permissible control of general constrained mechanical systems

8/9/2019 Permissible Control of General Constrained Mechanical Systems

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Journal of the Franklin Institute 347 (2010) 208227

Permissible control of general constrained

mechanical systems

Aaron D. Schutte

The Aerospace Corporation, 2310 E. El Segundo Blvd., El Segundo, CA 90245-4691, USA

Received 28 August 2009; accepted 5 October 2009

Abstract

This paper develops a unified approach for modeling and controlling mechanical systems that are

constrained with general holonomic and nonholonomic constraints. The approach conceptually

distinguishes and separates constraints that are imposed on the mechanical system for developing its

physical structure between constraints that may be used for control purposes. This gives way to a

general class of nonlinear control systems for constrained mechanical systems in which the controlinputs are viewed as the permissible control forces. In light of this view, a new and simple technique

for designing nonlinear state feedback controllers for constrained mechanical systems is presented.

The general applicability of the approach is demonstrated by considering the nonlinear control of an

underactuated system.

& 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: General constrained systems; Control of constrained mechanical systems; Nonlinear control;

Underactuated control; Multibody dynamics and control

1. Introduction

In Lagrangian mechanics, a constrained mechanical system is, in general, a nonlinear

system described by an n-dimensional second-order vector differential equation, which is

called its equation of motion. The dimension n depends upon the number of generalized

coordinates used to describe the configuration of the mechanical system. When constraints

are present, an additional generalized force of constraint is conceptualized to arise so that

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the mechanical system satisfies the imposed constraints at each instant of time. Most

practical mechanical systems involve constraints, and quite often the constraints are

employed for distinct theoretical purposes. For example, a holonomic constraint can be

employed to model the fixed distance of the mass of a simple pendulum from its pivot

point, yet a holonomic constraint may also be employed to control it at a desiredconfiguration such as requiring it to maintain a fixed location at some point along its

circular trajectory. Thus, the force of constraint can be utilized in distinct ways.

The goal of this paper is to show how this perspective can bring together the dynamic

modeling and the control of general constrained mechanical systems, and in turn yield

some new insights into the design of nonlinear feedback controllers. The framework

presented herein relies on some simple and fundamental results in analytical dynamics

obtained by Udwadia and Kalaba [17]. They obtain explicitly, general equations of

motion for constrained discrete dynamical systems in terms of the generalized coordinates

that describe the systems configuration. The utility of their formulation has been

demonstrated in both the areas of dynamical modeling and control [810], and also in

areas of practical interest such as in astrodynamics [11,12].

Throughout the paper, the control of constrained mechanical systems is approached by

showing that the constraints imposed on the mechanical system may be distinguished as

those that physically model the system and those that control the system. An explicit form

for the entire set of control forces that may be applied to a constrained system is derived

yielding a general class of nonlinear control systems, where the control inputs are viewed as

the permissible control forces. Using the formulation, the design of two types of nonlinear

state feedback controllers are presented. Under certain conditions, these two controllers

are shown to provide exact tracking and stabilization to the constrained system. Themethodology is easily adaptable to the modeling and to the design of feedback control for

complex systems that may have many constraints. In an example, we show its capability to

address issues that can occur in both the dynamic modeling and control of constrained

mechanical systems by considering the stabilization and tracking of an underactuated

surface vessel vehicle.

2. Modeling general mechanical systems

Consider an unconstrained mechanical system Sgiven by the nonlinear nonautonomous

second-order differential equation

Mq; t q Qq; _q; t; q0 q0; _q0 _q0: 1Eq. (1) can be obtained by using Lagrangian mechanics, where the n-vector qt is thegeneralized coordinate vector describing the configuration of the mechanical system and

the dots represent derivatives with respect to time. Here, we shall assume that the

symmetric n by n matrix M is positive definite. The n-vector Q contains the given

generalized forces, and it is a known function of its arguments. We call the system in

Eq. (1) unconstrained because the initial velocity _q0 may be arbitrarily assigned.

Depending on the particular mechanical system under study, the modeling effort may be

complete after arriving at Eq. (1). In many mechanical systems, however, constraintequations must be imposed to correctly model any configuration or motion restrictions

that are intrinsic to the problem at hand. This is particularly the case in multibody

problems. When constraints are imposed to the system described by Eq. (1), we must

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modify it so that it becomes the constrained mechanical system Sm given by

Mq Q Qmq; _q; t: 2The n-vector Qm (usually called the constraint force) is the additional force applied to S so

that the required motion, which satisfies the imposed constraints, is obtained. Thesubscript m is used to indicate that the additional force Qm arises due to a specific

classification of equality constraints. In this paper, we shall classify these constraints as the

modeling constraints. The modeling constraints are the necessary tools needed to arrive at

an accurate physical model of the system, and they may enter the modeling process as the

coordinate constraints

jk;mq; t 0; k 1; 2; . . . ; h1; 3and as the physical constraints

fk;mq; t

0; k

1; 2; . . . ; h2;

4

ck;mq; _q; t 0; k 1; 2; . . . ; h3: 5

Eqs. (3) and (4) are both called holonomic constraint equations since they are functions

of position and time only. They are distinguished from one another here because a

coordinate constraint arises due to the choice of a coordinate system with dependent

variables, whereas a physical holonomic constraint will arise due to real physical

restrictions on the configuration of the mechanical system. A common coordinate

constraint includes the unit norm constraint, which, for example, appears when unit

quaternions are used as a rotation coordinate for rigid bodies. Eq. (5) is called a physicalfirst-order nonholonomic constraint since it cannot be once integrated and put in the form

of Eq. (4). Another type of physical constraint (not typically used in the modeling process)

is given in the form

wk;mq; _q; q; t 0; k 1; 2; . . . ; h4; 6which is linear in the accelerations q. Eq. (6) is called a physical second-order

nonholonomic constraint since it cannot be once integrated to obtain the form of

Eq. (5), or twice integrated to obtain the form of Eq. (4). As we shall see later on, the

constraints in Eq. (6) can be used to model systems that are underactuatedsystems that

cannot physically actuate all of its degrees of freedom.The total number of modeling constraints required to model a given general mechanical

system is then simply

m X4i1

hi: 7

By assuming that Eqs. (3)(5) are sufficiently smooth functions of time, we can obtain all

of the m consistent modeling constraints as

Amq; _q; t q bmq; _q; t 8by appropriately differentiating Eqs. (3)(5) with respect to time and because Eq. (6) islinear in q. In Eq. (8), Am is an m by n matrix whose rank is rm, and bm is an mvector.Note that the general set of kinematical and dynamical modeling constraints contained in

Eq. (8) may be (1) nonlinear functions of position and velocity, (2) explicitly dependent on

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time, and (3) functionally dependent. We shall assume that these modeling constraints,

which we are imposing on the system S, may be nonideal. The nonideal nature of the

constraint force Qm may be specified by a sufficiently smooth n-vector, Cq; _q; t, so thatthe work done by the force of constraint under the virtual displacements dq at each instant

of time is [5]

Wmt dqTQmq; _q; t dqTCq; _q; t: 9When DAlemberts principal is assumed to be satisfied C 0 the right hand side ofEq. (9) is identically zero, and the force of constraint is then ideal. The force of constraint

that causes the mechanical system S to satisfy the general set of ideal, or nonideal,

modeling constraints in Eq. (8) is explicitly found by the fundamental equation of

constrained motion developed by Udwadia and Kalaba [1,6]. It is given as

Qm Qim Qnim : M1=2Bmbm AmM1Q I M1=2BmBmM1=2C; 10where the symbol denotes the MoorePenrose (MP) generalized inverse of thematrix Bm AmM1=2 and Idenotes the n by n identity matrix. The constraint force Qm isthus made-up of two additive parts, where Qim denotes the ideal part and Q

nim the nonideal

part. For the unfamiliar reader, the MP generalized inverse of a matrix is defined in the

following definition.

Definition 1 (MP Generalized Inverse [13]). The unique matrix B that satisfies thefollowing four conditions is called the MP generalized inverse of the arbitrary matrix

B2 Rn.

1. BBT BB.2. BBT BB.3. BBB B.4. BBB B.

The explicit acceleration of the modeled mechanical system Sm then becomes

q M1Q M1=2Bmbm AmM1Q M1I M1=2BmBmM1=2C: 11Thus, given a generalized displacement qt and a generalized velocity _qt at some instantof time t such that qt and _qt are compatible with the modeling constraints in Eq. (8),and by appropriately specifying the n-vector C describing the nonideal nature of themodeling constraints, the generalized acceleration in Eq. (11) will yield the unique

acceleration of the appropriately modeled mechanical system Sm at that instant of time. It

provides the most comprehensive description to date of the motion of a mechanical system

subjected to the general set of constraints given by Eq. (8).

3. Determination of the permissible control force

Consider now the constrained mechanical system Sm given by Eq. (2). We assume that

we have arrived at the system Sm by using the m modeling constraints contained in Eq. (8)so that the constraint force, Qm, is explicitly given by Eq. (10), where the n-vector C has

been properly prescribed by considering the condition of the modeling constraints along

with Eq. (9). Our task is now aimed at determining the manner in which this system may be



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controlled; i.e., what is the general form of the control force n-vector Qc so that the

controlled system Sc given by

Mq Q Qm Qc 12is compatible with the modeling constraints in Eq. (8)? We use the subscript c to indicatethat the additional force Qc arises due to an arbitrary set of control objectives that we

would like to impose on the system Sm. Conceptually, we presume that the modeling

constraints are absolute requirements, and that we must find this control force so that they

are never violated by the trajectories of the controlled system Sc. This simply means that

the system Sc given by Eq. (12) must evolve over time so that Eq. (8) is valid at each instant

of time. Thus, the modeling constraints must be intrinsic to the design of the control force

Qc. Note that Eq. (12) is a general nonlinear control system with no restrictions on its

structure such as being a control-affine system or utilizing any approximations. However,

the control force Qc is restricted in the sense that it cannot cause the trajectories of the

system Sc to violate any of the modeling constraints that are enforced by the constraint

force Qm. This leads us to the following proposition.

Proposition 1. The constrained mechanical system Sm can be controlled by means of an

n-vector control force Qc if and only if the modeling constraints given by Eq. (8) are satisfied

by the controlled system Sc. The entire set of control forces, QcDRn, that guarantee the

system Sc will satisfy the modeling constraints are called the permissible control forces.

To determine the set of permissible control forces Qc according to Proposition 1, we

begin by considering the system Sm so that

Mq Q Qm Q M1=2Bmbm AmM1Q I M1=2BmBmM1=2C I M1=2BmBmM1=2Q C M1=2Bmbm: 13

In Eq. (13), observe that the total force is made-up of the two n-vectors

Q I M1=2BmBmM1=2Q C : PQ C 14and

Qm M1=2Bmbm: 15

The n-vector^

Q shows the effect the modeling constraints have on the given generalizedforce n-vector Q, and also the prescribed n-vector C, while it is clear that the n-vector Qmarises solely due to the modeling constraints that we have imposed on the unconstrained

system S. We now state the following lemmas.

Lemma 1. The n-vector Q is the projection of the sum of the n-vectors Q and C by the n by n

projection matrix P. In general, this projection is nonorthogonal.

Proof. The n by n matrix P is a projection matrix since we can show that it is idempotent

(P2 P) by

P

2

I M1=2

BmBmM1=2

I M1=2

BmBmM1=2

I 2M1=2BmBmM1=2 M1=2BmBmBmBmM1=2

I M1=2BmBmM1=2 P; 16



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where either the third or the fourth MP condition in Definition 1 can be used in the second

equality. The matrix P is, in general, a nonorthogonal projection operator since

PT I M1=2BmBmTM1=2 I M1=2BmBmM1=2aP; 17where the first MP condition is used in the first equality. &

Lemma 2. The projection operator P projects any n-vector w 2 Rn onto the null space of thematrix AmM

1.

Proof. Pre-multiplying the matrixvector product Px by the matrix BmM1=2, we have

BmM1=2Pw BmM1=2I M1=2BmBmM1=2w

BmM1=2 BmBmBmM1=2w 0; 18where the third MP condition is used in the second equality. The result follows since

BmM1=2 AmM1. &

Suppose we now create a control system Sz by adding an arbitrary control input vector

z 2 Rn to Sm so that it becomesMq Q Qm z; 19

or equivalently

Mq Q Qm z: 20Throughout this paper, we will use the terms control input and control force

interchangeably. In terms of a control effort, we then imagine that the n-vector z denotesany control input to the system Sm. However, according to Proposition 1, we are actually

interested in a specific set of control inputs for the system Sm; namely, those control forces

z/Qc that cause the controlled system Sz/Sc so that the control inputs become

compatible with the modeling constraints. Therefore, the system Sz must be constrained by

the set of modeling constraints in Eq. (8). To determine the manner in which the system Szbecomes compatible with the modeling constraints, we take the following two steps: (1)

view Sz as an unconstrained system and (2) impose the modeling constraints (Eq. (8)) to

this so-called unconstrained system by applying the fundamental equation of constrained

motion [14]. We now state the following result.

Result 1. A general class of nonlinear control systems for any constrained mechanical

system Sm can be cast into the form

Mq Q Qm Pz : Fq; _q; t; z 21in which the arbitrary n-vector z denotes any freely chosen control input. The permissible

control forcethe actual control input to the systemis given by

Qc Pz I M1=2BmBmM1=2z; 22where Qc

2Null

AmM

1

. It is called the permissible control force because the constrained

system Sm must be controlled in the null space of the matrix AmM1. Applying a controlforce Qc that is different in form from Eq. (22) to Sm will invalidate the physical structure

of the system since in this case compatibility with the modeling constraints is not

guaranteed. When the mechanical system is correctly modeled without using any modeling



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constraints (Eq. (8)), the control system reduces to the obvious form

Mq Q z : Fq; _q; t; z 23so that the control force is simply

Qc z; 24where Qc 2 Rn.Proof. First, we view Eq. (20) as an unconstrained system. Since, in actuality, the system

Sm is constrained, we know that the addition of an arbitrary control input z to Sm may

invalidate the modeling constraints. Thus, we must enforce the modeling constraints to the

unconstrained system Sz to yield the correctly modeled dynamics. This is carried out by

utilizing the fundamental equation of constrained motion so that we have

Mq

Q

Qm

z

M1=2Bm

bm

Am

M1Q

Q

m z

PQ Qm z Qm: 25Using Lemma 1, we find that

PQ P2Q C PQ C Q 26and

PQm I M1=2BmBmM1=2M1=2Bmbm 0; 27where Eq. (27) follows from the fourth MP condition. Eq. (25) then reduces to

Mq Q Qm Pz; 28or equivalently to

Mq Q Qm Pz: 29By Proposition 1, Eq. (29) yields the permissible control force

Qc : Pz; 30where Qc 2 NullAmM1 by Lemma 2. To verify that Eq. (29) is indeed compatible withthe modeling constraints, we can pre-multiply Eq. (28) by the matrix AmM

1 and obtain

AmM1Mq AmM1Q AmM1Qm AmM1Pz;

Am q AmM1Qm;

Am q AmM1M1=2Bmbm;

Am q BmBmbm; 31where the first and third members on the right hand side of the first equality are zero by

Lemma 2. Since we assume that the mechanical system is modeled using a consistent set of

modeling constraints, we require that BmBmbm bm. Hence, the result. &

Corollary 1. When the control input z 2 ColM1=2Bm, the permissible control force Qc hasno effect on the system Sm.



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Proof. If z (nonzero) is in the column space of the matrix M1=2Bm, then

z M1=2Bmw; 32where w

2R

m is arbitrary. Substituting Eq. (32) into Eq. (22), we obtain

Qc I M1=2BmBmM1=2M1=2Bmw M1=2Bmw M1=2BmBmBmw 0; 33where the fourth MP condition is used in the second equality. Hence,

z 2 ColM1=2Bm ) Qc 0. &Corollary 2. The control input z causes the system Sc to become compatible with the

modeling constraints at each instant of time when it is chosen so that z 2 NullAmM1. Thepermissible control force is then defined as Qc z.Proof. If z is in the null space of the matrix AmM

1, then we have

AmM1z BmM1=2z Gmz 0: 34

The general solution to Eq. (34) when it is consistent is given by

z I Gm Gmw; 35where w 2 Rn is arbitrary. Substituting Eq. (35) into the permissible control force (Eq.(22)), we obtain

Qc I M1=2BmBmM1=2I Gm Gmw I M1=2BmGmI Gm Gmw

I

G

m

Gm

M1=2Bm

Gm

M1=2Bm

GmGm

Gm

w

I

Gm

Gmw

z;

36

where the third MP condition is used in the third equality above. Hence, the modeling

constraints are automatically satisfied if z 2 NullAmM1. &Result 1 and Corollaries 1 and 2 show that, in general, a control force n-vector Qc

applied to any constrained mechanical system Sm must have a precise form in order to

preserve the physical structure of the system. We can interpret this form as the set of

permissible control forces since they are the control forces that are confined to the null

space of the matrix AmM1. Though we are free to choose the control input z, it must be

chosen carefully so that it can yield, if possible, the desired motion (controlled trajectories)

when applied in the form of the permissible control force. These results underly the closeconnection that exists between analytical dynamics and control theory. For indeed, the

permissible control force given by Eq. (22) has the same form as that of the nonideal

component of the constraint force Qm, namely, the component Qnim.

In the following section, a control methodology is developed using the concepts

developed thus far to design general sets of controllers for the class of nonlinear control

systems given by Eq. (21).

4. Nonlinear feedback control of general constrained mechanical systems

Utilizing Result 1, the feedback control problem for the constrained mechanical system

Sm can now be interpreted as designing a state feedback control law

z zsq; _q; t; 37



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for static feedback, or in the case of dynamic feedback

z zdq; _q; t; l; 38_l

Zq;

_q;

t;l;

39

such that the controlled constrained mechanical system Sc satisfies a set of desired control

objectives, where zs, zd, and Z are continuously differentiable in their arguments. Although

we can posit similar control laws for the case of output feedback, we will only focus on

state feedback in this section. The control objectives considered herein are expressible by

any combination of the consistent and sufficiently smooth constraint equalities

fk;cq; t 0; k 1; 2; . . . ; h5; 40

ck;cq; _q; t 0; k 1; 2; . . . ; h6: 41

The control objectives are thus represented by what we shall call the control constraints.Analogous to the modeling constraints, Eqs. (40) and (41) include the general variety of

holonomic and nonholonomic constraint equations, and they yield a total of c constraintsgiven by

c X6i5

hi: 42

The concept of applying a control constraint to the system Sm in this context is equivalent

to requiring it to track a reference signalone of the primary control objectives in state

feedback control. We say that exact tracking is achieved when the system Sc satisfies Eqs.(40) and/or (41) at each instant of time. However, a systems initial conditions are generally

given such that fcq0; 0a0 and/or ccq0; _q0; 0a0. Therefore, we introduce asymptotictracking (stabilization) by altering the control constraints so that

fk;c! fk;c fkfk;c; _fk;c; k 1; 2; . . . ; h5; 43

ck;c! _ck;c gkck;c; k 1; 2; . . . ; h6: 44The tracking and/or stabilization efforts are then carried out by choosing the functions fkand gk such that the fixed points

fk;c;

_fk;c

0; 0

and ck;c

0 are asymptotically

stable. Hence, we can obtain all c control constraints given in the form of Eqs. (43)and (44) as

Acq; _q; t q bcq; _q; t; 45where the c by n matrix Ac has rank rc, and bc is an cvector.

Note that the control constraints as given by Eq. (45) have the same form as the

modeling constraints. Consider then the combined constraint matrix equation

A q :Am

Ac

" #q

bm

bc

" #: b: 46

Though we have assumed that both the control and the modeling constraints are

consistent, we do not require here that Eq. (46) is consistent; i.e.,

AAbab: 47



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This assumption permits the specification of control constraints that may not be

compatible with the modeling constraints. This aspect can be advantageous, especially

when attempting to control complex systems, because it is often difficult to derive the

control constraints that can both (1) satisfy the desired control objectives and (2) lie exactly

on the modeling constraint manifold. Thus, by assuming Eq. (47), we can permit the designof the control constraints such that

fk;c-0; k 1; 2; . . . ; h5 as t-1; 48and/or

ck;c-0; k 1; 2; . . . ; h6 as t-1; 49while

AAb-b as t-1: 50In terms of the permissible control force, we have already guaranteed the satisfaction of themodeling constraints for the controlled system Sc, and so this design process isolates the

control constraints from the modeling constraints allowing the specification of control

paths that may not coincide with the allowable paths of the physical system. Whether the

system Sc can actually achieve the control objectives or not therefore depends critically on

the choice of the control constraints. Their effect on the system Sm can be ascertained by

using the permissible control force as we shall see in the following.

In this paper, we propose the control constraints so that the functions fk and gk in

Eqs. (43) and (44), respectively, can take both the forms of static and dynamic feedback as

listed in Table 1. The type I and II functions in Table 1 are selected because it is

straightforward to choose the coefficients ak, bk, gk, and sk that create the asymptotically

stable fixed points fk;c; _fk;c 0; 0 and ck;c 0. We could also just as well choose anyother nonlinear, or linear, second order fk and first order gk ordinary differentialequations that produce a particularly desirable behavior about the same fixed points. Thus,

we have devised the control constraints that we would like to impose on the system Sm with

the goal of (1) satisfying some tracking objectives defined by the constraints fk;c and ck;cand (2) asymptotically approaching these constraints by appropriately selecting the

coefficients shown in Table 1. We now state the following result.

Result 2. The control design for any general constrained mechanical system Sm may be

carried out by

1. describing the desired control objectives in terms of the control constraints fk;c and ck;cindependently of the modeling constraints, where the control constraints are allowed to

be nonlinear functions of position and velocity, explicitly dependent on time, and

functionally dependent,

ARTICLE IN PRESS

Table 1

Function types for stabilization and tracking of the control constraints.

Type fk gk

I ak _fk;c bkfk;c gkck;cII ak _fk;c bkfk;c sk

Rt0fk;cq; t dt gkck;c sk

Rt0ck;cq; _q; t dt

A.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 217
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2. using the type I and II functions for fk and gk listed in Table 1 so that Eqs. (43) and (44)

will yield the asymptotically stable fixed points fk;c; _fk;c 0; 0 and/or ck;c 0,3. collecting each of the c altered control constraints (Eqs. (43) and (44)) into the form of

the constraint matrix equation given by Eq. (45).

In this form, the control constraints can be imposed on the system Sm using the permissible

control force by choosing the control input z so that

zs M1=2Bc bcq; _q; t AcM1Q Qm 51for the type I functions, or

zd M1=2Bc bcq; _q; t; l AcM1Q Qm; 52

Z

f1;c;f2;c; . . . ;fh5;c;c1;c;c2;c; . . . ;ch6;c

T

53

for the type II functions, where Bc AcM1=2. The permissible control force in either caseis then given by Eq. (22).

Corollary 3. When the control constraints are chosen so that the control laws

zs; zd 2 NullAmM1, 8tZt0, the system Sc will exactly satisfy both the modeling and thecontrol constraints.

Proof. Let Pz z, so that the system Sc becomesMq Q Qm z Fm z; 54

where the control input z M1=2

Bc bc AcM1

Fm. Pre-multiplying by M1=2

, we gets : M1=2 q M1=2Fm Bc bc AcM1Fm: 55

We also have

A q Bs Bm

Bc

" #s

bm

bc

" # b: 56

Substitute Eq. (55) into Eq. (56) and get

Bm

Bc" #s

BmM1=2Fm BmBc bc AcM1Fm

BcM1=2Fm BcBc bc AcM1Fm" #


BcBc bc

" #; 57

where the third MP condition is used in the first equality. But the right hand side of

Eq. (57) is also


BcBc bc

" #

bm

bc

" #: 58

Since we assume the control constraints are consistent, it is true that BcBc bc bc.

Furthermore, we have BmM1=2Fm AmM1Fm bm from Eq. (31), yielding

BmBc bc AcM1Fm AmM1z 0: 59

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Eq. (59) is true because Pz z3z 2 NullAmM1 by Corollary 2. Finally, since s inEq. (55) is indeed a solution to Bs A q b, the system Sc exactly satisfies both themodeling and control constraints. &

From a control design perspective, isolating the control constraints from the modelingconstraints allows the selection of stable trajectories that may asymptotically satisfy our

desired control objectives, but in the process violate the required modeling constraints. The

modeling constraints must be preserved at all times because they represent the physical

structure of the system, whereas the control constraints can be adaptable to the given

system at hand. Deriving the control constraints in this mannerwithout regard to the

modeling constraintsmay however make attaining Eq. (50) difficult, or even physically

impossible if the control constraints are poorly posed. By Corollary 3, we ultimately want

to design the control constraints such that when z zs, or z zd, the quantity

Pz

z

T

Pz

z

0:60

This guarantees satisfaction of both the modeling and control constraints because

z 2 NullAmM1. If this is not possible for a given control constraint set, it may befeasible to alleviate this requirement by only requiring

x : zTI P PTP PTz % 0; 61where x is a performance index measuring the ability of the control laws zs and zd to satisfy

the control constraints. Thus, when x 0 the system Sc exactly tracks the imposed controlconstraints.

In the last section, we show the wide applicability and simplicity of the approach

presented herein by designing a nonlinear state feedback controller for an underactuatedsystem.

5. Application to underactuated control

Consider the surface vessel vehicle in Fig. 1. The vehicle is shown relative to an inertial

coordinate frame fx; ^yg, where the unit vectors e1 and e2 denote the directions of the body-fixed coordinate frame of the vehicle. The angle y determines the orientation of the vehicle.

ARTICLE IN PRESS

m, J

x

y

12

Fig. 1. A surface vessel vehicle with mass m and rotational inertia J.

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Typically these types of vehicles are designed with no side thruster capability such that they

can only actuate in the direction of e1 and about the direction e1 e2 normal to thex; ^yplane. The vehicle is underactuated since it cannot actuate in the direction of e2.

The general dynamics of a surface vessel vehicle are developed in [15]. Here, we choose

the generalized coordinate vector, which describes the configuration of the vehicle, asq x;y; u0; u1T 2 R4. The position to the center of mass of the vehicle is given by the2-vector R x;yT, and the orientation of the vehicle is determined by the two parameterunit quaternion

u u0; u1T cosy=2; siny=2T; 62where y is the rotation angle shown in Fig. 1. For counterclockwise rotations y about the

origin, the 2 by 2 orthogonal rotation matrix Tu is given in terms of quaternions as

Tu T1; T2 u20 u21 2u0u12u0u1 u20 u21" #; 63

where T1 and T2 denote the columns of Tu. By a unit quaternion, we mean that

Nu : uTu 1: 64Its relationship to the vehicles body-fixed angular velocity is given by

o 2u1 _u0 2u0 _u1: 65Thus, the configuration of the vehicle is completely described by n 4 parameters and ithas three degrees of freedom. The advantage of using two coordinates to represent the

vehicles orientation will be shown later on.Without loss of generality, we assume the vehicle is modeled so that its body is

symmetric with respect to the axes e1 and e2, and also such that there is no viscous friction.

The unconstrained equation of motion of the vehicle assuming no impressed forces or

torques is then given by the system S as

Mq : mI2 00 4ETJE

R

u

" #

0

8 _ETJE_u 4J0N _uu

" #: Q; 66

where I2 is a 2 by 2 identity matrix and the scalar J040 is any positive number (see

Ref. [16]). The orthogonal matrix E and the augmented inertia matrix J in Eq. (66) aregiven by the 2 by 2 matrices

E u0 u1

u1 u0

" #67

and

J J0 00 J

: 68

We now define the necessary modeling constraints needed to correctly model the physicalstructure of the system. Since the unit quaternion is required to satisfy Eq. (64), we must

enforce the coordinate constraint

jm : Nu 1 0 69

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to the system S. Additionally, the vehicle cannot actuate in the direction of e2 such

that [15]

_v2 v1o 0; 70

where v1; v2 are the vehicles linear velocities in the ^e1, ^e2directions. In the inertialcoordinate frame, Eq. (70) yields the modeling constraint

wm : 2u0u1 x u20 u21 y TT2 R: 71Differentiating Eq. (69) twice with respect to time, we can then obtain the m 2 modelingconstraints as

Am q :0 uT

TT2 0

" #R

u

" # N _u

0

: bm: 72

We assume here that the modeling constraints are ideal C 0, and so DAlembertsprinciple is satisfied. Using Eq. (10), the modeling constraint force Qm is then computedexplicitly as

Qm 0

2Jo2u

: 73

The equation of motion of the constrained system Sm is then simply

Mq Q Qm 0

J0o2u

" #74

since 8 _ETJE_u 2Jo2u and N _u 1=4o2.The system Sm is now cast into the control system Sc by first computing the 4 by 4

matrix

P I4 M1=2BmBmM1=2 I2 1

TT2 T2T2T

T2 0

0 I2 uuT

264

375: 75

The underactuated surface vessel vehicle in permissible control form is then given explicitly

as

Mq Q Qm Pz I2 1

TT2 T2T2T

T2 zR

J0o2u I2 uuTzu

264

375; 76

where z zTR; zTu T is an arbitrary control input 4-vector. In this control problem, we seeka nonlinear state feedback control law that will stabilize the vehicle along a unit circle

trajectory in inertial space. This control objective is given by the control constraints

f1;c : x cost 0; 77

f2;c: y sint 0; 78

which essentially requires the vehicle to track a unit circle with period 2p in the

counterclockwise fashion. Note that because the vehicle is underactuatedit cannot

actuate in the direction of e2F we must also derive a steering objective involving the



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coordinates u0 and u1. In this situation, it is not readily apparent how to obtain the steering

objective so that both Eqs. (77) and (78) are satisfied. To obtain the steering constraint, we

use the permissible control force as follows. First, we collect the control constraints f1;cand f2;c into the preliminary form denoted by the superscript * as

Ac q : I2 0R

u

" #

costsint

" #: bc : 79

These control constraints are then imposed on the system Sm by using the control law zs,

which is explicitly given in Result 2 by Eq. (51). The control input needed to exactly satisfy

f1;c and f2;c without regard to the modeling constraints is computed as

z : zs mbc ; 0T: 80

However, by Corollary 3, we know zs 2 NullAmM1 guarantees that both the modelingand the control constraints are satisfied. Requiring that zs 2 NullAmM1, we have

AmM1zs 0; TT2 bc T 0: 81

The necessary steering objective is therefore given by the control constraint

f3;c : TT2 bc 2u0u1cost u20 u21sint 0: 82

This constraint, combined with the constraints given by Eqs. (77) and (78), ensures the

vehicle will track the unit circle. Physically, Eq. (82) demands that the e2direction ofthe vehicle is oriented normal to the circle trajectory so that the vehicle can generate the

necessary centripetal forces needed to track the circle. We can now utilize the type I

function fk listed in Table 1 to generate the altered control constraints (Eq. (43)) so that the

c 3 constraints f1;c, f2;c, and f3;c take the form

Ac q :1 0 0 0

0 1 0 0

0 0 2u0sint 2u1cost 2u0cost 2u1sint

2

64

3

75

x

y

u0

u1

2

6664

3

7775

cost a1 _f1;c b1f1;csint a2 _f2;c b2f2;c

b3;c a3 _f3;c b3f3;c

2664

3775 : bc; 83

where b3;c denotes the terms on the right-hand side generated from differentiating f3;c.

With the appropriate selection of the coefficients ak and bk, Eq. (83) determines the stable

design control paths

qc; _qc

we want to command the vehicle to follow. The reason for

using the two parameter quaternion u is now apparent. If we were to simply use thecoordinate y to represent the vehicles orientation, f3;c would become

f3;c sinycost cosysint siny t 0; 84



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and after differentiating, we would instead obtain the matrix

Ac 1 0 0

0 1 0

0 0 cost y

264

375: 85

Clearly, when y t kp=2, k71;72; . . . ; in Eq. (85), we see that the matrix Ac wouldchange rank. This may cause the design control path to become undefined. In contrast, the

matrix Ac in Eq. (83) does not suffer from this difficulty.

We now provide numerical results showing the capability of the methodology by applying

the control law zs (Eq. (51)) to the system Sc (Eq. (76)). We select the coefficients a13 ffiffiffi

3p

and b13 34, so that the fixed points f1;c; _f1;c f2;c; _f2;c f3;c; _f3;c 0; 0 areasymptotically stable. The mass of the vehicle is m 5 kg and its rotational inertia isJ 10kgm2. The initial conditions are specified by q0 x0;y0; u00; u10T 3; 3; cosp=4; sinp=4

T

and _q0 _x0; _y0; _u00; _u10T

0; 0; 0; 0T

so that thevehicle is sufficiently distant from the desired trajectory and at rest with an orientation

defined by y p=2. The numerical integration of the controlled system Sc is carried out fort 0; 45 s using ode 113 in MATLAB with a relative error tolerance of 1011 and anabsolute error tolerance of 1014. Fig. 2 shows the resulting position of the vehicle along withthe design position xc;yc we have specified by Eq. (83). The required control force andtorque is found by simply decomposing the permissible controller Qc so that

Qc Pz L

G

; 86

where L and G are 2-vectors. The control forces applied in the body frame of reference aregiven by

LB;1

LB;2

" # TTu L 87

ARTICLE IN PRESS

2 1 0 1 2 3 4

4

3

2

1

0

1

2

Fig. 2. The actual and design positions of the vehicle showing their paths as the vehicle converges to the unit circle

trajectory.

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and the body-fixed control torque is [16]

0

GB

" # 1

2EG: 88

Figs. 3(a and b) show the required body control force and the torque needed to converge to the

unit circle. As required by the modeling constraint wm, we see that the component LB;2 is zero

better than the relative accuracy used for integrating the system Sc. In Fig. 4, we plot the

performance index x over the integration. This indicates that the design paths determined by

the control constraints become satisfied when reaching the control objective even when they

are not satisfied initially. Finally, we illustrate the accuracy of the approach in Figs. 5(a and b).

In Fig. 5(a), the two parameter quaternion is unity (as demanded by the modeling constraintgiven by Eq. (69)) to an order better than the relative error tolerance used in the integration.

The satisfaction of the control constraints is shown in Fig. 5(b) demonstrating that the vehicle

has indeed met the control objectives.

ARTICLE IN PRESS

0 10 20 30 405

0

5

10

15

20

2

0

2

4

6

8

0 10 20 30 40

1.5

1

0.5

0

0.5

1

1.5x 10

14

Fig. 3. Control forces and the control torque in the body fixed frame computed by Eqs. (86) and (87),

respectively. (a) Control force in the e1direction and the control torque. (b) Control force in the e2direction.

A.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227224
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ARTICLE IN PRESS

0 10 20 30 40

1.5

1

0.5

0

0.5

1

1.5x 10

12

0 10 20 30 40

4

2

0

2

4

6

Fig. 5. The modeling and control constraints throughout the integration. (a) Error in the modeling constraintsjmand _jm. (b) Error in the control constraints f1;c, f2;c, and f3;c.

0 10 20 30 40

0

100

200

300

400

500

Fig. 4. Performance index x computed by Eq. (61).



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6. Conclusions

In this paper, a uniform and simple approach for the modeling and control of general

constrained mechanical systems is developed. The main contributions of the paper are as

follows:

1. The equality constraints considered in this paper include the general holonomic and

nonholonomic varieties. When applied to a mechanical system, they are conceptually

distinguished from one another by constraints that model the physical structure of the

system (including its coordinates), and constraints that control the system.

2. The idea that the control of a general constrained mechanical system can be cast into a

precise form so that the physical structurethe modeling constraintsof the system is

preserved. The control inputs to the system are the permissible control forces Qc Pz,which are the set of forces that are confined to the null space of the matrix AmM

1

forarbitrary control inputs z 2 Rn.

3. The control constraints are devised to represent the desired control objectives of the

modeled system Sm. Since they are derived independently of the modeling constraints,

they may specify control paths which are not consistent with the allowable paths of the

modeled system. This aspect can be beneficial, especially in the control of complex

systems, because the specification of the control paths can often become difficult when

lumping the modeling and control constraints together.

4. The control constraints, or the control objectives, are imposed to the constrained system

Sm by the control laws zs and zd. Their effect on the system is ascertained by using the

permissible control force. When zs; zd 2 NullAmM1, both the modeling and controlconstraints are exactly satisfied. This yields exact stabilization and/or tracking of the

constrained mechanical system.

5. The application of the methodology is demonstrated by designing a feedback controller

for an underactuated surfaced vessel vehicle required to stabilize and track a time

varying unit circle trajectory. The model utilizes a two parameter unit quaternion

requiring the satisfaction of a coordinate constraint in conjunction with a dynamical

underactuation constraint. The utility of casting the feedback control problem into

permissible control form and the accuracy of the approach to satisfy both the modeling

and control constraints are both substantiated by the numerical results.

References

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1722.



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[6] F.E. Udwadia, R.E. Kalaba, On the foundations of analytical dynamics, International Journal of Non-linear

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permissible control of general constrained mechanical systems

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