Passive Decomposition of Multiple Mechanical

Submitted to IEEE Transactions on Automatic Control 1

Passive Decomposition of Multiple Mechanical Systems under Motion Coordination Requirements Dongjun Lee and Perry Y. Li

Abstract

Consider a mechanical system whose Lagrangian dynamics evolves on a

n-dimensional

congura-

M with its kinetic energy as the Lagrangian and a sumbersion h : M → N from M to a m-dimensional manifold N (n ≥ m). In this paper, we reveal a fundamental property of the mechanical system in this setting s.t. its n-dimensional dynamics can be decomposed into 1) m-dimensional shape system describing the dynamics of the mapped point of h on N ; 2) locked system representing the (n − m)-dimensional dynamics on the level set of h; and 3) energetically conservative coupling between tion manifold

them. This coupling can be canceled out while preserving passivity of the open-loop mechanical system (i.e. passive with mechanical power and kinetic energy as supply rate and storage function). Moreover, the locked and shape systems also individually inherit passivity from the mechanical system. Due to these passivity properties, we name the presented decomposition passive decomposition. We exhibit and analyze intrinsic and geometric properties of the passive decomposition and the locked and shape systems. A control design example is also given for the decoupled locked and shape systems.

Index Terms

decomposition, differential geometry, Lagrangian systems, motion coordination, passivity

I. I NTRODUCTION Consider a group of multiple mechanical systems whose Lagrangian dynamics evolves on their

n-dimensional

(connected and complete) product conguration manifold

M

with the sum

of their kinetic energies as the Lagrangian. Suppose that a certain coordination aspect among

h(q) ∈ N , where h : Mn → N m manifold N (n ≥ m) and q ∈ M is

their motions can be described by from

M

to a

m-dimensional

is a smooth submersion [1] the system's conguration.

Submitted to IEEE Transactions on Automatic Control April 2007. Paper Type: Regular. Dongjun Lee is with the Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee at Knoxville, 502 Dougherty Hall, 1512 Middle Dr. Knoxville, TN 37996 USA 1-865-974-7688 . Phone: +1-865-974-5309, Fax: +1-865-974-5274, E-mail:

[email protected].

Perry Y. Li is with the Department of Mechanical Engineering, University of

Minnesota, 111 Church St. SE, Minneapolis, MN 55455 USA. E-mail:

[email protected].

Research supported in part by NSF CMS-9870013 and Doctoral Dissertation Fellowship of University of Minnesota.

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2

This description is equally applicable to a single mechanical system as well, with

h(q) specifying

a certain coordination aspect among its internal motion. The main contribution of this paper is to reveal a fundamental property of the mechanical systems in the above setting: its system whose

m-dimensional

n-dimensional

dynamics can be decomposed into 1) the shape

dynamics describes the evolution of

h(q)

on

N

(i.e. coordination

(n−m)-dimensional dynamics on the level set Hh(q) := {p ∈ M | h(p) = h(q)} containing the current conguration q (i.e. overall motion of coordinated group); and 3) the coupling between them, which is a function of (q, q) ˙ , quadratic in q˙, and denes energetically conservative internal energy shufing between the two systems. The

aspect); 2) the locked system which represents the

(decoupled) locked and shape systems also individually inherit passivity from the mechanical systems (i.e. skew-symmetric property in coordinates [2]), i.e. each of them is passive with its kinetic energy and mechanical power as storage function and supply rate [3]. This mathematical result is pertinent to numerous applications and has enabled us to generate new control frameworks for some of them: multirobot coordination and formation ying [4], [5], [6], [7] (internal group formation/overall group maneuver), multirobot cooperative grasping [8] (cooperative grasping/motion of the grasped object), and robotic teleoperation [9], [10] (masterslave coordination/dynamics of the coordinated teleoperator), where the two phrases in each parenthesis respectively point to what aspects the shape and locked systems can describe in each application with a suitably dened

h.

For these applications, it is desired and often necessary to

control the locked and shape systems separately and individually without any crosstalk between them. For instance, in the multirobot cooperative grasping, this crosstalk may cause the grasped object to be dropped as the robot team moves [11]. To avoid such harmful crosstalk, we need to cancel out the coupling and decouple the locked and shape systems from each other. One of unique and powerful properties of the presented decomposition is that this decoupling can be achieved while preserving passivity of the mechanical systems. This is a direct consequence of that the coupling is energetically conservative so that its cancellation does not require or generate any net energy. This passive decoupling property turns out to be very powerful for such applications as multirobot cooperative grasping [8] and robotic teleoperation [9], [10], in which we want, not only to decouple the locked and shape systems from each other, but also to enforce passivity of the closed-loop system so that its interaction with any passive humans/environments (with compatible supply rates) can be guaranteed to be stable without relying on their detailed models [12], [13]. Once the coupling is cancelled out, a variety of passivity-based control laws (e.g. proportional-derivative control) are readily applicable for controlling the decoupled locked and shape systems. This is possible, since both of them inherit passivity from the mechanical systems. Due to this passive decoupling property and the inherited passivity of the locked and shape systems, we call the presented decomposition passive decomposition. One obvious way to achieve the locked and shape decoupling is to cancel out all the (nonlinear)

April 19, 2007

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3

open-loop dynamics of the mechanical systems and replace it with a certain desired (usually linear) dynamics as done in [14], [15]. Passivity of the system would then be implied by the passivity of the implemented dynamics. For this, the external force is usually cancelled out rst and, after the cancellation of the open-loop dynamics, re-installed with the desired dynamics (See [16] for an example). This procedure generally requires unrealistic perfect-cancellation of the open-loop dynamics as well as expensive accurate force sensing. On the other hand, in [17], (passive) nonlinear mechanical systems are directly connected to a (passive) virtual object by (passive) springs/dampers, and the shape and locked behaviors are controlled by adjusting parameters of the springs/dampers and pulling the virtual object, respectively. Thus, although passivity of the closed-loop system is granted, the locked and shape systems' behaviors are in general perturbed by each other, since the coupling between them is left intact there. In contrast to these approaches, the passive decomposition presented here enables us to achieve the lockedshape decoupling and preserve passivity of the mechanical systems at the same time, by using sensings of only

(q, q) ˙

and utilizing nonlinear dynamics of the mechanical systems rather than

unnecessarily cancelling it out. The terms “locked” and “shape” are adopted from the reduced Lagrangian approach [18], [19] which provides a similar decomposition for the mechanical systems possessing principle bundle structure with symmetry. Although many mechanical systems have this symmetry, still many others do not (e.g. tele-robot consisting of two multi-degree-of-freedom revolute-joint robots). Moreover, since it is developed mainly for the robotic locomotion problem, it is not clear if this reduced Lagrangian approach is suitable (or how it can be used) for applications where the passivity aspects need to be considered (e.g. robotic teleoperation, multirobot grasping). In contrast, our passive decomposition does not require such symmetry, and comes with the powerful passivity properties: passive decoupling property and inherited passivity of the locked and shape systems. There is a vast body of literature on the constrained mechanical systems with holonomic constraints, that is,

h(q) = c

with a constant

c

[20], [21], [22]. Thus, they are not suitable

when we need to control the coordination aspect

h(q)

h(q)

itself. However, a direct control of

is often desired or required in many applications. Our passive decomposition enables

us to do so, by allowing us to directly control the shape system. For instance, consider the multirobot cooperative grasping with

h

describing the grasping shape among the robots. In the

aforementioned constrained schemes [20]-[22], it is necessary to somehow rigidly maintain the grasping shape to satisfy the condition

h(q) = c.

This often necessitates using a rigid xture

among the robots [14], [15], [20], which needs to be exchanged whenever the size/shape of the object to be grasped is changed. On the other hand, with the passive decomposition, we can directly adjust the grasping shape

h(q) by controlling the shape system. Thus, even without using

a rigid xture, we can achieve the grasping of objects with various size/shape (i.e. xture-less

April 19, 2007

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grasping [11]). Several results based on the passive decomposition have been previously reported in [4], [5], [6], [7], [8], [9], [10]. However, all of them are derived in coordinates and for specic applications. Thus, many of the intrinsic and geometric properties of the passive decomposition and the locked and shape systems could not be revealed there. In this paper, relying on the concepts from differential geometry, we rigorously exhibit and analyze such intrinsic and geometric properties of the passive decomposition and its related mathematical objects. A portion of this paper has been presented in [23]. The rest of the paper is organized as follows. Sec. II introduces some mathematical concepts needed for the development of the passive decomposition. In Sec. III, we dene the passive decomposition and elaborate its geometric/energetic properties along with those of the locked and shape systems. To give a avor of control design while manifesting the properties of the passive decomposition, we provide a control design example in Sec. IV, which stabilizes the shape system while guiding the locked system along a velocity eld. Sec. V contains some concluding remarks.

II. P ROBLEM F ORMULATION A. Mathematical Preliminary and Geometry of Single Mechanical System In this section, we present a geometric formulation of the dynamics of mechanical systems. For more details, refer to standard textbooks [1], [24], [25] and some recent thesis [26], [27], [28].

n-dimensional mechanical system a Lagrangian dynamical system, which evolves on a n-dimensional smooth conguration manifold M and whose Lagrangian is given solely by its kinetic energy. At every q ∈ M, we can dene the tangent space Tq M and ∗ ∗ its dual cotangent space Tq M. Both Tq M and Tq M are n-dimensional vector spaces. Then, at time t, the position of the mechanical system can be represented by a point q(t) ∈ M, while its d velocity v(t) = q(t) and external/control forces F (t), T (t) are given by entities in Tq M and dt ∗ Tq M, respectively. A smooth vector eld X (or, covector eld w , resp.) on M is a smooth assignment of a tangent vector X(q) ∈ Tq M (or, cotangent vector w(q), resp.) at each q ∈ M. As in [10], [29], In this paper, we mean by a

this vector eld can be used to encode a motion guidance objective by specifying a desired velocity at each position. We denote the set of all smooth vector elds (or, covector elds, resp.)

X(M) (or, by X∗ (M), resp.). We also denote the set of all real smooth functions on M by C ∞ (M). A (s, k)-type tensor eld T on M is a smooth multilinear mapping at each q ∈ M s.t. T : Tq∗ M × · · · × Tq∗ M × Tq M × · · · × Tq M → )⊥ = ∇q˙ = v˜j ∇q˙ + Lq˙ (˜ vj ) ∂ q˜j ∂ q˜j j=1 j=1 "n−m à n !#⊥ n X X X ∂ d˜ v ∂ j ˜ k (q) Γ = v˜i v˜j + ij ∂ q ˜ dt ∂ q˜j k j=1 i=1 k=1 =

n n−m X X

n X

˜ kij (q) v˜i v˜j Γ

i=1 j=1 k=n−m+1

Pn Pn

¶#⊥

∂ . ∂ q˜k

Pn−m

˜ k (q) ∂ . Therefore, implementation of the v˜i v˜j Γ ij ∂ q˜k requires sensings of only (q, q) ˙ , which are usually available in most practical applications. This also shows that the coupling is quadratic in q˙. d To present the passive decoupling property of T (q, q) ˙ , we need the following denitions. We (∇q˙ v ⊥ )> = d decoupling control T Similarly,

i=1

j=n−m+1

k=1

say that the closed-loop mechanical system (8) is (energetically) passive (i.e. passive with the mechanical power as the supply rate [3]), if there exists

Z

t

d∈
, vi dτ = 0 | {z } =0

(34)

from (28)

t ≥ 0. This implies that the decoupling control T d (q, q) ˙ satises controller passivity (31).1

Therefore, we can decouple the locked and shape systems from each other while enforcing passivity (30). In other words, the decoupling control (29) preserves passivity of the openloop mechanical systems, that is, even with the decoupling control (32) with only

T

replaced by

T? .

T d,

the system still satises

We call this property passive decoupling property. This

property is particularly useful when the system interacts with humans/external environments (e.g. teleoperation [10], multirobot cooperation [8]). This is because, by enforcing passivity (30), we can ensure stable interaction with any passive humans/environments without relying on their detailed models [12]. Of course, if we want additional useful behaviors of the closed-loop

1

This is still true even when an incorrectly estimated inertia is used in computing the decoupling control, since, in this case,

(28) still holds with

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M (q)

replaced by the incorrect one (see [10] for example).

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16

system on the top of the decoupling, we need an additional control

T?

in (29). In that case, from

(33) and (34), passivity (30) can still be ensured by enforcing controller passivity of only

T?

as

done in Sec. IV. Note also from (26)-(27) that, once the couplings are eliminated (i.e. without the last terms of (26)-(27)), both the locked and shape systems will possess property similar to (32), i.e., just like the mechanical system (8), each of them is passive with its kinetic energy (e.g. in (26)) and mechanical power (e.g.

hF > + T > , v > i

1 hhv > , v > ii 2

in (26)) as storage function and supply

rate [3]. Therefore, the locked and shape systems “inherit” passivity of the mechanical system. This inherited passivity is very useful for control design, since it makes various passivity-based control laws developed for usual mechanical systems readily applicable for the locked and shape systems. See Sec. IV for example. Due to this inherited passivity and the passive decoupling property explained above, we name the presented decomposition passive decomposition. From (23), the passive decoupling control (29) gives us the following decoupled dynamics:

M ∇dq˙ v = M (∇q˙ v > )> + M (∇q˙ v ⊥ )⊥ = T? + F where the decoupled connection

∇d

is dened s.t.: for

X, Y ∈ X(M),

∇dX Y := (∇X Y > )> + (∇X Y ⊥ )⊥ . Proposition 2 The decoupled connection Proof:

The decoupled connection

∇d

∇d

is afne and compatible w.r.t. the metric

M.

satises the properties of the afne connection in Sec.

II-A. Among them, here, we show only the third property, since others are easy to prove: for

f ∈ C ∞ (M)

and

X, Y ∈ X(M),

∇dX (f Y ) = (∇X (f Y )> )> + (∇X (f Y )⊥ )⊥ = (f ∇X Y > + LX (f )Y > )> + (f ∇X Y ⊥ + LX (f )Y ⊥ )⊥ = f (∇X Y > )> + f (∇X Y ⊥ )⊥ + LX (f )(Y > + Y ⊥ ) = f ∇dX Y + LX (f )Y. Also,

∇d

is compatible w.r.t. the metric

M,

since, for every

X, Y, X ∈ X(M),

we have

LX hhY, Zii = LX hhY > , Z > ii + LX hhY ⊥ , Z ⊥ ii = hh∇X Y > , Z > ii + hh∇X Y ⊥ , Z ⊥ ii + hhY > , ∇X Z > ii + hhY ⊥ , ∇X Z ⊥ ii = hh(∇X Y > )> + (∇X Y ⊥ )⊥ , Zii + hhY, (∇X Z > )> + (∇X Z ⊥ )⊥ ii = hh∇dX Y, Zii + hhY, ∇dX Zii.

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From the uniqueness of the Levi-Civita connection

∇

on

M,

this compatible

∇d

is generally

not torsion-free.

D. Shape System Dynamics: Connection over the Coordination Map

h

The coordination aspect of the mechanical system (8) is represented by the mapped point

d h(q) dt

= h∗ (v ⊥ ) ∈ Th(q) N from (21). Therefore, if we can nd the dynamics of the mapped point h(q) on N in a meaningful way, the coordination aspect of the system (8) may be completely described on the m-dimensional coordination manifold N . h In order for this, we dene the shape system connection ∇ to be a connection over the h coordination map h : M → N as below. Roughly speaking, ∇ enables us to map the original dynamics (8) from M to N in a natural way. For more details on the connection over a map, h see Appendix. For dening ∇ , we need the following proposition. h(q) ∈ N ,

while its velocity on

N

is given by

h, X h , which is a smooth map X h : M → T N s.t. X h (q) ∈ Th(q) N for every q ∈ M. Let Xh (M) denote the set of such h h smooth vector elds over h. Then, for each X ∈ X (M), there exists a unique smooth vector ⊥ ⊥ ⊥ ⊥ h eld X ∈ X(M) s.t. X ∈ ∆ and h∗ (X ) = X . Proposition 3 Consider a smooth vector eld over the map

X h ∈ Xh (M), we can nd a unique tangent vector Xq⊥ ∈ ∆⊥ (q) ⊂ Tq M at h∗q (Xq⊥ ) = X h (q), since 1) both ∆⊥ (q) and Th(q) N are m-dimensional vector

Proof: For a given every

q∈M

s.t.

spaces; and 2)

h∗q

h∗

is linear and surjective (as assumed in Sec. II-C). This actually means that

is bijective, if its domain is restricted to

over all

q ∈ M,

∆⊥ (q).

we can construct a unique vector eld

where smoothness of

X⊥

Xq⊥ X h ∈ Xh (M),

Thus, by collecting this tangent vector

follows from the smoothness

X ⊥ ∈ ∆⊥ associated to h h of h and X ∈ X (M).

∇h : X(M) × Xh (M) → Xh (M) ¡ ¢⊥ ∇hY X h := h∗ ∇Y X ⊥

We dene the shape system connection

s.t. (35)

X h ∈ Xh (M), h∗ is the push-forward of h, and X ⊥ ∈ X(M) is the h h smooth vector eld on M uniquely associated to X ∈ X (M) as constructed in Proposition h 3. This ∇ is a connection over the coordination map h (see Appendix). h h We also dene the induced metric on N s.t. for v1 , v2 ∈ Th(q) N ,

where

Y ∈ X(M)

and

hhv1h , v2h iiN := hhv1⊥ , v2⊥ iiM

(36)

hh , iiM is the inertia metric on M, and v1⊥ , v2⊥ ∈ Tq M are the uniquely-given counterparts v1h , v2h ∈ Th(q) N as constructed in Proposition 3 s.t. v1⊥ , v2⊥ ∈ ∆⊥ (q) and h∗q v1⊥ = v1h ,

where of

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18

h∗q v2⊥ = v2h . We denote this T h : Tq M × Tq M → Th(q) N

N by Mh (q). Let pair v1 , v2 ∈ Tq M,

induced metric on of

∇

h

s.t., for a

us also dene the torsion

T h (v1 , v2 ) = ∇hX h∗ Y − ∇hY h∗ X − h∗ [X, Y ]

(37)

X, Y ∈ X(M) are local extensions of v1 , v2 ∈ Tq M, and [, ] is the Lie-bracket on M. torsion Th (v1 , v2 ) only depends on v1 , v2 , and not on their extensions X, Y (i.e. it is a local

where This

property) [24, Sec. 5.10]. Theorem 1 The shape system connection

∇h

is afne and compatible w.r.t. the induced metric

Z ∈ X(M)

and

X h , Y h ∈ Xh (M),

(36) in the sense that: for all

LZ hhX h , Y h ii = hh∇hZ X h , Y h ii + hh∇hZ Y h , X h ii. Moreover, when restricted on

∆⊥ ,

its torsion

Th

(37) vanishes and

like) unique torsion-free and compatible connection over the map Proof:

∇h becomes the (Levi-Civita h w.r.t. the metric (36).

Among the properties of the afne connection in Appendix, we only prove the third

f ∈ C ∞ (M), Y ∈ X(M), and X h ∈ Xh (M), ¡ ¢⊥ ¡ ¢⊥ ∇hY f X h = h∗ ∇Y f X ⊥ = h∗ LY (f )X ⊥ + f ∇Y X ⊥ ¡ ¢⊥ = LY (f )h∗ (X ⊥ ) + f h∗ ∇Y X ⊥

one, since others are easy to prove: for every

= LY (f )X h + f ∇hY X h X h ∈ Xh (M) as found in Proposition 3. h h h The compatibility of ∇ can be shown as follows: for Z ∈ X(M) and X , Y ∈ Xh (M) ⊥ ⊥ with X , Y ∈ X(M) being their associated vector elds on M as in Proposition 3, we have, ¡ ¢⊥ ¡ ¢⊥ LZ hhX h , Y h ii = LZ hhX ⊥ , Y ⊥ ii = hh ∇Z X ⊥ , Y ⊥ ii + hh ∇Z Y ⊥ , X ⊥ ii ¡ ¢⊥ ¡ ¢⊥ = hhh∗ ∇Z X ⊥ , h∗ (Y ⊥ )ii + hhh∗ ∇Z Y ⊥ , h∗ (X ⊥ )ii

where

X ⊥ ∈ X(M)

is uniquely associated to

= hh∇hZ X h , Y h ii + hh∇hZ Y h , X h ii from the compatibility of The torsion

T

h

∇

and the denition of

∇h

in (35).

∇ is restricted on ∆⊥ , because, if X, Y ∈ ∆⊥ , ¢⊥ ¢⊥ ¡ ¡ T h (X ⊥ , Y ⊥ ) = h∗ ∇X ⊥ Y ⊥ − h∗ ∇Y ⊥ X ⊥ − h∗ [X ⊥ , Y ⊥ ] ¡ ¢ = h∗ ∇X ⊥ Y ⊥ − ∇Y ⊥ X ⊥ − [X ⊥ , Y ⊥ ] (37) vanishes if

h

we have

= h∗ (T (X ⊥ , Y ⊥ )) = 0 from the torsion-free property of

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∇,

where

T

is the torsion of

∇

dened in (3).

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19

¯h D

Finally, suppose that when restricted on

⊥

∆

h, which is compatible and torsion-free ¯ h (i.e. its Christoffel connections, this D

is a connection over the map

. Then, like usual Levi-Civita

h k symbols Γij (55)) is uniquely given by the following Koszul's formula:

¯ h ⊥ Y h , Z h ii =X ⊥ hhY h , Z h ii + Y ⊥ hhZ h , X h ii − Z ⊥ hhX h , Y h ii 2hhD X + hhY h , h∗ [Z ⊥ , X ⊥ ]ii + hhZ h , h∗ [X ⊥ , Y ⊥ ]ii − hhX h , h∗ [Y ⊥ , Z ⊥ ]ii where

⊥

h∗ ?

?⊥ ∈ ∆⊥ = ?h . This

and

?h ∈ Xh (M)

are uniquely associated according to Proposition 3 s.t.

Koszul's formula can be obtained by permutating the above compatibility

and torsion-free conditions restricted on

∆ ⊥ , ∇h the map h.

when restricted on connection over

If not restricted on

∆⊥ .

Since

∇h

also satises this,

¯ h = ∇h , D

i.e.

becomes this (Levi-Civita like) unique torsion-free and compatible

∆⊥ , this shape system connection ∇h

in general does not satisfy the above

¯ h . Using (35) D species ∇X Y when

Koszul's formula, thus, will not be the above unique connection over the map, and (36), we can show that the above Koszul's formula also (uniquely)

X, Y ∈ ∆⊥ . This means that ∇h directly inherits its structure from the Levi-Civita ∇ when restricted on ∆⊥ . Note also that if h = id (i.e. M = N ), the shape system ∇h in (35) will become the unique Levi-Civita connection ∇ on M.

connection connection

Using the denition of the shape system connection (35), we have

∇hq˙ h∗ (v ⊥ ) = h∗ (∇q˙ v ⊥ )⊥ . This shows that the shape system dynamics (24b) indeed describes the dynamics of

N, ∇h

if the coupling term

M (∇q˙ v > )⊥

h(q)

on

is cancelled out by the decoupling control (29). Since

is compatible (albeit not necessarily torsion-free), we can use many passivity-based control

techniques for controlling this (decoupled) shape system by considering it pretty much as a usual passive mechanical system on

N.

See Sec. IV and [4], [5], [10].

E. Projection of the Locked System on a (n

− m)-dimensional

Suppose that there exists a smooth submersion manifold

l

from

M

Manifold

to a (n

− m)-dimensional

smooth

L l : Mn → Ln−m l∗ : Tq M → Tl(q) L q ∈ M and v ∈ Tq M,

such that its push-forward similar to (21): for all

satises the following projectability condition

d l(q) = l∗ (v) = l∗ (v > + v ⊥ ) = l∗ (v > ) ∈ Tl(q) L dt

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(38)

(39)

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20

that is,

l∗ (v ⊥ ) = 0.

We call such a map

l

and a manifold

system manifold, respectively. We also call this pair If there exists such a pair

(l, L),

L,

locked system map and locked

(l, L) a projection pair of the locked system.

we can handle the locked system much like as we did the

l, X l ∈ Xl (M), we can > > > > l nd its unique counterpart X ∈ X(M) s.t. X ∈ ∆ and l∗ (X ) = X ; 2) we can dene the l l l locked system connection ∇ : X(M) × X (M) → X (M) as a connection over the map l s.t. l l for Y ∈ X(M) and X ∈ X (M), ¡ ¢> ∇lY X l := l∗ ∇Y X > (40)

shape system in Sec. III-D: 1) for a smooth vector eld over the map

where map

l,

∇

is the Levi-Civita connection on

M;

and 3) this

is compatible w.r.t. the induced metric on

L

∇l

is an afne connection over the

(dened similar to (36)), and will be the

(Levi-Civita like) unique torsion-free and compatible connection over the map

X, Y ∈ ∆

>

l

if we restrict

.

With this projection pair

(l, L),

the (decoupled) locked system dynamics can be projected on

(n−m)-dimensional manifold L, with l(q) ∈ L, dl(q)/dt = l∗ (v > ) ∈ Tl(q) L, and ∇lq˙ l∗ (v > ) = l∗ (∇q˙ v > )> as its position, velocity, and dynamics on L, respectively. Then, by controlling the m n−m (decoupled) shape and locked systems individually on their respective manifolds N and L ,

the

it will be possible to control the coordination aspect (e.g. cooperative grasping) and the overall behavior of the coordinated system (e.g. motion of the grasped object), simultaneously and separately. The decoupling control (29) is in general still needed here though, since the coupling terms in (23) generally exist in this case [26, Sec. 3.5]. As shown in the following theorem, integrability of the normal distribution

∆⊥

is a necessary

(l, L). This, however, is generally not granted > (see [10] for a counterexample). In contrast, note that the tangential distribution ∆ is integrable condition for the existence of such a projection pair

by its construction. Theorem 2 Suppose that there exists a projection pair normal distribution Proof:

⊥

∆

of the locked system. Then, the

is integrable.

l ⊂M

Since the locked system map

m-dimensional

(l, L)

submanifold

Gl(q)

is a smooth submersion, at every

q ∈ M,

there exists a

dened s.t.

Gl(q) := {p ∈ M | l(p) = l(q)} l:M→L ¡ ⊥¢ = 0 for all v ⊥ ∈ ∆⊥ (q), that is, ∆⊥ (q) ⊂ ker(l∗q ) = Tq Gl(q) satises that, at each q ∈ M, l∗q v for every q ∈ M, where l∗q is the push-forward of l at q ∈ M and ker(l∗q ) ⊂ Tq M is its kernel. ⊥ > ⊥ Suppose that ker(l∗q ) 6= ∆ (q) for some q ∈ M. Then, since Tq M = ∆ (q) ⊕ ∆ (q), there exists a tangent vector v ˜ ∈ ∆> (q) s.t. l∗q (˜ v ) = 0 (i.e. v˜ ∈ ker(l∗q )). However, this is impossible, where

p

is a dummy variable. Also, from the projectability condition (39), the map

April 19, 2007

DRAFT

21

since, following the same reasoning in Proof of Proposition 3, l∗q denes a bijective linear map > ⊥ between the two (n−m)-dim. vector spaces, ∆ (q) and Tl(q) L. Thus, ∆ (q) = ker(l∗q ) = Tq Gl(q) . ⊥ m This shows that ∆ is integrable having Gl(q) as its integral manifold at each q ∈ M.

Converse of this theorem holds only locally, since, even if single function

l:M→L

⊥

satisfying l∗ (∆

)=0

∆⊥

is integrable for all

q ∈ M,

a

in general exists only locally [32].

h is designed s.t. its foliation is “parallel” w.r.t. ∇, there exists a locked system projection pair (l, L) and the coupling terms in (23) vanish d (i.e. decoupling control T in (29) is not necessary). Next theorem says that if the coordination map

M of the mechanical system (8) is complete ∇ and simply-connected. Suppose further that ∆ is invariant by the parallel transport P (see ∇ > > Sec. II-A) s.t. Pγ(0)→γ(1) ∆ (γ(0)) = ∆ (γ(1)) for any curve γ(t) ∈ M, t ∈ [0, 1]. Choose a point qo ∈ Hh(qo ) ⊂ M, and denote a smooth curve from qo to a point q ∈ M by z(t), t ∈ [0, 1], s.t. z(0) = qo and z(1) = q . Dene its projection curve z 0 (t) on Hh(qo ) , t ∈ [0, 1], s.t. > ∇ z 0 (0) = qo and z˙ 0 (t) = Pz∇0 (0)→z0 (t) A(t), where A(t) := Pz(t)→z(0) (z(t)) ˙ , and construct a map π 0 : M → Hh(qo ) by π 0 (q) := z 0 (1). Then, (π 0 , Hh(qo ) ) denes a locked system projection pair.

Theorem 3 Suppose that the conguration manifold

>

Moreover, the coupling terms in (23) vanish. Proof:

Tq M ∆> and ∆⊥

Since the parallel transport is a linear map and

invariant by

P ∇.

From the invariance by

group Holq [33] dened s.t., for Holq

P ∇,

both

is a vector space,

∆⊥

is also

are invariant by the holonomy

q ∈ M,

∇ := {Pγ(0)→γ(1) |γ

q}

γ(t) ∈ M s.t. γ(0) = γ(1) = q . Then, following [34, Prop. 5.1, Ch. IV], ∆ is integrable and has a m-dimensional integral manifold Gq for every q ∈ M, which is complete and totally geodesic (i.e. every geodesic γ(t) of M with (γ(0), γ(0)) ˙ ∈ (Gq , Tγ(0) Gq ) stays in Gq all the time). Similarly, Hh(q) is also complete and totally geodesic for every q ∈ M. 0 Let us denote these two submanifolds stemming from qo by Gqo and Hh(qo ) . Similar to z (t), 00 00 00 ∇ dene z (t) ∈ Gqo , t ∈ [0, 1], s.t. z (0) = qo and z˙ (t) = Pz 00 (0)→z 00 (t) B(t), where B(t) := ⊥ ∇ 00 00 00 Pz(t)→z(0) (z(t)) ˙ , and construct a map π : M → Gqo s.t. π (q) := z (1). As shown in [34, 0 00 pp.187], both the maps, π (q) and π (q), depend only on q (i.e. the end-point of the curve z(t)) > ⊥ rather than on a particular choice of curve z(t). Furthermore, since ∆ and ∆ are invariant by the holonomy group and M is complete and simply-connected, from the de Rham decomposition 0 00 theorem, M is isometric to the direct product of Hh(qo ) ×Gqo , and the combined map π := (π , π ) denes an (bijective) isometry of M onto Hh(qo ) × Gqo [34, Th. 6.1, Ch. IV].

where we mean by a loop based at

q

is a loop based at

a curve

⊥

April 19, 2007

DRAFT

22

v

v

v h(q)=c

Coordination map : h

q

h (v)=h (v ) * * h(q)=c h(q)=c1 c1 l(q)=a

h(q)=c2

cd

l (v)=l (v ) * *

Coordination manifold N

Locked manifold L

Product manifold M Fig. 3.

3-dimensional

A visualization of Theorem 3 for a

1-dimensional

straight lines as the level sets

Since the construction of

π 0 (q)

Hh(q)

Euclidean space with the

and the normal submanifolds

Gq ,

uses only the tangential component

2-dimensional

planes and the

respectively.

> 0 (z(t)) ˙ , π

satises the

0

(π , Hh(qo ) ) denes a locked system projection pair. Vanishing of the coupling terms is a direct consequence of the product structure Hh(qo ) ×Gqo , i.e. from (11), ¢ ¢ ¡ ¡ ¯ X¯ Y¯2 = 0, with indexes 1, 2 corresponding ¯ X¯ Y¯1 = 0 and ∇ the coupling terms are given by ∇ 1 2 to the components in Hh(qo ) and Gqo , respectively. projectability condition (39), thus,

One example where the condition of Theorem 3 is trivially satised is when (e.g. point mass dynamics) and the level sets

Hh(q)

M

is Euclidean

are given by at planes (e.g. vectorial position

difference). See Fig. 3 for an illustration. This “at” property has been successfully used for the formation control [4], [5] and the ocking analysis [6], [7] of multiple agents with point mass dynamics. If there does not exist such a projection pair described on any (n 1) if we can

(l, L), the locked L˜, because

− m)-dimensional manifold nd only a map ˜ l : Mn → L˜n−m

system dynamics cannot be

which is not a submersion, some dimensions

of the locked system dynamics cannot be described on

n − m for some q ∈ M; or map ˜ l : Mn → L˜n−m which is

L˜n−m ,

since the Jacobian of

˜l then

has the rank less than 2) if we can nd a

a submersion but fails to satisfy the

˜l(q(t)) ∈ L˜ ˜n−m , achieved on L

projectability condition (39), the basic kinematic relation between the position and the velocity

April 19, 2007

˜l∗ (v > (t)) ∈ T˜ L˜ l(q)

of the locked system can not be

DRAFT

23

since, for some

q∈M

and

v ∈ Tq M ,

we have

d˜ l(q(t)) = ˜l∗ (v) = ˜l∗ (v > ) + ˜l∗ (v ⊥ ) 6= ˜l∗ (v > ) dt i.e. the position is not given by the integration of the velocity.

(l, L) does not exist, the locked system does not (n − m)-dimensional manifold. Thus, control objectives

An implication of this is that, if such a pair have a well-dened conguration on a

requiring the locked system's conguration (e.g. position tracking) would be infeasible. Even so, control objectives dened on the locked system's velocity is always feasible (e.g. velocity tracking), since its velocity is well-dened. In this case, we will be able to exploit the locked system's inherited passivity (26) for control design (e.g. passivity-based control). See Sec. IV for example. Even if there does not exist such a pair

(l, L),

once we achieve

h(q) = c, the locked system = {p ∈ M | h(p) = c}.

n−m dynamics becomes the Levi-Civita connection on the level set Hc Then, we can control it on position on

Hc .

Hc

much as we do the usual mechanical systems with a well-dened

See Remark 1.

IV. C ONTROL D ESIGN E XAMPLE : S HAPE S YSTEM S TABILIZATION

AND

L OCKED S YSTEM

V ELOCITY F IELD F OLLOWING In this section, we present a design example for the additional control

T? in (29). The presented

control consists of two separate controls: 1) PD (proportional-derivative) control to stabilize the shape system to a certain desired (constant) point

cd ∈ N ;

and 2) PVFC (passive velocity eld

Vl ∈ X(Hcd ) on the stabilized on Hcd by the PD-

control) [29] to guide the locked system to follow a certain velocity eld level set

Hcd := {p ∈ M | h(p) = cd },

once the system (8) is

control. We choose these PD-control and PVFC due to their passivity properties, with which we can manifest the passivity properties of the decomposition and the decoupled dynamics as shown below. For more control design examples and their applications, see [4], [5], [6], [7], [8], [9], [10], [11], [26]. We design the additional control

Tl ∈ Ω>

T?

in (29) such that

T? = Th + Tl ,

where

Th ∈ Ω⊥

and

will be designed to embed the PD-control and the PVFC, respectively. Then, from (23)

with (29), the closed-loop dynamics of the locked and shape systems are decoupled s.t.

M (∇q˙ v > )> = Tl + F >

and

M (∇q˙ v ⊥ )⊥ = Th + F ⊥

(41)

F > ∈ Ω> and F ⊥ ∈ Ω⊥ are the tangential and normal components of F . Therefore, designing Tl , Th , we can control the locked and the shape systems (41) individually and

where by

separately. This separation is useful when we want the coordination aspect (e.g. cooperative grasping) and the overall behavior of the coordinated system (e.g. motion of the grasped object) do not affect each other.

April 19, 2007

DRAFT

24

(h(q), h∗ (v ⊥ )) → (cd , 0), we rst design it back to M s.t. £ ¤ Th (q, v ⊥ ) := h∗ −Kh (h(q))h∗ v ⊥ − dϕh (h(q))

Following [35], [36], to achieve

N

control on

and pull

a stabilizing PD-

(42)

where 1)

2)

3)

Kh (c) : Tc N → Tc∗ N is a smooth dissipation eld on N s.t. ∀v h ∈ Tc N , hKd (c)(v h ), v h iN ≥ 0 with the equality hold if and only if v h = 0; ϕh : N → < is a non-negative smooth potential function designed on N s.t., if ϕh (c) < L with L > 0 being a nite constant, i) ϕh (c) = 0 if and only if c = cd ; and ii) its one-form, dϕh (c), vanishes, if and only if c = cd ; ∗ N → Tq∗ M is the pull-back map of h s.t. hh∗ wh , viM = hwh , h∗ (v)iN for all h∗ : Th(q) ∗ N . This implies that h∗ wh ∈ Ω⊥ (q), since, from (21), for any v ∈ Tq M and wh ∈ Th(q) ∗ N . Thus, Th (q, v ⊥ ) ∈ Ω⊥ (q). ∀v ∈ ∆> (q), hh∗ wh , viM = hwh , 0iN = 0 ∀wh ∈ Th(q)

(h(q), h∗ (v ⊥ )) → (cd , 0), the evolution of the mechanical system (8) is constrained on the level set Hcd . We suppose that a (smooth) velocity eld Vl ∈ X(Hcd ) encodes a desired behavior of this constrained system on Hcd . More precisely, on Hcd , Once the PD-control (42) stabilizes

we want

v > (t) → α(t)Vl (q(t)) α(t) ∈
(t)

This objective (43) can be easily achieved on the single level set

as the standard PVFC on

Tl

(43)

as the PVFC on

Vl

from

Hcd

Hcd )

while keeping the

to the (current) level set

Hh(q) .

This lifting idea implicitly assumes that the topology of

Hcd

and

Hh(q)

are more or less the

same. In our case, this is granted, because, following [37, Prop.3.2] with the completeness of

M

and the orthogonality of the decomposition (22), all the level sets of 2

each other.

h

are diffeomorphic to

Then, by concatenating such diffeomorphisms, we can construct lifting map

¯l : Mn → Hn−m cd 2

Even without resorting to [37], all the level sets are at least locally diffeomorphic to each other, since, from

submersion (see Sec. II-C), they are all locally diffeomorphic to

April 19, 2007

(44)

h

being a

(t) → α(t)Vlq (q(t))

(45)

(h(q), h∗ (v ⊥ )) → (cd , 0). (45) on each level set Hh(q) , we design Tl in (41) as the PVFC on Hh(q) µ ¶ wl (q, q) ˙ > − σl · v > c (Pl (q) ∧ pl (q, q)) ˙ Tl (q, q) ˙ := v c Pl (q) ∧ 2El

which converges to (43) if To achieve

σl ∈ < is a control gain, c and ∧ are respectively the (1)-(2), and Pl (q), pl (q, q), ˙ wl (q, q) ˙ ∈ Tq Hh(q) are dened by

where

s.t. (46)

contraction and the wedge product

pl (q, q) ˙ := M (q)v > Pl (q) := M (q)ξl (q)Vlq (q) wl (q, q) ˙ := M (q) (∇q˙ ξl (q)Vlq (q))> with

ξl (q)

being a scalar function dened to satisfy the constant energy condition s.t.

1 El = hhξl (q)Vlq (q), ξl (q)Vlq (q)ii 2 where El is a positive constant. Using the properties of c and ∧ (1)-(2), we that hTl (q, q), ˙ v 0 i = 0, ∀v 0 ∈ ∆⊥ (q). This shows that Tl (q, q) ˙ ∈ Ω> (q). For

can show from (46) more details on the

PVFC, refer to [29], [38]. Theorem 4 Consider the mechanical system (8) under the decoupling control PD-control

Th

in (42), and the PVFC

1) The total control

T

Tl

Td

in (29), the

in (46).

in (29) ensures controller passivity (31), and if the external environment

is passive in the sense that there exists a nite constant

Z

t

r∈
0

is bounded;

to have the following properties [36]: s.t.

∀v1h , v2h ∈ Th(q) N ,

a1 kv1h kkv2h k ≥ hKh v1h , v2h iN ≥ a2 kv1h kkv2h k April 19, 2007

q˙

(48)

DRAFT

26

b1 ≥ b2 > 0

• there exist constants

s.t. if

ϕh (h(q)) < L,

b1 kdϕh k2 ≥ ϕh ≥ b2 kdϕh k2 c1 , c2 ≥ 0

• there exist constants

kdϕh k :=

Suppose further that

v h ∈ Th(q) N

h h∇hq˙ dϕh , v h i ≤ c1 kv h k2 + c2 kqkkdϕ ˙ h kkv k

q where

s.t. for any

(49)

hdϕh , Mh−1 dϕh i F⊥ = 0

with

Mh

being the induced metric (36) on

and the environment (with

F = F >)

(50)

N.

satises (47). Dene

1 V 0 (t) := hhv ⊥ , v ⊥ ii + ϕh (h(q)). 2

(51)

V 0 (0) < L, (h(q), h∗ (v ⊥ )) → (cd , 0) exponentially. > 3) If F = 0, v > → βl ξl (q)Vlq (q) exponentially from almost all initial conditions (except zeroq > measure unstable equilibria v = −βl ξl (q)Vl (q)), where βl is a constant scalar given by βl = q Then, if

1 hhv > , v > ii/El . 2

sign(σl ) Proof:

hT, vi = hT d , vi + hTh , vi + hTl , vi, where h∗ v = h∗ v ⊥ from (21) and dϕh (h(q))/dt = hdϕh , h∗ v ⊥ i,

1) The total control power is given by

hT d , vi = 0

from (34); b) using

hTh , vi = h−Kh h∗ v ⊥ − dϕh , h∗ v ⊥ i = −hKh h∗ v ⊥ , h∗ v ⊥ i − c

and c) from the properties of

and

∧

a)

d d ϕh (h(q)) ≤ − ϕh (h(q)) dt dt

(1)-(2),

wl ), vi − σl hv > c(Pl ∧ pl ), vi 2El ¡ ¢ wl > wl , vi − hPl , vih , v i − σl hPl , v > ihpl , vi − hPl , vihpl , v > i = hPl , v > ih 2El 2El w l = hv > c(Pl ∧ ), v > i − σl hv > c(Pl ∧ pl ), v > i = 0. 2El

hTl , vi = hv > c(Pl ∧

Combining these three relations and integrating them with the fact that controller passivity (31):

Z

t

ϕh (h(q)) ≥ 0,

we obtain

∀t ≥ 0,

hT (τ ), v(τ )idτ ≤ −ϕh (h(q(t)) + ϕh (h(q(0)) ≤ ϕh (h(q(0)) =: c2 .

0 By combining the above inequality with (33) and (47), we have,

Z

Z

t

κ(t) = κ(0) +

hF (τ ), v(τ )idτ + 0

This proves that

q˙

t

is bounded

∀t ≥ 0,

hT (τ ), v(τ )idτ ≤ κ(0) + r2 + c2 .

0

∀t ≥ 0,

since

κ(t)

in (6) is positive-denite and quadratic in

q˙.

∗ ⊥ ⊥ ∗ ⊥ −1 ∗ 2. Note that, for any wh ∈ Th(q) N and v ∈ ∆ (q), we have hh wh , v i = hhM h wh , v ⊥ ii = hhh∗ M −1 h∗ wh , h∗ v ⊥ ii = hwh , h∗ v ⊥ i = hhMh−1 wh , h∗ v ⊥ ii, where M and Mh are the metric on M and its induced metric (36) on N , respectively. This shows that h∗ M −1 h∗ = Mh−1 . Using

April 19, 2007

DRAFT

27

this and the denition of

N

∇h

in (35), we can rewrite the closed-loop shape system dynamics on

by

Mh (q)∇hq˙ h∗ v ⊥ + Kh (h(q))h∗ v ⊥ + dϕh (h(q)) = Fh Fh := Mh h∗ M −1 F ⊥ . 0 Choose V in (51) as a Lyapunov

(52)

where

function. Then, if

F ⊥ = 0,

we have

d dV 0 (t) = hMh ∇hq˙ h∗ v ⊥ , h∗ v ⊥ i + ϕh (h(q)) = −hKh h∗ v ⊥ , h∗ v ⊥ i ≤ 0 (53) dt dt h ⊥ where we use (7), (36), the property of ∇ in Theorem 1, and dϕh (h(q))/dt = hdϕh , h∗ v i. 0 0 ⊥ Thus, V (t) ≤ V (0) < L and (h(q), h∗ v ) = (cd , 0) is Lyapunov stable. This also implies that 0 the condition (49) is ensured ∀t ≥ 0, since, with (51), ϕh (h(q(t)) ≤ V (t) < L. 0 To show the exponential convergence, we dene V by augmenting V (51) with a cross⊥ coupling term Vcr := hdϕh , h∗ v i [36], [39] s.t. Ã √ !T " #Ã √ ! √² 1 − ϕ ϕ h h 2 b2 V (t) := V 0 (t) + ²Vcr (t) ≥ 1 kh∗ v ⊥ k kh∗ v ⊥ k − 2√²b2 2 where the inequality comes from

hdϕh , h∗ v ⊥ i ≤ kdϕh kkh∗ v ⊥ k

and (49). Then, following (56),

dVcr (t) = h∇hq˙ dϕh , h∗ v ⊥ i + hdϕh , Mh−1 [−Kh h∗ v ⊥ − dϕh ]i dt ⊥ 2 ⊥ ≤ c1 kh∗ v ⊥ k2 + c2 kqkkdϕ ˙ h kkh∗ v k − kdϕh k + a1 kdϕh kkh∗ v k hdϕh , Mh−1 Kh h∗ v ⊥ i = hKh h∗ v ⊥ , Mh−1 dϕh i ≤ a1 kh∗ v ⊥ kkdϕh k, since kMh−1 dϕh k2 = hhMh−1 dϕh , Mh−1 dϕh ii = hdϕh , Mh−1 dϕh i = kdϕh k2 . 0 Combining dV (t)/dt and dVcr (t)/dt with (48), we have à √ #à √ !T " ! ² −²¯ p ϕh ϕ dV (t) h b1 ≤− dt kh∗ v ⊥ k kh∗ v ⊥ k −²¯ p a2 − ²c1 √ where p ¯ = (a1 + c2 kqk)/ ˙ 4b2 is bounded, since kqk ˙ is bounded (from item 1 of this Theorem). Thus, we can always nd a small enough ² > 0 so that V (t) and −dV (t)/dt are positive1 ² √ ⊥ 2 denite. Moreover, since V (t) ≤ ϕh + ||h∗ v || + √ ϕh ||h∗ v ⊥ ||, with this ² > 0, we can 2 b2 nd a constant λ > 0 s.t. dV (t)/dt ≤ −λV (t). This completes the proof. where we use (48), (50) and (52) with the fact that

3. Here, we provide only a sketch of proof, since it is similar to the standard proof of PVFC in [29]. First, we can show that

¶ µ wl =0 M (q) (∇q˙ ξl Vl ) − ξl Vl c Pl ∧ 2El q >

q

(54)

v ∈ Tq M, we have ¶ µ wl > wl wl q , vi = hwl , v > i − hPl , ξl Vlq ih , v i + hPl , v > ih , ξl Vlq i = 0 hwl − ξl Vl c Pl ∧ 2El 2El 2El

since, for all

April 19, 2007

DRAFT

28

> > where we use (1)-(2), the denitions of wl ∈ Ω and Pl ∈ Ω in the PVFC (46), and the facts q q that hPl , ξl Vl i = 2El and hwl , ξl Vl i = dEl /dt = 0. 1 > > > > > > Here, note that βl is indeed constant, since d( hhv , v ii)/dt = hM (v ) , v i = hTl , v i = 0 2 from the proof of the item 1. Let us multiply (54) by

βl . Then, subtracting it from the closed-loop

locked system dynamics (41) under the PVFC (46), we obtain the following error-dynamics:

µ

wl M (q) (∇q˙ eβ ) − eβ c Pl ∧ 2El

¶

>

where

eβ ∈ ∆> (q)

+ σl v > c (Pl ∧ pl ) = 0

is the velocity eld following error dened by

eβ (t) := v > (t) − βl ξl (q(t))Vlq (q(t)). Let us dene the Lyapunov function

Wβ (t)

s.t.

1 Wβ (t) := hheβ (t), eβ (t)ii. 2 Then, following the same steps in the proof of [29, Theorem 3], we can eventually show that

dWβ (t) = −4σl βl El µ(t)Wβ (t) ≤ −4σl βl El µ(0)Wβ (t) dt µ(t) := (2βl El + hhξl Vlq , v > ii)/(4βl El ) ≥ 0 by the Schwartz's inequality, and the last inequality is because µ(t) ˙ ≥ 0, which can be shown in a same way as [29, Prop.5]. Therefore, Wβ (t) → 0 globally exponentially except the initial condition that µ(0) = 0 which is given by q > the zero-measure unstable equilibria v = −βl Vl .

where

Similar to [36], the conditions (48)-(50) are imposed for exponential stability. Passivity assumption in item 2 of this Theorem is to ensure boundedness of This is trivially satised if

F =0

||q|| ˙

(thus,

h∇hq˙ dϕh , v h i =

m X j,k=1

(h1 , ..., hm )

Ã

M, N ,

are local coordinates of

N,

and

∂2ϕ

h

∂hj ∂hk

this is guaranteed by any smooth

The lifting map

¯l∗ (v ⊥ ) 6= 0).

n

∂ 2 ϕh h h X h j i h ∂ϕh v v − Γ q˙ v ∂hj ∂hk j k i=1 ki k ∂hj

Thus, the condition (50) can be ensured, if compact

¯l

−dV (t)/dt.

!

h k Γji is the Christoffel symbol (55) of ∇h . h j and Γki are bounded. With smooth h and

M

and

ϕh .

in (44) generally does not satisfy the projectability condition (39) (i.e.

A consequence of this is that, if we design the PVFC

simply use its pull-back

Tl = ¯l∗ To

h¯l∗ To , v ⊥ i = hTo , ¯l∗ v ⊥ i = 6 0

if

To ∈ Tq∗ Hcd

on

Hcd

and

instead of (46), it will interfere with the shape system and

the locked-shape separation (41) will break down. This is because

April 19, 2007

in

(i.e. pure motion control problem without interaction). From

(57), coordinate expression of (50) is given by

where

p¯)

¯l∗ (v ⊥ ) 6= 0.

Of course, if

¯l satises

¯l∗ To ∈ / Ω> ,

as we have

the projectability condition

DRAFT

29

(39) so that

(¯l, Hcd )

denes the projection pair, we can do so as we did with the shape system

control (42). However, such a pair generally does not exist (see Sec. III-E). The control design example given here highlights the key properties of the passive decomposition: 1) the locked and shape decoupling can be achieved while enforcing passivity of the closedloop system (i.e. passive decoupling property); and 2) the decoupled locked and shape systems inherit passivity from the mechanical system (8) so that passivity-based control techniques (e.g. PD-control, PVFC) are readily applicable to control them as done here.

V. C ONCLUSION As demonstrated in [4], [5], [6], [7], [8], [9], [10], the passive decomposition presented here provides new powerful frameworks for the control problems in many important applications where we need to control the coordination aspect (e.g. cooperative grasping) of the mechanical system as well as to modulate its overall coordinated dynamics (e.g. motion of the grasped object). Due to its passivity property, this passive decomposition is particularly useful when the closed-system is required to mechanically interact with humans and/or environments (e.g. human-robot mechanical interaction), where coupled stability and limited energy exchange (both the characteristics of safety) are critical. In this paper, we assume that the mechanical system is fully-actuated and does not have any nonholonomic constraints. Since these assumptions are not always granted in many important applications (e.g. cooperative material handling with multiple mobile robots), we believe that further investigation addressing these issues would be demanded as well as rewarding. Also, as the number of deployed agents increases, how information ows among them (i.e. information topology) becomes more important. How to incorporate such information-ow related issues into this presented framework is another direction for future research. For some preliminary results in this direction, see [5], [6], [7].

A PPENDIX Here, we summarize some concepts and results regarding to the connection over a map. For more details, refer to [24, Ch.5].

h : Mn → N m . Then, a smooth vector eld X h over h is a smooth h h map X : M → T N s.t. for every q ∈ M, X (q) ∈ Th(q) N . We denote the set of such smooth ∂ h vector elds by X (M). Let { (q)} be a local basis of vector elds in the neighborhood of ∂hi h(q) ∈ N . Then, X h ∈ Xh (M) can be written in coordinates by Consider a smooth map

X(q) =

m X i=1

h where Xi April 19, 2007

Xih (q)

∂ (q) ∂hi

∞

∈ C (M). DRAFT

30

h is an object Dh which assigns to each v ∈ Tq M an operator Dvh h h h that maps vector elds over h into Th(q) N , i.e. D : Tq M × X (M) → Th(q) N . We say that D 0 ∞ h h h is afne, if for all v, v ∈ Tq M, a, b ∈