- Email: [email protected]
Bond graph modelling of multibody dynamics and its symbolic scheme
Bond Graph Modelling Dynamics ~~TAKEHIKO
Department
KAWASE
of Multibody
and its Symbolic and HIROAKI
of Mechanical
Scheme
YOSHIMURA
Engineering,
Waseda
Unitiersity,
Tokyo
169, Japan
ARSTRACT: A bond graph method of modelling multibody dynamics is demonstrated. Spec$ically, a symbolic generation scheme which ,fullyutilizes the bond gruph irzjbrmation is presented. It is also demonstruted that structurul understanding and representation in bond gruph theorv is quite powerful ,for the modelling of’ such large scale systems, and thut the non-energic multiport of junction structure, which is u multiport expression of’ the system structure, plays an important role, as,first sug~qested by Paynter. The principal part qf the proposed symbolic scheme, that is, the elimination of excess varinhles, is done throqh tearing and interconnection in the sense of Kron using newly de$ned causal and causal co#icient arrays.
I. Introduction
Much research has been directed to multibody dynamics in the fields of robotics, space dynamics and bio-engineering of the human body. One of the principal concerns is no doubt the modelling of the dynamics. A major stumbling block in such modelling comes from the considerable increase in the degree of freedom as the number of bodies involved increases. This requires : (1) the formulation of a large number of system equations and (2) the fast numerical integration of the resultant nonlinear algebraic-differential equations. These difficulties imply that the modelling of such multibody dynamics inevitably requires symbolic manipulation of equations on the computer and hence to make any symbolic scheme for the generation of the system equations, categorization of the required kinematic and dynamic relations is needed. Various modelling methods have been reported, and those by Roberson and Wittenburg (l), Wittenburg (2) and Kane and Wang (3) are noteworthy. Roberson and Wittenburg utilized a circuit graph model to represent the kinematic structure of the system together with its associated algebra. Kane proposed a new concept of the partial velocity (4) which enabled us to derive the required equations of motion without any elimination ofconstraint forces which come from the kinematic constraints. The equivalent circuit model mentioned above certainly entails the kinematic information which comes from the geometric configuration of the bodies, and its associated algebra plays quite an important role in the symbolic scheme to generate the system equations on the computer. However, the circuit model entails only a part of the required kinematic relations and hence the method needs other kinematic relations besides the above graphical information, for example, those for mechanical joints. 917
T. KUMVLWand H. Yoshinzura
On the other hand, the method proposed by Kane requires a certain amount of skill in mechanics to find an appropriate partial velocity for it defines a locally admissible subspace which is normal to the local subspace defined by the constraint relations imposed u priori among the velocities. The bond graph, as a modelling tool, graphically entails all kinds of kinematic and dynamic information. That is, the physical properties of the system elements, the structural relations representing how the system elements are interconnected and the causal relations showing which variables are to be replaced by which variables. Bos and Tiernego (5) and Bos (6) proposed a systematic modelling method using the bond graph technique. However, any idea to systematically eliminate a large number of excess variables by using the bond graph information, specifically causality, was not explicitly given in their papers. The elimination of those excess variables is most tedious and painstaking work and this is where any symbolic manipulation is most required. In the present paper, fundamental aspects of bond graph modelling of multibody dynamics are demonstrated and special concern is focused upon how the bond graph information is incorporated into the proposed symbolic scheme. The present paper is thus tutorial and it is suggested that the reader refers to other papers by the present authors for more details of the method and also how it is applied to more general flexible multibody dynamics (7-10). II. Fundamentals
of Bond
Graph Concepts
In the modelling of large scale systems like multibody systems, the most important things are (1) to write all the necessary relations and equations down and (2) to determine which variables are to be replaced by which variables and also where it should be done to formulate a set of the system equations. To do this for multibody dynamics. some kind of categorization of all the kinematic and dynamic relations may be helpful. In the bond graph as first suggested by Paynter (ll), those relations are classified into the following three categories : (I) energic multiports : physical properties of the system elements. (2) non-energic multiports : structural information of how the system elements are interconnected. (3) causality: inputtoutput relations indicating which variables are to be replaced by which variables. The energic multiports to appear in the present paper are storage elements, that is, mass I : M and the moment of inertia = I : J of a body, and power sources IZ Se and L: Sf. One of the key concepts of bond graph theory is the non-energic multiport. The term non-energicness was first coined by Birkhoff (12) in his famous textbook on mechanics. Let q and R be the generalized coordinate and force of the system having II degrees of freedom. Then the system is said to be non-energic when the following relation
Bond Graph Modelling (&R)
of Multibody
=o
D~wzamics (1)
identically holds for every time. In the above, () indicates a column vector, (,) the scalar product of vectors, and (‘) the time derivative. We are going to see four kinds of such non-energic multiports in the present paper. 1. l-junction and O-junction The l-junction stands for D’Alembert’s principle in mechanics, that is, the balance of forces (torques) exerted upon a body and simultaneously it imposes the condition that all the velocities at the bonds associated to the l-junction are common. The O-junction indicates that all the forces (torques) at the bonds associated to the O-junction are common. 2. Modulated transformer (MTF) We will see various relations are modelled as MTFs. The first is the MTF model of the coordinate transformation first discovered by Karnopp (13). Let us write a coordinate transformation, integrable or non-integrable, in a form decribed below using an n x n coefficient matrix A($, de = ii(q) dq
or equivalently
i = A(q)i
(2)
where q and it indicate the old and new coordinates respectively. There may be a case where the coefficient matrix A contains time explicitly. In that case, we may introduce the (IZ+ 1)th coordinate q,,, , = t, then again we have the same expression at least formally. The power invariant under the coordinate transformation insists
(~,R)+(iQ=I)
=
0
(3)
holds identically. In the above fi denotes the force expressed along the new coordinate system. From the above relations, we easily have the relation for the force variables. R=
-Jrn
(4)
where ( )’ indicates the transpose of a matrix. The bond graph representation is given in Fig. 1 but note that the power orientation is different from the conventional method. This is because we want to follow the context of non-energicness (1). The second is the kinematical constaint like those imposed by kinematical loops.
I
l/s
--I
FIG. 1. Bond graph model of the coordinate Vol. 328, No 5!6, pp. 917-940. Prmed in Great Bmam
transformation.
1991
919
T. Kuwa.~e and H. Yoshimura Let us denote
a kinematical
constraint
a(y) dq = 0
as - - 1 a(q)q = 0
or equivalently
(5)
where the coefficient matrix a is tn x n of rank m(m < n). Let us first introduce coordinate transformation described below, i.e.
the
d77, = a(q) dy di& = p(q) dy
(6)
where 7c = (757,E:)’ is an n-vector representing the new coordinate and the coefhcient matrix p in the second relation is arbitrary except that it is so chosen that the transformation (6) itself is invertible. Thus the transformation is modelled as an MTF and in this case the first m ports representing the new coordinates are connected to an outside velocity source of i , = 0. Another important MTF model to appear in multibody dynamics is that of the mechanical joint. First take a look at Fig. 2 showing two bodies connected by an ideal mechanical joint. The mechanical joint drives the body 2 with the linear velocity fi and the angular velocity fi relative to the body 1. From the geometrical and kinematic relations, we have ri> = k, -7, where (‘) denotes
a 3-dimensional
+ 1;‘+J,
(7)
vector.
Joint Body 2
.
-
+
cb) I r2
Body
FIG. 2. Kinematic
920
relation.
_-_--
Bond Graph Mode&y By differentiating the above relation in the inertial relation among the linear velocities of the two bodies, kZ=
tv,-~,x~,+~+~,xti+~j?x72.
ofMultibody Dynamics frame,
we easily get the
(8)
In the above the symbol x indicates the vector product. On the other hand, the angular velocity is written as C& = 6, +Q.
(9)
For the description of the translational and rotational motions, we employ the body fixed frame as is conventional. Every component of the velocity is hence described in the above local frame. Thus from the above two relations (8) and (9), we finally have the matrix representation of the MTF model of the mechanical joint.
where
In the above, the variable v(O) denotes the linear (angular) velocity, observed in the local frame as mentioned above, F;(F) its dual variable, that is, force (torque). And also tI?(G) indicates the linear (angular) driving velocity at the joint observed in the local frame and, I?(?) its dual force (torque). Furthermore T,2 indicates the matrix representing the base vector transformation between the two bodies, r,,z the distance vectors between the hinge point and the center of mass of each body, and 7, the skew symmetric matrix representing the vector product like operation (i’ x ) described below
where r,,, ri2 and ri3 are the scalar components of V, observed in the local frame. The bond graph representation is then given in Fig. 3. Again referring to the MTF model of the coordinate transformation, we rewrite the variables as follows
Yol.
32X. No. 36.
Prmed
in Great
pp 917-940. &nun
1991
921
7’. Kuwase and H. Yoshimura
Rotational
Translational
FIG. 3. Ideal mechanical joint
where the subscripts tr and rot indicate that the corresponding variables represent the translational and rotational motions respectively. Using the matrix representation of (lo), we can easily make a catalogue of mechanical joints. which will be useful as a data base for the symbolic scheme to be proposed in the later chapter. Part of such a catalogue is shown in Table I.
Euleriun junction .structure (EJS) The final bond graph model of the non-energic multiport to appear in the paper is the so-called Eulerian junction structure first proposed by Karnopp (14). The inertia coupling torque, when the motion is observed from the body fixed frame whose axes coincide with the principal axes of the body, is expressed as
(11) where rdenotes the coupling symmetric matrix representing below
torque, i the moment of inertia and fir,, the skew the vector product like operation (9 x ) described
where pj, ps and ph are the scalar components of the angular momentum &, observed in the local frame. It is easily shown that the above torque satisfies the non-energicness and so it is a member of the non-energic multiport defined previously. In the latter expression, the angular momentum p,,, is used instead of angular velocity. This is only because we are going to employ the Hamilton form of the
g
.g
UJ DOF=2
RJ I DOF=l
TJ I DOF=I
Scheme
F
\
Symbol
I
a 0
-W
-spse -C@H co 0 0 0
-sqwo -C@YH -so 0 0 0
0 0
- SO
-cgco
-s*co
0 0
r&P -r,S@O
s*
-Cd
r,SH-r,,C4
CO
-c&se
-s&so
-r,S&%
0
-r,CO
SO CO
sosg
0
0 0
Cd - Cmp
0 0
0
(I, +r,,SO)C~
0
0
0 0
-r,COCCp
-r,CO
Vdt
CO
s0
-(r,,+r,SO)Scp
x = r,+
0
so
0
so
0
S4
- ccp
r,,SH-r,Cqb
r,SqKO r,Cf&CO
0 0
-co
CO
0
0
0
so
0
-(_u+r,SO)
0
so -cocg co SOCd,
0
00
0
0
0
1
r,+.uSO
-r,co 0
0
xc0
-
matrix
0
sosc#l
-cosd,
0
0
00
00
0
0
-co co so
r0IO so
Transformation
ofideal mechanicaljoints
TABLE I A part qfthe catalogue
0 0
0
w
-ccp
-r,Cs
1
1
0
0
0
-rI 0
3 -. 2
g
2
s-
G g
T. Kuwasr
and H. Yoshimuru
FIG. 4. Eulcrian junction
structure.
equation of motion. The bond graph representation ture is shown in Fig. 4.
of the Eulerian
junction
struc-
III. System Equations As to the form of equations of motion, various forms have hitherto been used by many researchers. The research report by Kane and Levinson (15) of a comparative study of these various forms of the equations is worthy of note. There are two aspects to be considered. To get the mathematical mode1 of the least order for fast numerical integration or estimation it is most important to reduce the number of algebraic manipulations. However, from the system design point of view, specifically when we deal with control related problems, it is much more important that the generation process of the system equations be well systematized so that the effects of any changes in design parameters like geometrical parameters and other kinematic and dynamic conditions are more clearly predicted. It is widely known that, for fast numerical estimation, the NewtonEuler form is more effective than the Lagrange or Hamilton form. However, from the viewpoint of systematizing the modelling process. to say anything definite about which form is most advantageous over others, further comparative study will be needed. Much of the research concern of the present authors has been directed to the latter aspect mentioned above. In the present paper, the Hamilton form is employed. Of course, the proposed method is applicable to any other form of the equation of motion. It is worthwhile to note the Hamilton form to be used in the paper, strictly speaking, is not derived from the Lagrange but from the BoltzmannHamel equation of motion. One more thing worth to be mentioned is that Kane’s method, which elegantly eliminates the excess constraint forces, is essentially equivalent to
Bond Graph Modelling
qf Multiho&
Dynumics
FIG. 5. Stanford manipulator.
the proposd bond graph method, of constraint forces.
which utilizes the causality
relation,
of elimination
III. 1. System equations qf the primitice svstem For simplicity of illustration, here we consider a Stanford manipulator shown in Fig. 5. Obviously the Stanford manipulator has no kinematic loops and so it is decomposed into a set of fundamental pairs each of which consists of a body and an ideal mechanical joint adjacent to it. Using the bond graph tools described in the previous section, we have the bond graph representation of such a fundamental pair as shown in Fig. 6(a). The upper half indicates the translational motion of the center of mass of the body and the lower half the rotational motion. Again we note that both the translational and rotational motions are observed in the body fixed frame and hence the translational motion of the center of mass has to be transformed into the inertial frame. This is shown by an MTF in the upper half. According to the boundary conditions imposed upon velocity and force, the causality at every bond is automatically determined. In the case of the Stanford manipulator, which has no redundancy, the velocity at every bond is determined by those of the driving sources at the joints together with the base boundary (body 0). On the other hand, the force at every bond is determined by that of the end effector and the external forces exerted upon the bodies. This assignment of the causality is also clearly shown in the figure. Now we come to the point to set up the system equations of such a fundamental pair. We write the equation of motion for a totally torn apart system of a fundamental pair [Fig. 6(b)] using the generalized coordinate 71= (&7&)’ and its conjugate momentum p [see Fig. 6(a)]. The pair (Z, ii) is a state vector. Vol.328.No. 516,pp. 917-940.1991 Prmted tn Grcat Bntain
925
T. Kawase and H. Yoshimura
926
Bond Graph Modelling qf Multibody
dP -= dt
-.f+,+-n
Dynamics
(12)
where i? = diag (m, m, m, I,, Z2, Z3)- ‘, j = (0, -mg, 0,0,0,0), and Ii denotes the constraint force representing the interacting force from the adjacent body, g the gravitational acceleration and f the inertia coupling force representing EJS [see Fig. 6(a)]
In the above,
Using the causal relation
of the mechanical
joint,
we have
;o = Cz*+& The coordinate
transformation
(13)
for the translational
motion
gives us the relation
iT,,,= A?&
(14)
where the coefficient matrix A represents the base vector transformation. From the constraint imposed by the l-junction in Fig. 6(a), we have ~=$i&O, Now we are going back to the causal relation (n*T, RT)’ is expressed as
of the mechanical
(15)
joint. The force
(16) Using the dual relations
where E indicates
for (13) through
(15), we have
the 3 x 3 unity matrix and
Combining all those equations, damental pair. Vol. 328, No. S/h, pp. 917-940, 1991 Printed in Great Britam
we finally
get the system equations
for the fun-
927
T. Kawase and H. Ymhirnura
joint constraint coord. transformation l-junction = 0.
-T C -7 C equation Further eliminating the variables r? and i associated the above equations, we finally have
(18)
of motion with the inner bonds from
= 0.
(19)
3.2. System equations
The final step is to set up the system equations process is decomposed into two steps described (1) interconnection of the fundamental (2) elimination of the excess variables.
of the interconnected below :
system. The
pairs ;
The interconnection of any two fundamental pairs is mathematically expressed by the continuity conditions of velocity and force between the adjacent bodies and is bondgraphically equivalent to a l-junction between the fundamental pairs. Those continuity relations are l=T_n*
itI
‘* n,+ I --+
-0 = (j,
(20)
Thus we have a large number of equations for the Stanford manipulator, and we easily see that large numbers of excess variables appear such as those associated with the inner bonds of each fundamental pair. We know that those large numbers of system equations are reduced to a set of system equations having much more smaller numbers of equations in terms of six independent variables in the case of the Stanford manipulator. These equations are algebraic-differential equations since the equations of motion are augmented by the kinematic constraints imposed by the mechanical joints, etc. The above means we have to eliminate many excess variables. The elimination of those excess variables is most tedious and painstaking and hence this is where the symbolic manipulation of equations by computer comes to play a vital role. Now we look at this elimination process of the excess variables from the modelling and numerical simulation point of view. There are two aspects worth
Bond Graph Modelling mentioning
?f’Multibod~~ Dynamics
:
(1) final form of the system equations of the interconnected (2) numerical integration or other numerical estimations.
system;
As to the first, there are many stages of the above mentioned elimination of excess variables and hence, according to the purpose of modelling, we have various forms of the mathematical equations. From the design point of view of such multibody dynamics, the process of equation formulation should be well systematized. This is required so that effects of any changes in design conditions, geometric parameters or kinematic and dynamic conditions are well predicted in the formulation process. This is much more important in the design stage of such multibody systems, especially its control related problems. This may not require one to reduce the set of the equations to the least order. As to the second, it is necessary to reduce the execution time of the numerical integration or other numerical estimation like an inverse dynamics for on-line control. This implies we need to reduce the order of the system equations as much as possible. To meet those rather different purposes, various mathematical forms may be useful. Here we show two cases which have different elimination orders. First we combine six sets of equations for the fundamental pairs (19) and the same number of the continuity relations of velocity and force. In other words. we eliminate 6*s and li?. Thus we have the following set of 54 equations. In the case of the Stanford manipulator, the driving function at each joint is a scalar function representing velocity. So the number of the equations is 54 rather than 84.
= 0.
Vol. 328, No 5:6. pp 917-940, 1991 Printed m Great Bntam
(21)
929
T. Kuwase und H. Yoshimura In the above, s, = b+,z-S, (i = I,. .6). Thus, in the above, the matrix C, (i = 1,. ,6) is a 6 x 1 matrix, and hence each required joint torque or force l?, (i = 1, ,6) is also a scalar function of time. Next, we obtain the system equations of the least order. First we rearrange (21) obtained above, assuming the base body is fixed to the inertial frame and the end effector moves freely under no external forces and torques.
where R = [l?‘:, R.1,. ,a:]‘, 3 = [s{,sq,. ,5:]‘, f = [.??T,$i,. . , l?:]’ /7 = [$,/?P _. . ,pT]’ and the coefficient matrices D, 7 and cl are given below
and
ii... &, a=p=
10 Putting
,
B3C(3c3..
s = b-t fivp -J
p, into (22), we have
p-fi&j I=” R-&L&?p+&
(23)
where
Using the last equation differentiating it, i.e.
of (23), we get the expression
for the momentum
p and by
i = y ‘($G+&fi), Substituting
the above relation
into (23), we finally have - (24)
This is the expression
which is usually
used in the inverse dynamics.
Bond Graph Modelling
of Multibo&
Dynamics
As is easily seen from (24), we cannot have integral causality for the Stanford manipulator, and the discussion so far implies that the resultant system equations become an implicit form as is well known. IV. Symbolic
Generation Scheme
So far we have described the basic idea to formulate the system equations using the bond graph information. As stated in the previous section, all kinds of information are entailed in the bond graph. Thus to make a symbolic generation scheme, we can directly start from drawing the bond graph of the system of our concern. As we have shown, the Stanford manipulator has a tree-type geometrical configuration and hence it suffices to start from a fundamental pair. When the system has kinematic loops which are usually constrained in a plane, the constraint relations imposed by those kinematic loops are modelled as an MTF as previously shown. The symbolic scheme is essentially extended to those systems. Now, as shown in Fig. 7, the pair is further decomposed into a totally separated state which might be called a primitive system in the sense of Kron. The tearing and interconnection is fully described by the junction structure of the pair. The symbolic generation scheme proposed in this paper is schematically shown in Fig. 8. The key part of the scheme is the elimination of excess variables as stated previously. Therefore, in the following, we will show how three kinds of bond graph information are incorporated into the scheme. The system structure is here recognized as a hierarchy shown in Fig. 9. Level 0 corresponds to the interconnected system and level 2 the totally torn apart system. The elimination process is understood as the inverse direction of the tearing. So we start from the level 2. In the present scheme, the language “C” is used. So in the following the method is described using P
The junction structure of a fundamental pair consists of seven non-energic multiports and all the non-energic multiports and associated multibonds are numbered as shown in Fig. 10. Thus the junction structure is recognized and stored in the computer by giving the data as follows : sub _ prm system[i] : the array to describe the ith junction are described by the following arrays.
structure
and its contents
structures contained in the junc(1) num system : number of the sub-junction tion structure sub prm system[i]. the bonds associated with sub prm system[i]. (2) num_bond : numberof (3) num port[ ] : number of the ports associated with sub prm system[i]. (4) portr] : bond number to which the port is absorbed by connection. and the number + 1 is given when power flows (5) power[ ] : power orientation into sub prm system[i] and - 1 when it comes out. of either of the velocity output or force output at the (6) causal[ ] :assignment port. (7) type : type of the junction structure sub prm system[i]. st%ctures contained in the junc(8) sub system[ ] : naming of the sub-junction tion structure sub _ prm _ system[i]. Vol. 328. No. 5/6, pp. 917-940. Printed in Great Britain
1991
931
T. Kuwase
and H.
Yoshimura
Stanford
manipulator
system
-----I
’ I
MTF
1 lb +_--_J
---__-_ I=
MTF F
; ’
’
/-
-1’ r
WI lr Sf
M .. Jl
7
\
.-----_
P 1 K=MGY i
+_-‘__--L_----
IF Sf
I
P
I
L’F
I IK=MGY
iT
’
I
IF
;
’
II ..
___-_--;_---A-_---A
II .. J
I J L_____’
I 1 I
I I
FIG. 7. Tearing and interconnection. (9) phy type[ ] : type of element (10) phylnum[ ] : suffix number designated.
connected to the port designated. given to the element connected
to the port
Illustrative example of the system description are shown below (see Fig. 9). First the decomposition of thejunction structure sub - prm _ system[O] is described below. sub prm system[O] +sub system[O] = 1 sub-prm-system[ 11. sub-system[O] = 2 subIprmIsystem[ 11* sub _ system[ I] = 3 .. .... ..... ..... .... .... ..... ..... ..... .... .... .... .... .... .... ... sub _ prm _ system[4] -sub _ system[4] = 10
932
Journal of ihe Frankhn lnstitule Pcrgamon Press plc
Bond Graph Modelling
Object
Dynamics
Model
u I
of Multibody
Bond graph
I
model
I
I (input) Bond graph << Hierarchical
Primitive << Tableau
/ 1
Coeff
matrix
Eliminating
Vol. 328, No. 5/6, pp. 917-940, 1991 Printed in Creal Bruin
information structure
>>
system form >>
1 ( Causal matrix unnecessary
variables
FIG. 8. Symbolic
generation
1j
scheme.
933
T. Kuu~a.w and H. Yoshimura Interconnected Level 0
0
-
y
y
_ :
_
f__.__-._-___.___--_-.-_-i-_-.‘“””--_ 4
6
7
8
I sub_prm_system
w,b-v~,-vy
10
9
(4)
11
12
,
,
,
i
,
,
,
13
14
15
16
17
16
19
Level 2
(6)
FIG. 9. Hierarchy
structure
of the junction
structure.
1111 11 [121 (9) 1 l-T= IlO1
PI6 (8)
MTF
(6)
Ii]
Port
(1)
Non-energic
0
FIG. 10. Junction
934
Absorbed
structure
multiport port
of the fundamental
pair. Journalof the Franklin Pergamo”
Institute Press plc
Bond Graph A4odelling oj”A4ultibody Dynamics
The kinematic and dynamic described below.
relations
of each non-energic
multiport
are then
sub prm system[6] - num system = 1 sub_prm_system[6] snurn-port[O] = 5 _
sub prm system[6] - num bond = 5 syb_prmIsystem[6] * pow&[01 = + 1 _
sub_prm_system[6] * port[O] = 1 sub_prm_system[6] * causal[ l] = “E” sub prm system[6] *causal[3] = “F” sub_prm_system[6] - type = “MTF” _ _ .... .. .... .... .... .... .... .... ..... .... .... ... ..... .... ..... ..
sub_prm_system[6] - causal[O] = “E” sub_prm_system[6] *causal[2] = “E” sub_prm_system[6] - causal[4] = “F” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . .. . . .,............................................................
-m
The total descriptions will thus give just a juxtaposition of the separated nonenergic multiports contained in the junction structure of sub prm system[O]. Next we consider the interconnection of those non-energic multiports. Let us take a look at the fundamental pair sub prm system[2] and let us demonstrate how the interconnection is described. To do this,we first consider sub prm system[4]. The connections occur, for example, at the bonds of sub_prm_syitem[6], [7] and [lo], and those are described below. sub prm system[4] - num system = 7 sub_prm_system[4] * num-bond = 14 _ _ sub _ prm _ system[4] *port[O] = 1 sub_prm_system[4] * port[3] = 4 sub prm system[4] *port[4] = 5 sub_prm_system[4] * port[5] = 4 _ _ sub _ prm _ system[4] - port[l3]
= 5
The above indicates that ports[3] and [5] absorbed in bond 4 by connection and similarly ports[4] and [13] in bond 5. The above operations directly imply the possibility of the replacement of variables in the set of equations describing the junction structure sub prm system[4]. For example, the velocity and force variables associated with firt[3]and [5] are replaced by the velocity and force associated with bond 4 respectively. Now we come to the point to eliminate, as the result of interconnection, those excess variables associated with those ports and originally given different symbols. To do this, we first define the incident array. The component of the incident array indicates a relation between a non-energic multiport and each power bond in the junction structure of our concern. For example, the junction structure sub prm system[4] has seven non-energic multiports and 14 multibonds. So we define the incident array as a 7 x 14 matrix. When a power orientation at the ith Vol 328, No. 5/6, pp. 917-940, Pnnted in Great Bntain
1991
935
T. Kuwase
and H. Yoshitwuru
bond is directed towards the ,jth non-energic multiport, then the (,j, i) element of the array is + 1, when directed reverse, it is - 1 and when the ith bond is not associated with the .jth multiport, it is null. Thus we have the incident array described below for sub _ prm _ system[4]. bonds
+I
+I 0
non-energ,c multiports 1
+I
-I 0
0
-I
0
0-I
+I
0
0
0
0
0
0
0
0
0
0
fl
0
0
0
0
0
0
0
0
0
0
0
O-I
0
0
0
0
0
0
0
0
0
0
-I 0
0
+I
0
+I
+I
0
0
0
0
I
0
0
0
0-l
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+
0 +I
+I 0
0 +I
O-l
0
0
0
0
-1
0
0
0
0 (26)
Next we define two more arrays called the causal and cuusul co@c’i~~i arrays. First we define the causal array. Here we note that the causality @ations at the bonds associated with each non-energic multiports determine which variable (output variable) is to be defined by which variables (input variables). In other words, the output variable at each port is described in a causal form in terms of input variables at the other ports. Now take a look at sub prm system[4]. We have, in this case, 20 such causal relations in terms of 28 va!tablei [2 x sub prm system[4] * num bond (= 14)]. So we define the causal array as a 20 x 28 matrix-Thus each row-corresponds to a causal relation at a bond and each column to a variable. Here we arrange those variables in the order (l=f,.A,,n2,i2 ,..., ii,,, i,4) and we consider the causal relation at the ith port. When thejth variable is output, the (i,,j) element of the array is + I and when input, it is - I. and otherwise it is null. For example, from the assignment of the causality shown in Fig. 9, the computer understands that sub prm system[4] - sub system[O] = 6 has three force outputs at ports[O], [I] and [21,&d two velocity outputs at ports[3]. [4]. This simply implies that forces at the ports[O], [l] and [2] are defined in terms of those at the ports[3] and [4]. And conversely, the velocities at the ports[3] and [4] are defined by those at the ports[O]. [I], and [2]. Thus every element of the causal array is symbolically obtained as shown below. variables [O]
Ltl [2] ports
PI ,41 [S]
+ I
0
0 0 0 0 -1 0 -1
0
0+1
0
0
0
-1
o-~l
0
0
0
-I
0-l
0
0
-I
0-I
0
0 0
0 -1
0
0
-I
O-l
fl
0
-1
0 O+l
fl
0
0 0
0
0
0
..,.,.,..................,. .
0 0
+I 0
-1
0
-I
0
__........._........_........___......,._.......,._........_....................................... [I91
.fl
-I............~
(27) 936
Journal of’ the Franklm Inwtute Pcrgamon Press plc
Bond Graph Modelling
oJ’Multibody
Dynamics
Again from the description of the type of each non-energic multiport, element of the causal coefficient array is also symbolically obtained.
every
variables [O]
‘Orts
-
T
0
0
0
0
0
-c:,
0
-CT,
0
[iI
0
0
i
0
0
0
-cyz
0
-c:,
0
121
0
0
0
0
i
0
-c;,
0
-c:,
0
pi
0
-c,,
0
-c,?
0
-I?,,
0
i
0
0
[41
0
-C,,
0
-C,,
0
-I?,,
0
0
0
i
0
i
0
0
0
p]
0 -i
0 . . . ..-i
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. [I$]
i-6
..__,,,..
(28) The signs f in the above array come from the incident array defined previously. For example, the first row of the incident array, which corresponds to the sub prm - system[6], is [fl
+I
fl
-1
-1
0
0
0
0
0
0
0
0
01.
This implies that two power bonds at ports[3] and [4] are so oriented that power flows out of sub prm system[6]. This means that the coefficient matrices of the causal expressionfor defining the force at port[O] have to have minus signs. Now using those arrays, we eliminate the excess variables. For example, take a look at the 7th column. The (6: 7) element of the causal array is + 1. Then we have the expression of defining lid in terms of ii, and li,, and hence we eliminate n, by substituting this expression into the relations defining the forces Ii ,, li2 and n, at the ports[O], [1] and [2]. By continuing the like operations, we can symbolically eliminate all the excess variables within the junction structure sub prm system[4]. Then the resultant junction structure sub prm system[2] has-the causal and causal coefficient arrays having 8 x 16 elements. only the causal array is shown below. variables
[()I ItI PI PI ports
+I 0 0 0
-I -1 -1 -1
+l
0 -I
0 -I
-1
0 -1
0
-1
0
-I
O-l
-1
0 -I
-I
0
-I
0
-I
0
-1
O-l
0
-I
-I
0
-I
0
-I
0
-1
O-l
0
0 -1 0 -I
fl O-I
0
,41
0 -1
0 -1
0 -I
O+I
0
[51 [61
0 -1 0 -1
0 -1
o-1
0
0 -1
O-l
0
0
0
0
0
[71 _
0 -I
0 -I
O-I
0
0
0
0
0+1
0
0
0
0
0
0
Ofl
0
0
0
0
0
0
0
0
0
0
0
0
0
0+10
0
0
0
0
0
0
0
0
0
fl 0
(29) We note that every column of the causal array (29) has no pair of + 1 and - 1s as in (27), and this implies that the elimination of variables is completed. We think that the above elimination process of excess variables is also processed by matrix operations of those arrays, and it will be presented elsewhere. Vol. 328, No. 516.pp. 917-940, 1991 Printed m Great Britam
937
T. Kuwase and H. Yoshimura TAHLE 11
Physical relutions Port[i]
Phy-typc[i]
0
SF(I)
1
W(2)
2 3
SF(3) M(1)
4
SE(I)
5
I(1)
6
SW2)
I
SE(3)
Physical properties
-*
RI,
I* nro1 i? : % 1 n1, 1 %,I + % + %“I
Now we introduce the physical relations of the elements connected to every port of sub prm system[2]. This is given in Table II. Using thecausal coefficient array so far obtained and the above table, we have the equations of the fundamental pair.
[causal coefficient
array]
The system equations of the interconnected same procedures for the set of the fundamental
-* 71,,t
= D
(30)
system are derived by applying pairs.
the
V. Conclusions In the present paper, the bond graph modelling of multibody dynamics is briefly demonstrated. Specifically, a symbolic generation scheme is proposed fully utilizing bond graph information, that is, the physical, structural and causal relations. In the method, the system is first reticulated into the set of separated energic and non-energic elements, which might be called a primitizje system. Then those energic elements are interconnected through the non-energic multiports using the causality information. Although briefly stated in Section III and IV, it is evident that the bond graph method is much more oriented towards the understanding of system structure and is quite powerful in the elimination of excess variables, which is another look at 938
Journal
of the Franklm Pcrgamon
Institute Press plc
Bond Graph Modelling
of Multibody
Dynamics
the above mentioned interconnection and an essential part of the proposed symbolic generation scheme. The proposed symbolic scheme, especially the elimination process mentioned above, is basically based on the hierarchical structure of the junction structure and this is symbolically done using the causal form of the non-energic multiports and the causality relations among the variables. The results show how the formulation process of the system equations is systematized. What the authors have in mind throughout this study is to keep as close as possible to the bond graph. This is obvious in Section IV. From the practical application point of view, further refinements will be required for the proposed symbolic generation scheme.
Acknowledgement The authors express their sincere gratitude to reviewers for their thorough work and valuable suggestions, and also special thanks to Professor Nankano of Shonan Institute of Technology and a former graduate student, Mr. Fujita. for their help.
References (1) R. E. Roberson and J. Wittenburg, “A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control”, 3rd IFAC Congress, 1966, Proceedings, London, 46D. (2) J. Wittenburg, “Dynamics of Systems of Rigid Bodies”, B. G. Teubner, Stuttgart, 1977. (3) T. R. Kane and C. F. Wang, “On the derivation of equations of motion”, J. Sot. Ind. Appl. Math., Vol. 13, p. 487, 1965. (4) T. R. Kane, “Dynamics of nonholonomic systems”, ASME J. Appl. Me&., Vol. 28, p. 574, 1961. (5) A. M. Bos and M. J. L. Tierncgo, “Formula manipulation in the bond graph modelhng and simulation of large mechanical systems”, J. Franklin znst., Vol. 319, p. 51, 1985. (6) A. M. Bos, “Modelling multibody systems in terms of multibond graph with application to a motorcycle”, Doctoral thesis, Twente University, 1986. (7) T. Kawase, “Towards the modelling of multibody systems”, private note, Waseda University, 1983. (8) T. Kawasc, H. Nakano and T. Magoshi, “Modclling of artificial manipulators and computer simulation of their dynamics”, Proceedings, RoManSy ‘84, CISMIFTOMM Symposium, Theory and Practice of Robots and Manipulators, Kogan Page, 1985. (9) T. Kawase, H. Nakano and M. Ohta, “Modelling of rigid-body systems via bond graph and computer simulation of their dynamics”, Proceedings, 1lth IMACS World Congress on System Simulation and Scientific Computation, Vol. 4, p. 305, 1986. (10) H. Yoshimura, K. Fujita, H. Nakano and T. Kawase, “Modelling of flexible multibody dynamics and a computer oriented method via the bond graph method”, Proceedings 12th CANCAM, Vol. 2, p. 866, 1989. “Analysis and Design of Engineering Systems”, M.I.T. Press, (11) H. M. Paynter, Cambridge, MA, 1961. (12) G. D. Birkoff, “Dynamical Systems”, AMS. New York, 1929. Vol. 328, No. 5/6. pp. 917-940, 1991 Pnnted in Great Britam
939
T. Kuwase and H. Yoshimura (13) D. C. Karnopp. “Power-conserving transformations : physical interpretations and applications using bond graphs”. J. Fvunklirz Inst.. Vol. 288, p. 175, 1969. (14) D. C. Karnopp, “The energic structure of multibody dynamic systems”, J. Frunklirz Inst., Vol. 306, p. 168, 1978. (15) T. R. Kant and A. D. Lcvinson, “Formulation of equations of motion for complex spacecraft”, J. Guidance Corztr-01, Vol. 3, p. 99, 1980.
940
Journal ofthe
FrankIm lnstttute Press plc
Pcrpmmn
Department
KAWASE
of Multibody
and its Symbolic and HIROAKI
of Mechanical
Scheme
YOSHIMURA
Engineering,
Waseda
Unitiersity,
Tokyo
169, Japan
ARSTRACT: A bond graph method of modelling multibody dynamics is demonstrated. Spec$ically, a symbolic generation scheme which ,fullyutilizes the bond gruph irzjbrmation is presented. It is also demonstruted that structurul understanding and representation in bond gruph theorv is quite powerful ,for the modelling of’ such large scale systems, and thut the non-energic multiport of junction structure, which is u multiport expression of’ the system structure, plays an important role, as,first sug~qested by Paynter. The principal part qf the proposed symbolic scheme, that is, the elimination of excess varinhles, is done throqh tearing and interconnection in the sense of Kron using newly de$ned causal and causal co#icient arrays.
I. Introduction
Much research has been directed to multibody dynamics in the fields of robotics, space dynamics and bio-engineering of the human body. One of the principal concerns is no doubt the modelling of the dynamics. A major stumbling block in such modelling comes from the considerable increase in the degree of freedom as the number of bodies involved increases. This requires : (1) the formulation of a large number of system equations and (2) the fast numerical integration of the resultant nonlinear algebraic-differential equations. These difficulties imply that the modelling of such multibody dynamics inevitably requires symbolic manipulation of equations on the computer and hence to make any symbolic scheme for the generation of the system equations, categorization of the required kinematic and dynamic relations is needed. Various modelling methods have been reported, and those by Roberson and Wittenburg (l), Wittenburg (2) and Kane and Wang (3) are noteworthy. Roberson and Wittenburg utilized a circuit graph model to represent the kinematic structure of the system together with its associated algebra. Kane proposed a new concept of the partial velocity (4) which enabled us to derive the required equations of motion without any elimination ofconstraint forces which come from the kinematic constraints. The equivalent circuit model mentioned above certainly entails the kinematic information which comes from the geometric configuration of the bodies, and its associated algebra plays quite an important role in the symbolic scheme to generate the system equations on the computer. However, the circuit model entails only a part of the required kinematic relations and hence the method needs other kinematic relations besides the above graphical information, for example, those for mechanical joints. 917
T. KUMVLWand H. Yoshinzura
On the other hand, the method proposed by Kane requires a certain amount of skill in mechanics to find an appropriate partial velocity for it defines a locally admissible subspace which is normal to the local subspace defined by the constraint relations imposed u priori among the velocities. The bond graph, as a modelling tool, graphically entails all kinds of kinematic and dynamic information. That is, the physical properties of the system elements, the structural relations representing how the system elements are interconnected and the causal relations showing which variables are to be replaced by which variables. Bos and Tiernego (5) and Bos (6) proposed a systematic modelling method using the bond graph technique. However, any idea to systematically eliminate a large number of excess variables by using the bond graph information, specifically causality, was not explicitly given in their papers. The elimination of those excess variables is most tedious and painstaking work and this is where any symbolic manipulation is most required. In the present paper, fundamental aspects of bond graph modelling of multibody dynamics are demonstrated and special concern is focused upon how the bond graph information is incorporated into the proposed symbolic scheme. The present paper is thus tutorial and it is suggested that the reader refers to other papers by the present authors for more details of the method and also how it is applied to more general flexible multibody dynamics (7-10). II. Fundamentals
of Bond
Graph Concepts
In the modelling of large scale systems like multibody systems, the most important things are (1) to write all the necessary relations and equations down and (2) to determine which variables are to be replaced by which variables and also where it should be done to formulate a set of the system equations. To do this for multibody dynamics. some kind of categorization of all the kinematic and dynamic relations may be helpful. In the bond graph as first suggested by Paynter (ll), those relations are classified into the following three categories : (I) energic multiports : physical properties of the system elements. (2) non-energic multiports : structural information of how the system elements are interconnected. (3) causality: inputtoutput relations indicating which variables are to be replaced by which variables. The energic multiports to appear in the present paper are storage elements, that is, mass I : M and the moment of inertia = I : J of a body, and power sources IZ Se and L: Sf. One of the key concepts of bond graph theory is the non-energic multiport. The term non-energicness was first coined by Birkhoff (12) in his famous textbook on mechanics. Let q and R be the generalized coordinate and force of the system having II degrees of freedom. Then the system is said to be non-energic when the following relation
Bond Graph Modelling (&R)
of Multibody
=o
D~wzamics (1)
identically holds for every time. In the above, () indicates a column vector, (,) the scalar product of vectors, and (‘) the time derivative. We are going to see four kinds of such non-energic multiports in the present paper. 1. l-junction and O-junction The l-junction stands for D’Alembert’s principle in mechanics, that is, the balance of forces (torques) exerted upon a body and simultaneously it imposes the condition that all the velocities at the bonds associated to the l-junction are common. The O-junction indicates that all the forces (torques) at the bonds associated to the O-junction are common. 2. Modulated transformer (MTF) We will see various relations are modelled as MTFs. The first is the MTF model of the coordinate transformation first discovered by Karnopp (13). Let us write a coordinate transformation, integrable or non-integrable, in a form decribed below using an n x n coefficient matrix A($, de = ii(q) dq
or equivalently
i = A(q)i
(2)
where q and it indicate the old and new coordinates respectively. There may be a case where the coefficient matrix A contains time explicitly. In that case, we may introduce the (IZ+ 1)th coordinate q,,, , = t, then again we have the same expression at least formally. The power invariant under the coordinate transformation insists
(~,R)+(iQ=I)
=
0
(3)
holds identically. In the above fi denotes the force expressed along the new coordinate system. From the above relations, we easily have the relation for the force variables. R=
-Jrn
(4)
where ( )’ indicates the transpose of a matrix. The bond graph representation is given in Fig. 1 but note that the power orientation is different from the conventional method. This is because we want to follow the context of non-energicness (1). The second is the kinematical constaint like those imposed by kinematical loops.
I
l/s
--I
FIG. 1. Bond graph model of the coordinate Vol. 328, No 5!6, pp. 917-940. Prmed in Great Bmam
transformation.
1991
919
T. Kuwa.~e and H. Yoshimura Let us denote
a kinematical
constraint
a(y) dq = 0
as - - 1 a(q)q = 0
or equivalently
(5)
where the coefficient matrix a is tn x n of rank m(m < n). Let us first introduce coordinate transformation described below, i.e.
the
d77, = a(q) dy di& = p(q) dy
(6)
where 7c = (757,E:)’ is an n-vector representing the new coordinate and the coefhcient matrix p in the second relation is arbitrary except that it is so chosen that the transformation (6) itself is invertible. Thus the transformation is modelled as an MTF and in this case the first m ports representing the new coordinates are connected to an outside velocity source of i , = 0. Another important MTF model to appear in multibody dynamics is that of the mechanical joint. First take a look at Fig. 2 showing two bodies connected by an ideal mechanical joint. The mechanical joint drives the body 2 with the linear velocity fi and the angular velocity fi relative to the body 1. From the geometrical and kinematic relations, we have ri> = k, -7, where (‘) denotes
a 3-dimensional
+ 1;‘+J,
(7)
vector.
Joint Body 2
.
-
+
cb) I r2
Body
FIG. 2. Kinematic
920
relation.
_-_--
Bond Graph Mode&y By differentiating the above relation in the inertial relation among the linear velocities of the two bodies, kZ=
tv,-~,x~,+~+~,xti+~j?x72.
ofMultibody Dynamics frame,
we easily get the
(8)
In the above the symbol x indicates the vector product. On the other hand, the angular velocity is written as C& = 6, +Q.
(9)
For the description of the translational and rotational motions, we employ the body fixed frame as is conventional. Every component of the velocity is hence described in the above local frame. Thus from the above two relations (8) and (9), we finally have the matrix representation of the MTF model of the mechanical joint.
where
In the above, the variable v(O) denotes the linear (angular) velocity, observed in the local frame as mentioned above, F;(F) its dual variable, that is, force (torque). And also tI?(G) indicates the linear (angular) driving velocity at the joint observed in the local frame and, I?(?) its dual force (torque). Furthermore T,2 indicates the matrix representing the base vector transformation between the two bodies, r,,z the distance vectors between the hinge point and the center of mass of each body, and 7, the skew symmetric matrix representing the vector product like operation (i’ x ) described below
where r,,, ri2 and ri3 are the scalar components of V, observed in the local frame. The bond graph representation is then given in Fig. 3. Again referring to the MTF model of the coordinate transformation, we rewrite the variables as follows
Yol.
32X. No. 36.
Prmed
in Great
pp 917-940. &nun
1991
921
7’. Kuwase and H. Yoshimura
Rotational
Translational
FIG. 3. Ideal mechanical joint
where the subscripts tr and rot indicate that the corresponding variables represent the translational and rotational motions respectively. Using the matrix representation of (lo), we can easily make a catalogue of mechanical joints. which will be useful as a data base for the symbolic scheme to be proposed in the later chapter. Part of such a catalogue is shown in Table I.
Euleriun junction .structure (EJS) The final bond graph model of the non-energic multiport to appear in the paper is the so-called Eulerian junction structure first proposed by Karnopp (14). The inertia coupling torque, when the motion is observed from the body fixed frame whose axes coincide with the principal axes of the body, is expressed as
(11) where rdenotes the coupling symmetric matrix representing below
torque, i the moment of inertia and fir,, the skew the vector product like operation (9 x ) described
where pj, ps and ph are the scalar components of the angular momentum &, observed in the local frame. It is easily shown that the above torque satisfies the non-energicness and so it is a member of the non-energic multiport defined previously. In the latter expression, the angular momentum p,,, is used instead of angular velocity. This is only because we are going to employ the Hamilton form of the
g
.g
UJ DOF=2
RJ I DOF=l
TJ I DOF=I
Scheme
F
\
Symbol
I
a 0
-W
-spse -C@H co 0 0 0
-sqwo -C@YH -so 0 0 0
0 0
- SO
-cgco
-s*co
0 0
r&P -r,S@O
s*
-Cd
r,SH-r,,C4
CO
-c&se
-s&so
-r,S&%
0
-r,CO
SO CO
sosg
0
0 0
Cd - Cmp
0 0
0
(I, +r,,SO)C~
0
0
0 0
-r,COCCp
-r,CO
Vdt
CO
s0
-(r,,+r,SO)Scp
x = r,+
0
so
0
so
0
S4
- ccp
r,,SH-r,Cqb
r,SqKO r,Cf&CO
0 0
-co
CO
0
0
0
so
0
-(_u+r,SO)
0
so -cocg co SOCd,
0
00
0
0
0
1
r,+.uSO
-r,co 0
0
xc0
-
matrix
0
sosc#l
-cosd,
0
0
00
00
0
0
-co co so
r0IO so
Transformation
ofideal mechanicaljoints
TABLE I A part qfthe catalogue
0 0
0
w
-ccp
-r,Cs
1
1
0
0
0
-rI 0
3 -. 2
g
2
s-
G g
T. Kuwasr
and H. Yoshimuru
FIG. 4. Eulcrian junction
structure.
equation of motion. The bond graph representation ture is shown in Fig. 4.
of the Eulerian
junction
struc-
III. System Equations As to the form of equations of motion, various forms have hitherto been used by many researchers. The research report by Kane and Levinson (15) of a comparative study of these various forms of the equations is worthy of note. There are two aspects to be considered. To get the mathematical mode1 of the least order for fast numerical integration or estimation it is most important to reduce the number of algebraic manipulations. However, from the system design point of view, specifically when we deal with control related problems, it is much more important that the generation process of the system equations be well systematized so that the effects of any changes in design parameters like geometrical parameters and other kinematic and dynamic conditions are more clearly predicted. It is widely known that, for fast numerical estimation, the NewtonEuler form is more effective than the Lagrange or Hamilton form. However, from the viewpoint of systematizing the modelling process. to say anything definite about which form is most advantageous over others, further comparative study will be needed. Much of the research concern of the present authors has been directed to the latter aspect mentioned above. In the present paper, the Hamilton form is employed. Of course, the proposed method is applicable to any other form of the equation of motion. It is worthwhile to note the Hamilton form to be used in the paper, strictly speaking, is not derived from the Lagrange but from the BoltzmannHamel equation of motion. One more thing worth to be mentioned is that Kane’s method, which elegantly eliminates the excess constraint forces, is essentially equivalent to
Bond Graph Modelling
qf Multiho&
Dynumics
FIG. 5. Stanford manipulator.
the proposd bond graph method, of constraint forces.
which utilizes the causality
relation,
of elimination
III. 1. System equations qf the primitice svstem For simplicity of illustration, here we consider a Stanford manipulator shown in Fig. 5. Obviously the Stanford manipulator has no kinematic loops and so it is decomposed into a set of fundamental pairs each of which consists of a body and an ideal mechanical joint adjacent to it. Using the bond graph tools described in the previous section, we have the bond graph representation of such a fundamental pair as shown in Fig. 6(a). The upper half indicates the translational motion of the center of mass of the body and the lower half the rotational motion. Again we note that both the translational and rotational motions are observed in the body fixed frame and hence the translational motion of the center of mass has to be transformed into the inertial frame. This is shown by an MTF in the upper half. According to the boundary conditions imposed upon velocity and force, the causality at every bond is automatically determined. In the case of the Stanford manipulator, which has no redundancy, the velocity at every bond is determined by those of the driving sources at the joints together with the base boundary (body 0). On the other hand, the force at every bond is determined by that of the end effector and the external forces exerted upon the bodies. This assignment of the causality is also clearly shown in the figure. Now we come to the point to set up the system equations of such a fundamental pair. We write the equation of motion for a totally torn apart system of a fundamental pair [Fig. 6(b)] using the generalized coordinate 71= (&7&)’ and its conjugate momentum p [see Fig. 6(a)]. The pair (Z, ii) is a state vector. Vol.328.No. 516,pp. 917-940.1991 Prmted tn Grcat Bntain
925
T. Kawase and H. Yoshimura
926
Bond Graph Modelling qf Multibody
dP -= dt
-.f+,+-n
Dynamics
(12)
where i? = diag (m, m, m, I,, Z2, Z3)- ‘, j = (0, -mg, 0,0,0,0), and Ii denotes the constraint force representing the interacting force from the adjacent body, g the gravitational acceleration and f the inertia coupling force representing EJS [see Fig. 6(a)]
In the above,
Using the causal relation
of the mechanical
joint,
we have
;o = Cz*+& The coordinate
transformation
(13)
for the translational
motion
gives us the relation
iT,,,= A?&
(14)
where the coefficient matrix A represents the base vector transformation. From the constraint imposed by the l-junction in Fig. 6(a), we have ~=$i&O, Now we are going back to the causal relation (n*T, RT)’ is expressed as
of the mechanical
(15)
joint. The force
(16) Using the dual relations
where E indicates
for (13) through
(15), we have
the 3 x 3 unity matrix and
Combining all those equations, damental pair. Vol. 328, No. S/h, pp. 917-940, 1991 Printed in Great Britam
we finally
get the system equations
for the fun-
927
T. Kawase and H. Ymhirnura
joint constraint coord. transformation l-junction = 0.
-T C -7 C equation Further eliminating the variables r? and i associated the above equations, we finally have
(18)
of motion with the inner bonds from
= 0.
(19)
3.2. System equations
The final step is to set up the system equations process is decomposed into two steps described (1) interconnection of the fundamental (2) elimination of the excess variables.
of the interconnected below :
system. The
pairs ;
The interconnection of any two fundamental pairs is mathematically expressed by the continuity conditions of velocity and force between the adjacent bodies and is bondgraphically equivalent to a l-junction between the fundamental pairs. Those continuity relations are l=T_n*
itI
‘* n,+ I --+
-0 = (j,
(20)
Thus we have a large number of equations for the Stanford manipulator, and we easily see that large numbers of excess variables appear such as those associated with the inner bonds of each fundamental pair. We know that those large numbers of system equations are reduced to a set of system equations having much more smaller numbers of equations in terms of six independent variables in the case of the Stanford manipulator. These equations are algebraic-differential equations since the equations of motion are augmented by the kinematic constraints imposed by the mechanical joints, etc. The above means we have to eliminate many excess variables. The elimination of those excess variables is most tedious and painstaking and hence this is where the symbolic manipulation of equations by computer comes to play a vital role. Now we look at this elimination process of the excess variables from the modelling and numerical simulation point of view. There are two aspects worth
Bond Graph Modelling mentioning
?f’Multibod~~ Dynamics
:
(1) final form of the system equations of the interconnected (2) numerical integration or other numerical estimations.
system;
As to the first, there are many stages of the above mentioned elimination of excess variables and hence, according to the purpose of modelling, we have various forms of the mathematical equations. From the design point of view of such multibody dynamics, the process of equation formulation should be well systematized. This is required so that effects of any changes in design conditions, geometric parameters or kinematic and dynamic conditions are well predicted in the formulation process. This is much more important in the design stage of such multibody systems, especially its control related problems. This may not require one to reduce the set of the equations to the least order. As to the second, it is necessary to reduce the execution time of the numerical integration or other numerical estimation like an inverse dynamics for on-line control. This implies we need to reduce the order of the system equations as much as possible. To meet those rather different purposes, various mathematical forms may be useful. Here we show two cases which have different elimination orders. First we combine six sets of equations for the fundamental pairs (19) and the same number of the continuity relations of velocity and force. In other words. we eliminate 6*s and li?. Thus we have the following set of 54 equations. In the case of the Stanford manipulator, the driving function at each joint is a scalar function representing velocity. So the number of the equations is 54 rather than 84.
= 0.
Vol. 328, No 5:6. pp 917-940, 1991 Printed m Great Bntam
(21)
929
T. Kuwase und H. Yoshimura In the above, s, = b+,z-S, (i = I,. .6). Thus, in the above, the matrix C, (i = 1,. ,6) is a 6 x 1 matrix, and hence each required joint torque or force l?, (i = 1, ,6) is also a scalar function of time. Next, we obtain the system equations of the least order. First we rearrange (21) obtained above, assuming the base body is fixed to the inertial frame and the end effector moves freely under no external forces and torques.
where R = [l?‘:, R.1,. ,a:]‘, 3 = [s{,sq,. ,5:]‘, f = [.??T,$i,. . , l?:]’ /7 = [$,/?P _. . ,pT]’ and the coefficient matrices D, 7 and cl are given below
and
ii... &, a=p=
10 Putting
,
B3C(3c3..
s = b-t fivp -J
p, into (22), we have
p-fi&j I=” R-&L&?p+&
(23)
where
Using the last equation differentiating it, i.e.
of (23), we get the expression
for the momentum
p and by
i = y ‘($G+&fi), Substituting
the above relation
into (23), we finally have - (24)
This is the expression
which is usually
used in the inverse dynamics.
Bond Graph Modelling
of Multibo&
Dynamics
As is easily seen from (24), we cannot have integral causality for the Stanford manipulator, and the discussion so far implies that the resultant system equations become an implicit form as is well known. IV. Symbolic
Generation Scheme
So far we have described the basic idea to formulate the system equations using the bond graph information. As stated in the previous section, all kinds of information are entailed in the bond graph. Thus to make a symbolic generation scheme, we can directly start from drawing the bond graph of the system of our concern. As we have shown, the Stanford manipulator has a tree-type geometrical configuration and hence it suffices to start from a fundamental pair. When the system has kinematic loops which are usually constrained in a plane, the constraint relations imposed by those kinematic loops are modelled as an MTF as previously shown. The symbolic scheme is essentially extended to those systems. Now, as shown in Fig. 7, the pair is further decomposed into a totally separated state which might be called a primitive system in the sense of Kron. The tearing and interconnection is fully described by the junction structure of the pair. The symbolic generation scheme proposed in this paper is schematically shown in Fig. 8. The key part of the scheme is the elimination of excess variables as stated previously. Therefore, in the following, we will show how three kinds of bond graph information are incorporated into the scheme. The system structure is here recognized as a hierarchy shown in Fig. 9. Level 0 corresponds to the interconnected system and level 2 the totally torn apart system. The elimination process is understood as the inverse direction of the tearing. So we start from the level 2. In the present scheme, the language “C” is used. So in the following the method is described using P
The junction structure of a fundamental pair consists of seven non-energic multiports and all the non-energic multiports and associated multibonds are numbered as shown in Fig. 10. Thus the junction structure is recognized and stored in the computer by giving the data as follows : sub _ prm system[i] : the array to describe the ith junction are described by the following arrays.
structure
and its contents
structures contained in the junc(1) num system : number of the sub-junction tion structure sub prm system[i]. the bonds associated with sub prm system[i]. (2) num_bond : numberof (3) num port[ ] : number of the ports associated with sub prm system[i]. (4) portr] : bond number to which the port is absorbed by connection. and the number + 1 is given when power flows (5) power[ ] : power orientation into sub prm system[i] and - 1 when it comes out. of either of the velocity output or force output at the (6) causal[ ] :assignment port. (7) type : type of the junction structure sub prm system[i]. st%ctures contained in the junc(8) sub system[ ] : naming of the sub-junction tion structure sub _ prm _ system[i]. Vol. 328. No. 5/6, pp. 917-940. Printed in Great Britain
1991
931
T. Kuwase
and H.
Yoshimura
Stanford
manipulator
system
-----I
’ I
MTF
1 lb +_--_J
---__-_ I=
MTF F
; ’
’
/-
-1’ r
WI lr Sf
M .. Jl
7
\
.-----_
P 1 K=MGY i
+_-‘__--L_----
IF Sf
I
P
I
L’F
I IK=MGY
iT
’
I
IF
;
’
II ..
___-_--;_---A-_---A
II .. J
I J L_____’
I 1 I
I I
FIG. 7. Tearing and interconnection. (9) phy type[ ] : type of element (10) phylnum[ ] : suffix number designated.
connected to the port designated. given to the element connected
to the port
Illustrative example of the system description are shown below (see Fig. 9). First the decomposition of thejunction structure sub - prm _ system[O] is described below. sub prm system[O] +sub system[O] = 1 sub-prm-system[ 11. sub-system[O] = 2 subIprmIsystem[ 11* sub _ system[ I] = 3 .. .... ..... ..... .... .... ..... ..... ..... .... .... .... .... .... .... ... sub _ prm _ system[4] -sub _ system[4] = 10
932
Journal of ihe Frankhn lnstitule Pcrgamon Press plc
Bond Graph Modelling
Object
Dynamics
Model
u I
of Multibody
Bond graph
I
model
I
I (input) Bond graph << Hierarchical
Primitive << Tableau
/ 1
Coeff
matrix
Eliminating
Vol. 328, No. 5/6, pp. 917-940, 1991 Printed in Creal Bruin
information structure
>>
system form >>
1 ( Causal matrix unnecessary
variables
FIG. 8. Symbolic
generation
1j
scheme.
933
T. Kuu~a.w and H. Yoshimura Interconnected Level 0
0
-
y
y
_ :
_
f__.__-._-___.___--_-.-_-i-_-.‘“””--_ 4
6
7
8
I sub_prm_system
w,b-v~,-vy
10
9
(4)
11
12
,
,
,
i
,
,
,
13
14
15
16
17
16
19
Level 2
(6)
FIG. 9. Hierarchy
structure
of the junction
structure.
1111 11 [121 (9) 1 l-T= IlO1
PI6 (8)
MTF
(6)
Ii]
Port
(1)
Non-energic
0
FIG. 10. Junction
934
Absorbed
structure
multiport port
of the fundamental
pair. Journalof the Franklin Pergamo”
Institute Press plc
Bond Graph A4odelling oj”A4ultibody Dynamics
The kinematic and dynamic described below.
relations
of each non-energic
multiport
are then
sub prm system[6] - num system = 1 sub_prm_system[6] snurn-port[O] = 5 _
sub prm system[6] - num bond = 5 syb_prmIsystem[6] * pow&[01 = + 1 _
sub_prm_system[6] * port[O] = 1 sub_prm_system[6] * causal[ l] = “E” sub prm system[6] *causal[3] = “F” sub_prm_system[6] - type = “MTF” _ _ .... .. .... .... .... .... .... .... ..... .... .... ... ..... .... ..... ..
sub_prm_system[6] - causal[O] = “E” sub_prm_system[6] *causal[2] = “E” sub_prm_system[6] - causal[4] = “F” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . .. . . .,............................................................
-m
The total descriptions will thus give just a juxtaposition of the separated nonenergic multiports contained in the junction structure of sub prm system[O]. Next we consider the interconnection of those non-energic multiports. Let us take a look at the fundamental pair sub prm system[2] and let us demonstrate how the interconnection is described. To do this,we first consider sub prm system[4]. The connections occur, for example, at the bonds of sub_prm_syitem[6], [7] and [lo], and those are described below. sub prm system[4] - num system = 7 sub_prm_system[4] * num-bond = 14 _ _ sub _ prm _ system[4] *port[O] = 1 sub_prm_system[4] * port[3] = 4 sub prm system[4] *port[4] = 5 sub_prm_system[4] * port[5] = 4 _ _ sub _ prm _ system[4] - port[l3]
= 5
The above indicates that ports[3] and [5] absorbed in bond 4 by connection and similarly ports[4] and [13] in bond 5. The above operations directly imply the possibility of the replacement of variables in the set of equations describing the junction structure sub prm system[4]. For example, the velocity and force variables associated with firt[3]and [5] are replaced by the velocity and force associated with bond 4 respectively. Now we come to the point to eliminate, as the result of interconnection, those excess variables associated with those ports and originally given different symbols. To do this, we first define the incident array. The component of the incident array indicates a relation between a non-energic multiport and each power bond in the junction structure of our concern. For example, the junction structure sub prm system[4] has seven non-energic multiports and 14 multibonds. So we define the incident array as a 7 x 14 matrix. When a power orientation at the ith Vol 328, No. 5/6, pp. 917-940, Pnnted in Great Bntain
1991
935
T. Kuwase
and H. Yoshitwuru
bond is directed towards the ,jth non-energic multiport, then the (,j, i) element of the array is + 1, when directed reverse, it is - 1 and when the ith bond is not associated with the .jth multiport, it is null. Thus we have the incident array described below for sub _ prm _ system[4]. bonds
+I
+I 0
non-energ,c multiports 1
+I
-I 0
0
-I
0
0-I
+I
0
0
0
0
0
0
0
0
0
0
fl
0
0
0
0
0
0
0
0
0
0
0
O-I
0
0
0
0
0
0
0
0
0
0
-I 0
0
+I
0
+I
+I
0
0
0
0
I
0
0
0
0-l
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+
0 +I
+I 0
0 +I
O-l
0
0
0
0
-1
0
0
0
0 (26)
Next we define two more arrays called the causal and cuusul co@c’i~~i arrays. First we define the causal array. Here we note that the causality @ations at the bonds associated with each non-energic multiports determine which variable (output variable) is to be defined by which variables (input variables). In other words, the output variable at each port is described in a causal form in terms of input variables at the other ports. Now take a look at sub prm system[4]. We have, in this case, 20 such causal relations in terms of 28 va!tablei [2 x sub prm system[4] * num bond (= 14)]. So we define the causal array as a 20 x 28 matrix-Thus each row-corresponds to a causal relation at a bond and each column to a variable. Here we arrange those variables in the order (l=f,.A,,n2,i2 ,..., ii,,, i,4) and we consider the causal relation at the ith port. When thejth variable is output, the (i,,j) element of the array is + I and when input, it is - I. and otherwise it is null. For example, from the assignment of the causality shown in Fig. 9, the computer understands that sub prm system[4] - sub system[O] = 6 has three force outputs at ports[O], [I] and [21,&d two velocity outputs at ports[3]. [4]. This simply implies that forces at the ports[O], [l] and [2] are defined in terms of those at the ports[3] and [4]. And conversely, the velocities at the ports[3] and [4] are defined by those at the ports[O]. [I], and [2]. Thus every element of the causal array is symbolically obtained as shown below. variables [O]
Ltl [2] ports
PI ,41 [S]
+ I
0
0 0 0 0 -1 0 -1
0
0+1
0
0
0
-1
o-~l
0
0
0
-I
0-l
0
0
-I
0-I
0
0 0
0 -1
0
0
-I
O-l
fl
0
-1
0 O+l
fl
0
0 0
0
0
0
..,.,.,..................,. .
0 0
+I 0
-1
0
-I
0
__........._........_........___......,._.......,._........_....................................... [I91
.fl
-I............~
(27) 936
Journal of’ the Franklm Inwtute Pcrgamon Press plc
Bond Graph Modelling
oJ’Multibody
Dynamics
Again from the description of the type of each non-energic multiport, element of the causal coefficient array is also symbolically obtained.
every
variables [O]
‘Orts
-
T
0
0
0
0
0
-c:,
0
-CT,
0
[iI
0
0
i
0
0
0
-cyz
0
-c:,
0
121
0
0
0
0
i
0
-c;,
0
-c:,
0
pi
0
-c,,
0
-c,?
0
-I?,,
0
i
0
0
[41
0
-C,,
0
-C,,
0
-I?,,
0
0
0
i
0
i
0
0
0
p]
0 -i
0 . . . ..-i
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. [I$]
i-6
..__,,,..
(28) The signs f in the above array come from the incident array defined previously. For example, the first row of the incident array, which corresponds to the sub prm - system[6], is [fl
+I
fl
-1
-1
0
0
0
0
0
0
0
0
01.
This implies that two power bonds at ports[3] and [4] are so oriented that power flows out of sub prm system[6]. This means that the coefficient matrices of the causal expressionfor defining the force at port[O] have to have minus signs. Now using those arrays, we eliminate the excess variables. For example, take a look at the 7th column. The (6: 7) element of the causal array is + 1. Then we have the expression of defining lid in terms of ii, and li,, and hence we eliminate n, by substituting this expression into the relations defining the forces Ii ,, li2 and n, at the ports[O], [1] and [2]. By continuing the like operations, we can symbolically eliminate all the excess variables within the junction structure sub prm system[4]. Then the resultant junction structure sub prm system[2] has-the causal and causal coefficient arrays having 8 x 16 elements. only the causal array is shown below. variables
[()I ItI PI PI ports
+I 0 0 0
-I -1 -1 -1
+l
0 -I
0 -I
-1
0 -1
0
-1
0
-I
O-l
-1
0 -I
-I
0
-I
0
-I
0
-1
O-l
0
-I
-I
0
-I
0
-I
0
-1
O-l
0
0 -1 0 -I
fl O-I
0
,41
0 -1
0 -1
0 -I
O+I
0
[51 [61
0 -1 0 -1
0 -1
o-1
0
0 -1
O-l
0
0
0
0
0
[71 _
0 -I
0 -I
O-I
0
0
0
0
0+1
0
0
0
0
0
0
Ofl
0
0
0
0
0
0
0
0
0
0
0
0
0
0+10
0
0
0
0
0
0
0
0
0
fl 0
(29) We note that every column of the causal array (29) has no pair of + 1 and - 1s as in (27), and this implies that the elimination of variables is completed. We think that the above elimination process of excess variables is also processed by matrix operations of those arrays, and it will be presented elsewhere. Vol. 328, No. 516.pp. 917-940, 1991 Printed m Great Britam
937
T. Kuwase and H. Yoshimura TAHLE 11
Physical relutions Port[i]
Phy-typc[i]
0
SF(I)
1
W(2)
2 3
SF(3) M(1)
4
SE(I)
5
I(1)
6
SW2)
I
SE(3)
Physical properties
-*
RI,
I* nro1 i? : % 1 n1, 1 %,I + % + %“I
Now we introduce the physical relations of the elements connected to every port of sub prm system[2]. This is given in Table II. Using thecausal coefficient array so far obtained and the above table, we have the equations of the fundamental pair.
[causal coefficient
array]
The system equations of the interconnected same procedures for the set of the fundamental
-* 71,,t
= D
(30)
system are derived by applying pairs.
the
V. Conclusions In the present paper, the bond graph modelling of multibody dynamics is briefly demonstrated. Specifically, a symbolic generation scheme is proposed fully utilizing bond graph information, that is, the physical, structural and causal relations. In the method, the system is first reticulated into the set of separated energic and non-energic elements, which might be called a primitizje system. Then those energic elements are interconnected through the non-energic multiports using the causality information. Although briefly stated in Section III and IV, it is evident that the bond graph method is much more oriented towards the understanding of system structure and is quite powerful in the elimination of excess variables, which is another look at 938
Journal
of the Franklm Pcrgamon
Institute Press plc
Bond Graph Modelling
of Multibody
Dynamics
the above mentioned interconnection and an essential part of the proposed symbolic generation scheme. The proposed symbolic scheme, especially the elimination process mentioned above, is basically based on the hierarchical structure of the junction structure and this is symbolically done using the causal form of the non-energic multiports and the causality relations among the variables. The results show how the formulation process of the system equations is systematized. What the authors have in mind throughout this study is to keep as close as possible to the bond graph. This is obvious in Section IV. From the practical application point of view, further refinements will be required for the proposed symbolic generation scheme.
Acknowledgement The authors express their sincere gratitude to reviewers for their thorough work and valuable suggestions, and also special thanks to Professor Nankano of Shonan Institute of Technology and a former graduate student, Mr. Fujita. for their help.
References (1) R. E. Roberson and J. Wittenburg, “A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control”, 3rd IFAC Congress, 1966, Proceedings, London, 46D. (2) J. Wittenburg, “Dynamics of Systems of Rigid Bodies”, B. G. Teubner, Stuttgart, 1977. (3) T. R. Kane and C. F. Wang, “On the derivation of equations of motion”, J. Sot. Ind. Appl. Math., Vol. 13, p. 487, 1965. (4) T. R. Kane, “Dynamics of nonholonomic systems”, ASME J. Appl. Me&., Vol. 28, p. 574, 1961. (5) A. M. Bos and M. J. L. Tierncgo, “Formula manipulation in the bond graph modelhng and simulation of large mechanical systems”, J. Franklin znst., Vol. 319, p. 51, 1985. (6) A. M. Bos, “Modelling multibody systems in terms of multibond graph with application to a motorcycle”, Doctoral thesis, Twente University, 1986. (7) T. Kawase, “Towards the modelling of multibody systems”, private note, Waseda University, 1983. (8) T. Kawasc, H. Nakano and T. Magoshi, “Modclling of artificial manipulators and computer simulation of their dynamics”, Proceedings, RoManSy ‘84, CISMIFTOMM Symposium, Theory and Practice of Robots and Manipulators, Kogan Page, 1985. (9) T. Kawase, H. Nakano and M. Ohta, “Modelling of rigid-body systems via bond graph and computer simulation of their dynamics”, Proceedings, 1lth IMACS World Congress on System Simulation and Scientific Computation, Vol. 4, p. 305, 1986. (10) H. Yoshimura, K. Fujita, H. Nakano and T. Kawase, “Modelling of flexible multibody dynamics and a computer oriented method via the bond graph method”, Proceedings 12th CANCAM, Vol. 2, p. 866, 1989. “Analysis and Design of Engineering Systems”, M.I.T. Press, (11) H. M. Paynter, Cambridge, MA, 1961. (12) G. D. Birkoff, “Dynamical Systems”, AMS. New York, 1929. Vol. 328, No. 5/6. pp. 917-940, 1991 Pnnted in Great Britam
939
T. Kuwase and H. Yoshimura (13) D. C. Karnopp. “Power-conserving transformations : physical interpretations and applications using bond graphs”. J. Fvunklirz Inst.. Vol. 288, p. 175, 1969. (14) D. C. Karnopp, “The energic structure of multibody dynamic systems”, J. Frunklirz Inst., Vol. 306, p. 168, 1978. (15) T. R. Kant and A. D. Lcvinson, “Formulation of equations of motion for complex spacecraft”, J. Guidance Corztr-01, Vol. 3, p. 99, 1980.
940
Journal ofthe
FrankIm lnstttute Press plc
Pcrpmmn