- Email: [email protected]
A cam mechanism for gravity-balancing
Mechanics Research Communications 36 (2009) 523–530
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
A cam mechanism for gravity-balancing Kenan Koser * Faculty of Mechanical Engineering, Istanbul Technical University, 34437 Gumussuyu, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 22 April 2008 Received in revised form 18 September 2008 Available online 24 December 2008
Keywords: Gravity Compensation Balancing Cam mechanism
a b s t r a c t This study is concerned with a cam type gravity compensation mechanism. A new type interior cam mechanism is introduced as an alternative gravity-balancing mechanism for robot arms. Cam profile of the mechanism is derived from static balancing condition of an unbalanced rotating arm which is combined with an interior cam. Configurational drawing and mechanical structure of the mechanism are presented. Finally, application of mechanisms on parallelogram version of 2R robot manipulator is given. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The weight force’s balancing is a classical problem in the theory of mechanisms and robotic systems. The weight forces have a high share in a category of resistance that driving system must overcome. Hence, kinetostatic design of mechanisms, robot arms, parallel mechanisms, and commercial flight simulators are an important issue. Whereas the most common passive systems are mass and spring counterbalancing. In spring counterbalancing systems, springs or elastic members assembled between moving links and fixed platform, are used to produce a counteracting torque which is a function of position. The gravity compensation of robot manipulators and mechanisms has been an important research topic for several decades. Gosselin and Laliberte (1999) studied static balancing of 3-DOF planar parallel mechanisms. Static balancing of parallel robots has been studied by Russo et al. (2004). Agrawal and Fattah (2004) investigated gravity-balancing of spatial robotic manipulators and at the same time Agrawal and Agrawal (2005) dealed with the design of gravity-balancing in leg orthosis using non-zero free length springs. In this paper, a interior cam based on weight balancing mechanism is addressed. Mechanical design solution of gravity compensation given in this study will be extremely different from the conventional ones, namely spring and mass counterbalancing. Static balancing formulation of a rotating arm combined with an interior cam is firstly derived and then structure of an alternative balancing mechanism is presented. Example of the balanced robot arms is given which includes well-known 2Rserial anthropomorphic robot arm’s parallelogram version. 2. Cam profile and mechanism structure In this section, we will develop an interior cam profile in order to balance weight force of an unbalanced rotating arm exactly and then propose a cam type mechanism for gravity compensation. Let us consider an unbalanced rotating arm * Tel.: +90 346 219 10 10x12 80. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.12.005
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K. Koser / Mechanics Research Communications 36 (2009) 523–530
combined with an interior cam with spring loaded knife-edge follower shown in Fig. 1a. As seen in figure, the mass distance from the cam rotation axis and the stiffness of linear counter balancing spring are noted as m, ‘ and k. Similarly, the angular position of the rotating arm q, displacement of knife-edge follower sðqÞ, measured from cam base circle’s with the radius R in direction to inside which is equal to compensation spring’s compression, are noted in Fig. 1b. It is well-known that a pressure angle varying in position appears at the point of contact between a cam and a knife-edge follower while cam is rotating. It is also known that this pressure angle causes a normal force acting on cam surface which creates a resistance torque acting on the cam shaft of a spring loaded radial cam mechanism (Chen, 1982). Pressure angle aðqÞ, normal counteracting force NðqÞ created by counterbalancing spring force and resultant counterbalancing torque M c ðqÞ at an angular cam position q are shown in Fig. 1b. Fig. 2 shows four different configuration of an unbalanced arm-interior cam system. As shown in Figs. 1b and 2, as the arm is rotating, a pressure angle appears at the contact point or higher pair of follower and interior cam which creates a normal reaction force NðqÞ. It can also be seen from the figure that the normal force NðqÞ causes a counteracting torque M c ðqÞ about the cam rotation axis. Note that the spring loaded knife-edge follower is supported only on fixed base by means of a slider pair but has no contact with interior cam except its tip point-higher pair. For the purpose of calculating pitch curve of an interior cam employed for gravity compensation of single degree freedom arm, we can write neutral equilibrium condition of arm-cam and spring loaded follower system illustrated in Fig. 1. Then we obtain for potential energy function VðqÞ as,
VðqÞ ¼
1 2 ks ðqÞ þ mg‘ cosðqÞ ¼ C 2
ð1Þ
where q is the angular position, measured from vertical, of the arm or the cam, and C is an arbitrary constant. Imposing initial compression of follower spring s0 ¼ sðq ¼ 0Þ to Eq. (1), we then have, for counter balancing spring deformation sðqÞ,
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mg‘ ð1 cos qÞ sðqÞ ¼ s20 þ k
ð2Þ
Having obtained follower deformation, we can determine pitch curve of a centric interior cam. Let the base circle radius of cam is R and the cartesian coordinates of the pitch curve of cam are xðqÞ; yðqÞ in a coordinate frame fixed to the cam. Then, coordinate vector pðqÞ ¼ ½xðqÞ; yðqÞT is obtained as
pðqÞ ¼ AðqÞ rðqÞ
ð3aÞ
where AðqÞ and rðqÞ are given by
AðqÞ ¼
cos q
sin q
sin q cos q
;
rðqÞ ¼
0
R sðqÞ
ð3bÞ
(Angeles and Cajun, 1991). Hence, cartesian coordinates of the pitch curve of interior cam xðqÞ, yðqÞ are readily computed by using Eqs. (2) and (3b)
Fig. 1. Unbalanced arm-interior cam system and counteracting forces. (a) q ¼ 0, aðqÞ ¼ 0 and (b) q > 0, aðqÞ–0.
K. Koser / Mechanics Research Communications 36 (2009) 523–530
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Fig. 2. Four different angular positions of unbalanced arm-interior cam system, counterbalancing normal forces NðqÞ and torques M c ðqÞ.
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ s20 þ ð1 cos qÞ sin q k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ yðqÞ ¼ R s20 þ ð1 cos qÞ cos q k xðqÞ ¼ R
ð4Þ
It is clear that a knife-edge follower is practically unsuitable, because of contact stress, should be replaced by a follower with roller. Based on the parametric equation of pitch curve (4), cartesian coordinates of cam profile for a roller type follower can be written as
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ XðqÞ ¼ R s20 þ ð1 cos qÞ sin q þ rt sin½q þ aðqÞ k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ YðqÞ ¼ R s20 þ ð1 cos qÞ cos q þ rt cos½q þ aðqÞ k
ð5Þ
where rt and aðqÞ are radius of follower’s roller and pressure angle of the follower. Pressure angle aðqÞ can readily computed using follower displacement and its derivative with respect to q as
tan aðqÞ ¼
s0 ðqÞ ¼ R sðqÞ
mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R s20 þ 2mg‘ ð1 cos qÞ s20 þ 2mg‘ ð1 cos qÞ k k
ð6Þ
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K. Koser / Mechanics Research Communications 36 (2009) 523–530
(Angeles and Cajun, 1991). Substitution of Eq. (6) into Eq. (5) yields
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 6 2mg‘ 1 XðqÞ ¼ R s20 þ ð1 cos qÞ sin q þ rt sin 6 4q þ tan k "
3 7 mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 R s20 þ 2mg‘ ð1 cos qÞ s20 þ 2mg‘ ð1 cos qÞ k k 3
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 6 2mg‘ 1 YðqÞ ¼ R s20 þ ð1 cos qÞ cos q þ r t cos 6 4q þ tan k "
7 mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 2mg‘ 2mg‘ 2 2 R s0 þ k ð1 cos qÞ s0 þ k ð1 cos qÞ ð7Þ
Now, we can calculate counterbalancing torque produced by cam mechanism. Let NðqÞ be normal force between the roller follower and the cam surface, then one obtains counterbalancing torque M c ðqÞ as
M c ðqÞ ¼ NðqÞ sin aðqÞ½R sðqÞ or
ð8Þ
M c ðqÞ ¼ ksðqÞ tan aðqÞ½R sðqÞ which can be written as
Mc ðqÞ ¼ ksðqÞs0 ðqÞ or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mg‘ sinðqÞk s20 þ 2mg‘ ð1 cos qÞ k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mg‘ sinðqÞ Mc ðqÞ ¼ 2mg‘ 2 k s0 þ k ð1 cos qÞ
ð9Þ
above the relation is expected. Follower displacement, pressure angle, pitch curve and cam profile curves are given in Eq. (7) which the gravity compensation mechanism are computed according to collected data and illustrated in Fig. 3. It may be noted that cam profile is symmetrical and follower displacement has a maximum value at the arm position q ¼ 180 . Since the pressure angle is zero at this position of the arm, counterbalancing torque M c ðqÞ is given in Eqs. (8) and (9) will also be zero.
a
b
0.07
0.3
α ( q)
s( q)
0
0.035
[rad]
[m]
0.3
0 q [deg]
c
q [deg]
360
360
0.16
y (q) Y(q)
0
[m]
0.16 0.13
0
0.13
x(q), X( q) [m] Fig. 3. Displacement of follower (a), pressure angle (b), pitch curve and cam profile curves, (c) of gravity compensation mechanism. m = 8 kg, ‘ ¼ 0:4 m, k = 30,000 N/m, g = 9.806, s0 = 0.02 m, and R = 0.15 m.
K. Koser / Mechanics Research Communications 36 (2009) 523–530
527
Fig. 4. Structure of gravity compensation mechanism.
Now, we will present configurational drawing and mechanical structure of a interior cam based gravity compensation mechanism. Fig. 4 illustrates the side view drawing and numbered components of a mechanism. The components are: (1) interior cam and connecting shaft for the unbalanced arm, (2) counterbalancing compliant spring package and follower, (3) follower roller, and (4) circular tray. Referring in Fig. 4 the interior cam is located on the circular tray and can rotate freely into it, around a common axis. Counter balancing spring and follower package are screwed on circular tray by means of spring tip points. Follower roller tracks cam profile contour while interior cam is rotating together with the unbalanced arm. Note that the interior cam is shown in Fig. 4 has a cusp which the follower roller cannot follow. If initial position, namely, initial displacement of compensation spring sðq ¼ 0Þ ¼ s0 is selected as zero, then pitch curve of the interior cam would be unacceptable curve and would have a cusp (see Eq. (6); s0 ð0Þ ¼ 0=0). One easy way to eliminate this defect is to assign a small initial value for displacement (s0 –0) of the compensation spring (see Figs. 2 and 3c contain no cusp calculated by non-zero initial spring deformation). Furthermore, there is no objection of the cusp since the arm will be of its up position, namely robot arm will be rarely operated arm up position.
3. Application on 2R robot arm So far, we have developed an interior cam machine element in order to balance weight forces of a single degree unbalanced rotating arm exactly and have presented mechanical design of a cam type gravity compensation mechanism containing an interior cam. In this section, a brief outline of applications of this gravity compensation mechanism on the selected mechanical systems is presented. Gravity compensation may become serious problem for robot manipulators and parallel platforms compared with the other systems. The reduction or elimination of the weight forces of the links and payload of
Fig. 5. Parallelogram version of 2R anthropomorphic robot arm.
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K. Koser / Mechanics Research Communications 36 (2009) 523–530
such systems leads to light-weight structures which require less powerful actuators. In industrial applications, two composition of anthropomorphic robot arm, serial and its parallelogram version are widely employed due to their performance and design simplicity. As a preliminary design strategy, we can assemble two individual gravity compensation mechanisms between moving links or waist of the serial type 2R robot arm. Clearly, this design strategy compensates weight forces partially, does not provide exact compensation of weight forces of 2R anthropomorphic robot arm. For the purpose of exact gravity compensation, let us recall weight forces’ gravity torques transmitted to the joint actuators of 2R robot arm’s parallelogram version illustrated in Fig. 5. Let the joint domain position vector (measured from vertical, because we also measured gravity compensation mechanism’s angular position from vertical) of 2R robot could be define q ¼ ðq1 ; q2 ÞT , and also the mass and link length define for back and fore arms respectively m1 ; ‘1 ; m2 ; ‘2 . Supposing the center of masses are in mid-point of each link and neglecting gravity torques of parallelogram arm’s, we could write for the joint gravity torques as sG1 ; sG2 , which the actuators must overcome, using Jacobians of 2R robot arm (s ¼ J T F) as
sG1 ¼ sGq1 þ sGq12 ; sG2 ¼ sGq12
ð10Þ
where sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ, sGq12 ¼ ‘2 S12 m2 g=2, and S12 ¼ sinðq1 þ q2 Þ. Now we are ready to develop a design solution in order to perform complete gravity-balancing of fore and back arm of anthropomorphic 2R robot manipulator’s parallelogram version.
Fig. 6. Gravity compensation mechanism CGM1 (M GCM1 ¼ sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ assembled between back arm (3) shaft and base (1).
K. Koser / Mechanics Research Communications 36 (2009) 523–530
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Fig. 7. Gravity compensation mechanism CGM2 ðM GCM2 ¼ sGq12 ¼ ‘2 S12 m2 g=2Þ assembled between parallelogram arm shaft and base.
As shown in Fig. 6a and b, gravity torque of back arm and gravity torque of fore arm of the robot manipulator is transferred to the back arm actuator can be eliminated by assembling a gravity compensation mechanism CGM1 (M GCM1 ¼ sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ between back arm and base. Note that circular tray component and counterbalancing compliant spring package and follower of compensation mechanism CGM1 are fixed on the ground link or base, but its interior cam is connected to the back arm shaft, while rotating together. Note that other gravity torque term M GCM2 ¼ sGq12 ¼ ‘2 S12 m2 g=2 appears from absolute rotation of fore arm and is transferred both to fore arm and back arm actuator. Reader can readily see that the absolute rotation of parallelogram arm (2) which is equal fore arm’s absolute rotation ðq1 þ q2 Þ. Thus, one another compensation mechanism CGM2 assembled between parallelogram arm (2) shaft and ground link base, of course other side of the robot structure, creates a counteracting torque which is equal to the gravity torque M GCM2 ¼ ‘2 S12 m2 g=2 (Fig. 7a and b). 4. Conclusions The present study deals with the cam based gravity compensation mechanism. A novel design of gravity compensation mechanism is proposed in this paper. From the point of kinematic synthesis, it seems feasible to construct an effective cam type gravity compensation mechanism. In this study, possible application of compensation mechanism on the 2R robot manipulator’s parallelogram version is presented. However, some essential design problems involving component design,
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assembled engineering drawings of compensation mechanism and the robot arms, design of actuator-compensation mechanism composition, are excluded of this research study. References Agrawal, A., Agrawal, S.K., 2005. Design of gravity balancing leg orthosis non-zero free length springs. Mech. Mach. Theory (40), 693–709. Agrawal, S.K., Fattah, A., 2004. Gravity-balancing of spatial robotic manipulators. Mech. Mach. Theory (39), 1331–1344. Angeles, J., Cajun, C.S.L., 1991. Optimization of Cam Mechanics. Kluwer Academic Publishers, Netherlands. Chen, Fan Y., 1982. Mechanics and Design of Cam Mechanics. Pergamon Press Inc. Gosselin, M., Laliberte, T., 1999. Static balancing of 3-DOF planar parallel mechanisms. IEEE/ASME Trans. Mech. (4), 363–371. Russo, A. et al, 2004. Static balancing of parallel robots. Mech. Mach. Theory (40), 191–202.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
A cam mechanism for gravity-balancing Kenan Koser * Faculty of Mechanical Engineering, Istanbul Technical University, 34437 Gumussuyu, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 22 April 2008 Received in revised form 18 September 2008 Available online 24 December 2008
Keywords: Gravity Compensation Balancing Cam mechanism
a b s t r a c t This study is concerned with a cam type gravity compensation mechanism. A new type interior cam mechanism is introduced as an alternative gravity-balancing mechanism for robot arms. Cam profile of the mechanism is derived from static balancing condition of an unbalanced rotating arm which is combined with an interior cam. Configurational drawing and mechanical structure of the mechanism are presented. Finally, application of mechanisms on parallelogram version of 2R robot manipulator is given. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The weight force’s balancing is a classical problem in the theory of mechanisms and robotic systems. The weight forces have a high share in a category of resistance that driving system must overcome. Hence, kinetostatic design of mechanisms, robot arms, parallel mechanisms, and commercial flight simulators are an important issue. Whereas the most common passive systems are mass and spring counterbalancing. In spring counterbalancing systems, springs or elastic members assembled between moving links and fixed platform, are used to produce a counteracting torque which is a function of position. The gravity compensation of robot manipulators and mechanisms has been an important research topic for several decades. Gosselin and Laliberte (1999) studied static balancing of 3-DOF planar parallel mechanisms. Static balancing of parallel robots has been studied by Russo et al. (2004). Agrawal and Fattah (2004) investigated gravity-balancing of spatial robotic manipulators and at the same time Agrawal and Agrawal (2005) dealed with the design of gravity-balancing in leg orthosis using non-zero free length springs. In this paper, a interior cam based on weight balancing mechanism is addressed. Mechanical design solution of gravity compensation given in this study will be extremely different from the conventional ones, namely spring and mass counterbalancing. Static balancing formulation of a rotating arm combined with an interior cam is firstly derived and then structure of an alternative balancing mechanism is presented. Example of the balanced robot arms is given which includes well-known 2Rserial anthropomorphic robot arm’s parallelogram version. 2. Cam profile and mechanism structure In this section, we will develop an interior cam profile in order to balance weight force of an unbalanced rotating arm exactly and then propose a cam type mechanism for gravity compensation. Let us consider an unbalanced rotating arm * Tel.: +90 346 219 10 10x12 80. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.12.005
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K. Koser / Mechanics Research Communications 36 (2009) 523–530
combined with an interior cam with spring loaded knife-edge follower shown in Fig. 1a. As seen in figure, the mass distance from the cam rotation axis and the stiffness of linear counter balancing spring are noted as m, ‘ and k. Similarly, the angular position of the rotating arm q, displacement of knife-edge follower sðqÞ, measured from cam base circle’s with the radius R in direction to inside which is equal to compensation spring’s compression, are noted in Fig. 1b. It is well-known that a pressure angle varying in position appears at the point of contact between a cam and a knife-edge follower while cam is rotating. It is also known that this pressure angle causes a normal force acting on cam surface which creates a resistance torque acting on the cam shaft of a spring loaded radial cam mechanism (Chen, 1982). Pressure angle aðqÞ, normal counteracting force NðqÞ created by counterbalancing spring force and resultant counterbalancing torque M c ðqÞ at an angular cam position q are shown in Fig. 1b. Fig. 2 shows four different configuration of an unbalanced arm-interior cam system. As shown in Figs. 1b and 2, as the arm is rotating, a pressure angle appears at the contact point or higher pair of follower and interior cam which creates a normal reaction force NðqÞ. It can also be seen from the figure that the normal force NðqÞ causes a counteracting torque M c ðqÞ about the cam rotation axis. Note that the spring loaded knife-edge follower is supported only on fixed base by means of a slider pair but has no contact with interior cam except its tip point-higher pair. For the purpose of calculating pitch curve of an interior cam employed for gravity compensation of single degree freedom arm, we can write neutral equilibrium condition of arm-cam and spring loaded follower system illustrated in Fig. 1. Then we obtain for potential energy function VðqÞ as,
VðqÞ ¼
1 2 ks ðqÞ þ mg‘ cosðqÞ ¼ C 2
ð1Þ
where q is the angular position, measured from vertical, of the arm or the cam, and C is an arbitrary constant. Imposing initial compression of follower spring s0 ¼ sðq ¼ 0Þ to Eq. (1), we then have, for counter balancing spring deformation sðqÞ,
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mg‘ ð1 cos qÞ sðqÞ ¼ s20 þ k
ð2Þ
Having obtained follower deformation, we can determine pitch curve of a centric interior cam. Let the base circle radius of cam is R and the cartesian coordinates of the pitch curve of cam are xðqÞ; yðqÞ in a coordinate frame fixed to the cam. Then, coordinate vector pðqÞ ¼ ½xðqÞ; yðqÞT is obtained as
pðqÞ ¼ AðqÞ rðqÞ
ð3aÞ
where AðqÞ and rðqÞ are given by
AðqÞ ¼
cos q
sin q
sin q cos q
;
rðqÞ ¼
0
R sðqÞ
ð3bÞ
(Angeles and Cajun, 1991). Hence, cartesian coordinates of the pitch curve of interior cam xðqÞ, yðqÞ are readily computed by using Eqs. (2) and (3b)
Fig. 1. Unbalanced arm-interior cam system and counteracting forces. (a) q ¼ 0, aðqÞ ¼ 0 and (b) q > 0, aðqÞ–0.
K. Koser / Mechanics Research Communications 36 (2009) 523–530
525
Fig. 2. Four different angular positions of unbalanced arm-interior cam system, counterbalancing normal forces NðqÞ and torques M c ðqÞ.
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ s20 þ ð1 cos qÞ sin q k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ yðqÞ ¼ R s20 þ ð1 cos qÞ cos q k xðqÞ ¼ R
ð4Þ
It is clear that a knife-edge follower is practically unsuitable, because of contact stress, should be replaced by a follower with roller. Based on the parametric equation of pitch curve (4), cartesian coordinates of cam profile for a roller type follower can be written as
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ XðqÞ ¼ R s20 þ ð1 cos qÞ sin q þ rt sin½q þ aðqÞ k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2mg‘ YðqÞ ¼ R s20 þ ð1 cos qÞ cos q þ rt cos½q þ aðqÞ k
ð5Þ
where rt and aðqÞ are radius of follower’s roller and pressure angle of the follower. Pressure angle aðqÞ can readily computed using follower displacement and its derivative with respect to q as
tan aðqÞ ¼
s0 ðqÞ ¼ R sðqÞ
mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R s20 þ 2mg‘ ð1 cos qÞ s20 þ 2mg‘ ð1 cos qÞ k k
ð6Þ
526
K. Koser / Mechanics Research Communications 36 (2009) 523–530
(Angeles and Cajun, 1991). Substitution of Eq. (6) into Eq. (5) yields
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 6 2mg‘ 1 XðqÞ ¼ R s20 þ ð1 cos qÞ sin q þ rt sin 6 4q þ tan k "
3 7 mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 R s20 þ 2mg‘ ð1 cos qÞ s20 þ 2mg‘ ð1 cos qÞ k k 3
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 6 2mg‘ 1 YðqÞ ¼ R s20 þ ð1 cos qÞ cos q þ r t cos 6 4q þ tan k "
7 mg‘ sinðqÞ=k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 2mg‘ 2mg‘ 2 2 R s0 þ k ð1 cos qÞ s0 þ k ð1 cos qÞ ð7Þ
Now, we can calculate counterbalancing torque produced by cam mechanism. Let NðqÞ be normal force between the roller follower and the cam surface, then one obtains counterbalancing torque M c ðqÞ as
M c ðqÞ ¼ NðqÞ sin aðqÞ½R sðqÞ or
ð8Þ
M c ðqÞ ¼ ksðqÞ tan aðqÞ½R sðqÞ which can be written as
Mc ðqÞ ¼ ksðqÞs0 ðqÞ or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mg‘ sinðqÞk s20 þ 2mg‘ ð1 cos qÞ k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mg‘ sinðqÞ Mc ðqÞ ¼ 2mg‘ 2 k s0 þ k ð1 cos qÞ
ð9Þ
above the relation is expected. Follower displacement, pressure angle, pitch curve and cam profile curves are given in Eq. (7) which the gravity compensation mechanism are computed according to collected data and illustrated in Fig. 3. It may be noted that cam profile is symmetrical and follower displacement has a maximum value at the arm position q ¼ 180 . Since the pressure angle is zero at this position of the arm, counterbalancing torque M c ðqÞ is given in Eqs. (8) and (9) will also be zero.
a
b
0.07
0.3
α ( q)
s( q)
0
0.035
[rad]
[m]
0.3
0 q [deg]
c
q [deg]
360
360
0.16
y (q) Y(q)
0
[m]
0.16 0.13
0
0.13
x(q), X( q) [m] Fig. 3. Displacement of follower (a), pressure angle (b), pitch curve and cam profile curves, (c) of gravity compensation mechanism. m = 8 kg, ‘ ¼ 0:4 m, k = 30,000 N/m, g = 9.806, s0 = 0.02 m, and R = 0.15 m.
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Fig. 4. Structure of gravity compensation mechanism.
Now, we will present configurational drawing and mechanical structure of a interior cam based gravity compensation mechanism. Fig. 4 illustrates the side view drawing and numbered components of a mechanism. The components are: (1) interior cam and connecting shaft for the unbalanced arm, (2) counterbalancing compliant spring package and follower, (3) follower roller, and (4) circular tray. Referring in Fig. 4 the interior cam is located on the circular tray and can rotate freely into it, around a common axis. Counter balancing spring and follower package are screwed on circular tray by means of spring tip points. Follower roller tracks cam profile contour while interior cam is rotating together with the unbalanced arm. Note that the interior cam is shown in Fig. 4 has a cusp which the follower roller cannot follow. If initial position, namely, initial displacement of compensation spring sðq ¼ 0Þ ¼ s0 is selected as zero, then pitch curve of the interior cam would be unacceptable curve and would have a cusp (see Eq. (6); s0 ð0Þ ¼ 0=0). One easy way to eliminate this defect is to assign a small initial value for displacement (s0 –0) of the compensation spring (see Figs. 2 and 3c contain no cusp calculated by non-zero initial spring deformation). Furthermore, there is no objection of the cusp since the arm will be of its up position, namely robot arm will be rarely operated arm up position.
3. Application on 2R robot arm So far, we have developed an interior cam machine element in order to balance weight forces of a single degree unbalanced rotating arm exactly and have presented mechanical design of a cam type gravity compensation mechanism containing an interior cam. In this section, a brief outline of applications of this gravity compensation mechanism on the selected mechanical systems is presented. Gravity compensation may become serious problem for robot manipulators and parallel platforms compared with the other systems. The reduction or elimination of the weight forces of the links and payload of
Fig. 5. Parallelogram version of 2R anthropomorphic robot arm.
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such systems leads to light-weight structures which require less powerful actuators. In industrial applications, two composition of anthropomorphic robot arm, serial and its parallelogram version are widely employed due to their performance and design simplicity. As a preliminary design strategy, we can assemble two individual gravity compensation mechanisms between moving links or waist of the serial type 2R robot arm. Clearly, this design strategy compensates weight forces partially, does not provide exact compensation of weight forces of 2R anthropomorphic robot arm. For the purpose of exact gravity compensation, let us recall weight forces’ gravity torques transmitted to the joint actuators of 2R robot arm’s parallelogram version illustrated in Fig. 5. Let the joint domain position vector (measured from vertical, because we also measured gravity compensation mechanism’s angular position from vertical) of 2R robot could be define q ¼ ðq1 ; q2 ÞT , and also the mass and link length define for back and fore arms respectively m1 ; ‘1 ; m2 ; ‘2 . Supposing the center of masses are in mid-point of each link and neglecting gravity torques of parallelogram arm’s, we could write for the joint gravity torques as sG1 ; sG2 , which the actuators must overcome, using Jacobians of 2R robot arm (s ¼ J T F) as
sG1 ¼ sGq1 þ sGq12 ; sG2 ¼ sGq12
ð10Þ
where sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ, sGq12 ¼ ‘2 S12 m2 g=2, and S12 ¼ sinðq1 þ q2 Þ. Now we are ready to develop a design solution in order to perform complete gravity-balancing of fore and back arm of anthropomorphic 2R robot manipulator’s parallelogram version.
Fig. 6. Gravity compensation mechanism CGM1 (M GCM1 ¼ sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ assembled between back arm (3) shaft and base (1).
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Fig. 7. Gravity compensation mechanism CGM2 ðM GCM2 ¼ sGq12 ¼ ‘2 S12 m2 g=2Þ assembled between parallelogram arm shaft and base.
As shown in Fig. 6a and b, gravity torque of back arm and gravity torque of fore arm of the robot manipulator is transferred to the back arm actuator can be eliminated by assembling a gravity compensation mechanism CGM1 (M GCM1 ¼ sGq1 ¼ ‘1 S1 ðm1 g=2 þ m2 gÞ between back arm and base. Note that circular tray component and counterbalancing compliant spring package and follower of compensation mechanism CGM1 are fixed on the ground link or base, but its interior cam is connected to the back arm shaft, while rotating together. Note that other gravity torque term M GCM2 ¼ sGq12 ¼ ‘2 S12 m2 g=2 appears from absolute rotation of fore arm and is transferred both to fore arm and back arm actuator. Reader can readily see that the absolute rotation of parallelogram arm (2) which is equal fore arm’s absolute rotation ðq1 þ q2 Þ. Thus, one another compensation mechanism CGM2 assembled between parallelogram arm (2) shaft and ground link base, of course other side of the robot structure, creates a counteracting torque which is equal to the gravity torque M GCM2 ¼ ‘2 S12 m2 g=2 (Fig. 7a and b). 4. Conclusions The present study deals with the cam based gravity compensation mechanism. A novel design of gravity compensation mechanism is proposed in this paper. From the point of kinematic synthesis, it seems feasible to construct an effective cam type gravity compensation mechanism. In this study, possible application of compensation mechanism on the 2R robot manipulator’s parallelogram version is presented. However, some essential design problems involving component design,
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assembled engineering drawings of compensation mechanism and the robot arms, design of actuator-compensation mechanism composition, are excluded of this research study. References Agrawal, A., Agrawal, S.K., 2005. Design of gravity balancing leg orthosis non-zero free length springs. Mech. Mach. Theory (40), 693–709. Agrawal, S.K., Fattah, A., 2004. Gravity-balancing of spatial robotic manipulators. Mech. Mach. Theory (39), 1331–1344. Angeles, J., Cajun, C.S.L., 1991. Optimization of Cam Mechanics. Kluwer Academic Publishers, Netherlands. Chen, Fan Y., 1982. Mechanics and Design of Cam Mechanics. Pergamon Press Inc. Gosselin, M., Laliberte, T., 1999. Static balancing of 3-DOF planar parallel mechanisms. IEEE/ASME Trans. Mech. (4), 363–371. Russo, A. et al, 2004. Static balancing of parallel robots. Mech. Mach. Theory (40), 191–202.