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Dexterous mechanisms for robot locomotion
PII:
Mech. Mach. Theory Vol. 33, No. 8, pp. 1153±1165, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00115-8
DEXTEROUS MECHANISMS FOR ROBOT LOCOMOTION K. N. UMESH Mechanical Department, P.E.S. College of Engineering, Mandya, 571 401, India AbstractÐThe analysis and design of isotropic legs for robot locomotion is the subject of this article. A study of existing closed loop structures is done by substituting pneumatic/hydraulic actuators by electrical actuators wherever necessary. General conditions relating gear ratios (lead ratios) and link parameters for isotropy of foot are derived. These expressions could be used for synthesizing dexterous mechanisms and identifying their dexterous postures. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
In robot locomotion it is desirable to adopt the leg postures which are most suitable for landing and setting o from rest. The best positions would be those from which one could move in any direction with the least eort. There are some positions in the workspace of a manipulator, wherein the end eector can move in all directions with equal ease [1]. These positions are known as isotropic points [2]. The postures of the mechanism corresponding to these positions may be known as Isotropic postures where the end eector is dexterous. There are dierent approaches for evaluating the dexterity of a given posture [3±5]. However, the isotropic Jacobian (whose condition number is unity) is used as an index of the most dexterous posture. There have been few studies on isotropy of open loop manipulators [2, 6]. This study of closed loop mechanisms which are in vogue for walking machines focuses on the dexterous postures. We know that legged robots are designed for ground too rough even for tracked vehicles. The task of design and control of walking machines is quite challenging and, hence, gaining more interest. Ohio State University has been a center of this kind of research for quite many years. Waldron and his associates have been involved in this research. There are several types of walking machines of which only few most popular ones are studied here. The well known types of legs are: (i) Waldron Leg used in ASV (Automatic Suspension Vehicle) of Oklahama State University (Fig. 1) [7±10]; (ii) Odetics Leg is a variable geometry robot used for remote handling (Fig. 2) [11, 12]; and (iii) Sirur Leg designed and developed at Indian Institute of Technology (IIT), Bombay, India (Fig. 3) [13]. A study of the linkages used for walking machine legs indicates that there are basically two types of mechanisms [11, 12]: (i) a 1DOF mechanism mainly a 4-bar linkage (Fig. 4), and (ii) a 2DOF mechanism mainly a 5-bar linkage (Fig. 5). A 4-bar linkage is primarily used to generate a secondary input to a sequential 5-bar linkage. In a majority of cases, the 4-bar linkage used for this purpose is an inverted slider crank mechanism. They are, in general, driven by a hydraulic or pneumatic actuator. Another type of 4-bar linkage, which recently found its use in the leg mechanism as a primary driving mechanism, is a Scott±Russel linkage (Fig. 4b). A 5-bar linkage actually guides the foot of the walking machine. It normally derives one input through a 4-bar linkage, which is mentioned earlier, and draws another input (primary input) directly from an independent actuator. The conversion of blended inputs is derived as output motion of the tip of leg or foot. The most commonly used linkage is a special 5-bar linkage which is a parallel bar mechanism with its ®xed link being of zero length as shown in Fig. 5(a±c). It is a pantograph mechanism. In the case of the pantograph application one of the 1153
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K. N. Umesh
Fig. 1. Waldron leg mechanism.
inputs eliminates the other [9, 14]. Each time when one input is ineective the mechanism behaves as a 1DOF system, i.e. the pantograph mechanism functions as a 4-bar linkage with either O or Q as the tracer point and P as the end point. ISOTROPY OF LEG
It is known that end eector of a mechanism will be at isotropic position when its Jacobian is isotropic. The isotropic positions of the end eector (in this case foot) could be visualised in the end eector velocity domain very easily since the velocities are Jacobian dependent at the given position. It is established that a planar 2DOF mechanism under consideration will be isotropic when the velocities of the point of interest has the following properties [15, 16]. If vp1(vp2) is the end point velocity at the isotropic position when actuator-1 (actuator-2) is moving with unit angular velocity and actuator-2 (actuator-1) is locked, then: (i) the magnitudes of vp1 and vp2 are equal, i.e. vp1=vp2, and
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Fig. 2. Odetics leg mechanism.
(ii) the velocities vp1 and vp2 are orthogonal to each other, i.e. vp1_vp2. In order to examine the isotropy of a leg we shall ®rst consider a schematic and then the original mechanism. For this purpose we lock one of the actuators alternatively and obtain the respective foot velocities vp2 and vp1 considering the resulting 1DOF mechanism by method of instant centers. Now we shall study the isotropy of the important mechanisms used for robot locomotion.
WALDRON LEG
This leg is named after Waldron, the leader of the group of researchers at Ohio State University and is as shown in Fig. 1. It is hydraulically actuated 2DOF system (n = 7, j = 8, F = 2). The inputs are applied at the hinges O and Q of the sliders 2 and 7 . One of the interesting feature of the leg is lines of action of the two sliders are mutually orthogonal and the x and y motions of the foot are decoupled. Hence, it is possible to obtain the orthogonal velocities at all con®gurations of the mechanism. The necessary condition for isotropy (i.e. condition of magnitudes vp1 and vp2 being equal) of the foot is derived here. When actuator-1 is moving and actuator-2 is locked Point Q is ®xed and the linkage QABC-OP behaves as a pantograph with O as the tracer point and P as the end point. Since O, Q and P are collinear we can obtain the scale k' of the pantograph as follows. QP OP ÿ OQ OP ÿ1 QO OQ OQ i:e: k0 k ÿ 1;
1
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K. N. Umesh
Fig. 3. Schematic of Sirur leg mechanism (both actuators electricÐnot shown).
where k = (OP/OQ) = (OB/OA), the scale of the pantograph when O is ®xed and Q becomes tracer point. From the de®nition of pantograph we can write the magnitude of foot velocity vp1 as vp1 k ÿ 1vo :
2
When actuator-1 is locked and actuator-2 is moving Point O is ®xed and the linkage OABC-QP behaves as a pantograph with Q as the tracer point and P as the end point. We can obtain the scale of the pantograph in this case as OP OB : 3 k OQ OA We know that the velocity of the end point P is merely a scaled output of velocity of point Q, i:e: vp2 kvq ;
4
where vp2 is the magnitude of foot velocity when actuator-2 is moving and actuator-1 is locked. Since vo is perpendicular to vq, the orthogonality condition is satis®ed, i.e. vp1_vp2( Fig. 1) and now we need for isotropy,
Dextrous mechanisms for robot locomotion
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Fig. 4. Primary linkage of walking machines (dierent types of 4-bar linkages).
v p1 v p2 ; i:e: kvq k ÿ 1vo vo k or : vq kÿ1
5
From the above equation it is obvious that equality of foot velocities vp1 and vp2 can be achieved only when the input velocities vo and vq are applied in the ratio given by Equation (5). Further, if the prismatic pairs are replaced by screw pairs one can compare the performance characteristics with other types of leg mechanisms which are driven by electric actuators in general. For such mechanisms the above expression can be written as follows. When actuator-1 is locked (y_ 1=0) and actuator-2 is moving we can write, L2 _ L2 y2 for y_ 2 1: vq 2p 2p When actuator-1 is moving and actuator-2 is locked (y2=0) we can write, L1 _ L1 y1 for y_ 1 1: vo 2p 2p Substituting for vo and vq in Equation (5) we get, vo L1 k : vq L2 kÿ1
6
It is obvious from the above expression that from a careful selection of leads of the screws, it is possible to obtain isotropic properties at all con®gurations of the Waldron leg mechanism.
Fig. 5. Secondary linkage of walking machines (dierent types of pantograph mechanisms).
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K. N. Umesh
ODETICS LEG
One of the most compact versatile remote controlled hexapod is Odetics walking machine. It consists of a leg mechanism shown in Fig. 2 and the planar kinematic equivalent with the screw pair replaced by a sliding pair is shown in Fig. 6. We consider this mechanism for further discussion of Odetics leg mechanism. The mechanism is a 2DOF system (n = 11, j = 11, F = 2) and consists of a modi®ed pantograph and an inverted slider crank mechanism. It should be noted that the sliders F and G receive separate inputs from individual electric actuators. A close examination of the mechanism [Fig. 7a(i) and b] indicates that the leg will be isotropic only when the mechanism is in a con®guration wherein: (i) OABC is in rectangular con®guration keeping vp2 along AB and perpendicular to BC (or OA) when y_ 1=0 and y_ 2=1. (ii) FQ must be parallel to PA such that respective path normals of points P and Q lie along FQ and AP, leading to the parallel paths when y_ 1=1 and y_ 2=0. The above conditions imply that slider oset of F must be equal to CQ. When actuator-1 is moving and actuator-2 is locked Link 9 (slider F) is ®xed and the linkage behaves as well constrained 1DOF system (n = 8, j = 10, F = 1). Point Q is constrained to move on circular path centered at F with a radius of FQ. The foot P copies the motion of point Q and, hence, moves on a path parallel to that of point P since OABC-QP functions as a pantograph [Fig. 7a(i)]. In this con®guration A (or 61) will be instant center of link 6 (ABP) w.r.t. link 1 and we can write, AP vb : v p1 AB Link FQ is pivoted at F and hence, Q can have motion perpendicular to FQ only. P being the end point, follows the motion of tracer point Q, i.e. vp1 is parallel to vq. The instant center of link 6 w.r.t. link 1 (61) lies at A, hence the link 6 has absolute rotation about A at the instant. Thus, we have vb parallel to vp1. The rectangular con®guration of the linkage leads to vb parallel to vc. vb=vc since AB = OC. The link 7 translates and, hence, we have vq=vb=vc. Thus we have velocities vb, vc, vq and vp1 parallel to each other.
Fig. 6. Schematic of Odetics leg mechanism.
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Points C and D lie on the same link 3 (CODÐa bell crank lever), which rotates about O. So we can write, OC vd vb ; since vc vb : vc OD Also we know that instant center R (or 13) of link 3 (DG) w.r.t. link 1 lies on the intersection of DO and a perpendicular to link 3 drawn through E. Then we can write, RD vg ; vd RG 3 where vg3 is magnitude of absolute velocity (vg3) of G as a point on link 3. On backward substitution of vb, vc and vd and simplifying we get, AP RD l6 RD v p1 vg vg l41 RG 3 OD RG 3
7
where l41 is length of the arm OD of bell crank lever 4. The absolute velocity of G as a point on link 3 could be obtained as a sum of two vectors as vg3 vg2 4 vg32 ;
8
where vg2 is absolute velocity G as a point on link 2 and vg32 is sliding velocity of point G on link 3 relative to a coincident point on link 2 which acts along EG [Fig. 7a(ii)]. In the actual linkage a screw pair driven by an electric actuator replaces the prismatic pair at G. Hence, the magnitude of the sliding velocity vg32 can be obtained as a result of actuator-1 velocity y_ 1 as L1 _ L1 y1 9 for y_ 1 1; _l2 say: vg32 2p 2p L1 is the lead of screw of actuator-1 that forms link 3 and l2 (=EG) is the magnitude of displacement of point G from the ®xed pivot E [Fig. 7a(iii)]. The magnitude of absolute velocity vg2 is given by vg2=l2y_ 2. It must be noted that here y_ 2 indicates the angular velocity of link 2, but the speed of actuator-2 (which is equal to zero in this case). An expression for y_ 2 could be obtained from the loop closure relation given by equation (A4) as y_ 2
_l2 l1 S y1 ÿ y2 f12 a=bg
10
where a = C(y1 ÿ y2) and b = {(l4/l1)2 ÿ S2(y1 ÿ y2)}1/2, and where Cyi=cos yi and Syi=sin yi. The magnitude of absolute velocity vector vg3 could be obtained as vg3 fv2g2 v2g32 g1=2 ;
11
since the vectors vg2 and vg32 are orthogonal to each other [Fig. 7a(ii)]. Alternatively, one could also obtain the magnitude of velocity vector vg3 from the velocity analysis as given in equation (A7) of the Appendix. vg3 fv2gx v2gy g1=2 ; where vgx= ÿ l2 sin y2+cos y2_l 2 and vgy= + l2 cos y2+sin y2_l 2. When actuator-1 is locked and actuator-2 is moving Link 4 (or OC) is ®xed and hence links 1-2(3)-4 behaves instantaneously as a structure (not shown). The linkage OABC behaves as a parallel 4-bar linkage (n = 6, j = 7, F = 1) with CB as the input link remotely driven by the actuator-2 at F (Fig. 7b). In this case the linkage AB translates since it forms the coupler of a parallel 4-bar linkage OABC and we have the velocities
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Fig. 7a. Odetics leg (actuator-1 moving and actuator-2 locked; link 4 ®xed).
vb=va=vp (or vp2). And the instant center of link 6 (BC) w.r.t. link 1 (i.e. 57) lies at C. Hence, we can write, CB 12 v q v p2 : vb CQ At the isotropic con®guration line of action of the screw driving slider 9 is collinear with FQ and angle FQC is a right angle. Therefore, the link FQ translates and, hence the end point velocities of link FQ are equal, i.e. vq=vf. We know that in the actual linkage the input is actually derived from an electric actuator for the slider 9. So we have magnitude of vf given by L2 _ y2 v q ; vf 2p where L2 is the lead of screw of actuator-2 that drives S2. Substituting for vq in Equation (12) we get, CB L2 _ L2 y2 k 13 for y_ 2 1; v p2 2p CQ 2p where k is scale of the pantograph OABC-QP when Q is the tracer point [Fig. 7a(i)]. It could be proved that scale of the pantograph OABC-QP,
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Fig. 7b. Odetics leg (actuator-1 locked and actuator-2 moving; link 9 ®xed).
k
OP CB l6 l42 OQ CQ
where l42 is length of the arm OC of bell crank lever 4. Substituting the value of scale (k) in Equation (13) we have, l6 L2 : vp2 l42 2p
14
For isotropy we can equate (7) and (14) and obtain an expression for lead of the actuator-2 as l42 RD vg 15 L2 2p l41 RG 3 assuming the lead of the actuator-1 speci®ed or known. However, in special case when the both arms of the bell crank lever 4 are equal in length the above expression gets further simpli®ed to get RD 16 vg : L2 2p RG 3
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K. N. Umesh
SIRUR LEG
This leg was developed at Indian Institute of Technology, Bombay (India) by Issac and his group and is named after the chief contributor to the design of the leg. It is driven by two independent electric actuators whose axes are along Y-axis (Fig. 3). It is a 2DOF system (n = 9, j = 11, F = 2). The inputs are applied at the hinges F and G of the sliders 4 and 9. The point to be noted here is that the horizontal motion of the foot is derived from a modi®ed Scott± Russel mechanism (Fig. 4b) and, hence, the mechanism has become more compact. An instantaneous kinematic equivalent is shown in Fig. 3. One of the interesting feature of the leg is lines of action of the two screws are coplanar and the x and y motions of the foot are decoupled as in the case of Waldron leg mechanism. Hence, it is possible to obtain orthogonal velocities at all con®gurations of the mechanism. The necessary condition for isotropy of the foot is derived here. When actuator-1 is moving and actuator-2 is locked Point G is ®xed and the linkage BCDE-FP behaves as a pantograph with B as the pivot. Similar to the earlier case, the foot point P lies on the extension of line joining F and B. The slider F is driven by the actuator-1 in the vertical direction and hence the motion of P is also vertical as shown in full line in Fig. 3. When actuator-2 is locked the Scott±Russel mechanism behaves as a structure and the pivot at G becomes immobile. In this case the linkage behavior is identical to the ®rst case of Waldron mechanism wherein actuator-2 is locked and actuator-1 is moving. Similar to the case of Waldron mechanism discussed earlier, a relation between the velocities of points P and F could be obtained as vp1 k ÿ 1vf ;
since
k0 k ÿ 1
17
where k and k' are scales of the pantographs derivable from Sirur leg mechanism. The respective scales of pantographs could be obtained as k = (FP/FB) when B acts as ®xed pivot and k' = (BP/BF) when F acts as ®xed pivot. The slider F is actuated by an electric motor with a screw of lead L1 the expression for velocity of point F is given by L1 _ L1 y1 ; for y_ 1 1: vf 2p 2p Substituting for vf in Equation (17) we get, vp1 k ÿ 1
L1 : 2p
18
When actuator-1 is locked and actuator-2 is moving Point F is ®xed. The foot point P lies on the extension of line joining F and B. As the point B is driven by the actuator-2 and the linkage BEDC-FP behaves as a pantograph. The directions of motion of the tracer point and end point are shown in dashed lines in Fig. 3. Now we can analyze the motion of the driving point B as an output of a Scott±Russel mechanism as follows. Scott±Russel mechanism It is used for deriving the horizontal motion from the vertical motion of actuator and is as shown in Fig. 8. The line of action of G (vertical) is perpendicular to that of line of action of point B. At any given instant the instant center I (or 13) of link 3 w.r.t. link 1 will lie at the intersection of perpendiculars drawn to the paths of G and B, i.e. IB remains perpendicular to IG and the right-angled triangles OBG and IGB so formed are similar. Hence, we can write:
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Fig. 8. Sirur legÐScott±Russel mechanism.
IG OB tan f IB OG
19
where f = OGÃB, is the angle made by link 3 with line of action of slider G. The velocities of the points B and P at the instant could be related by vp2 kvb :
20
Since I is the instant center of link 3 w.r.t. ground (Fig. 8) we can write, IG vb vg vg tan f: IB Substituting for vb in Equation (20) we get, vp2 kvg tan f:
21
In the actual mechanism slider G is driven by an electric actuator-2 (whose lead is L2) and actuator-1 is locked (y_ 1=0). Therefore, we can rewrite the above Equation (21) as, L2 L2 tan fy_ 2 k tan f; for y_ 2 1: 22 vp 2 k 2p 2p For isotropy we can equate (18) and (22) and obtain the expression L1 k tan f; L2 kÿ1
23
for lead ratios of the actuators. It can be observed from the above Equation (23) that a proper choice of leads for the actuator screws will enable one to obtain isotropic property at a speci®ed con®guration given by the angle f. Further the above equation can also be used to identify the isotropic con®guration of a given Sirur type linkage as follows. For a given set of lead screws and a chosen pantograph L1, L2 and k are constants. Therefore, an isotropic con®guration will exist when the link 3 (GB) is at an inclination of f to the line of action of slider G. The value of the con®guration angle is given by,
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K. N. Umesh
f tanÿ1
L1 L2
kÿ1 : k
24
Hence, there can be only one con®guration at which isotropy of foot can be obtained in the case of Sirur type of leg mechanism. CONCLUSIONS
This paper presents some investigations on the dexterous postures of the important mechanisms used for robot locomotion. The study reveals that leg of Waldron type exhibits isotropic property at all the con®gurations, when it is driven by an electric actuator with appropriate lead screws. Where as the leg developed at IIT Bombay (India) has only one isotropic con®guration and is dependent on the scales of the pantographs con®gured during motion. The Odetics leg indicates the possibility of isotropy at the rectangular con®guration wherein the linkage has to meet certain conditions based on link parameters. The individual expressions derived here helps in obtaining quick solutions for the synthesis of dexterous legs for walking machines. This study also presents a method to identify the best postures in order to set o and for landing as well. At these postures (or isotropic positions) as the error propagation is minimum, the transmission eciency will be maximum and hence the operation will be quite energy ecient. The adoption of isotropic postures in the robot locomotion has certain in¯uence on the gait. This fact opens up further topics of study such as the gaits with isotropic positions. AcknowledgementsÐThe author wishes to sincerely acknowledge the highly inspiring discussions he had with his research supervisor Professor C. Amarnath and his colleagues in the Robotics laboratory of Indian Institute of Technology, Bombay during this research work.
REFERENCES 1. Ghosal, A. and Roth, B., Trans. ASME J. Mechanisms, Transmissions and Automation in Design, 1987, 109, 107± 115. 2. Salisbury, J. K. and Craig, J. J., Int. J. Robo. Res., 1982, 1(1), 4±7. 3. Chiu, S. L., Int. J. Robo. Res., 1988, 109(5), 13±21. 4. Gosselin, C. M., IEEE Int. Conf. on Robotics and Auto, 1990, VI, 650±655. 5. Lee, M. Y., Erdman, A. G. and Gutman, Y., Mechanisms and Machine Theory, 1993, 28(5), 651±670. 6. Angles, J., Int. J. Robo. Res., 1992, 11(3), 196±201. 7. Waldron, K. J. and Kinzel, G. L., The relationship between actuator geometry and mechanical eciency in robots. 4th Symposium on Theory and Practices of Robots and Manipulators CISMÐIFTOMM, 1981, pp. 305±316. 8. Song, S. M., Waldron, K. J. and Kinzel, G. L., J. Mechanisms and Machine Theory, 1985, 20(6), 585±596. 9. Song, S. M. and Waldron, K. J., J. Mech. Transm. and Auto. in Design, 1987, 109, 21±28. 10. Vohnout, V. J., Alexander, K. S. and Kinzel, G. L., Mechanical design of prototype leg for a walking machine, in Proc of the 8th Oklahoma State University Applied Mechanisms Conf., St Louis, Missouri, 1983. 11. Todd, D. J., Walking MachinesÐAn Introduction to Legged Robots, Kogan Page Ltd, London, 1985. 12. Marsh, P., Robots, Salamander Books Ltd, U.K., 1985. 13. Sirur, S. V., Mechanical design of hexapod. B.Tech. Thesis, IIT, Bombay, India, 1991. 14. Yang, D. C. H. and Lin, Y. Y., J. Mechanisms and Machine Theory, 1985, 20(1), 115±122. 15. Umesh, K. N., Studies on instantaneous kinematic properties of planar two degrees-of-freedom robotic linkages. Ph.D. Thesis, Mechanical Engg Dept, IIT, Bombay, 1994. 16. Umesh, K. N. and Amarnath, C., Studies on instantaneous kinematic properties of planar two degrees-of-freedom linkages, Communicated to Trans ASME Mech. Design, 1994.
APPENDIX Expressions For Velocities Vg3 And vg32 Expressions for the velocity of point G on link 2 relative to link 3, vg32 could be obtained from loop closure relation of the inverted slider crank mechanism shown in Fig. 7a(iii). The loop closure equation could be obtained as: l 1 l2 l3 l 4 iy1
or l1 e
l2 eiy2 l3 eiy3 l4 eiy4 ;
A1
where yi indicates the position of ith link measured from the positive X0 axis. Separating real and imaginary parts and rearranging we get the following equations, l4 Cy4 l1 Cy1 ÿ l2 Cy2 ÿ l3 Cy3
A2
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l4 Sy4 l1 Sy1 l2 Sy2 ÿ l3 Sy3
A3
where Cyi=cos yi and Syi=sin yi. Squaring and adding both sides and rearranging the above equations we can write, u1 l22 2u2 l2 u3 0;
A4
where, u1=1, u2={l3ÿl1C(y1ÿy2)} and u3=l21+l23ÿl24ÿ2l1l3C(y1ÿy3). The solution of the above quadratic equation in l2 could be obtained in its simplest form after substituting y2 for y3 as l2 w1 2w2 ;
A5
w2={l24ÿl21S2(y1ÿy2)}1/2.
where, w1= ÿ {l3ÿl1C(y1ÿy2)} and The two solutions given by equation (A5) correspond to two dierent assemblies of the mechanism. Dierentiating further w.r.t. time we could obtain an expression for _l 2 in terms of y_ 1, y_ 2 and y_ 3. In the mechanism y1 indicates position of ®xed link 1 and hence the terms with y_ 1 could be eliminated. Orientations of links 2 and 3 are always aligned and hence y3=y2 for all positions of the mechanism, Fig. 7a. Therefore, we have y_ 3 equal to y_ 2 at every instant. We could use these facts to obtain an expression of _l 2 in its simplest form as C y1 ÿ y2 _l2 l1 12 A6 S y1 ÿ y2 y_ 2 : f l4 =l1 2 ÿ S2 y1 ÿ y2 g1=2 The above equation (A6) gives magnitude of the velocity vector vg32 and is a function of y_ 2, the angular velocity of link 2 of the equivalent planar linkage shown in Fig. 7a. We shall now obtain an expression for the absolute velocity of point G on link 3, vg3 as follows. The displacement of point G could be obtained as [Fig. 7a(iii)], pg fpgx ; pgy gT
A7
where pgx=l2Cy2 and pgy=l2Sy2. Dierentiating w.r.t. time we get absolute velocity of point G as point on link 3 as vg fvgx ; vgy gT vg3
A8
where vgx= ÿ l2Sy2y_ 2+Cy2_l 2 and vgy= + l2Cy2y_ 2+Sy2_l 2. The above equation could be rewritten as _ vg3 JY where the manipulator Jacobian is given by
J
ÿl2 Sy2 l2 Cy2
A9
Cy2 ; Sy2
_ and Y={ y_ 2 _l 2}T is the input velocity vector. It is known that _l 2 is a function of y_ 2 and can be obtained from equation (A6).
ZusammenfassungÐDie analyses und fabricat die isotropic fuÈss fuÈr robot locomotion ist die haupt thema. Die untersuchungen die existierte zu geschlossene lupe structure ist getan durch substituieren pneumatic/hydraulic actuators durch elektrische actuators jedoch moÈglich. Die generale bedingung uber die gear ratios (Lead ratios) und die link parameters fuÈr isotropy die fuÈss sind diskutiert. Diese parameters benutzen wir fuÈr die synthesis dexterous mechanisms und die identi®kation fuÈr dexterous postures. # 1998 Elsevier Science Ltd. All rights reserved
Mech. Mach. Theory Vol. 33, No. 8, pp. 1153±1165, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00115-8
DEXTEROUS MECHANISMS FOR ROBOT LOCOMOTION K. N. UMESH Mechanical Department, P.E.S. College of Engineering, Mandya, 571 401, India AbstractÐThe analysis and design of isotropic legs for robot locomotion is the subject of this article. A study of existing closed loop structures is done by substituting pneumatic/hydraulic actuators by electrical actuators wherever necessary. General conditions relating gear ratios (lead ratios) and link parameters for isotropy of foot are derived. These expressions could be used for synthesizing dexterous mechanisms and identifying their dexterous postures. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
In robot locomotion it is desirable to adopt the leg postures which are most suitable for landing and setting o from rest. The best positions would be those from which one could move in any direction with the least eort. There are some positions in the workspace of a manipulator, wherein the end eector can move in all directions with equal ease [1]. These positions are known as isotropic points [2]. The postures of the mechanism corresponding to these positions may be known as Isotropic postures where the end eector is dexterous. There are dierent approaches for evaluating the dexterity of a given posture [3±5]. However, the isotropic Jacobian (whose condition number is unity) is used as an index of the most dexterous posture. There have been few studies on isotropy of open loop manipulators [2, 6]. This study of closed loop mechanisms which are in vogue for walking machines focuses on the dexterous postures. We know that legged robots are designed for ground too rough even for tracked vehicles. The task of design and control of walking machines is quite challenging and, hence, gaining more interest. Ohio State University has been a center of this kind of research for quite many years. Waldron and his associates have been involved in this research. There are several types of walking machines of which only few most popular ones are studied here. The well known types of legs are: (i) Waldron Leg used in ASV (Automatic Suspension Vehicle) of Oklahama State University (Fig. 1) [7±10]; (ii) Odetics Leg is a variable geometry robot used for remote handling (Fig. 2) [11, 12]; and (iii) Sirur Leg designed and developed at Indian Institute of Technology (IIT), Bombay, India (Fig. 3) [13]. A study of the linkages used for walking machine legs indicates that there are basically two types of mechanisms [11, 12]: (i) a 1DOF mechanism mainly a 4-bar linkage (Fig. 4), and (ii) a 2DOF mechanism mainly a 5-bar linkage (Fig. 5). A 4-bar linkage is primarily used to generate a secondary input to a sequential 5-bar linkage. In a majority of cases, the 4-bar linkage used for this purpose is an inverted slider crank mechanism. They are, in general, driven by a hydraulic or pneumatic actuator. Another type of 4-bar linkage, which recently found its use in the leg mechanism as a primary driving mechanism, is a Scott±Russel linkage (Fig. 4b). A 5-bar linkage actually guides the foot of the walking machine. It normally derives one input through a 4-bar linkage, which is mentioned earlier, and draws another input (primary input) directly from an independent actuator. The conversion of blended inputs is derived as output motion of the tip of leg or foot. The most commonly used linkage is a special 5-bar linkage which is a parallel bar mechanism with its ®xed link being of zero length as shown in Fig. 5(a±c). It is a pantograph mechanism. In the case of the pantograph application one of the 1153
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K. N. Umesh
Fig. 1. Waldron leg mechanism.
inputs eliminates the other [9, 14]. Each time when one input is ineective the mechanism behaves as a 1DOF system, i.e. the pantograph mechanism functions as a 4-bar linkage with either O or Q as the tracer point and P as the end point. ISOTROPY OF LEG
It is known that end eector of a mechanism will be at isotropic position when its Jacobian is isotropic. The isotropic positions of the end eector (in this case foot) could be visualised in the end eector velocity domain very easily since the velocities are Jacobian dependent at the given position. It is established that a planar 2DOF mechanism under consideration will be isotropic when the velocities of the point of interest has the following properties [15, 16]. If vp1(vp2) is the end point velocity at the isotropic position when actuator-1 (actuator-2) is moving with unit angular velocity and actuator-2 (actuator-1) is locked, then: (i) the magnitudes of vp1 and vp2 are equal, i.e. vp1=vp2, and
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Fig. 2. Odetics leg mechanism.
(ii) the velocities vp1 and vp2 are orthogonal to each other, i.e. vp1_vp2. In order to examine the isotropy of a leg we shall ®rst consider a schematic and then the original mechanism. For this purpose we lock one of the actuators alternatively and obtain the respective foot velocities vp2 and vp1 considering the resulting 1DOF mechanism by method of instant centers. Now we shall study the isotropy of the important mechanisms used for robot locomotion.
WALDRON LEG
This leg is named after Waldron, the leader of the group of researchers at Ohio State University and is as shown in Fig. 1. It is hydraulically actuated 2DOF system (n = 7, j = 8, F = 2). The inputs are applied at the hinges O and Q of the sliders 2 and 7 . One of the interesting feature of the leg is lines of action of the two sliders are mutually orthogonal and the x and y motions of the foot are decoupled. Hence, it is possible to obtain the orthogonal velocities at all con®gurations of the mechanism. The necessary condition for isotropy (i.e. condition of magnitudes vp1 and vp2 being equal) of the foot is derived here. When actuator-1 is moving and actuator-2 is locked Point Q is ®xed and the linkage QABC-OP behaves as a pantograph with O as the tracer point and P as the end point. Since O, Q and P are collinear we can obtain the scale k' of the pantograph as follows. QP OP ÿ OQ OP ÿ1 QO OQ OQ i:e: k0 k ÿ 1;
1
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K. N. Umesh
Fig. 3. Schematic of Sirur leg mechanism (both actuators electricÐnot shown).
where k = (OP/OQ) = (OB/OA), the scale of the pantograph when O is ®xed and Q becomes tracer point. From the de®nition of pantograph we can write the magnitude of foot velocity vp1 as vp1 k ÿ 1vo :
2
When actuator-1 is locked and actuator-2 is moving Point O is ®xed and the linkage OABC-QP behaves as a pantograph with Q as the tracer point and P as the end point. We can obtain the scale of the pantograph in this case as OP OB : 3 k OQ OA We know that the velocity of the end point P is merely a scaled output of velocity of point Q, i:e: vp2 kvq ;
4
where vp2 is the magnitude of foot velocity when actuator-2 is moving and actuator-1 is locked. Since vo is perpendicular to vq, the orthogonality condition is satis®ed, i.e. vp1_vp2( Fig. 1) and now we need for isotropy,
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Fig. 4. Primary linkage of walking machines (dierent types of 4-bar linkages).
v p1 v p2 ; i:e: kvq k ÿ 1vo vo k or : vq kÿ1
5
From the above equation it is obvious that equality of foot velocities vp1 and vp2 can be achieved only when the input velocities vo and vq are applied in the ratio given by Equation (5). Further, if the prismatic pairs are replaced by screw pairs one can compare the performance characteristics with other types of leg mechanisms which are driven by electric actuators in general. For such mechanisms the above expression can be written as follows. When actuator-1 is locked (y_ 1=0) and actuator-2 is moving we can write, L2 _ L2 y2 for y_ 2 1: vq 2p 2p When actuator-1 is moving and actuator-2 is locked (y2=0) we can write, L1 _ L1 y1 for y_ 1 1: vo 2p 2p Substituting for vo and vq in Equation (5) we get, vo L1 k : vq L2 kÿ1
6
It is obvious from the above expression that from a careful selection of leads of the screws, it is possible to obtain isotropic properties at all con®gurations of the Waldron leg mechanism.
Fig. 5. Secondary linkage of walking machines (dierent types of pantograph mechanisms).
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K. N. Umesh
ODETICS LEG
One of the most compact versatile remote controlled hexapod is Odetics walking machine. It consists of a leg mechanism shown in Fig. 2 and the planar kinematic equivalent with the screw pair replaced by a sliding pair is shown in Fig. 6. We consider this mechanism for further discussion of Odetics leg mechanism. The mechanism is a 2DOF system (n = 11, j = 11, F = 2) and consists of a modi®ed pantograph and an inverted slider crank mechanism. It should be noted that the sliders F and G receive separate inputs from individual electric actuators. A close examination of the mechanism [Fig. 7a(i) and b] indicates that the leg will be isotropic only when the mechanism is in a con®guration wherein: (i) OABC is in rectangular con®guration keeping vp2 along AB and perpendicular to BC (or OA) when y_ 1=0 and y_ 2=1. (ii) FQ must be parallel to PA such that respective path normals of points P and Q lie along FQ and AP, leading to the parallel paths when y_ 1=1 and y_ 2=0. The above conditions imply that slider oset of F must be equal to CQ. When actuator-1 is moving and actuator-2 is locked Link 9 (slider F) is ®xed and the linkage behaves as well constrained 1DOF system (n = 8, j = 10, F = 1). Point Q is constrained to move on circular path centered at F with a radius of FQ. The foot P copies the motion of point Q and, hence, moves on a path parallel to that of point P since OABC-QP functions as a pantograph [Fig. 7a(i)]. In this con®guration A (or 61) will be instant center of link 6 (ABP) w.r.t. link 1 and we can write, AP vb : v p1 AB Link FQ is pivoted at F and hence, Q can have motion perpendicular to FQ only. P being the end point, follows the motion of tracer point Q, i.e. vp1 is parallel to vq. The instant center of link 6 w.r.t. link 1 (61) lies at A, hence the link 6 has absolute rotation about A at the instant. Thus, we have vb parallel to vp1. The rectangular con®guration of the linkage leads to vb parallel to vc. vb=vc since AB = OC. The link 7 translates and, hence, we have vq=vb=vc. Thus we have velocities vb, vc, vq and vp1 parallel to each other.
Fig. 6. Schematic of Odetics leg mechanism.
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Points C and D lie on the same link 3 (CODÐa bell crank lever), which rotates about O. So we can write, OC vd vb ; since vc vb : vc OD Also we know that instant center R (or 13) of link 3 (DG) w.r.t. link 1 lies on the intersection of DO and a perpendicular to link 3 drawn through E. Then we can write, RD vg ; vd RG 3 where vg3 is magnitude of absolute velocity (vg3) of G as a point on link 3. On backward substitution of vb, vc and vd and simplifying we get, AP RD l6 RD v p1 vg vg l41 RG 3 OD RG 3
7
where l41 is length of the arm OD of bell crank lever 4. The absolute velocity of G as a point on link 3 could be obtained as a sum of two vectors as vg3 vg2 4 vg32 ;
8
where vg2 is absolute velocity G as a point on link 2 and vg32 is sliding velocity of point G on link 3 relative to a coincident point on link 2 which acts along EG [Fig. 7a(ii)]. In the actual linkage a screw pair driven by an electric actuator replaces the prismatic pair at G. Hence, the magnitude of the sliding velocity vg32 can be obtained as a result of actuator-1 velocity y_ 1 as L1 _ L1 y1 9 for y_ 1 1; _l2 say: vg32 2p 2p L1 is the lead of screw of actuator-1 that forms link 3 and l2 (=EG) is the magnitude of displacement of point G from the ®xed pivot E [Fig. 7a(iii)]. The magnitude of absolute velocity vg2 is given by vg2=l2y_ 2. It must be noted that here y_ 2 indicates the angular velocity of link 2, but the speed of actuator-2 (which is equal to zero in this case). An expression for y_ 2 could be obtained from the loop closure relation given by equation (A4) as y_ 2
_l2 l1 S y1 ÿ y2 f12 a=bg
10
where a = C(y1 ÿ y2) and b = {(l4/l1)2 ÿ S2(y1 ÿ y2)}1/2, and where Cyi=cos yi and Syi=sin yi. The magnitude of absolute velocity vector vg3 could be obtained as vg3 fv2g2 v2g32 g1=2 ;
11
since the vectors vg2 and vg32 are orthogonal to each other [Fig. 7a(ii)]. Alternatively, one could also obtain the magnitude of velocity vector vg3 from the velocity analysis as given in equation (A7) of the Appendix. vg3 fv2gx v2gy g1=2 ; where vgx= ÿ l2 sin y2+cos y2_l 2 and vgy= + l2 cos y2+sin y2_l 2. When actuator-1 is locked and actuator-2 is moving Link 4 (or OC) is ®xed and hence links 1-2(3)-4 behaves instantaneously as a structure (not shown). The linkage OABC behaves as a parallel 4-bar linkage (n = 6, j = 7, F = 1) with CB as the input link remotely driven by the actuator-2 at F (Fig. 7b). In this case the linkage AB translates since it forms the coupler of a parallel 4-bar linkage OABC and we have the velocities
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Fig. 7a. Odetics leg (actuator-1 moving and actuator-2 locked; link 4 ®xed).
vb=va=vp (or vp2). And the instant center of link 6 (BC) w.r.t. link 1 (i.e. 57) lies at C. Hence, we can write, CB 12 v q v p2 : vb CQ At the isotropic con®guration line of action of the screw driving slider 9 is collinear with FQ and angle FQC is a right angle. Therefore, the link FQ translates and, hence the end point velocities of link FQ are equal, i.e. vq=vf. We know that in the actual linkage the input is actually derived from an electric actuator for the slider 9. So we have magnitude of vf given by L2 _ y2 v q ; vf 2p where L2 is the lead of screw of actuator-2 that drives S2. Substituting for vq in Equation (12) we get, CB L2 _ L2 y2 k 13 for y_ 2 1; v p2 2p CQ 2p where k is scale of the pantograph OABC-QP when Q is the tracer point [Fig. 7a(i)]. It could be proved that scale of the pantograph OABC-QP,
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Fig. 7b. Odetics leg (actuator-1 locked and actuator-2 moving; link 9 ®xed).
k
OP CB l6 l42 OQ CQ
where l42 is length of the arm OC of bell crank lever 4. Substituting the value of scale (k) in Equation (13) we have, l6 L2 : vp2 l42 2p
14
For isotropy we can equate (7) and (14) and obtain an expression for lead of the actuator-2 as l42 RD vg 15 L2 2p l41 RG 3 assuming the lead of the actuator-1 speci®ed or known. However, in special case when the both arms of the bell crank lever 4 are equal in length the above expression gets further simpli®ed to get RD 16 vg : L2 2p RG 3
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K. N. Umesh
SIRUR LEG
This leg was developed at Indian Institute of Technology, Bombay (India) by Issac and his group and is named after the chief contributor to the design of the leg. It is driven by two independent electric actuators whose axes are along Y-axis (Fig. 3). It is a 2DOF system (n = 9, j = 11, F = 2). The inputs are applied at the hinges F and G of the sliders 4 and 9. The point to be noted here is that the horizontal motion of the foot is derived from a modi®ed Scott± Russel mechanism (Fig. 4b) and, hence, the mechanism has become more compact. An instantaneous kinematic equivalent is shown in Fig. 3. One of the interesting feature of the leg is lines of action of the two screws are coplanar and the x and y motions of the foot are decoupled as in the case of Waldron leg mechanism. Hence, it is possible to obtain orthogonal velocities at all con®gurations of the mechanism. The necessary condition for isotropy of the foot is derived here. When actuator-1 is moving and actuator-2 is locked Point G is ®xed and the linkage BCDE-FP behaves as a pantograph with B as the pivot. Similar to the earlier case, the foot point P lies on the extension of line joining F and B. The slider F is driven by the actuator-1 in the vertical direction and hence the motion of P is also vertical as shown in full line in Fig. 3. When actuator-2 is locked the Scott±Russel mechanism behaves as a structure and the pivot at G becomes immobile. In this case the linkage behavior is identical to the ®rst case of Waldron mechanism wherein actuator-2 is locked and actuator-1 is moving. Similar to the case of Waldron mechanism discussed earlier, a relation between the velocities of points P and F could be obtained as vp1 k ÿ 1vf ;
since
k0 k ÿ 1
17
where k and k' are scales of the pantographs derivable from Sirur leg mechanism. The respective scales of pantographs could be obtained as k = (FP/FB) when B acts as ®xed pivot and k' = (BP/BF) when F acts as ®xed pivot. The slider F is actuated by an electric motor with a screw of lead L1 the expression for velocity of point F is given by L1 _ L1 y1 ; for y_ 1 1: vf 2p 2p Substituting for vf in Equation (17) we get, vp1 k ÿ 1
L1 : 2p
18
When actuator-1 is locked and actuator-2 is moving Point F is ®xed. The foot point P lies on the extension of line joining F and B. As the point B is driven by the actuator-2 and the linkage BEDC-FP behaves as a pantograph. The directions of motion of the tracer point and end point are shown in dashed lines in Fig. 3. Now we can analyze the motion of the driving point B as an output of a Scott±Russel mechanism as follows. Scott±Russel mechanism It is used for deriving the horizontal motion from the vertical motion of actuator and is as shown in Fig. 8. The line of action of G (vertical) is perpendicular to that of line of action of point B. At any given instant the instant center I (or 13) of link 3 w.r.t. link 1 will lie at the intersection of perpendiculars drawn to the paths of G and B, i.e. IB remains perpendicular to IG and the right-angled triangles OBG and IGB so formed are similar. Hence, we can write:
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Fig. 8. Sirur legÐScott±Russel mechanism.
IG OB tan f IB OG
19
where f = OGÃB, is the angle made by link 3 with line of action of slider G. The velocities of the points B and P at the instant could be related by vp2 kvb :
20
Since I is the instant center of link 3 w.r.t. ground (Fig. 8) we can write, IG vb vg vg tan f: IB Substituting for vb in Equation (20) we get, vp2 kvg tan f:
21
In the actual mechanism slider G is driven by an electric actuator-2 (whose lead is L2) and actuator-1 is locked (y_ 1=0). Therefore, we can rewrite the above Equation (21) as, L2 L2 tan fy_ 2 k tan f; for y_ 2 1: 22 vp 2 k 2p 2p For isotropy we can equate (18) and (22) and obtain the expression L1 k tan f; L2 kÿ1
23
for lead ratios of the actuators. It can be observed from the above Equation (23) that a proper choice of leads for the actuator screws will enable one to obtain isotropic property at a speci®ed con®guration given by the angle f. Further the above equation can also be used to identify the isotropic con®guration of a given Sirur type linkage as follows. For a given set of lead screws and a chosen pantograph L1, L2 and k are constants. Therefore, an isotropic con®guration will exist when the link 3 (GB) is at an inclination of f to the line of action of slider G. The value of the con®guration angle is given by,
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f tanÿ1
L1 L2
kÿ1 : k
24
Hence, there can be only one con®guration at which isotropy of foot can be obtained in the case of Sirur type of leg mechanism. CONCLUSIONS
This paper presents some investigations on the dexterous postures of the important mechanisms used for robot locomotion. The study reveals that leg of Waldron type exhibits isotropic property at all the con®gurations, when it is driven by an electric actuator with appropriate lead screws. Where as the leg developed at IIT Bombay (India) has only one isotropic con®guration and is dependent on the scales of the pantographs con®gured during motion. The Odetics leg indicates the possibility of isotropy at the rectangular con®guration wherein the linkage has to meet certain conditions based on link parameters. The individual expressions derived here helps in obtaining quick solutions for the synthesis of dexterous legs for walking machines. This study also presents a method to identify the best postures in order to set o and for landing as well. At these postures (or isotropic positions) as the error propagation is minimum, the transmission eciency will be maximum and hence the operation will be quite energy ecient. The adoption of isotropic postures in the robot locomotion has certain in¯uence on the gait. This fact opens up further topics of study such as the gaits with isotropic positions. AcknowledgementsÐThe author wishes to sincerely acknowledge the highly inspiring discussions he had with his research supervisor Professor C. Amarnath and his colleagues in the Robotics laboratory of Indian Institute of Technology, Bombay during this research work.
REFERENCES 1. Ghosal, A. and Roth, B., Trans. ASME J. Mechanisms, Transmissions and Automation in Design, 1987, 109, 107± 115. 2. Salisbury, J. K. and Craig, J. J., Int. J. Robo. Res., 1982, 1(1), 4±7. 3. Chiu, S. L., Int. J. Robo. Res., 1988, 109(5), 13±21. 4. Gosselin, C. M., IEEE Int. Conf. on Robotics and Auto, 1990, VI, 650±655. 5. Lee, M. Y., Erdman, A. G. and Gutman, Y., Mechanisms and Machine Theory, 1993, 28(5), 651±670. 6. Angles, J., Int. J. Robo. Res., 1992, 11(3), 196±201. 7. Waldron, K. J. and Kinzel, G. L., The relationship between actuator geometry and mechanical eciency in robots. 4th Symposium on Theory and Practices of Robots and Manipulators CISMÐIFTOMM, 1981, pp. 305±316. 8. Song, S. M., Waldron, K. J. and Kinzel, G. L., J. Mechanisms and Machine Theory, 1985, 20(6), 585±596. 9. Song, S. M. and Waldron, K. J., J. Mech. Transm. and Auto. in Design, 1987, 109, 21±28. 10. Vohnout, V. J., Alexander, K. S. and Kinzel, G. L., Mechanical design of prototype leg for a walking machine, in Proc of the 8th Oklahoma State University Applied Mechanisms Conf., St Louis, Missouri, 1983. 11. Todd, D. J., Walking MachinesÐAn Introduction to Legged Robots, Kogan Page Ltd, London, 1985. 12. Marsh, P., Robots, Salamander Books Ltd, U.K., 1985. 13. Sirur, S. V., Mechanical design of hexapod. B.Tech. Thesis, IIT, Bombay, India, 1991. 14. Yang, D. C. H. and Lin, Y. Y., J. Mechanisms and Machine Theory, 1985, 20(1), 115±122. 15. Umesh, K. N., Studies on instantaneous kinematic properties of planar two degrees-of-freedom robotic linkages. Ph.D. Thesis, Mechanical Engg Dept, IIT, Bombay, 1994. 16. Umesh, K. N. and Amarnath, C., Studies on instantaneous kinematic properties of planar two degrees-of-freedom linkages, Communicated to Trans ASME Mech. Design, 1994.
APPENDIX Expressions For Velocities Vg3 And vg32 Expressions for the velocity of point G on link 2 relative to link 3, vg32 could be obtained from loop closure relation of the inverted slider crank mechanism shown in Fig. 7a(iii). The loop closure equation could be obtained as: l 1 l2 l3 l 4 iy1
or l1 e
l2 eiy2 l3 eiy3 l4 eiy4 ;
A1
where yi indicates the position of ith link measured from the positive X0 axis. Separating real and imaginary parts and rearranging we get the following equations, l4 Cy4 l1 Cy1 ÿ l2 Cy2 ÿ l3 Cy3
A2
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l4 Sy4 l1 Sy1 l2 Sy2 ÿ l3 Sy3
A3
where Cyi=cos yi and Syi=sin yi. Squaring and adding both sides and rearranging the above equations we can write, u1 l22 2u2 l2 u3 0;
A4
where, u1=1, u2={l3ÿl1C(y1ÿy2)} and u3=l21+l23ÿl24ÿ2l1l3C(y1ÿy3). The solution of the above quadratic equation in l2 could be obtained in its simplest form after substituting y2 for y3 as l2 w1 2w2 ;
A5
w2={l24ÿl21S2(y1ÿy2)}1/2.
where, w1= ÿ {l3ÿl1C(y1ÿy2)} and The two solutions given by equation (A5) correspond to two dierent assemblies of the mechanism. Dierentiating further w.r.t. time we could obtain an expression for _l 2 in terms of y_ 1, y_ 2 and y_ 3. In the mechanism y1 indicates position of ®xed link 1 and hence the terms with y_ 1 could be eliminated. Orientations of links 2 and 3 are always aligned and hence y3=y2 for all positions of the mechanism, Fig. 7a. Therefore, we have y_ 3 equal to y_ 2 at every instant. We could use these facts to obtain an expression of _l 2 in its simplest form as C y1 ÿ y2 _l2 l1 12 A6 S y1 ÿ y2 y_ 2 : f l4 =l1 2 ÿ S2 y1 ÿ y2 g1=2 The above equation (A6) gives magnitude of the velocity vector vg32 and is a function of y_ 2, the angular velocity of link 2 of the equivalent planar linkage shown in Fig. 7a. We shall now obtain an expression for the absolute velocity of point G on link 3, vg3 as follows. The displacement of point G could be obtained as [Fig. 7a(iii)], pg fpgx ; pgy gT
A7
where pgx=l2Cy2 and pgy=l2Sy2. Dierentiating w.r.t. time we get absolute velocity of point G as point on link 3 as vg fvgx ; vgy gT vg3
A8
where vgx= ÿ l2Sy2y_ 2+Cy2_l 2 and vgy= + l2Cy2y_ 2+Sy2_l 2. The above equation could be rewritten as _ vg3 JY where the manipulator Jacobian is given by
J
ÿl2 Sy2 l2 Cy2
A9
Cy2 ; Sy2
_ and Y={ y_ 2 _l 2}T is the input velocity vector. It is known that _l 2 is a function of y_ 2 and can be obtained from equation (A6).
ZusammenfassungÐDie analyses und fabricat die isotropic fuÈss fuÈr robot locomotion ist die haupt thema. Die untersuchungen die existierte zu geschlossene lupe structure ist getan durch substituieren pneumatic/hydraulic actuators durch elektrische actuators jedoch moÈglich. Die generale bedingung uber die gear ratios (Lead ratios) und die link parameters fuÈr isotropy die fuÈss sind diskutiert. Diese parameters benutzen wir fuÈr die synthesis dexterous mechanisms und die identi®kation fuÈr dexterous postures. # 1998 Elsevier Science Ltd. All rights reserved