π2 , where n is an arbitrary integer. Therefore, under the assumption u1 ̸= 0, the variables X , Y , Z , and θ are configuration flat outputs on R 3 × SO(3) for φ ̸= nπ , θ ̸= (2n + 1) π2 , and ψ ̸= (2n + 1) π2 , where n is an arbitrary integer. We can verify in a straightforward manner that φ and ψ are configuration flat outputs. In fact, by (44) we can put φ and ψ in the form of (5) as given in Box IV. We note that under the assumption u1 ̸= 0, configuration flat outputs can be defined on R 3 × SO(3) with some exceptional sets of measure zero.
span ξk := a1k ∂∂x + a2k ∂∂x , k = 1, 2 , where det(ak12 )1≤k1 ,k2 ≤2 ̸= 0. 3 4 We wish to find new coordinates ( y1 , y2 , z 1 , z 2 ) such that y1 , y2 are configuration flat outputs. If such coordinates exist, xi , i = 1, . . . , 4, are functions of ( y1 , y2 , z 1 , z 2 ). Then, Eq. (19) with n = 4 and m = 2 is put in the form
5.2. Mass–spring system
Since det(ak12 )1≤k1 ,k2 ≤2 ̸= 0, each element of ∆2 equals zero if and
4 1
2 i =1
mi x˙ 2i −
3 ki i=1
2
∆2 =
a11 a12
a21 a22
m3 ∂ x3 ∂ z1 ∂x m4
4
∂ z1
∂ x3 ∂ z2 . ∂ x4 m4 2 ∂z m3
k
In this subsection, we give a system for which Assumption 1 does not hold. Although we cannot apply Theorem 3 to it, we can apply Theorem 4 to show that the system is configuration flat. Consider a mass–spring system of four degrees of freedom with two control inputs, which is shown in Fig. 2. Let xi , i = 1, . . . , 4, denote displacements of particles of mass mi from their equilibrium positions, respectively. The Lagrangian of this system is given by L :=
k
(xi+1 − xi )2 .
only if x3 and x4 are independent of z 1 , z 2 . It turns out that ∆2 = 0 if and only if x3 = x3 ( y1 , y2 ), x4 = x4 ( y1 , y2 ). Then, it follows from (47) that y1 and y2 are not differentially independent, which implies that y1 and y2 are not configuration flat outputs. Thus, we observe that Assumption 1 does not hold in this case. Hence, Theorem 3 does not apply to this system. However, this system is shown to be configuration flat as follows: Suppose that there are differentially independent functions y1 = y1 (x1 , x4 ), y2 = y2 (x1 , x4 ), and that there are functionally independent functions z 1 = z 1 (x3 ), z 2 = z 2 (x2 ), and that a11 = a22 = 1, a12 = a21 = 0. In addition, if this
K. Sato, T. Iwai / Systems & Control Letters 61 (2012) 334–342
φ = arccos ±
Z¨ + γ cos θ ·
X¨ 2 + Y¨ 2 + (Z¨ + γ )2
341
,
√
sin arccos ± cos θ · ψ = arcsin ±
Z¨ +γ
X¨ 2 +Y¨ 2 +(Z¨ +γ )2
X¨ 2 + Y¨ 2
¨X 2 + Y¨ 2 + (Z¨ + γ )2 ¨ + arctan Y X¨ Box IV.
transformation is invertible, that is, x1 , . . . , x4 are put in the form x1 = x1 ( y1 , y2 ), x2 = x2 (z 2 ), x3 = x3 (z 1 ), x4 = x4 ( y1 , y2 ), then W1 and W2 of (29) and (30) are expressed as
∂ x3 1 ∂ 2 x3 1 2 ¨ z + (˙ z ) + k2 x3 (z 1 ) − x2 (z 2 ) 2 1 1 ∂z ∂ z − k3 x4 ( y1 , y2 ) − x3 (z 1 ) = 0, ∂ x4 1 ∂ 2 x4 1 2 ∂ x4 2 ∂ 2 x4 2 2 ¨ ˙ ¨ ˙ + ) + + ) W2 = m4 y ( y y ( y ∂ y1 ∂ y2 ∂ y1 2 ∂ y2 2 1 2 1 + k3 x4 ( y , y ) − x3 (z ) = 0.
W1 = m3
Now, we perform a coordinate transformation of the form y = y(q1 , q2 , q3 ) and z i = z i (q1 , q2 , q3 ), i = 1, 2, where y is a i candidate of configuration flat output and z the other coordinates.
ηk := b1k ∂∂z 1 + b2k ∂∂z 2 , k = 1, 2 . By a straightforward calculation, the quantity G(ξ1 ∧ ξ2 , η1 ∧ η2 ) is put Then, one has Ker y∗ =
in the form G(ξ1 ∧ ξ2 , η1 ∧ η2 ) = q42 det
b11 b21
b12 b22
a11 det 1
Since ξ1 = ∂∂x and ξ2 = ∂∂x , and z 1 = z 1 (x3 ), z 2 = z 2 (x2 ) 3 4 and since the coefficients of the Levi-Civitá connection all vanish, Assumption 2 holds. Further, the following conditions hold
×
∂ W1 ∂ W2 , ̸= 0, ∂ z2 ∂ z1
a1 + det 11 a2
m3 m4 (4) m3 + m4 m4 x4 + + k2 k3 k2 k3 m4 (2) x3 = x + x4 . k3 4
x2 =
(2)
x4 + x4 ,
5.3. A gap between the necessary condition and the sufficient conditions In this subsection, we give a simple example which satisfies the necessary condition for a Lagrangian control system to be configuration flat, but does not satisfy the sufficient conditions. Given a Lagrangian with only a kinetic energy, we caneasily find such examples. Let Q := (q1 , q2 , q3 ) ∈ R 3 |q2 ̸= 0 be a configuration space and L : TQ → R a Lagrangian. Suppose that the Lagrangian L given by L := 12 g (˙q, q˙ ) has the Riemannian metric
g of the form (gij ) :=
q22
0
0
q22
0
0
0 0
. Consider a Lagrangian control
q22
system with one control input, d dt
∂L ∂ q˙ i
−
∂L = uFi , ∂ qi
i = 1, 2, 3,
where u is a control input and F1 = 0, F2 = F3 = 1. The codistribution F spanned by the control input is expressed as F := span {dq2 + dq3 }. The annihilator of F is easily found to be expressed as
∂ ∂ ∂ 2 + a2k + a3k a ∂ q1 ∂ q2 ∂ q3 k + a3k = 0, k = 1, 2 .
F⊥ = span
ξk := a1k
a2
∂ W2 = 0. ∂ z2
Hence, Theorem 4 is applied to show that y1 and y2 are configuration flat outputs. Note here that x1 and x4 are configuration flat outputs as well. This fact can be verified in a straightforward manner. In fact, we can put x2 and x3 in the form of (5) as follows:
∂q
1
a21 a22
1 · ∂z ∂ q2 ∂ z1 ∂q 1 a31 ∂ z1 · ∂ q3 a32 ∂ z1
∂ q1 ∂ z 2 ∂ q2 ∂ z2 ∂ q1 ∂ z 2 . ∂ q3 ∂ z2
Thus, if we choose y = q1 , z 1 = q2 and z 2 = q3 , we obtain G(ξ1 ∧ ξ2 , η1 ∧ η2 ) = 0, which is the necessary condition of Theorem 1 for a Lagrangian control system to be configuration flat. Hence, we verify that y = q1 serves as a configuration flat output. However, this system does not satisfy the sufficient conditions for any choice of y, z 1 and z 2 . In fact, by computing a part of the left-hand side of (23), we obtain l aik Γ12 gli = a1k q2 ,
k = 1, 2.
(48)
If a11 and a12 are equal to zero, ξ1 and ξ2 in F⊥ are not linearly independent at each point on Q . This is a contradiction to the fact that ξ1 and ξ2 are basis vectors in F⊥ . Thus, either a11 or a12 is not equal to zero. If this is the case, Eq. (48) never vanish, so that Eq. (23) does not hold on Q . Since (21), (23) and (32) are equivalent, this system does not satisfy the sufficient conditions stated in Theorems 3 and 4. Hence, the present system satisfies the necessary condition (17) for a Lagrangian control system to be configuration flat, but does not satisfy the sufficient conditions stated in Theorems 3 and 4. 6. Conclusion We have presented a necessary condition and a sufficient condition for configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom. In addition, we have given another sufficient condition for configuration flatness of Lagrangian control systems with n degrees of freedom and n − 2 controls. Finally, we have shown that the sufficient conditions provide a way to find configuration flat outputs with examples. Acknowledgments The authors would like to thank two anonymous referees for helpful comments, which served to improve this paper.
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