8/9/2019 Permissible Control of General Constrained Mechanical Systems
1/20
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
ARTICLE IN PRESS
www.elsevier.com/locate/jfranklin
0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfranklin.2009.10.002
E-mail address: [email protected]
http://www.elsevier.com/locate/jfranklinhttp://dx.doi.org/10.1016/j.jfranklin.2009.10.002mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.jfranklin.2009.10.002http://www.elsevier.com/locate/jfranklin8/9/2019 Permissible Control of General Constrained Mechanical Systems
2/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 209
8/9/2019 Permissible Control of General Constrained Mechanical Systems
3/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227210
8/9/2019 Permissible Control of General Constrained Mechanical Systems
4/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 211
8/9/2019 Permissible Control of General Constrained Mechanical Systems
5/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227212
8/9/2019 Permissible Control of General Constrained Mechanical Systems
6/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 213
8/9/2019 Permissible Control of General Constrained Mechanical Systems
7/20
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.
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227214
8/9/2019 Permissible Control of General Constrained Mechanical Systems
8/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 215
8/9/2019 Permissible Control of General Constrained Mechanical Systems
9/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227216
8/9/2019 Permissible Control of General Constrained Mechanical Systems
10/20
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
http://-/?-http://-/?-http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
11/20
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" #
BmM1=2Fm BmBc 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
BmM1=2Fm BmBc bc AcM1Fm
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227218
http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
12/20
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.
A.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 219
http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
13/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227220
http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
14/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 221
8/9/2019 Permissible Control of General Constrained Mechanical Systems
15/20
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
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227222
8/9/2019 Permissible Control of General Constrained Mechanical Systems
16/20
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.
A.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 223
http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
17/20
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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-8/9/2019 Permissible Control of General Constrained Mechanical Systems
18/20
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).
A.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 225
8/9/2019 Permissible Control of General Constrained Mechanical Systems
19/20
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
[1] F.E. Udwadia, R.E. Kalaba, A new perspective on constrained motion, Proceedings of the Royal Society of
London Series A 439 (1992) 407410.
[2] F.E. Udwadia, R.E. Kalaba, On motion, Journal of the Franklin Institute 330 (1993) 571577.
[3] R.E. Kalaba, F.E. Udwadia, Equations of motion for nonholonomic constrained dynamical systems via
Gausss principle, Journal of Applied Mechanics 60 (1993) 662668.[4] F.E. Udwadia, Equations of motion for mechanical systems: a unified approach, International Journal of
Nonlinear Mechanics 31 (1997) 951958.
[5] F.E. Udwadia, Nonideal constraints and Lagrangian dynamics, Journal of Aerospace Engineering 13 (2000)
1722.
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227226
8/9/2019 Permissible Control of General Constrained Mechanical Systems
20/20
[6] F.E. Udwadia, R.E. Kalaba, On the foundations of analytical dynamics, International Journal of Non-linear
Mechanics 37 (2002) 10791090.
[7] F.E. Udwadia, R.E. Kalaba, What is the general form of the explicit equations of motion for constrained
mechanical systems, Journal of Applied Mechanics 69 (2002) 335339.
[8] F.E. Udwadia, A new perspective on the tracking control of nonlinear structural and mechanical systems,
Proceedings of the Royal Society of London Series A 459 (2003) 17831800.
[9] F.E. Udwadia, Equations of motion for constrained multibody systems and their control, Journal of
Optimization Theory and Applications 127 (2005) 627638.
[10] F.E. Udwadia, Optimal tracking control of nonlinear dynamical systems, Proceedings of the Royal Society of
London Series A 464 (2008) 23412363.
[11] A.D. Schutte, B.A. Dooley, Constrained motion of tethered satellites, Journal of Aerospace Engineering 18
(2005) 242250.
[12] T. Lam, New approach to mission design based on the fundamental equation of motion, Journal of
Aerospace Engineering 19 (2006) 5967.
[13] F.A. Graybill, Matrices with Applications in Statistics, second ed., Wadsworth Publishing Company,
Belmont, CA, 1983.
[14] F.E. Udwadia, R.E. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press,Cambridge, 1996.
[15] T.I. Fossen, Guidance and Control of Ocean Vehicles, Wiley, New York, 1994.
[16] F.E. Udwadia, A.D. Schutte, An alternative derivation of the quaternion equations of motion for rigid-body
rotational dynamics, submitted for publication. Journal of Applied Mechanics, 2010.
ARTICLE IN PRESSA.D. Schutte / Journal of the Franklin Institute 347 (2010) 208227 227