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FormulaTion and Control of Space-based Flexible Robots with Slewing-deployable Links
PII:
Acta Astronautica Vol. 42, No. 9, pp. 519±531, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/98 $19.00 + 0.00 S0094-5765(98)00040-X
FORMULATION AND CONTROL OF SPACE-BASED FLEXIBLE ROBOTS WITH SLEWING-DEPLOYABLE LINKS SHINJI HOKAMOTO Department of Aeronautics and Astronautics, Kyushu University, 6-10-1 Hakozaki, Fukuoka, 812-8581, Japan
MASAMITSU KUWAHARA Nagasaki Shipyard and Machinery Works, Special Machinery Department, Mitsubishi Heavy Industries, 1-1 Ainoura, Nagasaki, 850-8610, Japan
VINOD MODI Department of Mechanical Engineering, University of British Columbia, 2324 Main Mall, Vancouver, B.C., Canada
and ARUN MISRA Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada, V6T 1Z4, H3A 2K6 (Received 25 November 1997) AbstractÐThe present study deals with a space-based variable geometry mobile manipulator with an arbitrary number of modules, each with two ¯exible links: one of them free to slew (revolute joint); and the other deployable (prismatic joint). The versatile manipulator has several attractive features: favorable obstacle avoidance, absence of singular con®gurations, reduced inertia coupling, relatively simpler inverse kinematics as well as governing equations of motion, to mention a few. To begin with, derivation of the governing equations of motion, using the Lagrangian procedure, is explained. As can be expected, the recursive equations are highly nonlinear, nonautonomous and coupled. This is followed by the development of a numerical algorithm leading to the solution for the inverse kinematics. Finally, some typical simulation results for trajectory control of the end-eector using the resolved acceleration approach are presented. They clearly emphasize importance of the control strategy based on the ¯exible manipulator model. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Space robotics is likely to be one of the key technologies playing signi®cant role in the development of future space projects. The proposed International Space Station, to be constructed by the turn of century, makes this strikingly evident through the mobile manipulator system, which will be used in its construction, operation and maintenance. The weight constraint requires the manipulator to be light and ¯exible. Dynamics and control of a ¯exible manipulator, operating on a ¯exible platform, represents a class of problems seldom encountered before. Most of the earlier studies [1-9] in space robotics are aimed at manipulators with revolute joints resulting in slewing motions. On the other hand, the present study deals with a space-based variable 519
geometry manipulator, composed of arbitrary number of modules, each with two ¯exible links: one of them is free to slew (revolute joint); while the other is permitted to deploy (prismatic joint). The system, which is referred to as Mobile Deployable Manipulator (MDM), represents the generalized manipulator model operating on a ¯exible space platform. The MDM has several attractive features: favorable obstacle avoidance, absence of singular con®guration, reduced inertia coupling, simpler governing equations and inverse kinematics, and others. The redundancy permits completion of a speci®ed task even when some of the joints are not operational. To begin with, inherent advantages of the MDM are brie¯y explained, compared to a conventional revolute joint based system. This is followed by the
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application of a recursive procedure in conjunction with the Lagrangian principle to obtain highly nonlinear, nonautonomous and coupled equations of motion. The governing equations of motion in the recursive form are particularly suitable for the inverse dynamics, because the total amount of calculations for the control torque is proportional to the number of links. Next, a numerical procedure for the inverse kinematics of the system is developed. Finally, some typical simulation results for trajectory control of the end-eector using the resolved acceleration approach are presented. They clearly emphasize importance of the control strategy based on the ¯exible manipulator model.
2. ADVANTAGES OF THE MDM
Figure 1 shows the system under investigation schematically: the Mobile Deployable Manipulator (MDM) operating on a ¯exible space platform. The MDM has several inherent advantages compared to a conventional revolute joint based manipulator as follows. (i) Integration of a desired number of modules can lead to a variable geometry system with favorable obstacle avoidance character. This is because the deployable link needs only a small work space during its operation. Thus, it is possible to reduce the attitude disturbance experienced by the platform when the MDM maneuvers around some obstacles. (ii) There are no static singular con®gurations. This can be understood quite readily. The prismatic and the revolute joints move the end-eector
Fig. 1. A schematic diagram of the MDM.
along and normal to the link directions, respectively. Thus, an appropriate combination of the two joint velocities can generate the end-eector motion in any desired direction. This feature is one of the most important advantages of the MDM, because it guarantees nonsingularity of the Jacobian matrix for the inverse kinematics. On the other hand, the revolute joint based manipulator has singular con®gurations when the joint angle is 08 or 1808. In the neighborhood of a singular con®guration, the manipulator may demand unrealistically large joint angular velocities. (iii) Mobile character of the manipulator can be modeled quite readily through the deployment feature. Therefore, the interaction between the manipulator motion and the platform motion can be treated easily and systematically. (iv) Inertia coupling eects due to deployment are relatively small compared with those for a slewing maneuver, resulting in simpler equations of motion.
3. DERIVATION OF KINETIC ENERGY AND POTENTIAL ENERGY
For trajectory control of a manipulator, the computed torque method based on the inverse dynamics is often used. To this end, it is desirable to reduce the number of calculations involved [10± 14]. Here the equations of motion in recursive form are derived for the MDM, and it is shown that they are suitable for the inverse dynamical computations. The model treated here is a variable geometry manipulator interconnected by n ¯exible slewing links and n ¯exible deployable links. Note, the system can serve as a model for a manipulator that is free to translate on a space platform. Figure 2 shows coordinate systems, associated with the MDM, used in the derivation. Consecutive numbers are assigned to the links from one end of the manipulator to the other, and the joint i connects the link i ÿ 1 to the link i. The coordinate system Fi is ®xed to the joint i at the link i with the xi-axis coincident with the link direction in the undeformed state. Note, the deployable link 2j and the slewing link 2j ÿ 1 are in contact at the joint 2j ÿ 1 only. Direction of the deployable link 2j in the undeformed state coincides with the direction of the slewing link 2j ÿ 1 in the deformed state. Therefore, the coordinate system F2j is a function of the elastic deformation of the slewing link 2j ÿ 1. An elastic deformation of a link is considered to be small and expressed by the expansion functions fij as
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. r 'i: position vector from the joint to a point on the link i(=Ri+r ri) while vectors in a deformed state are indicated with subscript f. Furthermore, d2j ÿ 1 expresses the deployed length of the link 2j ÿ 1, and aÄT implies the vector product a . The position and velocity vectors from the origin of the inertial system to an arbitrary point on the link i can be expressed as:
xi r0
iÿ1 X k1
vi r_ 0
iÿ1 X k1
A0k rfk A0i r 0fi ;
3
A0k ~rfk o k0 r_ fk A0;i ~ r 0fi o i0 r_ fi : 4
When the gravity force is not taken into consideration, the kinetic and potential energies of the system can be described as follows:
T Fig. 2. The coordinate system used in the derivation.
wi xi ; t
N X j1
U fij xi qij t f i xi qi t;
fij xi f^ ij xi mij x2i ;
Ui
i1
1
where N denotes the number of modes; fij(xi), the expansion function, referred to as the mode orthogonal to the translational deformation [12]; and qij(t) is the modal coordinate. The modal function fij(xi) is de®ned using the eigenfunction of a cantilever beam f^ ij(xi) as follows:
2n X
2n 1X hvT vi i ; 2 i1 i i
2n X Ei Ii @4 wi xi ; t dxi ; 2 i @x4i i1
5
6
where EiIi is the ¯exural rigidity of the link i. Considering the fact that, in most cases, an elastic deformation occurs in the plane orthogonal to the rotational axis, Z i and k 2j ÿ 1 represent unit vectors in the direction of the elastic deformation of the link i and the rotational axis of the revolute joint 2j ÿ 1, respectively. Now, the angular velocity of the joint relative to the inertial system can be expressed as:
2
where the constant mij satis®es that the mass integration equals zero, fi fij(xi)dmi=0. Hereafter the mass integration fi * dmi is denoted ash*ii. The following symbols are used for the links in the undeformed state: . Aij: transformation matrix from Fj to Fi; . o ij: angular velocity of Fi relative to Fj; . Ri: position vector from the joint i to the center of mass of the link i; . r i: position vector from the center of mass to a point on the link i; . ri: position vector from the joint i to the joint i + 1;
o 2i;0 A2i;2iÿ1 fo o2iÿ1;0 k 2iÿ1 f f 02iÿ1 q_ 2iÿ1 00 f 2iÿ1 q2iÿ1 d_ 2iÿ1 g;
7
o 2i1;0 A2i1;2i o o2i;0 k 2iÿ1 f 02i q_ 2i o 2i1;2i ; 8 fi/dxi and qÇ i=dqi/dt. Thus the vector where f 'i=df composed of the velocity of a point on the link i with respect to Fi can be expressed in the matrix form as v vT1 where
vT2
vT2n T L lHF_ ;
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F_ _rT0
o T10
2
I3 60 6 6 60 6 6 . H 6 6 .. 6 60 6 6 40 0
d_ 1
o T2n;2nÿ1
d_ 2nÿ1
0 I3 A31
0 0 I3
0 0 0
0 0 a3
.. .
.. .
..
.
.. .
.. .
A2nÿ1;1 0 0
A2nÿ1;3 0 0
I3 0 0
a 2nÿ1 In 0
q_ T1
q_ T2n ;
3
0 0 b3
7 7 7 7 7 .. 7 7 . 7; 7 b 2nÿ1 7 7 7 0 5 I2nN
00
00
a 2iÿ1 A2iÿ1;1 k 1 f 1 q1 A2iÿ1;2iÿ3 k 2iÿ3 f 2iÿ3 q2iÿ3 0 b 2iÿ1 A2iÿ1;1 k 1 f 01 A2iÿ1;2iÿ2 k 2iÿ3 f 02iÿ2 0 0; 2
0 60 6 6 6 l 6 ... 6 6 40
r~ f 1 ~ f 2 A21 r
0
0 0
0 r~ f 2 A21 k 1 f 01 q1
.. .
.. . 0
~ f ;2nÿ1 r
.. . 0
0
~ f ;2n A2n;2nÿ1 r
0
10
0 0
Zf1 r~ f 2 A21 k 1
.. . 0
1
0;
0 0
.. . 0
.. . 0
0
Z 2n f 2n
~ f ;2n A2n;2nÿ1 k 2nÿ1 f 02nÿ1 q2nÿ1 ; 1 r 2
A10 6 A 20 6 6 6 .. L 6 . 6 6 4 A2nÿ1;0 A2n;0
~ R ~ 2 A21 A21~rf 1 R
j j j j j
.. . A2nÿ1;1~rf 1 A2nÿ1;2~rf 2 A21 A2n;1~rf 1 A2n;2~rf 2 A21 0
0 0 .. .
~ 2nÿ1 R
~ 2n A2n;2nÿ1 A2n;2nÿ1~rf ;2nÿ1 R
~ 2 A21 k 1 f q1 A21 ex1 Z 1 f 01 q1 R 1 00
.. . 0 A2nÿ1;1 ex1 Z 1 f 1 q1 A2nÿ1;2~rf 2 A21 k 1 f 0 0 1 q1 00
A2n;1 ex1 Z 1 f 01 q1 A2n;2~rf 2 A21 k 1 f 1 q1 0 ~ 2 A21 k 1 f 0 A21 Z 1 f 1 R 1
j
.. .
0 0 .. .
A2nÿ1;1 Z 1 f 1 A2nÿ1;2~rf 2 A21 k 1 f 01 A2nÿ1;1 Z 1 f 1 A2nÿ1;2~rf 2 A21 k 1 f 01
A2nÿ1;2 Z 2 f 2 A2n;2 Z 2 f 2
0 0 .. . 0 2
j j j j j
0 0 .. . 0 3
3 0 07 7 7 .. 7; .7 7 7 05 0
~ 2n A2n;2nÿ1 k 2nÿ1 f 0 0 q2nÿ1 ; 2 A2n;2nÿ1 ex;2nÿ1 Z 2nÿ1 f 02nÿ1 q2nÿ1 R 2nÿ1 ~ 2n A2n;2nÿ1 k 2nÿ1 f 0 ; 3 A2n;2nÿ1 Z 2nÿ1 f 2nÿ1 R 2nÿ1
Ik=the unit matrix of k k. Therefore the kinetic energy of the system can be expressed as
j j j j
3 7 7 7 7 7; 7 7 5
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1 T F T HT LT ML G HF_ ; 2
11
where M diag m1 I3 ; m2 I3 ; ; m2n I3 ; 2 3 0 0 0 0 6 0 J B ST 7 6 7 G6 7; 4 0 BT C DT 5 0
S
D
K
J diag Jf 1 ; Jf 3 ; ; Jf ;2nÿ1 ; ~ f ;2iÿ1 i2iÿ1 A2iÿ1;2i h~ ~ f ;2i i2i A2i;2iÿ1 ; r Tf ;2iÿ1 r r Tf ;2i r Jf ;2iÿ1 h~ B diag B1 ; B3 ; ; B2nÿ1 ; 00
~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 2iÿ1 q2iÿ1 ; r Tf ;2i r B2iÿ1 A2iÿ1;2i h~ C diag c1 ; c3 ; ; c2nÿ1 ; 00
00
T T k 2iÿ1 A2iÿ1;2i h~ r Tf ;2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 2iÿ1 q2iÿ1 ; c2iÿ1 qT2iÿ1 f 2iÿ1
ST diag ST1
ST2 ; ST3
ST4 ; ; ST2nÿ1
ST2n ;
r Tf ;2iÿ1 Z 2iÿ1 f 2iÿ1 i2iÿ1 A2iÿ1;2i h~ r Tf ;2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; ST2iÿ1 h~ ~ f ;2i i2i ; ST2i A2iÿ1;2i h~ r Tf ;2i r DT diag DT1
DT2 ; DT3
DT4 ; ; DT2nÿ1
DT2n ;
00
T ~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; DT2iÿ1 qT2iÿ1 f 2iÿ1 KT2iÿ1 A2iÿ1;2i h~ r Tf ;2i r 00
T T DT2i qT2iÿ1 f 2iÿ1 k 2iÿ1 A2iÿ1;2i h~ r Tf ;2i Z 2i f 2i i2i ;
K diag K1 ; K3 ; ; K2nÿ1 ; " # T 2iÿ1;2iÿ1 K K 2i;2iÿ1 K2iÿ1 ; 2i;2iÿ1 2i;2i K K 0 T 2iÿ1;2iÿ1 hF ~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; FT2iÿ1 f 2iÿ1 i2iÿ1 f 2iÿ1 k T2iÿ1 A2iÿ1;2i h~ r Tf ;2i r K
2i;2iÿ1 hf K fT2iÿ1 Z T2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; 2i;2i hf fT2i f 2i i2i : K
4. EQUATIONS OF THE MOTION IN THE RECURSIVE FORM
Using the Lagrangian procedure, the equations of motion can be written as given in eqns (12)±(16). Note, the generalized velocity vector F_ of eqn (10) contains quasi-coordinates [15]: ^ r_ ; ^_ r0 G L 0
12
^ ~T ^ ^_ o2iÿ1;2iÿ2 o~ T ^ L 2iÿ1;0 Lo2iÿ1;2iÿ2 V2iÿ1 P2iÿ1 Go2iÿ1;2iÿ2 ;
13
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Shinji Hokamoto et al.
@A2iÿ1;2i ^ 2i ^ o2i1;2i Q A2i;2i1 L L_^ 2iÿ1 ÿ o T2i;0 A2i;2iÿ1 @d2iÿ1 000 00 T ^ 2i ^ o2i1;2i Q ÿ d_ 2iÿ1 qT2iÿ1 F 2iÿ1 q_ T2iÿ1 F 2iÿ1 k kT2iÿ1 A2iÿ1;2i A2i;2i1 L
ÿVT2i A2i;2iÿ1
@~rf ;2iÿ1 @A2iÿ1;2i ^ ^ 2i A2iÿ1;2i P P2i ÿ o T2iÿ1;0 @d2iÿ1 @d2iÿ1
00 T ^ 2i ÿ ÿ d_ 2iÿ1 qT2iÿ1 F 2iÿ1 q_ T2iÿ1 f 0T2iÿ1 Z ZT2iÿ1 A2iÿ1;2i P
@g G^ d_ 2iÿ1 ; @d2iÿ1
14
X @A2iÿ1;2i ^ o2i1;2i L_^ q_ 2iÿ1;j l2iÿ1; js q2iÿ1;s ÿ o T2i;0 A2i;2iÿ1 A2i;2i1 L @q2iÿ1; j s1 N
00 ^ o2i1;2i ÿ VT A2i;2iÿ1 @A2iÿ1;2i P ^ 2i ÿd_ 2iÿ1 f2iÿ1; j k T2iÿ1 A2iÿ1;2i1 L 2i @q2iÿ1;j
ÿo oT2iÿ1;0
@~rTf ;2iÿ1 T ^ 2i ÿ d_ 2iÿ1 f 0 ^ P 2iÿ1; j Z 2iÿ1 A2iÿ1;2i P2i @q2iÿ1; f
ÿo oT2i;0 A2i;2iÿ1 _ L^ q_ 2i;j
N X s1
l2i;js q2i;s ÿ VT2i1 A2i1;2i
ÿo oT2i;0 where
00 @A2iÿ1;2i ^ ^ 2i ÿ @g G^ q_ Q ÿ d_ 2iÿ1 f2iÿ1;j k T2iÿ1 A2iÿ1;2i Q ; 2iÿ1; j @q2iÿ1;j 2i @q2iÿ1; j
15
@A2i;2i1 ^ @A2i;2i1 ^ P2i1 ÿ o T2i;0 A2i1;2i Lo2i1;2i @q2i;j @q2i;j
@~rTf ;2i ^ 2i1 ÿ @g G^ q_ 2i;j ; A2i;2i1 P @q2i;j @q2i;j
16
2iÿ1;2iÿ1 @Jf ;2iÿ1 @g 1 @c2iÿ1 _ @K o T2iÿ1;0 o 2iÿ1;0 d_ 2iÿ1 q_ 2iÿ1 d2iÿ1 q_ T2iÿ1 @x 2 @x @x @x @B2iÿ1 _ @ST @ST @DT2iÿ1 @DT d2iÿ1 2iÿ1 q_ 2iÿ1 2i q_ 2i d_ 2iÿ1 q_ 2iÿ1 2i q_ 2i o T2iÿ1;0 @x @x @x @x @x q_ T2i
2i;2iÿ1 @K q_ 2iÿ1 @x
x d2iÿ1 ; q2iÿ1; j ; q2i; j ;
17
^ r_ , G^ _ , G^ q_ and G , and G^ q_ 2i; j are the generalized forces. In eqns (13)±(16) PÃi and QÃi denote 0 2iÿ1; j d2iÿ1 the translational and the rotational momentum of the subsystem from the link i; Vi, the velocity of the joint i; and LÃ, the generalized momentum. Note, these variables can be computed recursively as explained in the Appendix. The lijs is the eigenvalue of the link i according to the mode orthogonal to the translational deformation and is expressed as follows: d2 f^ ij d2 f^ is lijs l^ ij djs 2Ei Ii m 2m m dx mis dx 18 i i ; ij ij ij dx2i dx2i i i where l^ ij is the eigenvalue of the cantilever beam, and dij denotes the Kronecker's delta. Denoting the control force applied to the deployable joint 2i and the control torque applied to the slew 2i and t2i ÿ 1, respectively, the generalized forces can be described using the principle of ing joint 2i ÿ 1 as F virtual work as follows: ^ r_ F 0 ; 19 G 0 ^o G t 2iÿ1 ; 2iÿ1;2iÿ2
20
2i cosff f 02iÿ1 d2iÿ1 q2iÿ1 g; G^ d_ 2iÿ1 F
21
2i f2iÿ1; j d2iÿ1 sinff f 02iÿ1 d2iÿ1 q2iÿ1 g; G^ q_ 2i; j F
22
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Fig. 3. A ¯ow chart showing the oder n computational algorithm for the dynamical equations.
ZT2i A2i;2i1 t 2i1 i 1; 2; ; n ÿ1; ÿf2i;j r2i Z G^ q_ 2i; j 0 i 2n: 23 Fig. 3 shows the ¯ow chart for computation of the control force and torque recursively. The algorithm is composed of four steps: Step 1 Substitute values of the state variables. Step 2 Set i 3 1. Compute o i0, Vi, Pi and their time derivatives; and increase i 3 i + 1. Repeat this computation until i becomes 2n (forward recursion). Step 3 Set i 3 n. Compute PÃ2i ÿ 1, PÃ2i, LÃo2i ÿ 1,2i ÿ 2, LÃÇd2i ÿ 1, LÃÇq2i ÿ 1, LÃÇq2i and their time derivatives; and decrease i 3 i ÿ 1. Repeat this procedure until i becomes 1 (backward recursion). Step 4 Compute GÃo2i ÿ 1,2i ÿ 2, GÃÇd2i ÿ 1, G^ q_ 2iÿ1; j and G^ q_ 2i; j . Note, the number of iterations in Fig. 3 is equal to 3n; that is, the total amount of calculations for the control torque is proportional to the number of links. Thus the algorithm is computationally ecient.
5. INVERSE KINEMATICS
For a trajectory control of an end-eector, the time history of the joint variables must be determined so that the end-eector tracks the speci®ed trajectory (inverse kinematics). The dynamical equations presented in the previous section can be applied to compute the control forces and torques (inverse dynamics).
For a space platform based manipulator, its maneuver aects the platform's attitude. Thus it is not possible to determine the joint variables just from the geometrical relations between the manipulator and trajectory. Here a method of solution for the inverse kinematics of the space-based manipulator is developed in conjunction with the dynamical equations. The general equations of motion for the manipulator can be written as h F M F FF F; F_ t ;
24
where M denotes the inertia matrix; h, the contribution from centrifugal and Coriolis eects; and t , the control torque. The vector F is composed of the generalized coordinates and is de®ned in the same way as in eqn (10). Dividing the state vector into three parts: translation and rotation of the platform; joint variables of the manipulator; and the modal coordinates of the ¯exible links; and indicating the three parts by the subscripts S, M, and F, respectively, the equation can be rewritten as 2 32 3 2 3 2 3 S MS MSM MSF hS 0 F 4 MMS MM MMF 54 F M 54 hM 5 4 bM 5t M : F MFS MFM MF hF bFM F 25 Since the actuators to control the motion of the platform are not considered, bS=0; and the ®rst row gives the equation S MSM F M MSF F F hS 0: MS F
26
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Shinji Hokamoto et al.
This implies conservation of the total momentum of the system. On the other hand, the relation between the acceleration of the end-eector, PÈE, and the state variables can be described using the generalized Jacobian matrix J(F F) as J F _ FF_ : E J F FF P
27
Dividing the state variables in the same way as in eqn (25), the relation can be expressed as S JM F M JF F F Jh : E JS F P
28
Combining and arranging eqns (25) and (28), the relation between the acceleration of the end-eector and the joint acceleration of the manipulator can be derived as M F M J h F E J F F F; F_ : P
29
F) is a square matrix when the manipulaNote, JM(F tor does not have any redundant degree of freedom. The matrix is not singular in any posture because of the inherent advantage of the MDM. Thus the set of equations to solve for the inverse kinematics can be written as: ÿ1 F F; F_ g; M J F M F fPE ÿ Jh F
30
F I F F; F_ ; F FM F F M IFh F
31
M KSF F F KSh F S KSM F F F FF F; F
32
where M JM JS KSM JF JS KSF I ; J FM h Jh JS KSh JF JS KSF I ; J Fh
KSM
ÿMÿ1 S MSM ;
KSF ÿMÿ1 S MSF ; KSh ÿMÿ1 S MSh ; IFM ÿs sÿ1 1 MFM MFS KSM ÿ s 2 ; IFh ÿs sÿ1 1 hF MFS KSh ÿ s 3 ; s 1 MF MFS KSF ÿ bFM bÿ1 M MMF MMS KSF ; s 2 ÿbFM bÿ1 M MM MMS KSM ;
Fig. 4. Variation of kinetic and potential energy.
of the total energy of a MDM is checked in absence of control. The MDM has four ¯exible links: links 1 and 3 are free to slew; while links 2 and 4 are free to deploy. The mass, length and stiness of each link are m = 1.0 kg, l = 10 m, and EI = 50 Nm2, respectively; and the elastic deformation of the link is expressed by the ®rst and second modes (N = 2). At the beginning of the simulation, the condition of the manipulator is set as d1(0) = d3(0) = 5.0 m, angles of the rotational joint are 1808, and the ®rst and second time derivatives of these variables are zero. Figure 4 shows the variation of kinetic and potential energy, and the total energy is conserved during the typical disturbance of q41(0) = 1.0. 6.2. The end-eector trajectory control Figure 5 shows the block diagram for the resolved acceleration control, where P dE denotes the desired trajectory. To verify eectiveness of this control strategy, several dierent cases were simulated numerically. Here three typical sets of results are presented. Deformation of the links is expressed by the ®rst mode in the inverse kinematics, inverse dynamics, and direct dynamics. 6.2.1. Case 1. Consider a simple example of the manipulator with one ¯exible slewing link and one ¯exible deployable link. The manipulator is ®xed to
s 3 ÿbFM bÿ1 M hM MMS KSh :
6. NUMERICAL SIMULATION RESULTS
Planar performance of the MDM is illustrated here through some typical simulation results. 6.1. Energy conservation To verify the derived equations of motion and the associated computer program, the conservation
Fig. 5. Block diagram for the resolved acceleration control.
Space-based ¯exible robots
Fig. 6. The manipulator model and the commanded trajectory.
a large platform, for example the Canadarm attached to the Shuttle, and the translation of the manipulator is absent. Figure 6 shows the manipulator model and desired trajectory of the end-eector (dotted line). The mass, length, and stiness of the two links are m = 1.0 kg, l = 10 m, and EI = 3193 Nm2, respectively. The maneuver requires the end-eector to move 10 m in the y direction in 10 s, and the required velocity of the end-eector in the y direction is taken as a fourth order polynomial function in time. The solid lines in Figs 7±10 show the controlled response of the ¯exible manipulator, while the dashed±dotted lines present time histories of the desired velocity, acceleration, the desired deployed length, and slew angle. It is apparent that the controller is remarkably successful in following the speci®ed trajectory, even in the presence of ¯exibility. 6.2.2. Case 2. Consider the manipulator with four links in free-¯ying condition (i.e. g = 0). Figure 11 shows the manipulator model and its initial con®guration. The desired trajectory of the end-eector is speci®ed, as before, to be a straight line in the y direction. The maneuver requires the end-eector to move 0.5 m in 2 s, and the desired acceleration is de®ned by the sine function with the period of 2 s. The mass and stiness of the link 4 are m4=0.3172 kg and E4I4=0.006567 Nm2, respectively. The mass and stiness of the other links are expressed as follows: the mass ratios from link 1 to 4 are 100:20:2:1; and the ratios of the stiness are 1000:800:40:1. The length of all the links is the same, 1 m. The solid line in Figs 12 and 13 shows simulation results for the ¯exible manipulator model. Corresponding results for the rigid system are also included for comparison. Results clearly show small discrepancy in acceleration although the displacement time history correlates quite well. 6.2.3. Case 3. Consider the same manipulator model and desired trajectory as in Case 2, except for a change in speci®cations of the links. The
527
Fig. 7. Velocity of the end-eector.
Fig. 8. Acceleration of the end-eector.
Fig. 9. Time history of the deployed length.
Fig. 10. The history of the slew angle.
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Shinji Hokamoto et al.
Fig. 11. Manipulator model and the desired trajectory.
Fig. 14. Time history of the modal coordinates of the links.
Fig. 12. Comparison between the acceleration pro®les for the ¯exible and rigid systems.
Fig. 15. Time history of the deployed length of the link 2.
7. CONCLUDING REMARKS
Fig. 13. Comparison between the displacement pro®les for the ¯exible and rigd manipulators.
mass, length, and stiness of the four links are taken to be the same: m = 0.3172 kg, l = 1 m, and EI = 0.006567 Nm2. In this simulation, the deployed length of the link 2 exceeds the limitation of the deployment (0 < d2R1.0) when t = 0.45 s. Figure 14 shows time variation of the modal coordinates of the links, and Fig. 15 presents the deployed length time history for link 2. It is obvious that the ¯exible modes become unstable in this simulation. Note, the endeector can track the desirable acceleration until t = 0.45 s, although the assumption that the ¯exible deformation is small does not hold. In this simulation, the modal damping was purposely not included to emphasize the possibility of instability. Inherent modal damping or the damping supplied by the direct velocity feedback law [16] would stabilize the ¯exibility.
In this study, the equations of motion in the recursive form for the Mobile Deployable Manipulator have been derived. The equations are particularly suitable for the inverse dynamics, as the total computational eort for the control torque is proportional to the number of links. Furthermore, a numerical procedure for the inverse kinematics of the system is developed. The proposed control strategy is quite eective even in the presence of ¯exibility. However, due to the absence of damping, it does not guarantee the stability of the ¯exible modes when the commanded acceleration exceeds the critical value. This situation can be readily corrected by accounting for the structural damping and that introduced through the direct velocity feedback.
REFERENCES
1. Luh, J. Y. S., Walker, M. H. and Paul, R. P. C., IEEE Transactions on Automatic Control, 1980, AC25(3), 468±474. 2. Vafa, Z. and Dubowsky, S., in Proceedings of the IEEE International Conference on Robots and Automation, 1987, pp. 579±585. 3. Koningstein, R., Ullman, M. and Cannon, R. H. Jr, in NASA Conference Space Telerobotics, 1989, pp. 235±243. 4. Umetani, Y. and Yoshida, K., IEEE Transactions on Robotics and Automation, 1989, 5(3), 303±314. 5. Nenchev, D. N., The International Journal of Robotics Research, 1992, 11(6), 584±597.
Space-based ¯exible robots 6. Torres, M. A. and Dubowsky, S., Journal of Guidance, Control, and Dynamics, 1992, 15(4), 1010± 1017. 7. Koningstein, R. and Cannon, R. H., Jr., Journal of Guidance, Control, and Dynamics, 1995, 18(6), 1386± 1391. 8. Chen, Y. and Meirovitch, L., Journal of Guidance, Control, and Dynamics, 1995, 18(4), 756±766. 9. Fujii, H. A., Uchiyama, K. and Yoneoka, H., Journal of Guidance, Control, and Dynamics, 1996, 19(5), 985± 991. 10. Book, W. J., International Journal of Robotics Research, 1984, 3(3), 87±101. 11. Benati, M. and Morro, A., Transactions of the ASME Journal of Dynamic Systems, Measurement, and Control, 1994, 116, 81±88. 12. Tsuchiya, K. and Watanabe, S., Finite Elements in Analysis and Design, 1993, 14, 261±272. 13. Rodriguez, G., IEEE Transactions on Robotics and Automation, 1989, RA-5(4), 510±521. 14. Hokamoto, S., Modi, V. J. and Misra, A. K., AAS/ AIAA Astrodynamics Specialist Conference, AAS, 95± 322, also in Misra AK et al., eds. Advances in the Astronautical Sciences, eds. A. K. Misra et al. Univelt Inc. Publisher, San Diego, CA, pp. 339±357, 1995. 15. Meirovitch, L., Method of Analytical Dynamics, McGraw-Hill, New York, 1970, pp. 157±160. 16. Balas, M. J., Journal of Guidance and Control, 1979, 2(3), 252±253.
The vectors used in eqns (19)±(23) can be computed recursively as follows. Generalized velocity with respect to the inertial system ^ o
o T10
o T2nÿ1;0
d_ 1 d_ 2nÿ1
Acceleration of the joints: V_ ~ 10 V1 A10r0 ; V_ 1 o ~ 2i;2iÿ1 V2i A2i;2iÿ1 V_ 2iÿ1 ~rf ;2iÿ1 o 2iÿ1;0 V_ 2i o ~rf ;2iÿ1 o_ 2iÿ1;0 ex;2iÿ1 d2iÿ1 Z 2iÿ1 w 2iÿ1 ; ~ 2i1;2i V2i1 A2i1;2i V_ 2i ~r_ f ;2i o 2i;0 V_ 2i1 o ~rf ;2i o 2i;0 Z 2i w 2i :
Translational momentum of the links: P ~ i o i0 Vi Pi mi R
q_ T1
q_ T2n T
HF_ :
i 1; 2; ; 2n:
Time derivative of the translational momentum of the links: P_ ~ i o_ i0 V_ i P_ i mi R
i 1; 2; ; 2n:
Translational momentum of the set from the one link to the ^ end-eector: P ^ i1 ^ i Pi Ai;i1 P P
APPENDIX
_rT0
529
i 1; 2; ; 2n ÿ 1;
^ 2n P2n : P
Time derivative of the translational momentum of the set from the one link to the end-eector: PÃ ^ i1 P ^_ i1 i 1; 2; ; 2n ÿ 1; ^_ i P_ i Ai;i1 o ~ Ti1;i P P
Velocity of the joints: F o 10 ; o 10 o o 2i;0 A2i;2iÿ1 o o 2iÿ1;0 k 2iÿ1 w_ 02iÿ1 ;
o 2i;0 k 2iÿ1 w_ 02i o 2i1;2i : o 2i1;0 A2i1;2i o Acceleration of the joints: F o_ 10 o_ 10 ; ~ 2i;2iÿ1 o 2i;0 A2i;2iÿ1 o_ 2iÿ1;0 k 2iÿ1 w _o 2i;0 o 02iÿ1 ; 0 _o 2i1;0 o ~ 2i1;2i o 2i;0 A2i1;2i o_ 2i;0 k 2iÿ1 w 2i o_ 2i1;2i :
Velocity of the joints: V V1 A10 r_ 0 ; V2i A2i;2iÿ1 V2iÿ1 ~rf ;2iÿ1 o 2iÿ1;0 V2i1
ex;2iÿ1 d_ 2iÿ1 Z 2iÿ1 w_ 2iÿ1 ; A2i1;2i V2i ~rf ;2i o 2i;0 Z 2i w_ 2i :
^_ 2n P_ 2n : P
Angular momentum of the set from the one link to the endeector: QÃ ^i R ^ i1 ~ T Pi ~rT Ai;i1 P Q i fi
i 1; 2; ; 2n ÿ 1;
^ 2n R ~ T P2n : Q 2n Time derivative of the angular momentum of the set ^_ from the one link to the end-eector: Q ^_ i R ~ T P_ i ~r_ T Ai;i1 P ^_ i1 ^ i1 ~rT Ai;i1 o ^ i1 P ~ Ti1;i P Q fi i fi i 1; 2; ; 2n ÿ 1; ^_ 2n R ~ T P_ 2n : Q 2n
Generalize momentum of the links: L
530
Shinji Hokamoto et al. h iT L LTr_ 0 LTo10 LTo2nÿ1;2nÿ2 Ld_ 1 Ld_ 2nÿ1 LTq_ 1 LTq_ 2n ; ^1; Lr_ 0 A01 P ^ 2iÿ1 A2iÿ1;2i Q ^ 2i Jf ;2iÿ1 o 2iÿ1;0 B2iÿ1 d_ 2iÿ1 ST q_ 2iÿ1 ST q_ 2i ; Lo2iÿ1;2iÿ2 Q 2iÿ1 2i ^ 2i BT o 2iÿ1;0 c2iÿ1 d_ 2iÿ1 DT q_ 2iÿ1 DT q_ 2i ; ^ 2i w k T A2iÿ1;2i Q Ld_ 2iÿ1 eTx;2iÿ1 w 02iÿ1 Z T2iÿ1 A2iÿ1;2i P 2iÿ1 2iÿ1 2i 2iÿ1 2iÿ1 00
T
^ 2i S2iÿ1 o 2iÿ1;0 D2iÿ1 d_ 2iÿ1 K 2iÿ1;2iÿ1 q_ 2iÿ1 K ^ 2i f T k T A2iÿ1;2i Q _ 2i ; Lq_ 2iÿ1 f T2iÿ1 Z T2iÿ1 A2iÿ1;2i P 2iÿ1 2iÿ1 2i;2iÿ1 q 0
^ 2i1 S2i o 2iÿ1;0 D2i d_ 2iÿ1 K 2i;2iÿ1 q_ 2iÿ1 K 2i;2i q_ 2i ; Lq_ 2i f T2i Z 2i A2i;2i1 P 2n;2nÿ1 q_ 2nÿ1 K 2n;2n q_ 2n : Lq_ 2n S2n o 2nÿ1;0 D2n d_ 2nÿ1 K
Time derivative of the generalized momentum of the links: L_ ^_ 1 ; ~ T10 Lr_ 0 A01 P L_ r_ 0 o ^_ 2iÿ1 A2iÿ1;2i o ^_ 2i J_ f ;2iÿ1 o 2iÿ1;0 Jf ;2iÿ1 o_ 2iÿ1;0 B_ 2iÿ1 d_ 2iÿ1 ^ 2i Q ~ T2i;2iÿ1 Q L_ o2iÿ1;2iÿ2 Q T B2iÿ1 d2iÿ1 ST2iÿ1 q_ 2iÿ1 ST2iÿ1 q2iÿ1 S_ 2i q_ 2i ST2i q2i ; 0 T ^ 2i eT ^ 2i P ^_ 2i ~ T2i;2iÿ1 P L_ d_ 2iÿ1 w_ 02iÿ1 Z T2iÿ1 A2iÿ1;2i P x;2iÿ1 w 2iÿ1 Z 2iÿ1 A2iÿ1;2i o 00 ^_ 2i ^ 2i w 0 0 k T A2iÿ1;2i o ^ 2i Q ~ T2i;2iÿ1 Q w_ 2iÿ1 k T2iÿ1 A2iÿ1;2i Q 2iÿ1 2iÿ1
T
T
_ _ 2iÿ1 DT2iÿ1 q2iÿ1 B_ 2iÿ1 o 2iÿ1;0 BT2iÿ1 o_ 2iÿ1;0 c_ 2iÿ1 d_ 2iÿ1 c2iÿ1 d2iÿ1 D 2iÿ1 q _ T q_ 2i DT q2i ; D 2i 2i 0T ^ 2i ^ f T Z T A2iÿ1;2i o ^ 2i P ^_ 2i d_ 2iÿ1 f 0 0 T k T A2iÿ1;2i Q ~ T2i;2iÿ1 P Z T2iÿ1 A2iÿ1;2i P L_ q_ 2iÿ1 d_ 2iÿ1 f 2iÿ1 2i 2iÿ1 2iÿ1 2iÿ1 2iÿ1 0 T ^_ 2i S_ 2iÿ1 o 2iÿ1;0 S2iÿ1 o_ 2iÿ1;0 D ^ 2i Q _ 2iÿ1 d_ 2iÿ1 ~ T2i;2iÿ1 Q f 2iÿ1 k T2iÿ1 A2iÿ1;2i o
2iÿ1;2iÿ1 q2iÿ1 K _ 2iÿ1;2iÿ1 q_ 2iÿ1 K _ _ 2i K 2i ; D2iÿ1 d2iÿ1 K 2i;2iÿ1 q 2i;2iÿ1 q T
T
^ 2i1 P _ 2i d_ 2iÿ1 ^_ 2i1 S_ 2i o 2iÿ1;0 S2i o_ 2iÿ1;0 D ~ T2i1;2i P L_ q_ 2i f T2i Z T2i A2i;2i1 o 2i;2iÿ1 q2iÿ1 K 2i;2i q2i ; _ 2i;2iÿ1 q_ 2iÿ1 K _ 2i;2i q_ 2i K D2i d2iÿ1 K _ 2n d_ 2nÿ1 D2n d2nÿ1 K 2n;2nÿ1 q2nÿ1 K 2n;2n q2n : _ 2n;2nÿ1 q_ 2nÿ1 K _ 2n;2n q_ 2n K L_ q_ 2n S_ 2n o 2nÿ1;0 S2n o_ 2nÿ1;0 D
Generalized momentum of the set from the one link to the end-eector: LÃ h T ^ L ^ L r_ 0
^T L o10
^T L o2nÿ1;2nÿ2
L^ d_ 1
L^ d_ 2nÿ1
^ r_ 0 Lr_ 0 ; L ^ o2i1;2i ; ^ o2iÿ1;2iÿ2 Lo2iÿ1;2iÿ2 A2iÿ1;2i1 L L ^ o2nÿ1;2nÿ2 Lo2nÿ1;2nÿ2 ; L 00 ^ o2i1;2i ; L^ d_ 2iÿ1 Ld_ 2iÿ1 w2iÿ1 k T2iÿ1 A2iÿ2i1 L
L^ d_ 2nÿ1 Ld_ 2nÿ1 ; ^ o2i1;2i ; ^ q_ 2iÿ1 Lq_ 2iÿ1 f 0 T k T A2iÿ1;2i1 L L 2iÿ1 2iÿ1 ^ q_ 2i Lq_ 2i f 0 T k T A2i;2i1 L ^ o2i1;2i ; L 2i 2i ^ q_ 2nÿ1 Lq_ 2nÿ1 ; L ^ q_ Lq_ : L 2n 2n
^T L q_ 1
^T L q_ 2n
iT
;
Space-based ¯exible robots ^_ Time derivative of the generalized momentum of the set from the one link to the end-eector: L ^_ r_ L_ r_ ; L 0 0 ^_ o2i1;2i ; ^ o2i1;2i A2iÿ1;2i1 L ^_ o2iÿ1;2iÿ2 L_ o2iÿ1;2iÿ2 A2iÿ1;2i o ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L L ^_ o2nÿ1;2nÿ2 L_ o2nÿ1;2nÿ2 ; L 00 ^ o2i1;2i L_^ d_ 2iÿ1 L_ d_ 2iÿ1 w_ 2iÿ1 k T2iÿ1 A2iÿ1;2i1 L 00 ^_ o2i1;2i g; ^ o2i1;2i A2iÿ1;2i1 L ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L w2iÿ1 L_ q_ 2iÿ1 fA2iÿ1;2i o
L_^ d_ 2nÿ1 L_ d_ 2nÿ1 ; ^_ q_ 2iÿ1 L_ q_ 2iÿ1 d_ 2iÿ1 f 0 0 T k T A2iÿ1;2i1 L ^ o2i1;2i f 0 T k T fA2iÿ1;2i o ^ o2i1;2i ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L L 2iÿ1 2iÿ1 2iÿ1 2iÿ1
^_ q_ 2i L
^_ o2i1;2i g; A2iÿ1;2i1 L 0 ^_ o2i1;2i ; ^ o2i1;2i L ~ T2i1;2i L L_ q_ 2i f 2iT k T2i A2i;2i1 o
^_ q_ 2nÿ1 L_ q_ 2nÿ1 ; L ^_ q_ L_ q_ : L 2n 2n
531
Acta Astronautica Vol. 42, No. 9, pp. 519±531, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/98 $19.00 + 0.00 S0094-5765(98)00040-X
FORMULATION AND CONTROL OF SPACE-BASED FLEXIBLE ROBOTS WITH SLEWING-DEPLOYABLE LINKS SHINJI HOKAMOTO Department of Aeronautics and Astronautics, Kyushu University, 6-10-1 Hakozaki, Fukuoka, 812-8581, Japan
MASAMITSU KUWAHARA Nagasaki Shipyard and Machinery Works, Special Machinery Department, Mitsubishi Heavy Industries, 1-1 Ainoura, Nagasaki, 850-8610, Japan
VINOD MODI Department of Mechanical Engineering, University of British Columbia, 2324 Main Mall, Vancouver, B.C., Canada
and ARUN MISRA Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada, V6T 1Z4, H3A 2K6 (Received 25 November 1997) AbstractÐThe present study deals with a space-based variable geometry mobile manipulator with an arbitrary number of modules, each with two ¯exible links: one of them free to slew (revolute joint); and the other deployable (prismatic joint). The versatile manipulator has several attractive features: favorable obstacle avoidance, absence of singular con®gurations, reduced inertia coupling, relatively simpler inverse kinematics as well as governing equations of motion, to mention a few. To begin with, derivation of the governing equations of motion, using the Lagrangian procedure, is explained. As can be expected, the recursive equations are highly nonlinear, nonautonomous and coupled. This is followed by the development of a numerical algorithm leading to the solution for the inverse kinematics. Finally, some typical simulation results for trajectory control of the end-eector using the resolved acceleration approach are presented. They clearly emphasize importance of the control strategy based on the ¯exible manipulator model. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Space robotics is likely to be one of the key technologies playing signi®cant role in the development of future space projects. The proposed International Space Station, to be constructed by the turn of century, makes this strikingly evident through the mobile manipulator system, which will be used in its construction, operation and maintenance. The weight constraint requires the manipulator to be light and ¯exible. Dynamics and control of a ¯exible manipulator, operating on a ¯exible platform, represents a class of problems seldom encountered before. Most of the earlier studies [1-9] in space robotics are aimed at manipulators with revolute joints resulting in slewing motions. On the other hand, the present study deals with a space-based variable 519
geometry manipulator, composed of arbitrary number of modules, each with two ¯exible links: one of them is free to slew (revolute joint); while the other is permitted to deploy (prismatic joint). The system, which is referred to as Mobile Deployable Manipulator (MDM), represents the generalized manipulator model operating on a ¯exible space platform. The MDM has several attractive features: favorable obstacle avoidance, absence of singular con®guration, reduced inertia coupling, simpler governing equations and inverse kinematics, and others. The redundancy permits completion of a speci®ed task even when some of the joints are not operational. To begin with, inherent advantages of the MDM are brie¯y explained, compared to a conventional revolute joint based system. This is followed by the
520
Shinji Hokamoto et al.
application of a recursive procedure in conjunction with the Lagrangian principle to obtain highly nonlinear, nonautonomous and coupled equations of motion. The governing equations of motion in the recursive form are particularly suitable for the inverse dynamics, because the total amount of calculations for the control torque is proportional to the number of links. Next, a numerical procedure for the inverse kinematics of the system is developed. Finally, some typical simulation results for trajectory control of the end-eector using the resolved acceleration approach are presented. They clearly emphasize importance of the control strategy based on the ¯exible manipulator model.
2. ADVANTAGES OF THE MDM
Figure 1 shows the system under investigation schematically: the Mobile Deployable Manipulator (MDM) operating on a ¯exible space platform. The MDM has several inherent advantages compared to a conventional revolute joint based manipulator as follows. (i) Integration of a desired number of modules can lead to a variable geometry system with favorable obstacle avoidance character. This is because the deployable link needs only a small work space during its operation. Thus, it is possible to reduce the attitude disturbance experienced by the platform when the MDM maneuvers around some obstacles. (ii) There are no static singular con®gurations. This can be understood quite readily. The prismatic and the revolute joints move the end-eector
Fig. 1. A schematic diagram of the MDM.
along and normal to the link directions, respectively. Thus, an appropriate combination of the two joint velocities can generate the end-eector motion in any desired direction. This feature is one of the most important advantages of the MDM, because it guarantees nonsingularity of the Jacobian matrix for the inverse kinematics. On the other hand, the revolute joint based manipulator has singular con®gurations when the joint angle is 08 or 1808. In the neighborhood of a singular con®guration, the manipulator may demand unrealistically large joint angular velocities. (iii) Mobile character of the manipulator can be modeled quite readily through the deployment feature. Therefore, the interaction between the manipulator motion and the platform motion can be treated easily and systematically. (iv) Inertia coupling eects due to deployment are relatively small compared with those for a slewing maneuver, resulting in simpler equations of motion.
3. DERIVATION OF KINETIC ENERGY AND POTENTIAL ENERGY
For trajectory control of a manipulator, the computed torque method based on the inverse dynamics is often used. To this end, it is desirable to reduce the number of calculations involved [10± 14]. Here the equations of motion in recursive form are derived for the MDM, and it is shown that they are suitable for the inverse dynamical computations. The model treated here is a variable geometry manipulator interconnected by n ¯exible slewing links and n ¯exible deployable links. Note, the system can serve as a model for a manipulator that is free to translate on a space platform. Figure 2 shows coordinate systems, associated with the MDM, used in the derivation. Consecutive numbers are assigned to the links from one end of the manipulator to the other, and the joint i connects the link i ÿ 1 to the link i. The coordinate system Fi is ®xed to the joint i at the link i with the xi-axis coincident with the link direction in the undeformed state. Note, the deployable link 2j and the slewing link 2j ÿ 1 are in contact at the joint 2j ÿ 1 only. Direction of the deployable link 2j in the undeformed state coincides with the direction of the slewing link 2j ÿ 1 in the deformed state. Therefore, the coordinate system F2j is a function of the elastic deformation of the slewing link 2j ÿ 1. An elastic deformation of a link is considered to be small and expressed by the expansion functions fij as
Space-based ¯exible robots
521
. r 'i: position vector from the joint to a point on the link i(=Ri+r ri) while vectors in a deformed state are indicated with subscript f. Furthermore, d2j ÿ 1 expresses the deployed length of the link 2j ÿ 1, and aÄT implies the vector product a . The position and velocity vectors from the origin of the inertial system to an arbitrary point on the link i can be expressed as:
xi r0
iÿ1 X k1
vi r_ 0
iÿ1 X k1
A0k rfk A0i r 0fi ;
3
A0k ~rfk o k0 r_ fk A0;i ~ r 0fi o i0 r_ fi : 4
When the gravity force is not taken into consideration, the kinetic and potential energies of the system can be described as follows:
T Fig. 2. The coordinate system used in the derivation.
wi xi ; t
N X j1
U fij xi qij t f i xi qi t;
fij xi f^ ij xi mij x2i ;
Ui
i1
1
where N denotes the number of modes; fij(xi), the expansion function, referred to as the mode orthogonal to the translational deformation [12]; and qij(t) is the modal coordinate. The modal function fij(xi) is de®ned using the eigenfunction of a cantilever beam f^ ij(xi) as follows:
2n X
2n 1X hvT vi i ; 2 i1 i i
2n X Ei Ii @4 wi xi ; t dxi ; 2 i @x4i i1
5
6
where EiIi is the ¯exural rigidity of the link i. Considering the fact that, in most cases, an elastic deformation occurs in the plane orthogonal to the rotational axis, Z i and k 2j ÿ 1 represent unit vectors in the direction of the elastic deformation of the link i and the rotational axis of the revolute joint 2j ÿ 1, respectively. Now, the angular velocity of the joint relative to the inertial system can be expressed as:
2
where the constant mij satis®es that the mass integration equals zero, fi fij(xi)dmi=0. Hereafter the mass integration fi * dmi is denoted ash*ii. The following symbols are used for the links in the undeformed state: . Aij: transformation matrix from Fj to Fi; . o ij: angular velocity of Fi relative to Fj; . Ri: position vector from the joint i to the center of mass of the link i; . r i: position vector from the center of mass to a point on the link i; . ri: position vector from the joint i to the joint i + 1;
o 2i;0 A2i;2iÿ1 fo o2iÿ1;0 k 2iÿ1 f f 02iÿ1 q_ 2iÿ1 00 f 2iÿ1 q2iÿ1 d_ 2iÿ1 g;
7
o 2i1;0 A2i1;2i o o2i;0 k 2iÿ1 f 02i q_ 2i o 2i1;2i ; 8 fi/dxi and qÇ i=dqi/dt. Thus the vector where f 'i=df composed of the velocity of a point on the link i with respect to Fi can be expressed in the matrix form as v vT1 where
vT2
vT2n T L lHF_ ;
9
522
Shinji Hokamoto et al.
F_ _rT0
o T10
2
I3 60 6 6 60 6 6 . H 6 6 .. 6 60 6 6 40 0
d_ 1
o T2n;2nÿ1
d_ 2nÿ1
0 I3 A31
0 0 I3
0 0 0
0 0 a3
.. .
.. .
..
.
.. .
.. .
A2nÿ1;1 0 0
A2nÿ1;3 0 0
I3 0 0
a 2nÿ1 In 0
q_ T1
q_ T2n ;
3
0 0 b3
7 7 7 7 7 .. 7 7 . 7; 7 b 2nÿ1 7 7 7 0 5 I2nN
00
00
a 2iÿ1 A2iÿ1;1 k 1 f 1 q1 A2iÿ1;2iÿ3 k 2iÿ3 f 2iÿ3 q2iÿ3 0 b 2iÿ1 A2iÿ1;1 k 1 f 01 A2iÿ1;2iÿ2 k 2iÿ3 f 02iÿ2 0 0; 2
0 60 6 6 6 l 6 ... 6 6 40
r~ f 1 ~ f 2 A21 r
0
0 0
0 r~ f 2 A21 k 1 f 01 q1
.. .
.. . 0
~ f ;2nÿ1 r
.. . 0
0
~ f ;2n A2n;2nÿ1 r
0
10
0 0
Zf1 r~ f 2 A21 k 1
.. . 0
1
0;
0 0
.. . 0
.. . 0
0
Z 2n f 2n
~ f ;2n A2n;2nÿ1 k 2nÿ1 f 02nÿ1 q2nÿ1 ; 1 r 2
A10 6 A 20 6 6 6 .. L 6 . 6 6 4 A2nÿ1;0 A2n;0
~ R ~ 2 A21 A21~rf 1 R
j j j j j
.. . A2nÿ1;1~rf 1 A2nÿ1;2~rf 2 A21 A2n;1~rf 1 A2n;2~rf 2 A21 0
0 0 .. .
~ 2nÿ1 R
~ 2n A2n;2nÿ1 A2n;2nÿ1~rf ;2nÿ1 R
~ 2 A21 k 1 f q1 A21 ex1 Z 1 f 01 q1 R 1 00
.. . 0 A2nÿ1;1 ex1 Z 1 f 1 q1 A2nÿ1;2~rf 2 A21 k 1 f 0 0 1 q1 00
A2n;1 ex1 Z 1 f 01 q1 A2n;2~rf 2 A21 k 1 f 1 q1 0 ~ 2 A21 k 1 f 0 A21 Z 1 f 1 R 1
j
.. .
0 0 .. .
A2nÿ1;1 Z 1 f 1 A2nÿ1;2~rf 2 A21 k 1 f 01 A2nÿ1;1 Z 1 f 1 A2nÿ1;2~rf 2 A21 k 1 f 01
A2nÿ1;2 Z 2 f 2 A2n;2 Z 2 f 2
0 0 .. . 0 2
j j j j j
0 0 .. . 0 3
3 0 07 7 7 .. 7; .7 7 7 05 0
~ 2n A2n;2nÿ1 k 2nÿ1 f 0 0 q2nÿ1 ; 2 A2n;2nÿ1 ex;2nÿ1 Z 2nÿ1 f 02nÿ1 q2nÿ1 R 2nÿ1 ~ 2n A2n;2nÿ1 k 2nÿ1 f 0 ; 3 A2n;2nÿ1 Z 2nÿ1 f 2nÿ1 R 2nÿ1
Ik=the unit matrix of k k. Therefore the kinetic energy of the system can be expressed as
j j j j
3 7 7 7 7 7; 7 7 5
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523
1 T F T HT LT ML G HF_ ; 2
11
where M diag m1 I3 ; m2 I3 ; ; m2n I3 ; 2 3 0 0 0 0 6 0 J B ST 7 6 7 G6 7; 4 0 BT C DT 5 0
S
D
K
J diag Jf 1 ; Jf 3 ; ; Jf ;2nÿ1 ; ~ f ;2iÿ1 i2iÿ1 A2iÿ1;2i h~ ~ f ;2i i2i A2i;2iÿ1 ; r Tf ;2iÿ1 r r Tf ;2i r Jf ;2iÿ1 h~ B diag B1 ; B3 ; ; B2nÿ1 ; 00
~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 2iÿ1 q2iÿ1 ; r Tf ;2i r B2iÿ1 A2iÿ1;2i h~ C diag c1 ; c3 ; ; c2nÿ1 ; 00
00
T T k 2iÿ1 A2iÿ1;2i h~ r Tf ;2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 2iÿ1 q2iÿ1 ; c2iÿ1 qT2iÿ1 f 2iÿ1
ST diag ST1
ST2 ; ST3
ST4 ; ; ST2nÿ1
ST2n ;
r Tf ;2iÿ1 Z 2iÿ1 f 2iÿ1 i2iÿ1 A2iÿ1;2i h~ r Tf ;2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; ST2iÿ1 h~ ~ f ;2i i2i ; ST2i A2iÿ1;2i h~ r Tf ;2i r DT diag DT1
DT2 ; DT3
DT4 ; ; DT2nÿ1
DT2n ;
00
T ~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; DT2iÿ1 qT2iÿ1 f 2iÿ1 KT2iÿ1 A2iÿ1;2i h~ r Tf ;2i r 00
T T DT2i qT2iÿ1 f 2iÿ1 k 2iÿ1 A2iÿ1;2i h~ r Tf ;2i Z 2i f 2i i2i ;
K diag K1 ; K3 ; ; K2nÿ1 ; " # T 2iÿ1;2iÿ1 K K 2i;2iÿ1 K2iÿ1 ; 2i;2iÿ1 2i;2i K K 0 T 2iÿ1;2iÿ1 hF ~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; FT2iÿ1 f 2iÿ1 i2iÿ1 f 2iÿ1 k T2iÿ1 A2iÿ1;2i h~ r Tf ;2i r K
2i;2iÿ1 hf K fT2iÿ1 Z T2i r~ f ;2i i2i A2i;2iÿ1 k 2iÿ1 f 02iÿ1 ; 2i;2i hf fT2i f 2i i2i : K
4. EQUATIONS OF THE MOTION IN THE RECURSIVE FORM
Using the Lagrangian procedure, the equations of motion can be written as given in eqns (12)±(16). Note, the generalized velocity vector F_ of eqn (10) contains quasi-coordinates [15]: ^ r_ ; ^_ r0 G L 0
12
^ ~T ^ ^_ o2iÿ1;2iÿ2 o~ T ^ L 2iÿ1;0 Lo2iÿ1;2iÿ2 V2iÿ1 P2iÿ1 Go2iÿ1;2iÿ2 ;
13
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Shinji Hokamoto et al.
@A2iÿ1;2i ^ 2i ^ o2i1;2i Q A2i;2i1 L L_^ 2iÿ1 ÿ o T2i;0 A2i;2iÿ1 @d2iÿ1 000 00 T ^ 2i ^ o2i1;2i Q ÿ d_ 2iÿ1 qT2iÿ1 F 2iÿ1 q_ T2iÿ1 F 2iÿ1 k kT2iÿ1 A2iÿ1;2i A2i;2i1 L
ÿVT2i A2i;2iÿ1
@~rf ;2iÿ1 @A2iÿ1;2i ^ ^ 2i A2iÿ1;2i P P2i ÿ o T2iÿ1;0 @d2iÿ1 @d2iÿ1
00 T ^ 2i ÿ ÿ d_ 2iÿ1 qT2iÿ1 F 2iÿ1 q_ T2iÿ1 f 0T2iÿ1 Z ZT2iÿ1 A2iÿ1;2i P
@g G^ d_ 2iÿ1 ; @d2iÿ1
14
X @A2iÿ1;2i ^ o2i1;2i L_^ q_ 2iÿ1;j l2iÿ1; js q2iÿ1;s ÿ o T2i;0 A2i;2iÿ1 A2i;2i1 L @q2iÿ1; j s1 N
00 ^ o2i1;2i ÿ VT A2i;2iÿ1 @A2iÿ1;2i P ^ 2i ÿd_ 2iÿ1 f2iÿ1; j k T2iÿ1 A2iÿ1;2i1 L 2i @q2iÿ1;j
ÿo oT2iÿ1;0
@~rTf ;2iÿ1 T ^ 2i ÿ d_ 2iÿ1 f 0 ^ P 2iÿ1; j Z 2iÿ1 A2iÿ1;2i P2i @q2iÿ1; f
ÿo oT2i;0 A2i;2iÿ1 _ L^ q_ 2i;j
N X s1
l2i;js q2i;s ÿ VT2i1 A2i1;2i
ÿo oT2i;0 where
00 @A2iÿ1;2i ^ ^ 2i ÿ @g G^ q_ Q ÿ d_ 2iÿ1 f2iÿ1;j k T2iÿ1 A2iÿ1;2i Q ; 2iÿ1; j @q2iÿ1;j 2i @q2iÿ1; j
15
@A2i;2i1 ^ @A2i;2i1 ^ P2i1 ÿ o T2i;0 A2i1;2i Lo2i1;2i @q2i;j @q2i;j
@~rTf ;2i ^ 2i1 ÿ @g G^ q_ 2i;j ; A2i;2i1 P @q2i;j @q2i;j
16
2iÿ1;2iÿ1 @Jf ;2iÿ1 @g 1 @c2iÿ1 _ @K o T2iÿ1;0 o 2iÿ1;0 d_ 2iÿ1 q_ 2iÿ1 d2iÿ1 q_ T2iÿ1 @x 2 @x @x @x @B2iÿ1 _ @ST @ST @DT2iÿ1 @DT d2iÿ1 2iÿ1 q_ 2iÿ1 2i q_ 2i d_ 2iÿ1 q_ 2iÿ1 2i q_ 2i o T2iÿ1;0 @x @x @x @x @x q_ T2i
2i;2iÿ1 @K q_ 2iÿ1 @x
x d2iÿ1 ; q2iÿ1; j ; q2i; j ;
17
^ r_ , G^ _ , G^ q_ and G , and G^ q_ 2i; j are the generalized forces. In eqns (13)±(16) PÃi and QÃi denote 0 2iÿ1; j d2iÿ1 the translational and the rotational momentum of the subsystem from the link i; Vi, the velocity of the joint i; and LÃ, the generalized momentum. Note, these variables can be computed recursively as explained in the Appendix. The lijs is the eigenvalue of the link i according to the mode orthogonal to the translational deformation and is expressed as follows: d2 f^ ij d2 f^ is lijs l^ ij djs 2Ei Ii m 2m m dx mis dx 18 i i ; ij ij ij dx2i dx2i i i where l^ ij is the eigenvalue of the cantilever beam, and dij denotes the Kronecker's delta. Denoting the control force applied to the deployable joint 2i and the control torque applied to the slew 2i and t2i ÿ 1, respectively, the generalized forces can be described using the principle of ing joint 2i ÿ 1 as F virtual work as follows: ^ r_ F 0 ; 19 G 0 ^o G t 2iÿ1 ; 2iÿ1;2iÿ2
20
2i cosff f 02iÿ1 d2iÿ1 q2iÿ1 g; G^ d_ 2iÿ1 F
21
2i f2iÿ1; j d2iÿ1 sinff f 02iÿ1 d2iÿ1 q2iÿ1 g; G^ q_ 2i; j F
22
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525
Fig. 3. A ¯ow chart showing the oder n computational algorithm for the dynamical equations.
ZT2i A2i;2i1 t 2i1 i 1; 2; ; n ÿ1; ÿf2i;j r2i Z G^ q_ 2i; j 0 i 2n: 23 Fig. 3 shows the ¯ow chart for computation of the control force and torque recursively. The algorithm is composed of four steps: Step 1 Substitute values of the state variables. Step 2 Set i 3 1. Compute o i0, Vi, Pi and their time derivatives; and increase i 3 i + 1. Repeat this computation until i becomes 2n (forward recursion). Step 3 Set i 3 n. Compute PÃ2i ÿ 1, PÃ2i, LÃo2i ÿ 1,2i ÿ 2, LÃÇd2i ÿ 1, LÃÇq2i ÿ 1, LÃÇq2i and their time derivatives; and decrease i 3 i ÿ 1. Repeat this procedure until i becomes 1 (backward recursion). Step 4 Compute GÃo2i ÿ 1,2i ÿ 2, GÃÇd2i ÿ 1, G^ q_ 2iÿ1; j and G^ q_ 2i; j . Note, the number of iterations in Fig. 3 is equal to 3n; that is, the total amount of calculations for the control torque is proportional to the number of links. Thus the algorithm is computationally ecient.
5. INVERSE KINEMATICS
For a trajectory control of an end-eector, the time history of the joint variables must be determined so that the end-eector tracks the speci®ed trajectory (inverse kinematics). The dynamical equations presented in the previous section can be applied to compute the control forces and torques (inverse dynamics).
For a space platform based manipulator, its maneuver aects the platform's attitude. Thus it is not possible to determine the joint variables just from the geometrical relations between the manipulator and trajectory. Here a method of solution for the inverse kinematics of the space-based manipulator is developed in conjunction with the dynamical equations. The general equations of motion for the manipulator can be written as h F M F FF F; F_ t ;
24
where M denotes the inertia matrix; h, the contribution from centrifugal and Coriolis eects; and t , the control torque. The vector F is composed of the generalized coordinates and is de®ned in the same way as in eqn (10). Dividing the state vector into three parts: translation and rotation of the platform; joint variables of the manipulator; and the modal coordinates of the ¯exible links; and indicating the three parts by the subscripts S, M, and F, respectively, the equation can be rewritten as 2 32 3 2 3 2 3 S MS MSM MSF hS 0 F 4 MMS MM MMF 54 F M 54 hM 5 4 bM 5t M : F MFS MFM MF hF bFM F 25 Since the actuators to control the motion of the platform are not considered, bS=0; and the ®rst row gives the equation S MSM F M MSF F F hS 0: MS F
26
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Shinji Hokamoto et al.
This implies conservation of the total momentum of the system. On the other hand, the relation between the acceleration of the end-eector, PÈE, and the state variables can be described using the generalized Jacobian matrix J(F F) as J F _ FF_ : E J F FF P
27
Dividing the state variables in the same way as in eqn (25), the relation can be expressed as S JM F M JF F F Jh : E JS F P
28
Combining and arranging eqns (25) and (28), the relation between the acceleration of the end-eector and the joint acceleration of the manipulator can be derived as M F M J h F E J F F F; F_ : P
29
F) is a square matrix when the manipulaNote, JM(F tor does not have any redundant degree of freedom. The matrix is not singular in any posture because of the inherent advantage of the MDM. Thus the set of equations to solve for the inverse kinematics can be written as: ÿ1 F F; F_ g; M J F M F fPE ÿ Jh F
30
F I F F; F_ ; F FM F F M IFh F
31
M KSF F F KSh F S KSM F F F FF F; F
32
where M JM JS KSM JF JS KSF I ; J FM h Jh JS KSh JF JS KSF I ; J Fh
KSM
ÿMÿ1 S MSM ;
KSF ÿMÿ1 S MSF ; KSh ÿMÿ1 S MSh ; IFM ÿs sÿ1 1 MFM MFS KSM ÿ s 2 ; IFh ÿs sÿ1 1 hF MFS KSh ÿ s 3 ; s 1 MF MFS KSF ÿ bFM bÿ1 M MMF MMS KSF ; s 2 ÿbFM bÿ1 M MM MMS KSM ;
Fig. 4. Variation of kinetic and potential energy.
of the total energy of a MDM is checked in absence of control. The MDM has four ¯exible links: links 1 and 3 are free to slew; while links 2 and 4 are free to deploy. The mass, length and stiness of each link are m = 1.0 kg, l = 10 m, and EI = 50 Nm2, respectively; and the elastic deformation of the link is expressed by the ®rst and second modes (N = 2). At the beginning of the simulation, the condition of the manipulator is set as d1(0) = d3(0) = 5.0 m, angles of the rotational joint are 1808, and the ®rst and second time derivatives of these variables are zero. Figure 4 shows the variation of kinetic and potential energy, and the total energy is conserved during the typical disturbance of q41(0) = 1.0. 6.2. The end-eector trajectory control Figure 5 shows the block diagram for the resolved acceleration control, where P dE denotes the desired trajectory. To verify eectiveness of this control strategy, several dierent cases were simulated numerically. Here three typical sets of results are presented. Deformation of the links is expressed by the ®rst mode in the inverse kinematics, inverse dynamics, and direct dynamics. 6.2.1. Case 1. Consider a simple example of the manipulator with one ¯exible slewing link and one ¯exible deployable link. The manipulator is ®xed to
s 3 ÿbFM bÿ1 M hM MMS KSh :
6. NUMERICAL SIMULATION RESULTS
Planar performance of the MDM is illustrated here through some typical simulation results. 6.1. Energy conservation To verify the derived equations of motion and the associated computer program, the conservation
Fig. 5. Block diagram for the resolved acceleration control.
Space-based ¯exible robots
Fig. 6. The manipulator model and the commanded trajectory.
a large platform, for example the Canadarm attached to the Shuttle, and the translation of the manipulator is absent. Figure 6 shows the manipulator model and desired trajectory of the end-eector (dotted line). The mass, length, and stiness of the two links are m = 1.0 kg, l = 10 m, and EI = 3193 Nm2, respectively. The maneuver requires the end-eector to move 10 m in the y direction in 10 s, and the required velocity of the end-eector in the y direction is taken as a fourth order polynomial function in time. The solid lines in Figs 7±10 show the controlled response of the ¯exible manipulator, while the dashed±dotted lines present time histories of the desired velocity, acceleration, the desired deployed length, and slew angle. It is apparent that the controller is remarkably successful in following the speci®ed trajectory, even in the presence of ¯exibility. 6.2.2. Case 2. Consider the manipulator with four links in free-¯ying condition (i.e. g = 0). Figure 11 shows the manipulator model and its initial con®guration. The desired trajectory of the end-eector is speci®ed, as before, to be a straight line in the y direction. The maneuver requires the end-eector to move 0.5 m in 2 s, and the desired acceleration is de®ned by the sine function with the period of 2 s. The mass and stiness of the link 4 are m4=0.3172 kg and E4I4=0.006567 Nm2, respectively. The mass and stiness of the other links are expressed as follows: the mass ratios from link 1 to 4 are 100:20:2:1; and the ratios of the stiness are 1000:800:40:1. The length of all the links is the same, 1 m. The solid line in Figs 12 and 13 shows simulation results for the ¯exible manipulator model. Corresponding results for the rigid system are also included for comparison. Results clearly show small discrepancy in acceleration although the displacement time history correlates quite well. 6.2.3. Case 3. Consider the same manipulator model and desired trajectory as in Case 2, except for a change in speci®cations of the links. The
527
Fig. 7. Velocity of the end-eector.
Fig. 8. Acceleration of the end-eector.
Fig. 9. Time history of the deployed length.
Fig. 10. The history of the slew angle.
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Fig. 11. Manipulator model and the desired trajectory.
Fig. 14. Time history of the modal coordinates of the links.
Fig. 12. Comparison between the acceleration pro®les for the ¯exible and rigid systems.
Fig. 15. Time history of the deployed length of the link 2.
7. CONCLUDING REMARKS
Fig. 13. Comparison between the displacement pro®les for the ¯exible and rigd manipulators.
mass, length, and stiness of the four links are taken to be the same: m = 0.3172 kg, l = 1 m, and EI = 0.006567 Nm2. In this simulation, the deployed length of the link 2 exceeds the limitation of the deployment (0 < d2R1.0) when t = 0.45 s. Figure 14 shows time variation of the modal coordinates of the links, and Fig. 15 presents the deployed length time history for link 2. It is obvious that the ¯exible modes become unstable in this simulation. Note, the endeector can track the desirable acceleration until t = 0.45 s, although the assumption that the ¯exible deformation is small does not hold. In this simulation, the modal damping was purposely not included to emphasize the possibility of instability. Inherent modal damping or the damping supplied by the direct velocity feedback law [16] would stabilize the ¯exibility.
In this study, the equations of motion in the recursive form for the Mobile Deployable Manipulator have been derived. The equations are particularly suitable for the inverse dynamics, as the total computational eort for the control torque is proportional to the number of links. Furthermore, a numerical procedure for the inverse kinematics of the system is developed. The proposed control strategy is quite eective even in the presence of ¯exibility. However, due to the absence of damping, it does not guarantee the stability of the ¯exible modes when the commanded acceleration exceeds the critical value. This situation can be readily corrected by accounting for the structural damping and that introduced through the direct velocity feedback.
REFERENCES
1. Luh, J. Y. S., Walker, M. H. and Paul, R. P. C., IEEE Transactions on Automatic Control, 1980, AC25(3), 468±474. 2. Vafa, Z. and Dubowsky, S., in Proceedings of the IEEE International Conference on Robots and Automation, 1987, pp. 579±585. 3. Koningstein, R., Ullman, M. and Cannon, R. H. Jr, in NASA Conference Space Telerobotics, 1989, pp. 235±243. 4. Umetani, Y. and Yoshida, K., IEEE Transactions on Robotics and Automation, 1989, 5(3), 303±314. 5. Nenchev, D. N., The International Journal of Robotics Research, 1992, 11(6), 584±597.
Space-based ¯exible robots 6. Torres, M. A. and Dubowsky, S., Journal of Guidance, Control, and Dynamics, 1992, 15(4), 1010± 1017. 7. Koningstein, R. and Cannon, R. H., Jr., Journal of Guidance, Control, and Dynamics, 1995, 18(6), 1386± 1391. 8. Chen, Y. and Meirovitch, L., Journal of Guidance, Control, and Dynamics, 1995, 18(4), 756±766. 9. Fujii, H. A., Uchiyama, K. and Yoneoka, H., Journal of Guidance, Control, and Dynamics, 1996, 19(5), 985± 991. 10. Book, W. J., International Journal of Robotics Research, 1984, 3(3), 87±101. 11. Benati, M. and Morro, A., Transactions of the ASME Journal of Dynamic Systems, Measurement, and Control, 1994, 116, 81±88. 12. Tsuchiya, K. and Watanabe, S., Finite Elements in Analysis and Design, 1993, 14, 261±272. 13. Rodriguez, G., IEEE Transactions on Robotics and Automation, 1989, RA-5(4), 510±521. 14. Hokamoto, S., Modi, V. J. and Misra, A. K., AAS/ AIAA Astrodynamics Specialist Conference, AAS, 95± 322, also in Misra AK et al., eds. Advances in the Astronautical Sciences, eds. A. K. Misra et al. Univelt Inc. Publisher, San Diego, CA, pp. 339±357, 1995. 15. Meirovitch, L., Method of Analytical Dynamics, McGraw-Hill, New York, 1970, pp. 157±160. 16. Balas, M. J., Journal of Guidance and Control, 1979, 2(3), 252±253.
The vectors used in eqns (19)±(23) can be computed recursively as follows. Generalized velocity with respect to the inertial system ^ o
o T10
o T2nÿ1;0
d_ 1 d_ 2nÿ1
Acceleration of the joints: V_ ~ 10 V1 A10r0 ; V_ 1 o ~ 2i;2iÿ1 V2i A2i;2iÿ1 V_ 2iÿ1 ~rf ;2iÿ1 o 2iÿ1;0 V_ 2i o ~rf ;2iÿ1 o_ 2iÿ1;0 ex;2iÿ1 d2iÿ1 Z 2iÿ1 w 2iÿ1 ; ~ 2i1;2i V2i1 A2i1;2i V_ 2i ~r_ f ;2i o 2i;0 V_ 2i1 o ~rf ;2i o 2i;0 Z 2i w 2i :
Translational momentum of the links: P ~ i o i0 Vi Pi mi R
q_ T1
q_ T2n T
HF_ :
i 1; 2; ; 2n:
Time derivative of the translational momentum of the links: P_ ~ i o_ i0 V_ i P_ i mi R
i 1; 2; ; 2n:
Translational momentum of the set from the one link to the ^ end-eector: P ^ i1 ^ i Pi Ai;i1 P P
APPENDIX
_rT0
529
i 1; 2; ; 2n ÿ 1;
^ 2n P2n : P
Time derivative of the translational momentum of the set from the one link to the end-eector: PÃ ^ i1 P ^_ i1 i 1; 2; ; 2n ÿ 1; ^_ i P_ i Ai;i1 o ~ Ti1;i P P
Velocity of the joints: F o 10 ; o 10 o o 2i;0 A2i;2iÿ1 o o 2iÿ1;0 k 2iÿ1 w_ 02iÿ1 ;
o 2i;0 k 2iÿ1 w_ 02i o 2i1;2i : o 2i1;0 A2i1;2i o Acceleration of the joints: F o_ 10 o_ 10 ; ~ 2i;2iÿ1 o 2i;0 A2i;2iÿ1 o_ 2iÿ1;0 k 2iÿ1 w _o 2i;0 o 02iÿ1 ; 0 _o 2i1;0 o ~ 2i1;2i o 2i;0 A2i1;2i o_ 2i;0 k 2iÿ1 w 2i o_ 2i1;2i :
Velocity of the joints: V V1 A10 r_ 0 ; V2i A2i;2iÿ1 V2iÿ1 ~rf ;2iÿ1 o 2iÿ1;0 V2i1
ex;2iÿ1 d_ 2iÿ1 Z 2iÿ1 w_ 2iÿ1 ; A2i1;2i V2i ~rf ;2i o 2i;0 Z 2i w_ 2i :
^_ 2n P_ 2n : P
Angular momentum of the set from the one link to the endeector: QÃ ^i R ^ i1 ~ T Pi ~rT Ai;i1 P Q i fi
i 1; 2; ; 2n ÿ 1;
^ 2n R ~ T P2n : Q 2n Time derivative of the angular momentum of the set ^_ from the one link to the end-eector: Q ^_ i R ~ T P_ i ~r_ T Ai;i1 P ^_ i1 ^ i1 ~rT Ai;i1 o ^ i1 P ~ Ti1;i P Q fi i fi i 1; 2; ; 2n ÿ 1; ^_ 2n R ~ T P_ 2n : Q 2n
Generalize momentum of the links: L
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Shinji Hokamoto et al. h iT L LTr_ 0 LTo10 LTo2nÿ1;2nÿ2 Ld_ 1 Ld_ 2nÿ1 LTq_ 1 LTq_ 2n ; ^1; Lr_ 0 A01 P ^ 2iÿ1 A2iÿ1;2i Q ^ 2i Jf ;2iÿ1 o 2iÿ1;0 B2iÿ1 d_ 2iÿ1 ST q_ 2iÿ1 ST q_ 2i ; Lo2iÿ1;2iÿ2 Q 2iÿ1 2i ^ 2i BT o 2iÿ1;0 c2iÿ1 d_ 2iÿ1 DT q_ 2iÿ1 DT q_ 2i ; ^ 2i w k T A2iÿ1;2i Q Ld_ 2iÿ1 eTx;2iÿ1 w 02iÿ1 Z T2iÿ1 A2iÿ1;2i P 2iÿ1 2iÿ1 2i 2iÿ1 2iÿ1 00
T
^ 2i S2iÿ1 o 2iÿ1;0 D2iÿ1 d_ 2iÿ1 K 2iÿ1;2iÿ1 q_ 2iÿ1 K ^ 2i f T k T A2iÿ1;2i Q _ 2i ; Lq_ 2iÿ1 f T2iÿ1 Z T2iÿ1 A2iÿ1;2i P 2iÿ1 2iÿ1 2i;2iÿ1 q 0
^ 2i1 S2i o 2iÿ1;0 D2i d_ 2iÿ1 K 2i;2iÿ1 q_ 2iÿ1 K 2i;2i q_ 2i ; Lq_ 2i f T2i Z 2i A2i;2i1 P 2n;2nÿ1 q_ 2nÿ1 K 2n;2n q_ 2n : Lq_ 2n S2n o 2nÿ1;0 D2n d_ 2nÿ1 K
Time derivative of the generalized momentum of the links: L_ ^_ 1 ; ~ T10 Lr_ 0 A01 P L_ r_ 0 o ^_ 2iÿ1 A2iÿ1;2i o ^_ 2i J_ f ;2iÿ1 o 2iÿ1;0 Jf ;2iÿ1 o_ 2iÿ1;0 B_ 2iÿ1 d_ 2iÿ1 ^ 2i Q ~ T2i;2iÿ1 Q L_ o2iÿ1;2iÿ2 Q T B2iÿ1 d2iÿ1 ST2iÿ1 q_ 2iÿ1 ST2iÿ1 q2iÿ1 S_ 2i q_ 2i ST2i q2i ; 0 T ^ 2i eT ^ 2i P ^_ 2i ~ T2i;2iÿ1 P L_ d_ 2iÿ1 w_ 02iÿ1 Z T2iÿ1 A2iÿ1;2i P x;2iÿ1 w 2iÿ1 Z 2iÿ1 A2iÿ1;2i o 00 ^_ 2i ^ 2i w 0 0 k T A2iÿ1;2i o ^ 2i Q ~ T2i;2iÿ1 Q w_ 2iÿ1 k T2iÿ1 A2iÿ1;2i Q 2iÿ1 2iÿ1
T
T
_ _ 2iÿ1 DT2iÿ1 q2iÿ1 B_ 2iÿ1 o 2iÿ1;0 BT2iÿ1 o_ 2iÿ1;0 c_ 2iÿ1 d_ 2iÿ1 c2iÿ1 d2iÿ1 D 2iÿ1 q _ T q_ 2i DT q2i ; D 2i 2i 0T ^ 2i ^ f T Z T A2iÿ1;2i o ^ 2i P ^_ 2i d_ 2iÿ1 f 0 0 T k T A2iÿ1;2i Q ~ T2i;2iÿ1 P Z T2iÿ1 A2iÿ1;2i P L_ q_ 2iÿ1 d_ 2iÿ1 f 2iÿ1 2i 2iÿ1 2iÿ1 2iÿ1 2iÿ1 0 T ^_ 2i S_ 2iÿ1 o 2iÿ1;0 S2iÿ1 o_ 2iÿ1;0 D ^ 2i Q _ 2iÿ1 d_ 2iÿ1 ~ T2i;2iÿ1 Q f 2iÿ1 k T2iÿ1 A2iÿ1;2i o
2iÿ1;2iÿ1 q2iÿ1 K _ 2iÿ1;2iÿ1 q_ 2iÿ1 K _ _ 2i K 2i ; D2iÿ1 d2iÿ1 K 2i;2iÿ1 q 2i;2iÿ1 q T
T
^ 2i1 P _ 2i d_ 2iÿ1 ^_ 2i1 S_ 2i o 2iÿ1;0 S2i o_ 2iÿ1;0 D ~ T2i1;2i P L_ q_ 2i f T2i Z T2i A2i;2i1 o 2i;2iÿ1 q2iÿ1 K 2i;2i q2i ; _ 2i;2iÿ1 q_ 2iÿ1 K _ 2i;2i q_ 2i K D2i d2iÿ1 K _ 2n d_ 2nÿ1 D2n d2nÿ1 K 2n;2nÿ1 q2nÿ1 K 2n;2n q2n : _ 2n;2nÿ1 q_ 2nÿ1 K _ 2n;2n q_ 2n K L_ q_ 2n S_ 2n o 2nÿ1;0 S2n o_ 2nÿ1;0 D
Generalized momentum of the set from the one link to the end-eector: LÃ h T ^ L ^ L r_ 0
^T L o10
^T L o2nÿ1;2nÿ2
L^ d_ 1
L^ d_ 2nÿ1
^ r_ 0 Lr_ 0 ; L ^ o2i1;2i ; ^ o2iÿ1;2iÿ2 Lo2iÿ1;2iÿ2 A2iÿ1;2i1 L L ^ o2nÿ1;2nÿ2 Lo2nÿ1;2nÿ2 ; L 00 ^ o2i1;2i ; L^ d_ 2iÿ1 Ld_ 2iÿ1 w2iÿ1 k T2iÿ1 A2iÿ2i1 L
L^ d_ 2nÿ1 Ld_ 2nÿ1 ; ^ o2i1;2i ; ^ q_ 2iÿ1 Lq_ 2iÿ1 f 0 T k T A2iÿ1;2i1 L L 2iÿ1 2iÿ1 ^ q_ 2i Lq_ 2i f 0 T k T A2i;2i1 L ^ o2i1;2i ; L 2i 2i ^ q_ 2nÿ1 Lq_ 2nÿ1 ; L ^ q_ Lq_ : L 2n 2n
^T L q_ 1
^T L q_ 2n
iT
;
Space-based ¯exible robots ^_ Time derivative of the generalized momentum of the set from the one link to the end-eector: L ^_ r_ L_ r_ ; L 0 0 ^_ o2i1;2i ; ^ o2i1;2i A2iÿ1;2i1 L ^_ o2iÿ1;2iÿ2 L_ o2iÿ1;2iÿ2 A2iÿ1;2i o ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L L ^_ o2nÿ1;2nÿ2 L_ o2nÿ1;2nÿ2 ; L 00 ^ o2i1;2i L_^ d_ 2iÿ1 L_ d_ 2iÿ1 w_ 2iÿ1 k T2iÿ1 A2iÿ1;2i1 L 00 ^_ o2i1;2i g; ^ o2i1;2i A2iÿ1;2i1 L ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L w2iÿ1 L_ q_ 2iÿ1 fA2iÿ1;2i o
L_^ d_ 2nÿ1 L_ d_ 2nÿ1 ; ^_ q_ 2iÿ1 L_ q_ 2iÿ1 d_ 2iÿ1 f 0 0 T k T A2iÿ1;2i1 L ^ o2i1;2i f 0 T k T fA2iÿ1;2i o ^ o2i1;2i ~ T2i;2iÿ1 o ~ T2i1;2i A2i;2i1 L L 2iÿ1 2iÿ1 2iÿ1 2iÿ1
^_ q_ 2i L
^_ o2i1;2i g; A2iÿ1;2i1 L 0 ^_ o2i1;2i ; ^ o2i1;2i L ~ T2i1;2i L L_ q_ 2i f 2iT k T2i A2i;2i1 o
^_ q_ 2nÿ1 L_ q_ 2nÿ1 ; L ^_ q_ L_ q_ : L 2n 2n
531