use of reduced basis technique in the inverse dynamics of large space cranes

Computing Systems in Engineering Vol. 1, Nos 2~,, pp. 577-589, 1990 0956-0521/90 $3.00 + 0.00 Printed in Great Britain. © 1990 Pergamon Press plc

USE OF REDUCED BASIS TECHNIQUE IN THE INVERSE DYNAMICS OF LARGE SPACE CRANES

S. K. DAS,~" S. UTKU'~ and B. K. WADA~

tDepartment of Civil and Environmental Engineering, Duke University, Durham, NC 27706, U.S.A. *Applied Technologies Section, Jet Propulsion Laboratory, Pasadena, CA 91109, U.S.A.

(Received 20 April 1990)

Abstract--The inverse dynamics of adaptive structures used as space cranes can prove computationally expensive in the case of large structures, due to the large number of degrees of freedom involved. Consequently, reduced basis techniques (reduction techniques) are frequently used to reduce the problem size to a time manageable level (for possible use in real time control). A reduced basis technique is proposed which is different from, but related to, the path-derivatives reduction technique. A linearly independent set of deflection n-tuples is used, chosen at the beginning of the time range in which it is wished to reduce the equations, in whose subspace it is assumed that the deflection vectors of the unreduced problem will lie (approximately). Some theory, the implications of using this sort of a reduced basis technique as compared to the path-derivatives technique, and some results, are given.

NOMENCLATURE

Boldface entities are used to denote matrices (upper case letters) and tuples (lower case letters). Non-boldface entities are used to denote scalars.

A , B , D

C* t g

J

K M P

p*, (p*)'

q r s, h u U(O I)

X

At

matrices associated with the inertia matrix in space crane,, dynamical equations, with dimensions p x ;,, p x n and n x n, respectively time varying solution (m-tuple) of reduced order system (u is the time varying solution of original unreduced system) m-tuple list of Taylor series coefficients p-tuple list of length adjustable bar forces n-tuple of forces (which are function of r) in the nodal equilibrium equations of the space crane matrix associated with length adjustable bar forces in upper partition of space crane dynamical equations stiffness matrix (size n x n) mass matrix (size n × n) forcing function (size n x 1) ith derivative of forcing function, p, evaluated at time point to forcing functions which lie entirely outside the column space of X p-tuple of angles between sections of space crane p-tuple of coriolis/centrifugal forces n-tuples of coriolis/centrifugal forces n-tuple of flexible deflections ith derivative of flexible deflection n-tuple, u, evaluated at time point t o matrix (size n × m, m ~ n) defining the reduced basis of dimension m small time interval (in simulation time range) between n-tuples constituting the reduced basis, in suggested scheme

INTRODUCTION

The motivat ion for this work arises from the problem of performing the inverse dynamics of large space

cranes in real time. The high computat ional complexity of such a procedure necessitates either the use of concurrent computat ion or the use of a reduction procedure, whereby the problem size is reduced for real time purposes. In this work we propose a reduction procedure which can be used for reducing the size of large dynamical systems.

The space crane can be imagined as very similar to conventional multi-link robots, with the solid links replaced by truss sections and length-adjustable mem- bers instead of the conventional torque generating motors (see Fig. 1). Elongation and contraction of the length adjustable bars give rise to any desired configuration. The truss sections are directly attached to each other at common points termed " joints" as well as by length adjustable bars. The line passing through the joints shared by two neighboring sections constitutes the axis of rotation about which one section rotates with respect to its neighboring section; the length adjustable bar/bars provide the requisite motion. Just as in conventional solid link robots, the last truss section of the open chain linkage system has an attached end effector capable of grasping and limited motion. In our work the space crane is considered flexible, i.e. any gross mot ion will be accompanied by flexibility induced motion. ~

A significant structural advantage of such adaptive structures over conventional robots lies in the fact that large link lengths are possible without the large masses that solid links would have entailed. These reasonably lightweight structures are expected to be widely used in N A S A ' s upcoming space station activities, especially so in space construction.

While the topic of this paper deals with the use of reduced bases in the dynamical equations of the

577

578 S.K. DAs e t al.

end effector

length-adju,~ bar 2

section 3

joint 3 /

length-adjustable bar 1 : \

joint 2 length-adjustable bar 3

section 1 ~ origin of global co-ordinate system

also joint 1

base structure (inertially fixed)

Fig. 1. Schematic of adaptive structure used as space crane.

space crane, the methodology behind obtaining the dynamical equations should be outlined.

Just as in conventional solid link robots, a path planning procedure is implemented to chart out the spatial path of the end effector. This is followed by an inverse kinematics procedure to obtain the time variation of angles between neighboring truss sections to achieve the desired motion; the inverse kinematics procedure assumes the links (sections) are rigid, and does not account for the flexibility induced motion. The inverse dynamics procedure consists of obtaining the time variation of length adjustable bar forces so as to maintain the time variation of angles between sections. Of course, maintaining the angles prescribed by the rigid body inverse kinematics procedure in a flexible space crane will mean that the end effector will oscillate (in some sense) about the prescribed path (we will not concern ourselves with this oscillation here; Das e t a l . 2 deal with angle adjustment to position the tip exactly on path).

The geometry of the space crane at any instant of time can be completely described by the p-tuple of angles between sections (q) and the n-tuple of flexible displacements (u). The flexible displacement vectors are defined as the vectors joining the hypothetical positions of the nodes of the space crane, if the space crane were rigid with angles q between sections, with the actual flexible body position of the nodes of the space crane (with angles q between sections). We define nodes as being attached to the vertices of bars.

The total set of dynamical equations obtained, quantifying the motion of the flexible space crane, are as follows (see Das e t a L l for details):

A B ~l - - [ B r D ] { i i } + { : } + { g ~ = O for t > O (la)

with initial conditions

u0=0

and

at t = 0 (lb)

q = q ( t ) is prescribed for t~>0. (lc)

The upper partition of the dynamical equations is the dynamical equilibrium equations associated with the generalized coordinates q l , q 2 . . , qp, while the lower partition represents the nodal equilibrium equations. The vectors r and s contain the coriolis/ centrifugal forces and stiffness effects; the vectors J-i" and g contain the contribution of the length adjustable bar forces (-f is the list of length adjustable bar forces). The inertia matrix (associated with the second order time derivatives) is symmetric.

The functional dependencies of the matrices/vectors constituting the dynamical equations are as follows:

A = A ( q )

B = B ( q )

D = constant (function of masses)

r = r(q, u,/I, ~)

s = s ( q , u, q)

g = g ( t q)

J = J ( q ) . (2)

Inverse dynamics of large space cranes 579

An algorithm for performing the inverse dynamics (obtaining the time variation of length adjustable bar forces ~) and simultaneously simulating for the flexible deflections, u, is given by Das e t a l J The algorithm essentially consists of an ordinary differential equation solver (Gear's fifth order method) to simulate the lower partition of the dynamical equations starting with known initial conditions. Simul- taneously, the upper partition of the equations is used to obtain the length adjustable bar forces. The upper and lower partitions of the equations are not independent; the length adjustable bar forces, T, are required to obtain g (in the lower partition) and the flexible deflections, u, are required to obtain r and B d2u/dt 2 in the upper partition.

The computational complexity of performing the inverse dynamics is O((no. of sections) 5, (no. of nodes) a) (see Das3). As a result, the expense involved in performing the inverse dynamics in real time (for control purposes) might be too high, unless we resort to parallel computation or basis reduction. In this work we deal with basis reduction to reduce the problem size.

With this fi'amework of preliminary knowledge we can put the principle of basis reduction into perspec- tive. We essentially assume that the flexible deflection vector u, at any instant of time, lies in a subspace of dimension less than the order of u; by so doing we reduce the size of the problem, and consequently gain in saved computational time. Real time control appli- cations can be thought of as possible candidates for basis reduction. The choice of reduced basis and the frequency with which we update it (in the simulation time range) greatly affects the accuracy of the results and whether or not simulation instability results.

Various reduction schemes such as the Ritz vectors method 4'5 Guyan-I ron 's reduction, 6,7 eigenvectors- expansion reduction (as explained in Clemente 8) and, more recently, the path-derivatives reduction technique, 9'1° have been suggested. A comparative study conducted 8'H showed the path-derivatives technique to arrive at better approximate results, in a variety of problems, than do the other three methods.

The method we propose here can be considered as quite similar to the path-derivatives technique in a number of respects. The advantages and disadvantages of our method and the path-derivatives method are evaluated in the following sections.


u(0)=Uo

fl(0) = 6o, (3b)

where M and K are the mass and stiffness matrices (assumed constant in time) of size n x n, and p is the time varying forcing function (size n x 1); u is the n-tuple of nodal deflections with a prescribed initial value and initial derivative value.

Let us now assume that the flexible deflection u lies (approximately) in the subspace of matrix X (size n x m , m <n) , i.e.

u~-Xe, (4)

where c is an m-tuple of time varying multipliers. Incorporating Eq. (4) into Eq. (3a), and pre- multiplying both sides by X ~, yields

(x~4x)~ + ( x ~ x ) c = XTp. (5)

The initial conditions of the unreduced problem (given by u 0 and du0/dt ) transform to c 0 and de0/dt in the reduced problem, where

C 0 = (xTx) - IXTu0

~0 = (X rX) 'Xru0 • (6)

The question then is: how good an approximat ion-- at any instant of t ime--is Xe to the flexible deflection n-tuple u. The answer lies in the choice of reduced basis and the frequency with which it is updated in the simulation time range.

In order to better understand how the dynamical equations of the reduced system [Eqs (5) and (6)] are related to the dynamical equations of the unreduced system [Eqs (3a) and (3b)], consider the dynamical equations

M~(t) + K~(t) = X(XVX)-tXrp(t) + p'(t); (7a)

~(t) belongs in subspace of X

X r p ' ( t ) = 0 (7b)

THE IDEA BEHIND BASIS REDUCTION

Basis reduction, as mentioned before, reduces the problem size, in turn leading to lower computational times. If properly utilized, reduction can keep the solution error to a minimum.

Let us, for example, consider the simple multi- degree of freedom forced dynamical problem, described by the time dependent equation

Mii + Ku = p (3a)


~(o) = x(x"x)-~XTuo

~(o) = x(xrX)-~xT,io. (7c)

~(t) is, then, nothing but the time varying solution of the reduced system [Eqs (5) and (6)], (Note that X(XrX)-~Xrp is the component of the forcing function in the reduced basis space; moreover, the only

580 S.K. DAS et al.

information we have about p' is that it is totally in the subspace orthogonal to the reduced basis space.) To verify this we can replace ~ and d2~/dt 2 by Xc and X(d2c/dt 2) and then premultiply both sides of the equation by X r to give Eq. (5).

In fact, we could fully imitate the behavior of the reduced system [Eqs (5) and (6)] by the sum of two unreduced systems, viz.

M~(t) + Kw(t) = X(XrX) - lXTp(t)

w(O) = x(x~x)-'Xruo

,(0) = x(x~x) 'XT~o (8)

where a bracketed superscript, say (i), refers to the ith derivative of the flexible deflection n-tuple.

The n-tuples u~ 2) through u~ m- l) can be generated recursively by repeatedly differentiating Eq. (3a), starting with the initial conditions of Eq. (3b). The n-tuple derivatives (of order >/2) can be expressed as follows:

k - I

u~ 2k)= E (--M-'K)k-'-~M-'p(o2° 0

+ (--M-IK)ku~ °) k = 1, 2 . . .

and

Mi~(t) + Kv(t) = p'(t)

Xrp'(t) = 0

k - I

U(o z~+')= E ( - M - ' K ) k - ' - ' M - ' P ~ i+1) 0

+ (-M-tK)ku<ol) k = 1, 2 . . . . (12)

v(O)=O

¢(o)=o (9)

The expressions in Eq. (12) show the following obvious advantages that path-derivatives have over the Ritz vector and modal reduction techniques;

with the restriction that

w + v belongs to the subspace of X. (10)

The n-tuple w + v will now have the same variation as Xe, e being the time varying solution of Eqs (5) and (6).

In the next subsection we outline the path- derivatives scheme and subsequently explain how our reduction scheme is related to it.

Path-derivatives reduction

Let us assume that we desire to simulate the reduced set of dynamical equations [of Eqs (5) and (6)] in the time interval to to to + T (T > 0). At time t o the n-tuple defining the flexible deflection and its derivative are denoted by u0 and duo/dt. (Note that Uo and du0/dt could have been obtained from the reduced basis simulation for time t < to, from the simulation of the unreduced equation set [Eqs (3a) and (3b)], or as initial conditions; the fashion in which we obtain Uo and duo/dt is of little importance in the immediate discussion to follow, since only the simulation error in the interval to to to + T .is considered.) The reduced initial condition problem is then defined by Eqs (5) and (6).

The path-derivatives technique uses as the columns of matrix X the flexible deflection n-tuple and its first rn - 1 derivatives, evaluated at time to; this reduction matrix will be used until time t o + T. In other words

X=[u~o°),uCol),u~o2)...U~ m-l)] rn <n , (11)

(1)

(2)

(3)

(-M-IK)kU<o°) and (-M-JK)ku~ 1) bring out higher mode information (with increasing k), which is similar to the Ritz vector expansion;

in addition, it also takes into account load derivative information, which the Ritz/eigenvalue- expansions do not account for (the significance of this point will be discussed soon);

the eigenvalues-expansion reduction technique does not account for the effect of initial conditions/ conditions at the beginning of the simulation time range (u~ °) and u~ 1)) in obtaining the reduced basis.

The m-tuple e(t), at any instant of time t > to, can be thought of as a replacement for the Taylor series coefficients {1, (t -- t0)/l!, (t - to)2/2! . . . (t - to)m-l~ (m - 1).t} r. The solution accuracy in obtaining c(t) from Eqs (5) and (6) may be better than the Taylor series approximation and at the same time have a larger radius of convergence/° Thus, it would seem that the path-derivative method should be more successful than other reduction schemes (Ritz vector method, eigenvectors expansion method, Guyan- Iron's method) and, indeed, a numerical survey 8'n has shown it to be so (for a range of dynamical problems).

In order to understand how basis reduction using path-derivatives affects the solution, let us differentiate Eq. (3a) at time to successively (up to a maximum of m - 1 times), multiply each differentiated equation by the Taylor series coefficients and add them together. We then obtain


M [11~0 ) + u~l)(t - to) u~m) ( t - - to) m l) ( t -- to) "+ ' ] - - - - ' i~ ! + ' ' ' + m! +U~'+ -(m + l--~! -J

+ K[u~00) + ~1) (t - t o ) + + U~0m_ l ) ( / - - / 0 ) m-l ] 1! ' " ( m ~ ~- . r _]

= P~o°) + P~o') ~ + . - . P(o" l)(t - - . t o ) ' - ' • (m -- 1)!

(13)

If we are to represent

e * r = F l , ( t - - t ° ) ( t - - t o ) 2 / i-! ' 2/

and

(t_to)m l ] (14) - - . . . j

(t -- to) (t -- t0)"- I P* = P~°°) + P~°') ~ + " " + p~0m--') (m -- 1)/ (15)

then Eq. (13) can be rewritten as

MX e*(t) + KX e*(t) = p*(t). (16)

The asterisk superscripts in c* and p* indicate their relevance in the discussion using the actual Taylor series coefficients; the solution of the reduced system of Eqs (5) ancl (6) gives e(t), which is different from c*(t).

If we now premult iply both sides of Eq. (16) by X r we obtain

XrMX ~ * ( t ) + X r K X c * ( t ) = X r p * ( t ) . (17)

Imagine now that, for some reason, we seek to obtain the time varying solution of the reduced equation set

XrMX ~ ( t ) + X r K X c(t)=Xrp*(t) (18)

with initial condit ions

Co = (XrX)-IXru0

~0 = (X rX) JXqi0 (19)

(note that u 0 = Xeo and duo/dt = X dc0/dt are satisfied exactly, since X contains u0 and duo/d0. We can then say, based on the discussions of the previous section, that the variations of c and c* may or may not be the same. The reasoning is as follows. Just as in the previous section, we can expand Eq. (18) as

MX i~(t) + KX e(t) = X(XrX)- 'Xrp*(t) + (p*(t))',

(20)

However, if we are to expand Eq. (17) in the same fashion

MX i~*(t) + KX e*(t)

= X(XTX) 'Xrp * "4- (I -- x ( x r x ) - I x r ) p *. (22)

Since (p*)', which lies outside the subspace of X, and ( I - x ( x r x ) - IXr)p*, which also lies outside the subspace of X, are not necessarily the same, it follows that e and e* are not necessarily the same.

If we are to ensure that c and e* have more or less the same variation, we must ensure that X(XrX)-IXrp and p have more or less the same variation. In other words, if the out of basis component of p is very small, then, even though (p*)' may still be different from ( I - - x ( x T x ) - l x T ) p *, we can be assured that II (P*)' II (ll (p*)' II is used to denote the norm of the n-tuple (p*)') is small, causing c to deviate very little from e*.

If ( I - X(XrX) I Xr)p* is to be small (i.e. to have a small norm), a large component of p* should somehow lie in the subspace of X. Keeping in mind the definition of p*(t) [Eq. (15)] and then looking at the definition of the component vectors of the reduced basis [Eq. (12)] we can see that the load information is contained. However, the load vector and its derivatives (at time to) are premultiplied by M - i. As a result, the subspace of X may just contain a small component of p*(t). To rectify this we can rewrite the dynamical equations as follows:

ii + M - IKu = M - lp, (23)

and write the equations for the reduced system [note the previous form of the dynamical equations in Eqs (5) and (6)] as

xrx i~ + X r ( M - I K ) X c = X r M - I p (24)

with initial condit ions

co = ( x r x ) iXru0

6o = ( x r x ) - i Xr~o. (25)

where

X~(p*)'=o. (21)

With this modification we can restate our goal: make the out of basis component of M ~p*, i.e. (I - x ( x r x ) I X r) M - i p . , as small as possible. Using

582 S.K. DAS et al.

path derivatives, with the modified equation sets of Eqs (23) and (24), satisfies the goal to a large extent; Eq. (12) contains M-lp~ °) and its derivatives (M-lp~ 1), M-lp~o2) . . . M-lp(o~ - 1)).

By modifying the reduced system, from Eqs (5) and (6) to Eqs (23) and (24), we have made the reduced system solution variation--using path-derivatives reduction--match the Taylor series coefficients (i.e. the unreduced system solution variation) to the best possible extent, within the range of convergence of the Taylor series. Outside the range of convergence of the Taylor series the modified reduced system solution variation will be better than that obtained from Eqs (5) and (6).

In the next subsection we suggest a method that has its roots in the path-derivatives method.

Suggested method o f basis reduction

The path-derivatives method, even though generally superior to other reduction schemes, s has the draw- back that its applicability could prove computationally expensive. For example, in the case of the inverse dynamics of space cranes the stiffness matrix is time varying, in addition to which evaluating the load vector and its derivatives is computationally expensive. Consequently, evaluating the path derivatives [the expressions in Eq. (12) have to be modified to account for time varying K] could prove very expensive.

In the suggested reduction method we use the flexible deflection n-tuples (of the unreduced system) at the beginning of the simulation time range (to to to + T), where the deflection n-tuples are chosen at successive time points separated by a small time interval (At) starting at to. A linearly independent set

of m such n-tuples would then constitute the reduced basis for the rest of the simulation time range.

This choice of reduced basis has two major advantages: (a) we do not waste time (i.e. computations) evaluating the derivatives (furthermore, obtaining the reduced basis vectors is a natural part of the simulation/inverse dynamics); and (b) the path-derivatives method picks out the first m - 1 derivatives (while a significant contribution might even come from derivatives of order > m - 1), whereas the suggested method uses n-tuples which contain within themselves the most significant derivative information required for the rest of the simulation time range.

A disadvantage of the suggested method is that obtaining a linearly independent set of m n-tuples might take more than m time intervals of size At. However, an analogous problem is also encountered in the path-derivatives method where the first m path derivatives may not necessarily be linearly independent, requiring that we obtain more derivatives to constitute an independent set.

If we consider the ith reduced basis n-tuple (in X), where xi is evaluated at time t = to -I- (i - 1) At, we have

x, = u~ °) + (i - 1)At u~l) + (i - 1)2At 2 u~2) + . . .

I! 2/

for i ~< m - 1. (26)

Since there are m independent n-tuples, we can obtain an m-tuple of multiplicative constants e** (different from c, which is obtained as the time varying solution of the reduced system; different from e*, which is the

100

9O

~ 70 uJ

~ 60

) ~ 4o

30

~ 2O

10

0 r ~ ~ f "r ' r ,~

0 1 2 3 4 5 6 7 8

DERIVATIVE #

Fig. 2. Variation of percentage of coefficients (with respect to Taylor series coefficients) of u~ °}, ~)... in Eq. (35) for increasing k, for the case m = 3.


Fig. 3.

M7 [ I t u 7

4

INERTIALLY FIXED BASE

Seven degree of freedom unforced spring-mass system.

list of Taylor (k > m) such that

Xe** ut0 °) k At k 2At z U{o2 ~ : +-i7-.' , U~o') + ~ + . . .

A m - 1At,. l _~_ U(O m -1) ..~ RmU(O m)

(m -- 1)!

series coefficients) at time to + k At

+ Rm+ lu{0 "+ t )+ . . . . (27)

where R m through R~ will be different from the actual Taylor series coefficients. Taking an example where m = 3 we have plotted the variation of the coefficients of u{0 °) and its derivatives [see Eq. (27)] for increasing k (Fig. 2). What we can infer is that the suggested reduction method has the capability of behaving like the path-derivatives method, and the likeness increases with increasing k (with increasing k derivatives of order > m - 1 have rapidly diminishing contributions). As to whether it actually behaves like the path- derivatives technique, or better, cannot be concluded.

As an illustrative example of the application of the suggested method, and its numerical behavior in comparison to that of the path-derivatives method, consider Fig. 3. The multi-degree of freedom spring mass system has seven degrees of freedom. We can write the governing dynamical equations of motion (free vibration) as

Mii + Ku = 0 (28)


u0~ = [l, 0, 0, o, 0, 0, o]

ti0 T = [0, 0, 1, 0, 0, 0, 0], (29)

where u is defined as

uT = [Ul ' U2' /'/3' /'/4' U5' /'/6' U7]" ( 3 0 )

The individual entries of the mass and stiffness matrices are defined as (see Fig. 3)

mid : M i

mij=O for i = j (31)

B

1.0

0.9

0.8

0.7

0.6

0.5

0.4

03

02

OA

#,

i I r i I i I

0 0.2 0.4 0.6 08 1.0 12 14 16 18 20

TIME (SECONDS/

Fig. 4 Variation of norm of displacement n-tuple, from reduced and unreduced systems, for seven degree of freedom spring mass system. (Dimension of reduced basis = 2; dimension of unreduced basis = 7;

4-I- = path-derivatives; - - 0 - - 0 - - = suggested method; = actual.)

CSE 124~P

584 S.K. DAS et al.

1.0

0.9

0.8

5 F-- 0.7

~; 0.6

0.5

0.4 rc O Z

0.3

0.2

/

I i i i i i i i i 1 i i i i i i i i i

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

TIME (SECONDS)

Fig. 5. Variation of norm of displacement n-tuple, from reduced and unreduced systems, for seven degree of freedom spring mass system. (Dimension of reduced basis = 3; dimension of unreduced basis = 7;

legend as for Fig. 4.)

1.2

1.1

1.0

0.9 n

z 0.8

Lid w 0.7 t..) 5 ~ 0.6 a

0.5 n- O z

0.4

0.3 I I I I I I I I I 1 I I I I I I I I I

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

TIME (SECONDS)

Fig. 6. Variation of norm of displacement n-tuple, from reduced and unreduced systems, for seven degree of freedom spring-mass system. (Dimension of reduced basis = 4; dimension of unreduced basis = 7;


and

ki, i = g i .-l- g i + l f o r / = 1 , 2 . . . 6

k7, 7 = g 7

ki, i+ 1 ..~-. g i + 1 for i = 1 , 2 . . . 6

k i + l , i = g i + l for i = 1 , 2 . . . 6 . (32)

In other words, the mass matrix is diagonal and the stiffness matrix is tridiagonal. We have assumed, for the sake of numerical simulation, that M~ = 1.0 kg (i = 1, 2 . . . . 7) and Ki = 50.0 N/m (i = 1, 2 . . . 7).

The results of simulating Eqs (28) and (29) and the corresponding reduced problems (m = 2, 3, 4, 5, and 6) are shown in Figs 4 through 8. In all cases the reduced bases were updated every 0.4 s; the 7-tuples constituting the reduced bases were

1 .0

0.9

2~

o_

z L; ~J

c)

03 E)

b E2

0.8

0 7

0 6

0 5

0 4 I f I I I I I I I ] I I I I


I !<

l 0 0.2 0 . 4 0.6 0.8 1.0 1 2 1.4 1.6 1 8 2.0

TIME (SECONDS)

Fig. 7. Variation of norm of displacement n-tuple, from reduced and unreduced systems, for seven degree of freedom spring mass system. (Dimension of reduced basis = 5: dimension of unreduced basis = 7;


1 0

0 9

v uj 0.8 &

2: F- ~.] 0 7

C)

a. 0 6 6"1 ffl LL 0

C) 0 5 z

0 4

f

I I I I I I I I I I I i I [ ] I I i I

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 0

TIME (SECONDS)

Fig. 8;. Variation of norm of displacement n-tuple, from reduced and unreduced systems, for seven degree of freedom spring-mass system. (Dimension of reduced basis = 6; dimension of unreduced basis = 7:


separated from each other by 0.005 s in time (i.e. At = 0.005 s).

We can see that the solution variations o f the reduced problems converge to that o f the unreduced problem with increasing basis size, i.e. increasing m. It is also seen that the solution variation using the suggested method is better than that o f the path- derivatives method; the solution is only marginally better in most cases. While nothing general can be

concluded from this simple example, i.e. as to which method is superior, what we can say is that the suggested method is definitely competit ive (with respect to solution accuracy and ease o f implementation) .

In the next section we discuss the application o f our suggested method to the dynamical equat ions governing the flexible mot ion of space cranes.

586 S.K. DAS e t al.

APPLICATION OF THE SUGGESTED METHOD TO SPACE CRANES

Since our primary aim was to develop a computationally efficient method for use in space cranes, this section deals with a numerical space crane example.

Before we proceed to the description of this specific problem, and the results, it would be of interest to see how the dynamical equations of the space crane [Eq. (1)] differ from those of the multi-degree of freedom spring-mass system discussed earlier [Fig. 3;

Eq. (26)]. The n-tuple s, in the lower partition of Eq. (1), consists of stiffness and coriolis/centrifugal effects. It can be written as

s = - Ku + h, (33)

where K is the time varying stiffness matrix and h is a time varying coriolis/centrifugal force vector. For convenience of discussion we also denote D as M, i.e.

M = D (34)

0.5 •,tll .............................................. UI~.

joint 2 i

iength_adjustable / . . . ~ / / / ~ . . i

i / ~ / length-adjustable bar 2

/ . no.

• ,~ti ..................................................................... I0,.

1.3 Fig. 9. Two section adaptive structure with six nodes.

flexible body position

"'% :":" "*..........

"".. .:" " " ~ flexible body end

.............. ...X. _.2>..,? /': ~ q .... '-~.i'i~q i . ..... ........ ~ J t,p displacement

rigid body ./" ~l .... "~ /~.¢. position .... / ~ .......... ~ ._

. . . . . . . . 'y t"'l ...... q / ". ..... / t 2 ~ rigid body end

Fig, lO. Graphical definition of tip displacement. (Solid line: hypothetical rigid body position; dashed line: actual flexible body position.)


since D is, in fact, a time constant mass matrix. Rewriting the lower partition of the dynamical equations [Eq. (1)], we have

Mii -I- Ku + ( - h + BTti -- g(-f')) = O, (35)

where ( h - BTd2q/dt2+ g(]')) can be thought of as a forcing function, its contributions coming from the rigid body motion inertia (Brd2q/dt2), the length adjustable ba~r forces g(-f), and the rigid and flexible coriolis/centrifugal force contributions h. The length adjustable bar force variations are not simple pre-

determined quantities; their variations are obtained from the upper partition of Eq. (1) (inverse dynamics procedure). The complex dependencies of the forcing function make Eq. (35) quite different from Eq. (28).

The planar adaptive structure of Fig. 9 was used to apply our method; it consists of two sections with two length adjustable bars and a uniformly distributed bar mass. The angles q~ and qz between sections is as shown. By controlling the length adjustable bars we can effect movement of the entire structure in any fashion dictated by the variations of q~ and q:. (We

0 0.2

0.08

0.07 / ~

0.06

0.05

0.04

003

0.02

0.01

000 I I I I I

1.6

I I I i I

0.4 0.6 0.8

\

1.0 1.2 1.4 1.8

Z LU LU (3

O_ r.,o

O_

TIME (SEC)

F i g . l 1. Comparison of tip displacements, from reduced and unreduced systems, for two section adaptive structure. (Reduced basis dimension = 4 ; unreduced basis dimension = 8; 4 - + = reduced system variation;

- actual variation.)

0.08

0.07

0 06

~" 0.05

004

& 0 03

[2,

002

001

000

0 0.2 0.4 0.6 0.8 l l l l l l l l l

1.0 1.2 1.4 1.6 1.8

TIME (SEC)

Fig. 1:2. Comparison of tip displacements, from reduced and unreduced systems, for two section adaptive structure, (Reduced basis dimension = 5; unreduced basis dimension = 8; legend as for Fig. 11.)

7~ ql = ~ + 0.1(1 -- c o s ( x t ) )

We assume, for the sake of convenience, that a

path has been planned for the end node and, in addition, a r ig id b o d y inverse kinematics p r o c e d u r e

for this path yields

q2 = ~ - 0.1(1 - c o s (/~t)). (36)

0.09

We also assume that all the bars have the same total mass and stiffness.

0.08

0 .07

0 .06 v

z w 0 .05

w 0 "~ 0 . 0 4 (3. co

o 0 .03

i -

0.01

0 .02

0 .00

/ I

0

0 .09

I I I I I I I I

0.2 0.4 0.6 0.8 1.0

define nodes as being attached to the vertices of bars; in this case we have six nodes, of which two are fully constrained. Each node possesses two independent degrees of freedom.) We also define a point called "end node" (see Fig. 9) and, in future, we will use the term "tip displacement" to mean the Euclid norm of the flexible displacement of the end node. Figure 10 illustrates this concept; the solid line position is the hypothetical position of the space crane if it were rigid (with angles qx and q2 between sections) and the dotted line position is the actual flexible body position of the space crane (with angles q~ and q2 between sections).

I I

1.2

I I I I I

1.4 1.6 1.8

TIME (SEC)

Fig. 13. Comparison of tip displacements, f rom reduced and unreduced systems, for two section adaptive structure. (Reduced basis dimension = 6; un reduced basis dimension = 8; legend as for Fig. 11.)

0.08

0 .07

0 .02

0 .06

w

0.05 o

0 ,04 a g.

0.03

0.01

0 . 0 0 i i i ~ i i i i i

0.2 0.4 0.6 0.8 1.0

TIME (SEC)

f r i i

1.2

588 S.K. DAS e t al.

1.4 1 6 1.8

Fig. 14. Comparison of tip displacements, f rom reduced and unreduced systems, for two section adaptive structure. (Reduced basis dimension = 7; un reduced basis dimension = 8; legend as for Fig. 11.)


Looking at Fig. 9, we can see that there are eight independent ,degrees of freedom, i.e. two for every unconstrained node. Consequently, in our study we have used reduced bases with m = 4, m = 5, m = 6 and m = 7 vectors.

Figures 11 through 14 illustrate the application of our suggested reduced basis technique. It is seen that increasing m results in bettering the reduced basis solution. The: small jump discontinuities seen in the reduced basis solutions occur at time points where the reduced basis is updated. The most we could do to minimize such discontinuities was start the updated reduced basis solution not at the time point where the previous solution left off, but at some time point a little before that. In other words, we back- tracked a little before continuing to march out in time.

CONCLUSIONS

In this paper we have proposed a reduced basis method which derives from the path-derivatives reduction technique. The motivat ion for arriving at this technique stems from its application to the inverse dynamics of large space cranes possessing many degrees of freedom. It has been shown that this method has potential for behaving like the path- derivatives technique, and maybe even better. Its attraction lies in the simplicity and computat ional ease with which the reduced basis n-tuples are obtained. A sim pie seven degree of freedom spring-mass system was used to compare the two methods; the suggested method proved better than the path-derivatives technique, though no general conclusions can be arrived at based on the result of this example. A numerical application of the suggested method to an example space crane problem was given.

Acknowledgements--This work is partially supported by the Jet Propulsion Laboratory, California Institute of Technol- ogy, under contract NAS7-100 of the National Aeronautics and Space Administration.

REFERENCES

I. S. K. Das, S. Utku and B. K. Wada, "Inverse dynamics of adaptive structures used as space cranes," Journal of Intelligent Material Systems and Structures I, 50 75 (1990).

2. S. K. Das, S. Utku and B. K. Wada, "Inverse dynamics of adaptive space cranes with tip point adjustment," Proceedings of the A1AA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamics and Materials Conference, Long Beach, CA, 2-4 April, 1990, part 4, pp. 2367 2374.

3. S. K. Das, "Dynamics of open-chain flexible adaptive truss structures used as mechanical manipulators," Doctoral thesis report, Duke University, Durham, NC, 1990.

4. E. L. Wilson, M.-W. Yuan and J. M. Dickens, "Dynamic analysis by direct superposition of Ritz vectors," Journal of Earthquake Engineering and Structural Dynamics 10, 813-821 (1982).

5. E. L. Wilson and T. Itoh, "An eigensolution strategy for large systems," Computers & Structures 16, 259 265 (1983).

6. R. J. Guyan, "Reduction of stiffness and mass matrices," A1AA Journal 3, 380 (1965).

7. B. Irons, "Structural eigenvalue problems: elimination of unwanted variables," A1AA Journal 3, 961 962 (1965).

8. J. L. M. Clemente, "A study of reduction methods in nonlinear dynamics," Doctoral thesis report, Duke University, Durham, NC, 1984.

9. A. K. Noor, "Recent advances in reduction methods for nonlinear problems," Computers & Structures 13, 31 44 (1982).

10. A. K. Noor, "On making large non-linear problems small," Computer Methods in Applied Mechanics and Engineering 34, 955-985 (1982).

11. S. Utku, J. L. M. Clemente and M. Salama, "On errors in reduction methods," Computers & Structures 21, 1153 1157 (1985).

use of reduced basis technique in the inverse dynamics of large space cranes

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