M. Hemami
Department of Electrical Engineering.
B. F. Wyman
Department of Mathematics.
The Ohio State University,
Columbus, Ohio
43210
Indirect Control of the Forces of
Constraint in Dynamic Systems1
In this paper two problems are investigated; how to control a dynamic system such
that holonomic constraints are maintained and further the forces of constraint are a
priori specified. Two cases of the latter are considered: constant forces of constraint
and forces that are functions of the state. The dynamic system is linearized about an
operating point and linear feedback is exploited for the solution of both problems.
A methodology for computing the feedback gains is developed and applied to a
nonhuman biped model that possesses ankle torques in the frontal plane. Simulation results are carried out for the nonlinear biped model to maintain the vertical
force of constraint constant under the foot. Applications to locking of joints in
natural biological systems is noted.
1
Introduction
The main objective of this paper is to develop an understanding and a methodology for the control of constrained
dynamic systems. The understanding involves developing
dynamic models, and the control problems involve
1) control of the system in such a way that specified
kinematic constraints are maintained or deliberately removed
and 2) control of the forces of constraint such that both the
state and the forces of constraint follow desirable specified
trajectories. Dynamic systems with constraints are discussed
in [1, 2]. An understanding of these dynamic systems is
provided by imbedding the system in another system of higher
dimension, and thereby calculating the forces of constraint
[2]. For systems with holonomic constraints of special type
[3], an alternative method is provided in [4] by which the
forces of constraint are directly derived as functions of the
state (position and velocities) and the inputs (applied torques,
forces, etc.). The latter method results in a dynamic model
that is valid both for when the constraints are maintained or
violated. The implementation of various feedback algorithms
that afford the control of such systems make them a special
case of systems with variable kinematic structure. These
systems have been discussed in [4, 5] and a thorough review is
available in [6]. The need for understanding and control of
constrained dynamic systems arises in many fields: fine
motion of manipulator [7], design of a force feedback chatter
control system [8], control a a boring bar operation [8, 10],
control of biped locomotion [4, 11, 12], quadruped and
hexapod locomotion [13] and more importantly in biological
systems [14, 15, 16, 17]. The imposition of the constraint in
biological systems is by locking a joint via the action of
certain muscles [14]. The imposition of the constraint, in such
cases, may be conscious as in locking the extended knee or
elbow, or unconscious, as in locking of the knee in certain
phases of walking. The imposition of the constraint, further,
This work was in part supported by NSF Grant No. ENG-7824440.
Contributed by the Dynamic Systems and Control Division for publication in
the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript
received at ASME Headquarters, October 12, 1979.
allows the main muscle group that is involved in the motion of
the joint to rest and not be involved in maintaining the locked
state. Also the locking affords a certain reduction in
dimensionality and affords simpler control.
In [4], the forces of constraint were not directly controlled,
but feedback algorithms were developed such that holonomic
constraints were maintained in the state space, i.e. in the
vicinity of an operating point a set of linear constraints were
imposed on the state of the system. Alternatively one may
maintain the holonomic constraints by specifying the forces
of constraint as functions of time or functions of state or
functions of both time and state. Seeking solutions for the
control problem u = u(t) results in two point boundary value
problems.
A heuristic proof of this statement is as follows: suppose
the forces of constraint Y are specified as functions of time.
Since T is a function of the state x(t) and the control u{t),
and since x(t) is a function of «(r) for T < t, it follows that
specification of Y(t) amounts to constraining values of U(T)
and consequently x (r) for r < t. This observation means that
specifying later values of Y constraints prior values of u(t)
and x(t). Thus the problem is a two-point boundary value
problem.
The two point boundary value nature of the problem
allows other approaches to control of constrained dynamic
systems: singular perturbation [18, 19, 20], dynamic systems
in descriptor form [21] and optimal programming [22, 23].
Thus the problems of controlling the transient behavior or
point to point transition of the state of the constrained system
could also be solved by optimal control theory with appropriate choice of a performance index and a set of Lagrange
multipliers [24, 25, 26], and with penalty functions [27, 28].
Optimal control could also be applied to cases where the
system's trajectories involve both an unconstrained portion
and a constrained portion [29, 30, 31]
On the other hand if feedback solutions u = u[x(t), t] are
allowed, as will be shown later, the problems of maintaining
the constraint is solvable without recourse to two point
boundary value problems. This approach appears to
Journal of Dynamic Systems, Measurement, and Control
December 1979, Vol. 101 / 355
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II
Constrained Dynamic Systems
Consider a nonlinear dynamic system described by equation
(1), evolving on a state space X of dimension n, with input
space U of dimension m.
x=f(x,
u, T)
(1)
where T is a vector of dimension r corresponding to r constraints
C,(x)=0
(2)
and x is a generalized position vector that consists of a set of
positions z and velocities i; and C, defines the holonomic
constraints. According to [4] the constraint forces are given as
explicit functions of the state x and input u.
Y = Y(x, u)
(3)
This representation results in the feedback model of Fig. 1
where the Y feedback loop is open (off) when the constraint is
violated and is closed (on) when the constraint is maintained.
In order to maintain the constraint either the state of the
system must be restricted to C, or the forces of constraint
should be controlled. For the latter situation vector Y of
equation (2) must satisfy certain inequalities. The simplest of
these inequalities are only functions of Y. For example, in
order for a biped foot not to slide horizontally on a surface
with coefficient of dry friction ^ the total horizontal component of force acting on the foot must be less than - /ry, in
magnitude. This total horizontal component of force in
general is a sum of the horizontal reaction force and appropriate inputs to the system.
Thus for the nonlinear dynamic system of equation (1) one
may state the following:
A sufficient condition for maintaining the constraint Cx (x)
= 0 in state space is to control Y.
A special case of this is when the vector Y is held
constant, and such that the corresponding inequalities
are satisfied.
Thus equality constraints (holonomic constraints) on the
state result in inequalities in the space of A: and u. Therefore to
maintain the constraint, either the state is controlled [u] or the
forces of constraint are controlled as is shown later. The
maintenance of constraints in the state space results is an
initial value problem with somewhat difficult computations
for the feedback gains [4]. The specification of the forces of
356 / Vol. 101, December 1979
i
•
1
'
Li_
correspond to nature's way of control of biological systems,
namely the use of some control inputs to maintain the constraint and others to stabilize the system or move the joints.
This approach is particularly suited for studying mechanical
systems where acceptable values for the forces of constraints,
and their relative values can be determined fairly easily from
laws of mechanics rather than continuity of boundary conditions and existence of optimal solutions. As a consequence
of this approach trajectories for when the constraints are
maintained and when the constraints are violated must be
separately analyzed.
In Section II constrained dynamics systems are introduced
via a feedback on-off model. The control problem is discussed
in Section III. A three segment planar example is considered
in Section IV with some applications. One application,
motion of a biped of human size is considered in detail. For
the biped the feedback signals necessary for maintaining
stability and constant forces of constraint are derived.
Finally, local behavior of the nonlinear biped system with
linear feedback is simulated in Section V. The global behavior
of the nonlinear system could also be considered in this
context by a sequence of linearized models, and consequently
linear time-varying feedback would be applied. However, the
latter extension is not explored.
• -
/
,
'
K
^
y\
Fig. 1
Control of constrained dynamic system
constraint as functions of state and time makes the design of
the feedback gains easier provided that a sufficient number of
inputs exist.
This contrast of the two methods applies as well to control
problems where the constraint is to be deliberately violated.
The approach in [4] to this problem involves moving the
system into a subspace orthogonal to C, (x) = 0.
In the context of control of the forces of constraint the
removal or violation of constraint amounts to the control of
inputs u and consequently state x such that Y(x, u) tends to
zero. Letting
Y(x,u)=0
(4)
may allow some u's to be computed as functions of x and the
remaining u's.
a, = « , ( * > "2)
(5)
Equation (5) is the feedback control law that instantaneously
brings the force Y to zero, and hence permits the constraint to
be violated. Alternatively, computation of dY/dx and dY/du
(Appendix) may allow sensitivity studies to determine which
of the state x or control inputs u or combinations of the two is
more effective in reducing Y to zero.
The reverse problem of how an unconstrained system
becomes instantaneously constrained by collision or gradually
constrained by contact is discussed in [32].
Ill The Control Problem
As was stated before, the control problem involves either
maintaining equalities in the state of x or inequalities in the
space of x and w. Hence, the problem may be formulated as
an optimal control problem [22, 25, 26]. Here it is assumed
that feedback solutions u{x(t),t) are allowed for maintaining
the constraint, and further that one can specify the forces of
constraint as functions of the state and time such that the
mentioned inequalities are satisfied: Y = T, (x(t),t).
Therefore solutions for input u must satisfy
Y(x, K ) = r , (*(/), t)
(6)
At this point one may again resort to optimal control where
equality (6) replaces the equalities in space of x or inequalities
the space of x and u and/or Y and w.
In what follows, however, equation (6) is solved for some
inputs as functions of state to maintain the constraint. The
remaining inputs are used for stability and movement under
constraint.
Substitution of (3) in (1) and linearization around a
stationary point x0, u0, T0 yields
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x=
dx
o(X~X°)+lfu
r -ar
o
"° )+ aT
o<«-
(x-x0)+
(r-r„)
—
3K
0
Let
df
dx
0
df
dT
0
=A;
dr
3«
=5
0
ar
Tx
d~u
0
0
F
and for notational convenience replace x — xQ by x, u — u0 by
u, T — T0 by y. The computation of matrices E and F is
developed in the appendix.
With the above notation
x=Ax+Bu+My
(8)
The point x0 is assumed to be in the constraint set, so
evaluation of the original constraint equations yields a linear
constraint:
Cx = 0
(9)
The system (6) evolves within the constraint subspace if and
only if Cx = 0, which gives
uV)=uja(t)+K(x(t)-xMV))
(11)
where A' is a feedback gain matrix and xie! (t) and Ule[ (t) are
given functions of time. It follows that
-xref (0 )
(12)
I f 7 = 0 f o r a l l x ( 0 , then
(,E+FK)x=0
(13)
F(Uref-Axref) = 0
(14)
The control objective is to solve the following problem:
given input signals uref(t) and uref(,t), derive a feedback
matrix such that the closed loop system x = (A + BK)x is
stable and the constraints (13) and (14) are satisfied.
Case 2: Forces of Constraint as Functions of State. Suppose
the forces of constraint are specified as functions of x, and are
linearized about an operating point
r=rw=r, + ^
(x-
f)
(15)
Linearization of T (x, u) also renders
r 0 =E(x-xlet)
(16)
+ F [Href +K(x-
or if equations (15) or (16) are equated
r0-r1+F(«ref-AArref)+^
ar,
dx
=0
-Exref=0
(17)
and
Journal of Dynamic Systems, Measurement, and Control
(18)
Therefore equations (17) and (18) in this case replace
equations (13) and (14) respectively.
By implementation of equation (14) or (18) in the dynamic
system
x=(A+BK)x
(19)
a new system is arrived at where
x=Alx+Blul
(10)
It is shown in [4] that for real physical systems CM is a square
invertible matrtix. If follows that:
For every state x such that Cx = 0 and every input u, there
is a unique y such that equation (10) holds.
Case 1: Constant Forces of Constraint. Suppose Fig. 1 is
used to control the dynamic system with the control inputs
u(t) comprised of an open loop component and a feedback
component:
+F(uK( (t) +K(x(t)
E + FK =
or
CMy = C(Ax+Bu)
y=Ex(t)
A three link dynamic system
(7)
y = Ex + Fu
C(Ax + Bu+My) = 0,
Fig. 2
(20)
where ux is a subset of u's.
The system of equation (20) can be stabilized by conventional pole assignment methods [32, 33, 34].
ul=K{x
(21)
In order to improve the performance of the system, one
may derive AT, such that the resulting system
x=(Al+BKl)x
(22)
satisfies the linearized holonomic constraint Cx = 0. Define
W={x\nX:
Cx=0j
(23)
Then one requires that
xe W implies {A, + Bt Kx )xe W
(24)
Equation (24) means that not only the system is stabilized
but also a given subspace W is maintained invariant. While
the pole assignment problem is treated in [33, 34], the additional requirement of keeping W invariant must be considered. It was shown in [4] that condition 24 can be satisfied
for some K, if and only if
A.WQW+B
(25)
If equation (25) is satisfied, the set of admissible Kx 's forms a
nonempty affine subset of set of all feedback matrices K that
map x onto u. If equation (25) is not satisfied the controlproblem has no solution.
Assuming that equation (25) is satisfied a "primal"
algorithm for the computation of suitable matrices Kt was
presented in [4]. In this paper, a "dual" method is used.
Let Wl be the orthogonal complement of W, so that Wk is
the column space of CT. One knows that Wis (At + i^A",)
- invariant if and only if WLis (Ax + B{K\)T - invariant.
The design procedure reported in Section IV below
proceeds as follows:
Dual Design Procedure:
(a) Choose "dual eigenvalues" \ , . . , \ , r = rank C.
(b) Find a feedback A", and vectors y,, v2
vr in the
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column space of CT such that
T
(Al+BlKl) vi
= Xivi
(26)
Note that if X, are distinct, the y, automatically span the
column space of CT.
(c) It follows that for the K{ chosen in (b),
(AL +
BlKl)(W)QW.
The methods of [4] can be applied to assign poles within W.
In comparison with the primal algorithm step (b) of this
method is rather ad hoc, but gives good results for small r. In
particular, the affine constraints on Kt appear in a convenient
form.
It may be also argued that if the system is linear one only
has to establish relations among the eigenvalues such that the
constraints are maintained and then assign the eigenvalues
such that all have negative real parts. While this last method is
conceptually feasible, in practice it seems to require solution
of nonlinear algebraic equations and nonlinear programming
methods. The primal algorithm of [4] and the dual algorithm
here exploit the affine property and solution of nonlinear
algebraic equations are avoided.
IV Biped in the Frontal Plane
Consider the three segment model of Fig. 2. This model
may be used to analyze the biped motion when one or two feet
are on the ground. This model also could be used to represent
three link manipulation in the plane and a three link hand
model for holding. The state of the system is xT = [0,, B2, 0 3 ,
0,, 0 2 , 6\] where the angles are measured with respect to the
vertical. The inputs are three torques uu u2 and « 3 . It is
assumed that linkage 1 touches the ground at the origin. Using
Newtonian or Lagrangian Dynamics one derives the equations
of motion
x=fx(x,u,F4,G4)
constraint force G4 under the limb labelled 3 is controlled.
The ankle torque ux is not available in human beings.
Therefore the side to side motion with a constant reaction
force cannot correspond to an actual human movement.
However this motion is more realistic in the design of robots
and manipulators where a constant reaction force may be
necessary which the robot, or manipulator moves in a
specified manner. Also in robots and manipulators usually a
torque is available at every joint. The linearization is about
the vertical stance (0, = d2 = 0, 03 = 180). From the appendix, matrices E and Fcan be computed:
E=
dT
Yx
_ar
1.17
."[ -0.026
du
0
0
0
19.076
-32.768
26.725
A=
x=f2(x,u,Fl,Gl)
(28)
B--
1 0
0 0
x=f(x,u,Flt0,FA,0)
(30)
In what follows only the biped is considered. The equation
of motion are given by equation (27) and V = (F4, G4)T. The
parameters of the model are selected to be roughly those of a
human with an ankle torque «, and hip torque, u2 and « 3 .
The objective is to control the side to side motion of this biped
model with both feet on the ground and such that the vertical
-6.17
0.026
0, 0, 01 T
0, 0, 0J
(31)
1 .171
1
(32)
0
0
0
-2.1542
3.7005
-16.762
0
1
0
0
0
0
(33)
0
0
0
-0.077172
0.25526
-0.10812
0
0
0
0.088559
-0.27483
0.37788
- 1 0
0
0 1 0
0
-1
(34)
(35)
A", =5.27(^ 2 -A- 3 ) + (-622.22 587.18
-622.22 0 0 0)x
(36)
With equation (36) used in (19), the new yl, and B, can be
computed. Carrying out steps 2, 23 and 24 for X] = - 5 and
X2 = - 7 one obtains eigenvectors
£>,=(1, 0, - 1 , 1/7, 0, -1/7) 7 "
(37)
0, - 1 , 1/5, 0, - l / 5 ) r
(38)
v2=(l,
(29)
w h e r e / = 1/2 (/, + f 2 ) . The biped may conveniently be
modelled by equation (29) with both feet on the ground. If the
right leg is lifted F, = G, = 0. If the left leg is lifted F4 = G4
= 0. For modelling a hand assume the two segments 1 and 3
are supposed to press against an object horizontally and hold
it with some force. In this case the model is
728,
-337,
Let the columns of B be labelled, respectively, BUB2, and B 3 .
Let the rows of A' be, respectively, labelled KUK2, and A"3.
With the imposition of constraint (12) on T, the vertical
reaction component of Y, the following constraint on K
results
Addition of equations (28) and (29) renders
x=f(x,u,Fl,Gl,F4,G4)
0
0
0
•6.699
28.359
-9.3854
0
0
0
0.028395
-0.048777
0.039782
(27)
where the forces of constraint are T = [F4, G4]T. These
equations are derived in [4] and are not repeated here. If one
is working with a manipulator, attached at the origin,
equation (27) is adequate for all its motions. If point B is free
of the surface F4 = G4 = 0, if point B slides on the surface F4
= 0, and if point B slides on a vertical wall G4 = 0. For study
of the biped and finger models, the model must be symmetric
with respect to the two segments labelled 1 and 3. The
segments correspond to feet for the biped and fingers for a
hand model. In this case equations of motion must also be
written with point B as the origin.
-687,
0 ,
728,
-337,
.-[
The new affine constraint on the two remaining rows of A'is
(-0.02906 = 0.229313)A"=(-34.43, 4, 13.307,
- 1 2 , 0 12)
(39)
With the implementation of the latter constraint, the system is
x=A3x+Blui
The poles of A3 are
-5,
-7,
+5.8547,
±0.18474y
(40)
Now elements of K3 can be computed [33] to assign the poles
of the system at
- 5 , - 7 , -5.8547, - 4 , - 3 , - 2 . 5
(41)
The final result is:
K=
358 / Vol. 101, December 1979
-780.43
106.66
136.68
-562.34
-209.07
9.0518
-5464.8
-867.04
51.847
-379.65
-17.493
54.547
-496.72
-8-3r653
10.601
-754.21
-173.46
-30.348
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the tilted biped configuration is statically stable:
"ref,
"ref 2
"ref 3
I
2
Vertical force of constraint
I
2
Seconds
Fig. 4
Horizontal force of constraint
I
2
3
Angle 63 as a function of time
I
2
Acknowledgment
The authors would like to acknowledge support and
sustained encouragement of this work by Professor H. C. Ko,
Chairman of the Department of Electrical Engineering, The
Ohio State University.
3
Seconds
Fig. 6
(44)
VI Conclusions
In this paper the control structures for a constrained
dynamic system are investigated that:
1) provide constant or prespecified forces of constraint
2) maintain a kinematic constraint.
Both objectives are achieved without direct sensing or
measurement of the force of constraint or involving this force
in the feedback mechanism.
Numerical calculations are carried out for an example of a
biped for maintaining a constant force of constraint while a
human like body sways five degrees to the vertical. The
analysis of this paper may shed light on some mechanisms
that are employed by humans and other living organisms in
maintaining their joints in locked states or maintain or
remove support with the ground.
Seconds
Fig. 5
U) =+ 30--15/
(0 = + 30--15/
In this respect it is to be noted that the reference signals uK(
are functions of A'.
With these specifications, the nonlinear system representing
the biped was simulated. The results of the stimulation are
shown in Figs. 3, 4, 5, and 6. It is noted that the force of
constraint did not remain at 363 Newtons but dropped to 358
Newtons during the transition (Fig. 3). This corresponds to an
error of less than 1.5 percent. Also this force was neither
sensed nor used as an additional external feedback signal. The
magnitude of the horizontal force component was not controlled, and its magnitude evolved from 57 Newtons initially
to zero (Fig. 4). Figs. 5 and 6 show evolution of 03 and 6} in
time.
3
Seconds
Fig. 3
(/) = - 5 3 . 8 + 26.9/
Angle 01 as a function of time
These feedback control gains have the necessary characteristics as is shown in the simulation later, and should be
compared to those in [4].
V
Simulation of the Biped Movement
The objective of the simulation is to move the biped from a
tilted state X(0) = ( - 5 , 0 , 175, 0, 0, 0) to the vertical stance
in two seconds while both feet maintain contact with the
ground and the constraint force under limb 3 is maintained at
about 1/2 the weight of the biped. In addition to K above, the
reference inputs xK(it) and «ref(/) have to be specified.
Ramp signals were selected for all components. To inhibit
high velocities i ref (/) = 0
APPENDIX 1
Partial derivatives of the forces of constraint F(x, u) with
respect to x and u, i.e. dT/du, are necessary in 1) linearization
of the system dynamics, 2) design of constant and timevarying feedback, and 3) sensitivity analysis and 4) design of
nonlinear feedback for such systems. It is shown below that
these partial derivatives can conveniently be derived as
functions of x, F, and u.
The forces of constraint for a nonlinear dynamic system are
given by [4, equation (8)]:
Idz
/ac/_,ac
(A-l)
V dz
dz
Since the forces of constraint are linear in input
T
dT _ r dC , bC '
(A-2)
w
Tz
~a*TJ
dz
The partial derivative of T with respect to individual components of x: z\, z2, • • •, z„, i\, • • ., z„ is computed as
follows. Let one element of x be xt. It is convenient to rewrite
equation (A-l) in a compact form:
=
X
<^t)
=+
W0(-5+52n
*ref2(0=0
* - , (0 = ^ ( 1 7 5 + 5 , )
(43)
v
*ref,
ref, ( 0 = 0
*ref2 ( 0 = 0
*ref3 ( 0 = 0
The reference signals uKt are also ramp signals such that
initially (at / = 0) equation (14) is satisfied and that initially
Journal of Dynamic Systems, Measurement, and Control
£'-£)••-/**•<
(A-3)
December 1979, Vol.101 / 359
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Differentiating equation (A-3) with respect to *, gives
A
^-.i^
W^
/ - . ^dz)/-9x,^ *3*,-- (A-4)
dXf \dz
dz 1
V dz
or
3x,
V dz
dz J
I
dXj \dz
dz
)
Equation (A-5) has to be computed In times for every element
ofx
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