Conventionally kinematical constraints in multibody systems are treated similar to geometrical constraints and are modeled by constraint reaction forces which are perpendicular to constraint surfaces. However, in reality, one may want to achieve the desired kinematical conditions by control forces having different directions in relation to the constraint surfaces. The conventional equations of motion for multibody systems subject to kinematical constraints are generalized by introducing general direction control forces. Conditions for the selections of the control force directions are also discussed. A redundant robotic system subject to prescribed end-effector motion is analyzed to illustrate the methods proposed

Ider, Sitki Kemal

English

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NgO:e3os 
NODELING OF CONTROL FORCES FOR KINEHATICAL CONSTRAINTS
IN THE DYNANICS OF HULTIBODY SYSTENS-A NEW APPROACH
SITKI KENAL IDER
Assistant Professor
Department of Mechanical Engineering
Middle East Technical University, Ankara, Turkey
ABSTRACT
Conventionally kinematical constraints in multibody systems are treated
similar to geometrical constraints and are modeled by constraint reaction
forces which are perpendicular to constraint surfaces. However, in
reality, one may want to achieve the desired kinematical conditions by
control forces having different directions in relation to the constraint
surfaces. In this paper the conventional equations of motion for multibody
systems subject to kinematical constraints are generalized by introducing
general direction control forces. Conditions for the selections of the
control force directions are also discussed. A redundant robotic system
subject to prescribed end-effector motion is analyzed to illustrate the
methods proposed.
I. I_ION
In many applications of multibody systems certain points are desired to
follow prescribed paths, such as the end-effector in a robotic system.
Such kinematical conditions are treated constraint equations to
determine the system motion and the control forces.
In this paper those constraints which arise from geometrical
restrictions such as closed loops and physical guides are termed
geometrical constraints. On the other hand, kinematical constraints are
defined as those conditions which represent desired motions or desired
paths of certain points or bodies.
In the conventional methods of analysis, regardless of the
fundamental dynamic equations (Newton-Euler, Lagrange, Kane, etc.) used,
the constraints are modeled by constraint reaction forces which are
perpendicular to the constraint surfaces. (See Arnold [I], Hemami and
Weimer [2], Kamman and Huston [3], Wehage and Haug [4], Nikravesh [5],
Kim and Vanderploeg [13], Amirouche and Jia [6].)
However kinematical constraints do not have to be satisfied by
constraint reaction forces, and usually have to be realized by control
forces applied by the actuators in the system. Hence the conventional
382
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solution procedure imposes an arbitrary restriction to the directions of
the control forces. Depending on the places of the actuators in the system
one may want to achieve the desired kinematical conditions by control
forces having different directions in relation to the constraint surfaces.
In this paper the conventional equations of motion are generalized by
inroducing general direction control forces for kinematical constraints,
that replaces the constraint force representation. And the dynamic
equations for multibody systems subject to geometrical and kinematical
constraints are developed. By the proposed method of solution the control
forces and the system motion are solved simultaneously.
This paper is divided into five sections. After the introduction, the
second section outlines the conventional equations of motion for
constrained multibody systems. In the third section the general direction
control forces for kinematical constraints are introduced and the
conditions for the control force directions are discussed. In section four
simulations of a redundant manipulator by the conventional and the
proposed methods are presented.Conclusions form the last section.
2. COI_IONAL EQUATIONS OF NOTION
Consider a mechanical system where q1,...,qn are a set of generalized
coordinates chosen for convenience to specify the configuration of the
system. Let the system be subject to c constraints. Kane's equations for
an arbitrary system of particles and rigid bodies can be expressed as
(Kamman and Huston [3], Baumgarte [7]),
where
F* + F + Fc = 0 (I)
F_ = Xi a_---JL i=l,...,c , p=1,...,n (Z)
ayp
F*, F and Fc are the vectors of generalized inertia, external and
constraint forces respectively. In equation (2) fi=O, i=1,...,c are the
constraint equations in the acceleration level, Ai are undetermined
multipliers, and y1,...,yn are the generalized speeds of the system chosen
for convenience as independent linear combinations of _p. The
tranformation between _p and yp, e.g. Euler angle derivatives and relative
angular velocity components can be expressed as
qh = Thp yp h,p=l,...,n (3)
where Thp are functions of qp (Kane and Levinson [8]).
The generalized inertia forces can be expressed in the following form
(Huston and Passarello [9]),
383
F* =M_+Q (4)
where M is the generalized mass matrix of the unconstrained system being
functions of qp, and Q contains the quadratic velocity terms.
The holonomic and nonholonomic constraint equations can be expressed
in the acceleration level as below
Bip yp - hi i=l,...,c (5)
In eq. (5) B is cxn constraint matrix, and Bip and hi are
functions of qp and yp.
Note that for holonomic constraints _i(qp,t)=O,
- _i Thp
Bip - a_ h
and for velocity level nonholonomic constraints_i(qp,yp,t)=O,
_i
Bip = --
ayp
Then, using eq. (2), the generalized constraint forces are
in general
Fc = BT X (6)
The undetermined multipliers _i represent the restraining constraint
forces and moments generated by the constraints at the points of
application (Ider and Amirouche [11]).
Equations (I) and (5) represent the governing dynamical equations.
Combining these and making use of equations (4) and (6), we have
MB
(T)
The accelerations obtained from eq. (7) are then used for numerical
integration for the time history of yp and, through the use of eq. (3),
the generalized coordinates qp.
Lagrange multipliers could be eliminated to reduce the equations for
computational efficiency. To this end let C represent a nx(n-c) matrix
which is orthogonal complement to B, obtained either by Singular value
decomposition (Singh and Likins [12]), Zero eigenvalue method (Kamman and
Huston [3]), Q-R decomposition (Kim and Vanderploeg [13], Amirouche and
Jia [6]), or row equivalence transformation (Ider and Amirouche [10]).
Premultiplying eq. (I) by CT yields
CT(F*-F) = 0 (8)
Combining equations (8) and (5) and utilizing eq. (4), we obtain the
reduced equations
384
[CTM]B[CTFQ]h (9)
Equations (9) and (3) form a set of 2n first order ordinary differential
equations that can be numerically integrated to obtain the time history of
yp and qp.
When relative joint coordinates are selected as the generalized
coordinates and the corresponding partial velocity vectors are developed
using recursive multibody kinematics (Huston and Passarello [9], Ider and
Amirouche [10]), constraint equations for joint connections are
automatically eliminated. Hence, in this paper an open tree-like system
represents an unconstrained system where n is the total number ofthe free
joint degrees of freedom.
3. CONTROL FORCES FOR KINENATICkL CONSTRAINTS
Now consider that a tree-like multobody system is subject to geometrical
and kinematical constraints. Kinematica] constraints represent desired
motions or desired paths of certain points or bodies. They are the
conditions that have to be realized by the actuators in the system. The
desired motions could be specified at position, velocity or acceleration
levels and could be holonomic or nonholonomic.
Whether one uses Newton-Euler, Lagrange or Kane's equations or other
variations of these, in the conventional approach the constraints in the
system are modeled by constraint reaction forces which are perpendicular
to the constraint surfaces. In the case of geometrical constraints
perpendicular reaction forces at the application points are generated,
hence the above approach is necessary. On the other hand kinematical
constraints could be achieved by a number of alternative control forces
whose directions in the generalized space can be be selected by physical
considerations. Therefore the conventional equations of motion should be
generalized by considering general direction control forces for
kinematical constraints.
Consider c constraint equations (5), and let cI of the constraints in
the system be geometrical and the remaining c2 (c2=c-ci) be kinematical.
The constraint matrix B and the vector of constraint force magnitudes X
can be partitioned such that
B = [ BGT BKT] T (10)
and
X = [XGT _KT] T (11)
where BG is a clxn matrix, BK is a c2 matrix, XG is a c_ dimensional
vector and XK is a c2 dimensional vector.
385
Addition of control forces to the equations of motion yields
M # + Q + BGTX G + BKrX K + ArM - F (12)
where A is a czxn control force matrix where each row represents the
direction of the control force for each kinematical constraint in the
generalized space, and _ is c2 dimensional vector of control force
magnitudes. Now assume that the control force directions and magnitudes
are selected such that the restraining constraint forces _G become zero.
This leads to
M _ + Q + BG'r XG + A'rp = F (13)
Eq. (13) can be written in the following form
M _ + Q + ZTu = F (14)
where Z is the augmented matrix of constraint and control force directions
Z = [B GT AT] T (15)
and is the vector of constraint and control force magnitudes,
v = [_G T _T]T (16)
Once the control force directions A are selected by physical
considerations eq. (14) needs to be solved together with eq. (5) to
determine the control force magnitudes simultaneously with the generalized
accelerations• Hence the augmented equations of motion are
(17)
Alternatively the equations could be reduced in a manner similar to
Section 2. To this end, let O be a nx(n-c) matrix orthogonal complement to
Z. Premultiplying eq. (14) by _T and augmenting with eq. (5) leads to
CT (F-Q)
h
(18)
In the case when the reduced equations are used v can be obtained from eq.
(14) utilizing the computed accelerations, as
v = (Z ZT) -I Z (F-My-Q) (19)
386
Note that A should be selected such that rank Z=c, because otherwise there
will be less than cz control forces to control cz kinematical conditions.
The augmented equations have a solution if and only if the augmented
mass matrix is nonsingular, in which case the prescribed conditions are
achieved with the corresponding control forces. On the other hand if it is
not physically possible to realize the kinematical constraints with the
selected control force directions, this reveals itself as a singular ( or
near singular) augmented mass matrix. Therefore the condition for the
existence of solution could be expressed as follows: Directions A should
be chosen such that a linear combination of the rows of OTM should not be
a linear combination of the rows of B. In other words the vector space
spanned by the rows of _TM and the vector space spanned by the rows of B
should be nonintersectin9 (except the zero vector). Since the dimension of
the vector space spanned by the rows of B is c, and that of _TM is n-c,
the possibilities for CTM are infinitely many (provided that n-c>O). Hence
one can construct various vector spaces for OTM by different selections of
the control force directions A. CTM that correspond to directions B is
only one of them.
For redundant systems, i.e. when c<n, it has been observed that one
usually has several physically meaningful control force directions to
satisfy the given kinematical conditions.
In the special case when A is selected such that its rows are linear
combinations of the rows of BK, then since _ is the same as C in the
conventional model, y becomes the same as the conventional case and ZTv
becomes equal to BT_. However vi may be different than _i depending on A.
Similarly for nonredundant systems, i.e. n=c, B is nxn, and rows of Z
are necessarily linear combinations of the rows of B. In this case C is
null matrix and the above procedure reduces to the conventional method
where ZTv is equal to BTX.
It should be noted that OTM and B may form nonintersecting vector
spaces even if CT and B do not. Hence realization of the prescribed
motions is possible even in the extreme case when a control force
direction is tangent to the corresponding constraint surface. This is due
to the inertia coupling between the generalized coordinates.
4. SIIIU.ATIONSOF A REDUNDANT ROBOTIC SYSTEM
In the three link manipulator shown in Figure I, the configuration of the
system can be described by three generalized coordinates el, 82, es. The
generalized speeds yp are defined as
yl : el , y2 : ()2 , y3 : (_s
The data used are LI=L2=Ls=I.0m, ml=3Okg, m2=m3=18kg, I1=10 kg.m 2,
I2=I3=8.64 kg.m 2.
The end-effector (point A) is desired to move in the horizontal
direction with a constant velocity vA. Hence the constraint equations in
the system are
387
yL2
1.0 m
Figure 1. Three link manipulator
Lie1 + L2clz + L3c13 = vat + 1.9088
Llsl + Lzs12 + L3s13 = 0.9893
(20)
where cl=cosel, c12=cos(el+ez), c_3=cos(el+e2+e3), and similarly sl=sinel,
s_2=sin(e,+e2), s_3=sin(e_+ez+e3). At the acceleration level the
constraint equations are given by eq. (5) where B and h are
r L1 Sl+L2s12+L3s1 3
B = |[ L1 c_ +L2c1 2+L3 cl 3 L2s12+L3s13 L3s13 ]L2cl 2+L3c13 L3c13
(21)
and
h =[ -Lls_y12-Lzslz(yl+yz)2-L3s13(yl+y2+y_)2]
-L1 cl yl 2-L2 cl 2 (yl +y2 )2-L3cl 3 (yl +yz+y3 )2
(22)
Initially the system is at the configuration 81=60 ° , e2=-I0 °,
e3=-90 o. The initial generalized speeds are yi=0.386 rad/sec, yz=O,
y3=-0.9618 rad/sec, which correspond to vA= -I m/sec.
388
First the system is simulated using the conventional method. The
generalized constraint forces can be expressed from equations (6) and (21)
as,
F_
F_
F_
,_1 (Llsl+Lzslz+L3s13) + >_2 (Llcl+L2Clz+L3c13)
_,1 (L2Slz+L3s13) + 42 (L2c12+L3c13)
>_1 L3s13 + )_z L3c13
(23)
In particular we wish to determine joint moments denoted as Mi, i=1,2,3
that would achieve the desired kinematica] conditions. F_, i=1,2,3
represent the required joint moments. It is seen from eq. (23) that all
three joint moments are nonzero, i.e. motors are required at all three
joints.
_e 4
(do_)
20 _ l
2
5
-40 _
t
Figure 2. Displacements.
Conventional method : I. A01, 2. z_e2, 3. z_03
control force method- 4. AB1, 5. A(_2, 6. z_03
389
0.5
-0.5
-1.0
0.5
I
• lme (,e¢)
1.0
!
ff ft ""
Figure 3. Velocities.
Conventional method : 1. 91, 2. Oz, 3. 03
Control force method: 4. B1, 5. B2, 6. es
The simulation is performed for I sec., until the end effector moves
Im in -x direction. Ael and 9i, i=1,2,3 are plotted in Figures 2 and 3
respectively. The joint moments Mi required to obtain the desired motion
of the end effector as obtained by the conventional method are shown in
Figure 4.
Second the system is resimulated by the proposed control force
approach. As an illustration the control force directions are selected
such that no moments are needed at the lower joint of link I. The
corresponding control force directions are
A=IO 1 O]0 0 1
(24)
Note that since there are no geometrical constraints in this system, the
3gO
-5
Joint
Moment
Nom
-10
/ / 1
/ /
/
/
/
/
/
/
/
/
/
/
/
Figure 4. Joint moments.
Conventional method : I. MI, 2. Mz, 3. M3
Control force method: 4. Mz, 5. M3
matrix Z is identical to A, and the vector v is identica] to H. The
control forces ZTv are
ZTv
0
-- Vl 1
0
0
V1
¥2
(25)
Hence, in this case, the required joint moments for the prescribed end
effector motion are MI=O, Mz=vl, Ma=vz.
The augmented mass matrix was observed to be full rank as expected.
ei and Oi, i=1,2,3 for a simulation of I sec. are plotted in Figures 2 and
3. The joint moments are plotted in Figure 4. Note that in this case a
motor is not needed at the lower joint of link I.
391
Initial configu_atlon
Figure 5. Final configurations: I. Conventional
method; 2. Control force method.
The system's motion differ in the above two approaches although in
both cases the end effector performs the same motion. The configurations
at t=1 sec. corresponding to the conventional model and the control force
model are shown in Figure 5.
5. CONCLUSIONS
This paper presented a general procedure for the dynamic modeling of
multibody systems subject to kinematical constraints. General direction
control forces have been introduced that replace the conventional
constraint reaction forces, hence increasing the ways of realization of
the prescribed motions. It is shown that the possible conrol force
directions are more than one, and the criteria for the existence of
solution have been presented.
The method proposed in this paper involves selecting the control
force directions in the generalized space by physical considerations, and
then solving their magnitudes simultaneously with the corresponding system
motion. As a result one can design alternative control forces that can be
applied by the actuators in the system. The method is expected to be
especially useful to control the extra degrees of freedom in systems that
have joint flexibility or joint clearance.
392
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Modeling of control forces for kinematical constraints in the dynamics of multibody systems: A new approach

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