An optimal trajectory planning problem for a single-link, flexible joint manipulator is studied. A global feedback-linearization is first applied to formulate the nonlinear inequality-constrained optimization problem in a suitable way. Then, an exact and explicit structural formula for the optimal solution of the problem is derived and the solution is shown to be unique. It turns out that the optimal trajectory planning and control can be done off-line, so that the proposed method is applicable to both theoretical analysis and real time tele-robotics control engineering

Chen, Guanrong

English

NASA Technical Reports Server

N91 "20667
Exact and Explicit Optimal Solutions
for Trajectory Planning and Control of
Single-Link Flexible-Joint Manipulators,
Guanrong Chen t
Department of Electrical and Computer Engineering
Rice University
Houston, Texas 77251
Abstract. In this paper, an optimM trajectory planning problem for a single-link t_exible-
joint manipulator is studied. A global feedback-linearization technique is first applied to
formulate the nonlinear inequality-constrained optimization problem in a suitable way.
Then, an exact and explicit structural formu/a for the optimal solution of the problem is
derived and the solution is shown to be unique. It turns out that the optimal trajectory
planning and control can be done off-line, so that the proposed method is applicable to
both theoretical analysis and reM-time tele-robotics control engineering.
1. Introduction
An optimal inequality-constrained trajectory plan-
ning problem for a standard single-link flexible-joint ma-
nipulator is studied in this paper.
From a structural point of view, a robot arm is a
weakly-coupled multi-link mechanical transmission chain.
Hence, the study of a single-link manipulator (a unit of
a robot arm or an independent mechanism) is of funda-
mental importance.
It is well known that a trajectory planning problem
for a flexible-joint manipulator has a nonlinear model.
If we consider such a trajectory planning problem under
certain additional optimality criterion, then we will en-
counter a constrained nonlinear optimization problem.
No analytic closed-form optimal solution can be found
for such problems in general. However, for a single-
link flexible-joint manipulator with single control input,
Marino and Spong (1986) shown that a nonlinear feed-
back configuration can be designed to linearize the non-
* Supported by ONR Contract N00014-88-K-0127.
t The author will join the Department of Electrical Engineering, University of Houston
in September, 1990.
201
https://ntrs.nasa.gov/search.jsp?R=19910011354 2020-03-19T19:06:57+00:00Z
linear system globally. The basic idea is, roughly speak-
ing, that one can find a nonlinear feedback to "cancel"
the nonlinearity of the system and obtain a linear plant
and a linear feedback, leaving the nonlinearity to an ex-
plicit transformation. The advantage of this approach
is that the final result is exact (no linearization error)
after a nonlinear inverse transform. This mathematical
technique is of course well known in nonlinear control
theory (see, for example, Isidori (1989)). Nevertheless,
based on this result, we show in this paper that if we con-
sider a minimum control-energy criterion for the trajec-
tory planning (with inequality-constraints) of such ma-
nipulators, then an explicit fornmlation of the optimal
solution for the overall nonlinear constrained optimiza-
tion problem can be obtained in closed form.
The proposed new approach for obtaining an ex-
act optimal solution explicitly for such an inequality-
constrained nonlinear optimization problem is novel in
mathematics and very useful in robotics engineering since
it provides us an analytic solution before the control pro-
cess is started, so that no on-line computer is needed
in the real-time applications (unless the environment is
changing and needs to be adapted), which is sometimes
impossible in certain control processing such as in some
tele-robotics control in aerospace engineering. Another
advantage of closed-form solutions over numerical solu-
tions is the convenience in theoretical analysis of the op-
timal trajectory planning. Even if in the case that the
resultant analytic optimal trajectory cannot be traced
by actual control inputs, we know the exact optimal tra-
jectory to be approximated.
This paper is organized as follows: We first describe
the optimal trajectory planning problem for a single-
link flexible-joint nmnipulator. Then, we use a stan-
dard global feedback-linearization technique to formulate
the nonlinear inequality-constrained optimization prob-
lem in a suitable way. Based on this nmthematical model,
we finally give an explicit structural solution for the prob-
lem in a closed-form.
2. Description of the Problem
Consider a single-link flexible-joint manipulator as
shown in Figure 1.
Figure 1. A single-link flexible-joint manipulator
Damping will be ignored in this system for simplicity.
The joint is assumed to be of revolute type and the link
is assumed to be rigid with inertia/1 about the axis of
rotation. Let 81 be the link-angular variable and 82 the
actuator-shaft angle. Suppose that the rotor inertia of
the actuator is 12. Assume also that the flexible joint is
modeled as a linear spring of stiffness K. Then, by the
Euler-Lagrange equations we have the following motion
equations for this manipulator:
{ I181 + MgLsin(81) + K(S1 - 82) = 0 (1)I2_2 -- K(@I -- 82) : u,
where M is the total mass of the link, L the distance
from the mass-center of the link to the axis of the ro-
tation, g the acceleration constant of gravity, and u the
(generalized) force-input applied to the shaft by the ac-
tuator.
Since from a mathematical point of view there is no
difference between bend and swivel joints (see, for exam-
ple, Section 5.3 in Nagy and Siegler (1987)), the problem
under investigation has rather wide applications.
We will consider an optimal point-to-point trajec-
tory planning problem for this model. To describe the
problem more precisely, let p = p(t), v = v(t), a =
a(t), j = j(t) be the position, velocity, acceleration,
and jerk of the link, respectively, which are functions
of the time variable t 6 [0, T] for some fixed terminal
time T < oo. The first objective is to design a control
input u = u(t) to drive the link such that
202
p_/< p(ti) < Pi, v--i < v(ti) < gi,
(2)
< a(ti) < gi, _ < j(ti) < ji,
i = 0, 1,..., n, for some pre-assigned constants
P---i' Pi, v-V-/, _i, ai, a--i, k d, _i : i = 0,1,...,n,
at the pre-desired time instants
0<t0<tx< "" <tn<_T.
It can be easily seen from the above trajectory con-
stralnts that we will have infinitely many solutions that
satisfy the requirements. We want to find an optimal
one from them. For this purpose, we consider the prob-
lem of controlling the link to satisfy the above trajectory
constraints while minimizing certain control-energy to be
described precisely later in the next section.
This consideration is especially important when the
link is heavy with a large mass M and the control en-
ergy (power source) is limited, such as in some aerospace
engineering applications.
A direct approach for formulating and solving such
a nonlinear inequality-constrained optimization problem
does not seem to be easy, unless numerically. However, as
mentioned above, numerical solutions are undesirable if
analytic solutions can be easily obtained, in particularly,
for the purpose of analysis of the control system. In
the following two sections, we will first formulate the
problem in a suitable way and then derive an closed-
form structure for the optimal solution. The resultant
optimal solution is actually exact in the sense that no
approximation will have been applied.
3. Mathematical Formulation of the Problem
tions so that it can be linearized globally by a feedback,
in the sense that an equivalent but linear closed-loop
results. All analyses given in this section are standard
in nonlinear systems control theory (see, again, Isidori
(1989)) and, in fact, have been done in Marino and Spong
(1986) (see, also, Spong and Vidyasagar (1989)) for this
particular manipulator model. This technique was also
used by Tam et al (1987).
Let
zl = 01, x2 = 01, za = 02, z4 = 02,
so that equations (1) can be rewritten as
= f(x)+g(x)u, (3)
where x = [xl z2 x3 x,] T, g(x) = [0 0 O /_-I]T, and
I x2 ]f(x) = -IllMgLsin(xl)- I-1I_(Xl -- x3) .
For this nonlinear system, the vector fields f(x) and
g(x) are smooth, the corresponding Lie brackets [f, g] :=
O0_xf - of7xg are given by
{ g, If, g], [f, [f, g]l, [f[f, If, gl]] }([0] 010 O0 I_-II_IK= 01 1 0
I_ , -I_2K
the vector fields
{ g, If, g], [f,If, g]] }
are constant and hence form an involutive set, and more-
over the vector fields
In order to formulate the above-described nonlinear
inequality-constrained optimization problem in a suit-
able way, we first rewrite the motion equations in a state-
vector setting and then verify that the resultant nonlin-
ear system satisfies some necessary and sufficient condi-
{ g, [f, g], [f, [f, g]], [f[f, [f, g]]] }
are linearly independent for all 0 < K,/1,/2 < oo. Hence,
it follows from a result of Su (1981) that the nonlinear
system (3) is globally feedback-linearizable, in the sense
203
that an equivalent linear feedback system with an ex-
plicit nonlinear inverse transform exists. More precisely,
we have the following analysis:
Let _7(.) : R --* R 4 be the gradient vector of the
scalar-valued argument and (., .) a standard inner-product
of vector-valued functions. Set y = [Yl Y2 ys V4]T with
Yl = Xl
v_ = (v(vl), f) = x_
Y3 = (V(y2), f) = -I1-1MgL sin(xl)
(4)
- IllK(za - x3)
Y4 = (V(y3), f) = -IllMgLx2 cos(x,)
- IliK(x2 - x4).
Then, with the linearizing feedback control of the form
u = F(x) + IiI2K-lv, (5)
where
F(x) = I_l MgLsin(xl)[x_ + I_l MgLcos(xx) + I_I K]
+ I;-1K(xa - xa)[(I_ -1 + I_1)I( + I_-lMgLcos(xa)],
the nonlinear system (3) has been linearized as
A
where
9 = Ay + by, (6)
0 0 1 and b =
0 0 0
0 0 0
with the following physical meanings:
Yl = Xl = position of the link
y2 = x2 = velocity of the link
Ya = Y2 = acceleration of the link
Y4 = Y3 = jerk of the link.
The original nonlinear control system and the equiva-
lent closed-loop and linerized feedback configuration of
the overall system are shown and compared in Figure 2
below,
nonlinear
nonlinearlinear
,ran  or 
Figure 2. Equivalent Feedback Loops
Here, it is important to point out that if we only
consider the trajectory planning (constraints (2)), then
the systems shown in Figure 2 are equivalent in the sense
that if the trajectory of the linearized feedback system
can be controlled to satisfy the constraints (2), then the
same can be done for the original nonlinear system by
inverting the nonlinear transform (4). For this reason,
from now on we can leave the original nonlinear system
and work on the linear system (6) together with the non-
linear feedback (5) instead.
Note that in the linearized feedback system, v is the
only external and active control input, as can be seen
from Figure 2 above. Hence, in the study of the trajec-
tory planning for the linearized feedback system instead
of the original nonlinear system, we may consider to min-
imize the total control-energy of this executive input v.
Based on this point of view, we formulate an optimal
trajectory planning problem as follows:
Problem:
rain v2(t)dt (7a)
vEL2(O,T)
subject to the linear system
jr = Ay + bv (7b)
and the trajectory constraints
<_p(td < _, _ <_v(t,) <_v,, (7c)
a i<a(ti)<_i, j, <_ j(ti) <_ Ti,
i = 0,1,...,n, whereO<to<tl <... < tn <T< oo.
204
Here, L_(0, T) denotes the standard Hilbert space
of square-integrable real-valued functions defined on the
time-interval [0, T]. Once we have solved the linear con-
strained optimization problem (7a-c) for optimal v* and
y. = [y_ y_ y_ y_]T, we obtain an optimal solution for
the original problem from the inversion of (4); namely,
from the following formulas:
x*= Y_ ]
Y; + I1K-I[Y_ + I_-lMgLsin(y;)] | (8)
* I K -1 * -1 . •Y2 + 1 [Y4 + Ii MgLy2 c°s(Yl )] j
u* = F(x*) + IlI2K-lv * .
In the next section, we will show a closed-form structure
for the optimal solution of the inequality-constrained op-
timization problem (7a-c).
4. Closed-Form Optimal Solution
In this section, we show the closed-form structure of
the optimal solution for the inequality-constrained op-
timal trajectory planning problem formulated in (7a-c)
above. As a result, the optimal solution for the origi-
nal nonlinear inequality-constrained optimization prob-
lem turns out to be exact in an explicit closed-form.
In order to state our result precisely, we need some
new notations. In addition to the notations used before,
set the matrix-valued exponential function
[0 ']
c := E3(t-r) E4(t r) '
0 < v,t < T, in which each submatrix Ei(t - r),i =
1,2,3,4, is a 4 x 4 block. Then, using the notation 1 :=
[1 1 1 1] T, define
h(t - r) = E_(t - r)l
and
f E2( t T)I
h+(, - _) =
t 0
Moreover, let
h(t - r) = Oh(t - r)
and {°gih( t - ,9 t >_,-h+(t T)
0 t<r.
Our main result can now be stated as follows:
Theorem.
given by{
The optimal solution for Problem (7a-c) is
y*(t) = Colh+!t- to)+ Co2h+(t- to)
n--i
.+ _ C_h+(t- t_)
v*(t) 0,(t),
(9)
where Col, Co2, Ci, i = 1,..., n - 1, are all 4 x 4 diagonal
constant matrices which are uniquely determined by the
trajectory constraints (7c) from the given data set
_, Pi, _, _i, a_, -ai, _, _i, : i= 0,1,...,n. (10)
Consequently, the optimal solution ( u*, x*) for the origi-
nal problem is obtained via (8) from the optimM sohtion
(v*, y*) given by (9).
We remark that the determination of the constant
coefficient matrices of y*(t) is simple, which can be done
easily by using any standard quadratic programming al-
gorithm even before the manipulator control processing
is started, so that no on-line computer is needed for
this optimal nonlinear trajectory planning problem un-
less adaptive control is necessary. More precisely, we
demonstrate this procedure as follows: First, we observe
that the minimization problem
min v2(t)dt
vEL2(O,T)
is equivalent to either
r[min bv]-r [bv]dt
vEL2(O,T)
or
rain fT
yeU,(O,T) J0 [Y -- Ay]T[:_ -- Ay]dt, (11)
where Hi(0, T) is the sta'ndard first order Sobolev space.
If we can solve the minimization problem (11) for y*,
205
then we can find the optimal solution v* -- _ (see (6)).
Secondly, we notice that the minimization problem (11)
together with the inequality-constraints (10) can be re-
formulated as the following quadratic programming prob-
lem:
rain cTwc (12)
co_,Co2,{c_}
subject to
& < p(ti) < _, v_a< v(t_) < %,
a_a< a(ti) < Hi, J-i < j(ti) < 3"i,
i = 0,1,..., n, where C is a constant vector consisting of
all elements of the diagonal matrices Col, Cos, C1,..., C,
and W is a constant matrix consisting of the integra-
tions of all elements of the functions h+ and h+. Both
C and W have simple explicit expressions as can be eas-
ily seen and derived from formulas (9) and (11). This
standard quadratic programming problem can be solved
by some existing computer routines, which will provide
us the unique optimal solution.
5. Conclusions
In this paper, we have studied an optimal trajec-
tory planning problem of a standard single-link flexible-
joint manipulator. We first used a standard feedback-
linearization technique to formulate the nonlinear inequa-
lity-constrained optimization as a minimum control-energy
problem. Then, we have derived an exact structural for-
mula in closed-form for the optimal solution of the prob-
lem and showed that this solution is unique. The pro-
posed approach is applicable to both theoretical analysis
and real-time robotics control engineering.
REFERENCES
[1] A.Isidori (1989): Nonlinear Control Systems: An
Introduction, second edition, Springer-Verlag, New
York.
[2] R.W.Marino and M.W.Spong (1986): Nonlinear con-
trol techniques for flexible joint manipulators: A sin-
gle link case study, Proc. 1986 IEEE Conf. Robotics
and Automation, San Francisco, April 1986, pp.1030-
1036.
[3] F.N-Nagy and A.Siegler (1987): Engineering Foun-
dation_ of Robotics, Prentice-Hall, Englewood Cliffs,
New Jersey.
[4] M.W.Spong and M.Vidyasagar (1989): Robot Dy-
namics and Control, Wiley, New York.
[5] R.Su (1981): On the linear equivalents of nonlinear
systems, Sys. Contr. Letts., Vol.2, pp.48-52.
[6] T.J.Tarn, A.K.Bejczy, and X.Yun (1987): Nonlin-
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Exact and explicit optimal solutions for trajectory planning and control of single-link flexible-joint manipulators

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