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Gravity-balancing of spatial robotic manipulators
Mechanism and Machine Theory 39 (2004) 1331–1344
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Gravity-balancing of spatial robotic manipulators Sunil K. Agrawal *, Abbas Fattah Mechanical Systems Laboratory, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Accepted 25 May 2004 Available online 2 October 2004
Abstract This paper describes the underlying theory of gravity balanced spatial robotic manipulators through a hybrid strategy which uses springs in addition to identification of the center of mass using auxiliary parallelograms. A significant contribution of this paper is to show that springs with fixed ends are sufficient to gravity balance a spatial mechanism if the hybrid method of gravity balancing is used where the center of mass is identified first through auxiliary parallelograms. Also, the system remains gravity balanced even if the orientation of the base is changed, i.e., the direction of the gravity is changed with respect to the base. Although the method can be applied to n link spatial serial manipulators, we apply the method for gravity compensation of two and three degrees-of-freedom (DOF) spatial manipulators. A prototype with the underlying principles of this paper was fabricated at the University of Delaware. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction A machine is said to be gravity balanced if joint actuator inputs are not needed to keep the system in equilibrium at any configuration of the machine. The system essentially behaves as if its motion is in a zero-gravity environment. Over the years, gravity balanced machines have been
*
Corresponding author. Tel.: +1 302 831 8049; fax: +1 302 831 3619. E-mail address: [email protected] (S.K. Agrawal).
0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.019
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realized through clever designs using counterweights, springs, and auxiliary parallelograms. These designs have been introduced to linkages [2,9,7], serial robotic manipulators [8,11] and parallel manipulators [4,10]. A number of mathematical descriptions can be given for gravity balanced machines such as (i) system center of mass remains inertially fixed during motion; (ii) potential energy remains invariant with configuration of the system; (iii) system has counter masses that balance the machine in every configuration. These mathematical conditions have been physically realized through clever engineering such as: (a) countermass on each body of the machine is used to inertially fix the center of mass of the system [2,10], (b) springs are used at appropriate places in the machine such that the sum total of the gravitational potential energy and spring potential energy together becomes invariant with configuration [5,3,10], (c) auxiliary parallelograms based on knowledge of geometry and inertia property are used to physically determine the center of mass of the machine [1]. While each of these methods have their scientific core along with advantages and disadvantages, this paper focuses on a hybrid methodology for gravity balancing of spatial robotic manipulators that combines the underlying fundamentals between these different methods. In the methods presented in the literature using springs, gravity balancing is achieved for a fixed direction of the gravity vector. However, using the method presented in this paper, under valid assumption, balancing is independent of the direction of the gravity vector. In other words, the base of the machine can be oriented in any direction with respect to the direction of the gravity vector. This paper uses the notion of auxiliary parallelograms to first locate the center of mass. In addition, springs are connected through this point to make the potential energy invariant with configuration. For these reasons, this hybrid two-step approach is desirable over approaches which do not consider finding the center of mass first. At this time, only a few studies exist on gravity balancing of spatial manipulators. A spatial two-dof serial manipulator was studied for gravity balancing in [11]. The authors proved in this study that for a two-dof manipulator, the conditions of gravity balancing requires that the end points of the springs move relative to the mechanism, i.e., be independently actuated. This will require extra actuators to be mounted on the system and is undesirable. In this paper, we show that there is no need for the end points of the springs to be actively moved if the hybrid method presented in this paper is used. The organization of this paper is as follows: Section 2 describes the underlying theory of gravity balanced robotic manipulators through hybrid strategy which uses springs in addition to identification of the center of mass using auxiliary parallelograms. Next, we present the method for gravity compensation of two and three degrees-of-freedom (DOF) spatial robotic manipulators.
2. Gravity balancing: a hybrid method The underlying idea behind the hybrid method is as follows: (i) determine the center of mass of the manipulator by adding auxiliary links to machine; (ii) select springs to connect to the center of mass such that the total potential energy of the system is invariant with configuration. This method allows one to physically determine the center of mass of the manipulator and connect this point to the inertially fixed frame through springs.
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2.1. Location of system center of mass Consider an n-link articulated serial manipulator with revolute joints. The Deneavit–Hartenberg (DH) parameters for two successive links, i.e., hi, di, ai and ai, shown in Fig. 1, which characterizes the motion of link i with respect to link i 1 in terms of the following transformation matrix. 1 0 sin ai sin hi ai cos hi cos hi cos ai sin hi B sin h cos ai cos hi sin ai cos hi ai sin hi C i C B i1 ð1Þ Ti ¼ B C: @ 0 sin ai cos ai di A 0
0
0
1
Here, i = 1, . . ., n and hi is the variable joint angle, while all other DH parameters are constants. The location of the system center of mass C for this manipulator from the reference point (origin) O is n 1 X mi rOCi ; ð2Þ rOC ¼ M i¼1 where M¼
n X
ð3Þ
mi :
i¼1
Here, M is the total mass of the manipulator, mi is the mass of link i, and rOCi is the location of the center of mass of link i, Ci, from point O. The vector rOCi can be written as rOCi ¼ rOOi þ rOi Ci ;
ð4Þ
where Oi is the origin of the frame attached to link i. Using Fig. 1, rOOi is rOOi ¼
i X
ðd j zj1 þ aj xj Þ
j¼1
Fig. 1. Denavit–Hartenberg frame assignment for link i.
ð5Þ
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and the location of center of mass of link i from point Oi in its local frame is expressed as rOi Ci ¼ bix xi þ biy yi þ biz zi :
ð6Þ
Upon substitution of Eqs. (5) and (6) into Eq. (4) and inserting the result thus obtained into Eq. (2), we obtain " # n i X 1 X ð7Þ mi ðd j zj1 þ aj xj Þ þ bix xi þ biy yi þ biz zi rOC ¼ M i¼1 j¼1 or in compact form as n X rOC ¼ ðcix xi þ ciy yi þ ciz zi Þ:
ð8Þ
i¼1
Here, it is assumed d1 = 0. Note that cix, ciy, ciz are factors of geometry and mass distribution of the links and are usually denoted by the terminology ‘‘scaled lengths’’. Having these scaled lengths, one can determine the location of the system center of mass by adding appropriate auxiliary links to machine. Here, we assume the masses of auxiliary links to be negligible as compared to the masses of the links. This is an acceptable assumption. However, the masses of auxiliary links can be also considered using the method presented in [1]. Vector rOC can be expressed in any coordinate frame by expressing the vectors xi, yi and zi in this coordinate frame. For example, in the inertially fixed reference frame can be written as rOC ¼ d1 x0 þ d2 y0 þ d3 z0 ;
ð9Þ
where di, i = 1, 2, 3 are configuration dependent, functions of scaled lengths, and also DH parameters of all links. 2.2. Springs selection Upon locating the system center of mass, the potential energy of the system due to the gravity can be written as V g ¼ Mgg rOC ;
ð10Þ
where g is the unit vector along the gravity. It can be expressed in the inertial frame as g ¼ n1 x0 þ n2 y0 þ n3 z0 ;
ð11Þ
where all ni, i = 1,2, 3 are constants. For gravity compensation, we attach a spring of stiffness k between the system center of mass C and a point P along the gravity direction at a distance d from origin, O shown in Fig. 2. The strain potential energy due to the spring is 1 V s ¼ kx2 ; 2 2 where x is
!
! x2 ¼ k PC k k PC k:
ð12Þ
ð13Þ
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Fig. 2. n-link robotic manipulator with spring connections.
! Here, PC can be written as
! PC ¼ rOC rOP
ð14Þ
and rOP ¼ dðn1 x0 þ n2 y0 þ n3 z0 Þ:
ð15Þ
Upon substitution of Eqs. (15) and (9) into Eq. (14) and inserting the result thus obtained into Eq. (13) and then substituting this into Eq. (12), we obtain 1 V s ¼ k½ðd1 dn1 Þ2 þ ðd2 dn2 Þ2 þ ðd3 dn3 Þ2 : ð16Þ 2 Upon substitution of Eqs. (11) and (9) into Eq. (10) and expanding the result thus obtained, we get Vg as V g ¼ Mgðd1 n1 þ d2 n2 þ d3 n3 Þ:
ð17Þ
Therefore, the total potential energy of the system is the sum of potential energies due to gravity and the spring 1 V ¼ V g þ V s ¼ ðMg kdÞd1 n1 þ ðMg kdÞd2 n2 þ ðMg kdÞd3 n3 þ kðn21 þ n22 þ n23 Þd 2 2 1 þ kðd21 þ d22 þ d23 Þ: ð18Þ 2 The total potential energy is invariant with configuration if the coefficients of configuration variable terms vanish. Hence, the stiffness of spring k is determined by setting the coefficients of dini, i = 1, 2, 3 to zero, i.e., Mg : ð19Þ k¼ d The only configuration varying term remaining in the total potential energy is 1 kðd21 þ d22 þ d23 Þ or simply 12 krOC rOC . It may be noted that this term is function of all joint 2 angles, i.e., hi, i = 1, . . ., n. The strategy is to add appropriate springs to the system such that the total potential energy of the system becomes configuration invariant. To this end, we should select springs such that the configuration dependent terms in potential energy vanish.
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First, we derive and classify these terms in terms of variable joint angles and then we try to attach springs to appropriate points such that they can cancel out the variable terms in total potential energy. The number of extra springs depends on the number of variable terms in total potential energy expression. If we change the base of the manipulator with respect to the gravity vector, the unit gravity vector g can be re-expressed in the inertial frame as g ¼ n01 x0 þ n02 y0 þ n03 z0 ;
ð20Þ
n0i ,
i = 1, 2, 3 are constants. This is the only change that we should make when the base where all has a different orientations with respect to the gravity vector. It is clearly seen from the procedure mentioned in this section that the gravity balancing of the manipulator does not depend on the coefficients n0i , i = 1, 2, 3. Therefore, this method results in gravity balancing for all directions of the gravity vector. In other words, the base of machine can be installed in any direction with respect to gravity vector. However, the point P in Fig. 2 remains always on the axis of the gravity vector. In other words, OP is kept in the direction of the gravity when the base of the platform is rotated. In this work, it is assumed that the undeformed length of the spring is zero. In other words, the spring force is zero when the deformation of the spring is also zero. In the physical implementation of zero free length, non-zero free length spring can be used behind the pulley where the spring force can be transmitted through the wire [6]. The method of gravity balancing mentioned in this section is applied to n-link spatial serial manipulators. However, to show some examples, we apply the method to gravity balancing of two and three degrees-of-freedom (DOF) spatial manipulators in the next section. The method can be extended to consider example of more dof serial manipulators.
3. Examples 3.1. Two DOF robotic manipulator Consider a two DOF spatial manipulator, i.e., n = 2 in Fig. 2. Here, the axis of joint 1 (z0) is not parallel to the axis of joint 2 (z1), i.e., a1 5 0. However, we assume z1 is parallel to the axis of joint 3 (z2), i.e., a2 = 0. The location of the system center of mass of two DOF manipulator can be derived using n = 2 in Eq. (8) as follows: rOC
2 X ¼ ðcix xi þ ciy yi þ ciz zi Þ;
ð21Þ
i¼1
where the coefficients are constants and can be computed inserting n = 2 into Eq. (7). The resulting coefficients are m1 ð22Þ c1x ¼ a1 þ b1x ; M c1y ¼
m1 b ; M 1y
ð23Þ
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c1z ¼
m1 m2 b1z þ d 2 ; M M
ð24Þ
c2x ¼
m2 ða2 þ b2x Þ; M
ð25Þ
c2y ¼
m2 b ; M 2y
ð26Þ
m2 b : M 2z The expression 12 krOC rOC for two-DOF manipulator is c2z ¼
ð27Þ
1 1 krOC rOC ¼ kðc21x þ 2c1x c2x x1 x2 þ 2c1x c2y x1 y2 þ 2c1x c2z x1 z2 þ c21y þ 2c1y c2x y1 2 2 x2 þ 2c1y c2y y1 y2 þ 2c1y c2z y1 z2 þ c21z þ 2c1z c2x z1 x2 þ 2c1z c2y z1 y2 þ 2c1z c2z z1 z2 þ c22x þ c22y þ c22z Þ:
ð28Þ
Upon substitution of i = 2, a2 = 0 in the transformation matrix given by Eq. (1), we obtain the angle between the axes of the local frames of two successive links 1 and 2 shown in Table 1. Then, inserting the results from Table 1 into Eq. (28), one obtains the non-zero configuration varying terms in 12 krOC rOC . These terms are kðc1x c2x þ c1y c2y Þ cos h2 and kðc1x c2y þ c1y c2x Þ sin h2 . We add two springs to compensate for the two configuration varying terms: a spring with stiffness k1 between the axes x1 and x2 and a second spring k2 between the axes y1 and x2. The connection points d 0i , i = 1, . . ., 4 are shown in Fig. 3. The expression for the potential energy of the two springs and configuration varying terms remaining in the total potential energy is written as
Table 1 The angle between the axes of the local frames at links 1 and 2 x2 y2 z2
x1
y1
z1
cos h2 sin h2 0
sin h2 cos h2 0
0 0 1
Fig. 3. Spring connections for 2-link manipulator.
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1 02 0 0 V r ¼ kðc1x c2x þ c1y c2y Þ cos h2 þ kðc1x c2y þ c1y c2x Þ sin h2 þ k 1 ðd 02 1 þ d 2 2d 1 d 2 Þ cos h2 2 1 02 0 0 þ k 2 ðd 02 ð29Þ 3 þ d 4 2d 3 d 4 Þ sin h2 : 2 The values of k1 and k2 are selected such that Eq. (29) becomes configuration invariant. The expressions for these two stiffnesses are k1 ¼
kðc1x c2x þ c1y c2y Þ ; d 01 d 02
ð30Þ
k2 ¼
kðc1x c2y þ c1y c2x Þ : d 03 d 04
ð31Þ
Therefore, the two DOF spatial manipulator is completely gravity balanced by using three from the center springs shown in Fig. 4. These three springs are: (i) a spring with stiffness k ¼ Mg d of mass of the manipulator to a point on a line from the origin of the inertially fixed frame parallel to the gravity vector, (ii) two more springs with stiffnesses k1 and k2. Please note that the location of the center of mass of the manipulator can be identified using auxiliary parallelograms based on Eq. (21). However, the location of the center of mass has a simpler expression when described with respect to point O1 instead of O and is given as rO1 C ¼ rOC rOO1 ¼ rOC a1 x1 ;
ð32Þ
where rOC was defined in Eq. (21). 3.1.1. Two DOF manipulator: special case We obtain the conditions for gravity balancing of a 2-link spatial manipulator when the center of mass of the links are located on xizi plane, i.e., ciy = 0 in Eq. (21). Using these simplifications in Eqs. (30) and (31), one gets
Fig. 4. 2-link gravity balanced spatial manipulator with spring connections.
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Fig. 5. 2-link gravity balanced spatial manipulator with spring connections: special case.
k1 ¼
kc1x c2x ; d 01 d 02
k 2 ¼ 0:
ð33Þ ð34Þ
Therefore, the 2-link spatial manipulator is gravity balanced using only two springs—one from the center of mass of the manipulator to an inertially fixed point along the gravity direction passand the second spring of stiffness k1 between the axis ing through the origin, with stiffness k ¼ Mg d x1 and x2, shown in Fig. 5. 3.1.2. Prototype of two DOF manipulator An engineering prototype based on the design explained in Section 3.1.1 was fabricated at University of Delaware and is shown in Fig. 6. This prototype is made of wooden parts and it has the following DH and inertia parameters. a1 = 0.5 m, d1 = 0, a1 = p/6 rad, d2 = 0.3 m, a2 = 0, a2 = 0.4 m, m1 = 4.188 kg and m2 = 3.903 kg. It verifies the conceptual idea of spatial gravity balancing of two DOF manipulator and was tested on different configurations of end-effector and also different orientations of the base (see http://mechsys4.me.udel.edu/movies/rehrob/spabalmov/). Adding auxiliary links results in some limitation for the motion of the prototype. As an example, the second link of the manipulator cannot be fully extended because of the auxiliary links. However, this limitation can be alleviated by changing the shape of these links. Also the manipulator cannot be rotated fully (360°) about the base because of the spring which is connected to the system center of mass. This is not a problem since the usual range of the motion for the joints are much smaller than 360°. 3.2. Three DOF spatial manipulator Consider a three DOF spatial manipulator, i.e., n = 3, as shown in Fig. 7. Here, the three joint axes 1, 2, and 3 axes are not parallel to each other. However, we assume that joint axis 3 (z2) is
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Fig. 6. Photographs of prototype of the two DOF spatial gravity balanced manipulator at different orientations of the base.
Fig. 7. A 3-link spatial manipulator.
parallel to joint axis 4 (z3), i.e., a3 = 0. The location of the system center of mass of three DOF manipulator can be derived using n = 3 in Eq. (8) as follows: rOC ¼
3 X ðcix xi þ ciy yi þ ciz zi Þ;
ð35Þ
i¼1
where the coefficients are constants and can be computed by inserting n = 3 into Eq. (7). Here, we assume the center of mass of the links are located on xizi plane, i.e., ciy = 0 in Eq. (21). The nonzero terms of expression 12 krOC rOC for the three-DOF manipulator are
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1 1 krOC rOC ¼ kðc21x þ c1x c2x x1 x2 þ c1x c2z x1 z2 þ c1x c3x x1 x3 þ c1x c3z x1 z3 þ c21z 2 2 þ c1z c2x z1 x2 þ c1z c2z z1 z2 þ c1z c3x z1 x3 þ kc1z c3z z1 z3 þ c22x þ c2x c3x x2 x3 þ c2x c3z x2 z3 þ c22z þ c2z c3x z2 x3 þ c2z c3z z2 z3 þ c23x þ c23z Þ:
ð36Þ
Upon substituting i = 2, 3 and a3 = 0 in transformation matrix given by Eq. (1), we obtain the angles between the different axes of the coordinate frames as shown in Table 2. Inserting the results from Table 2 into the non-zero terms of Eq. (36) and noting that c3z = 0 because the center of mass of link 3 is located on the z2x3 plane, one obtains the remaining non-zero terms to be as kc1x c2x cos h2 þ kc1x c2z sin a2 sin h2 þ kc1x c3x x1 x3 þ kc1z c3x sin a2 sin h3 þ kc2x c3x cos h3 . We add four springs to compensate for these five terms: (i) a spring k1 between axis x1 and x3, (ii) a spring k2 between x2 and x3, (iii) a spring k3 between y2 and x3, and (iv) a spring k4 between axis x1 and x2 with connection points d 0i , i = 1, . . ., 8, respectively, as shown in Fig. 8.
Table 2 The dot products of the axes of the local coordinate frames x2 y2 z2
x3 y3 z3
x1
y1
z1
cos h2 sin h2cos a2 sinh2sin a2
sin h2 cos h2cos a2 cos h2sin a2
0 sin a2 cos a2
x2
y2
z2
cos h3 sin h3 0
sin h3 cos h3 0
0 0 1
Fig. 8. 3-link gravity balanced spatial manipulator with springs connection.
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The expression for the potential energy of the four springs can be written as 1
!
! V k 1 ¼ k 1 b1 b2 b1 b2 ; 2
ð37Þ
1 02 0 0 V k 2 ¼ k 2 ðd 02 3 þ d 4 d 3 d 4 cos h3 Þ; 2
ð38Þ
1 02 0 0 V k 3 ¼ k 3 ðd 02 5 þ d 6 d 5 d 6 sin h3 Þ; 2
ð39Þ
1 02 0 0 V k 4 ¼ k 4 ðd 02 ð40Þ 7 þ d 8 d 7 d 8 cos h2 Þ; 2
! where b1 b2 is given by
! ð41Þ b1 b2 ¼ d 01 x1 þ d 2 z1 þ a2 x2 þ d 3 z2 þ d 02 x3 :
! Upon substitution of b1 b2 into Eq. (37) and inserting the angles between the different axes from Table 2 into the result thus obtained, one can obtain V k1 as 1 1 2 2 02 0 0 2 V k 1 ¼ k 1 ðd 02 1 þ d 2 þ a2 þ d 3 þ d 2 þ 2d 2 d 3 cos a2 Þ þ k 1 ð2d 1 a2 cos h2 2d 1 d 3 sin h2 sin a2 2 2 2d 01 d 02 x1 x3 þ 2d 2 d 02 sin h3 sin a2 þ 2a2 d 02 cos h3 Þ: ð42Þ Finally, the expression for the potential energy of the four springs and configuration varying terms remaining in the total potential energy is written as V r3 ¼ kc1x c2x cos h2 þ kc1x c2z sin a2 sin h2 þ kc1x c3x x1 x3 þ kc1z c3x sin a2 sin h3 þ kc2x c3x cos h3 1 þ C þ k 1 ð2d 01 a2 cos h2 2d 01 d 3 sin h2 sin a2 2d 01 d 02 x1 x3 þ 2d 2 d 02 sin h3 sin a2 2 þ 2a2 d 02 cos h3 Þ k 2 d 03 d 04 cos h3 k 3 d 05 d 06 sin h3 k 4 d 07 d 08 cos h2 ; ð43Þ where C is the constant. This expression can be classified in terms of variable joint angles as V r3 ¼ ðkc1x c2x k 1 d 01 a2 k 4 d 07 d 08 Þ cos h2 þ ðkc1x c2z k 1 d 01 d 3 Þ sin h2 sin a2 þ ðkc1x c3x k 1 d 01 d 02 Þx1 x3 þ ðkc2x c3x þ k 1 a2 d 02 k 2 d 03 d 04 Þ cos h3 þ ðkc1z c3x sin a2 þ k 1 d 2 d 02 sin a2 k 3 d 05 d 06 Þ sin h3 :
ð44Þ
The values of ki for i = 1, . . ., 4 are derived to make the potential energy invariant with configuration. The expressions for the spring stiffnesses are kc c ð45Þ k 1 ¼ 1x 02z ; d 3d 1 d 02 ¼
kc1x c3x ; k 1 d 01
ð46Þ
k2 ¼
kc2x c3x þ k 1 a2 d 02 ; d 03 d 04
ð47Þ
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Fig. 9. Fractional mechanism to locate system center of mass.
k3 ¼
kc1z c3x sin a2 þ k 1 d 02 d 2 sin a2 ; d 05 d 06
ð48Þ
k4 ¼
kc1x c2x k 1 d 01 a2 s: d 07 d 08
ð49Þ
Note that all d 0i , i = 1,. . .,8, are arbitrary except d 02 , specified by Eq. (47). The location of the system center of mass can be realized using the following procedure: (i) locate the center of mass of the first two links C12 using the method shown in Section 3.1; (ii) locate the center of mass of the third link C3; (iii) locate the system center of mass C using a fractional mechanism shown in Fig. 9. A fractional mechanism works in the following way: If the masses m1 + m2 and m3 are located at points C12 and C3, respectively, we use two links of arbitrary 1 þm2 l1 lengths l1 and l2 connected by a revolute joint. Two auxiliary links of lengths g1 l1 ¼ m1mþm 2 þm3 m3 and g2 l2 ¼ m1 þm2 þm3 l2 are attached to the parent links to form a parallelogram with revolute joints so that all joints of the fractional mechanism are parallel to each other. g1l1 and g2l2 form a parallelogram and locate the system center of mass C if this mechanism is attached to the manipulator at points C12 and C3 using two spherical joints. This method can be extended to include more links within the spatial manipulator.
4. Conclusion The paper described the underlying theory of gravity balanced for spatial robotic manipulators through a hybrid strategy which uses springs in addition to identifying the center of mass using parallelograms and auxiliary links. In the literature, it has been proved that springs alone cannot gravity balance a spatial multi-degree-of-freedom mechanism unless the attachment points of the springs are actively changed during motion of the manipulator. This paper showed that springs with ends fixed on the mechanism are sufficient to gravity balance a spatial mechanism if a hybrid method for gravity balancing is used, where the center of mass is identified first through auxiliary parallelograms. Also, the system remains gravity balanced even if the orientation of the base is
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changed, i.e., the direction of the gravity is changed with respect to the base. The method for gravity compensation was illustrated by two and three degrees-of-freedom (DOF) spatial manipulators. Currently, a prototype with the underlying principles of this paper is under fabrication.
Acknowledgments The support of NIH grant # 1 RO1 HD38582-01A2 is gratefully acknowledged. The authors acknowledge the help of Mark Deaver in the construction of the demonstration prototype.
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Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Gravity-balancing of spatial robotic manipulators Sunil K. Agrawal *, Abbas Fattah Mechanical Systems Laboratory, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Accepted 25 May 2004 Available online 2 October 2004
Abstract This paper describes the underlying theory of gravity balanced spatial robotic manipulators through a hybrid strategy which uses springs in addition to identification of the center of mass using auxiliary parallelograms. A significant contribution of this paper is to show that springs with fixed ends are sufficient to gravity balance a spatial mechanism if the hybrid method of gravity balancing is used where the center of mass is identified first through auxiliary parallelograms. Also, the system remains gravity balanced even if the orientation of the base is changed, i.e., the direction of the gravity is changed with respect to the base. Although the method can be applied to n link spatial serial manipulators, we apply the method for gravity compensation of two and three degrees-of-freedom (DOF) spatial manipulators. A prototype with the underlying principles of this paper was fabricated at the University of Delaware. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction A machine is said to be gravity balanced if joint actuator inputs are not needed to keep the system in equilibrium at any configuration of the machine. The system essentially behaves as if its motion is in a zero-gravity environment. Over the years, gravity balanced machines have been
*
Corresponding author. Tel.: +1 302 831 8049; fax: +1 302 831 3619. E-mail address: [email protected] (S.K. Agrawal).
0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.019
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realized through clever designs using counterweights, springs, and auxiliary parallelograms. These designs have been introduced to linkages [2,9,7], serial robotic manipulators [8,11] and parallel manipulators [4,10]. A number of mathematical descriptions can be given for gravity balanced machines such as (i) system center of mass remains inertially fixed during motion; (ii) potential energy remains invariant with configuration of the system; (iii) system has counter masses that balance the machine in every configuration. These mathematical conditions have been physically realized through clever engineering such as: (a) countermass on each body of the machine is used to inertially fix the center of mass of the system [2,10], (b) springs are used at appropriate places in the machine such that the sum total of the gravitational potential energy and spring potential energy together becomes invariant with configuration [5,3,10], (c) auxiliary parallelograms based on knowledge of geometry and inertia property are used to physically determine the center of mass of the machine [1]. While each of these methods have their scientific core along with advantages and disadvantages, this paper focuses on a hybrid methodology for gravity balancing of spatial robotic manipulators that combines the underlying fundamentals between these different methods. In the methods presented in the literature using springs, gravity balancing is achieved for a fixed direction of the gravity vector. However, using the method presented in this paper, under valid assumption, balancing is independent of the direction of the gravity vector. In other words, the base of the machine can be oriented in any direction with respect to the direction of the gravity vector. This paper uses the notion of auxiliary parallelograms to first locate the center of mass. In addition, springs are connected through this point to make the potential energy invariant with configuration. For these reasons, this hybrid two-step approach is desirable over approaches which do not consider finding the center of mass first. At this time, only a few studies exist on gravity balancing of spatial manipulators. A spatial two-dof serial manipulator was studied for gravity balancing in [11]. The authors proved in this study that for a two-dof manipulator, the conditions of gravity balancing requires that the end points of the springs move relative to the mechanism, i.e., be independently actuated. This will require extra actuators to be mounted on the system and is undesirable. In this paper, we show that there is no need for the end points of the springs to be actively moved if the hybrid method presented in this paper is used. The organization of this paper is as follows: Section 2 describes the underlying theory of gravity balanced robotic manipulators through hybrid strategy which uses springs in addition to identification of the center of mass using auxiliary parallelograms. Next, we present the method for gravity compensation of two and three degrees-of-freedom (DOF) spatial robotic manipulators.
2. Gravity balancing: a hybrid method The underlying idea behind the hybrid method is as follows: (i) determine the center of mass of the manipulator by adding auxiliary links to machine; (ii) select springs to connect to the center of mass such that the total potential energy of the system is invariant with configuration. This method allows one to physically determine the center of mass of the manipulator and connect this point to the inertially fixed frame through springs.
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2.1. Location of system center of mass Consider an n-link articulated serial manipulator with revolute joints. The Deneavit–Hartenberg (DH) parameters for two successive links, i.e., hi, di, ai and ai, shown in Fig. 1, which characterizes the motion of link i with respect to link i 1 in terms of the following transformation matrix. 1 0 sin ai sin hi ai cos hi cos hi cos ai sin hi B sin h cos ai cos hi sin ai cos hi ai sin hi C i C B i1 ð1Þ Ti ¼ B C: @ 0 sin ai cos ai di A 0
0
0
1
Here, i = 1, . . ., n and hi is the variable joint angle, while all other DH parameters are constants. The location of the system center of mass C for this manipulator from the reference point (origin) O is n 1 X mi rOCi ; ð2Þ rOC ¼ M i¼1 where M¼
n X
ð3Þ
mi :
i¼1
Here, M is the total mass of the manipulator, mi is the mass of link i, and rOCi is the location of the center of mass of link i, Ci, from point O. The vector rOCi can be written as rOCi ¼ rOOi þ rOi Ci ;
ð4Þ
where Oi is the origin of the frame attached to link i. Using Fig. 1, rOOi is rOOi ¼
i X
ðd j zj1 þ aj xj Þ
j¼1
Fig. 1. Denavit–Hartenberg frame assignment for link i.
ð5Þ
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and the location of center of mass of link i from point Oi in its local frame is expressed as rOi Ci ¼ bix xi þ biy yi þ biz zi :
ð6Þ
Upon substitution of Eqs. (5) and (6) into Eq. (4) and inserting the result thus obtained into Eq. (2), we obtain " # n i X 1 X ð7Þ mi ðd j zj1 þ aj xj Þ þ bix xi þ biy yi þ biz zi rOC ¼ M i¼1 j¼1 or in compact form as n X rOC ¼ ðcix xi þ ciy yi þ ciz zi Þ:
ð8Þ
i¼1
Here, it is assumed d1 = 0. Note that cix, ciy, ciz are factors of geometry and mass distribution of the links and are usually denoted by the terminology ‘‘scaled lengths’’. Having these scaled lengths, one can determine the location of the system center of mass by adding appropriate auxiliary links to machine. Here, we assume the masses of auxiliary links to be negligible as compared to the masses of the links. This is an acceptable assumption. However, the masses of auxiliary links can be also considered using the method presented in [1]. Vector rOC can be expressed in any coordinate frame by expressing the vectors xi, yi and zi in this coordinate frame. For example, in the inertially fixed reference frame can be written as rOC ¼ d1 x0 þ d2 y0 þ d3 z0 ;
ð9Þ
where di, i = 1, 2, 3 are configuration dependent, functions of scaled lengths, and also DH parameters of all links. 2.2. Springs selection Upon locating the system center of mass, the potential energy of the system due to the gravity can be written as V g ¼ Mgg rOC ;
ð10Þ
where g is the unit vector along the gravity. It can be expressed in the inertial frame as g ¼ n1 x0 þ n2 y0 þ n3 z0 ;
ð11Þ
where all ni, i = 1,2, 3 are constants. For gravity compensation, we attach a spring of stiffness k between the system center of mass C and a point P along the gravity direction at a distance d from origin, O shown in Fig. 2. The strain potential energy due to the spring is 1 V s ¼ kx2 ; 2 2 where x is
!
! x2 ¼ k PC k k PC k:
ð12Þ
ð13Þ
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Fig. 2. n-link robotic manipulator with spring connections.
! Here, PC can be written as
! PC ¼ rOC rOP
ð14Þ
and rOP ¼ dðn1 x0 þ n2 y0 þ n3 z0 Þ:
ð15Þ
Upon substitution of Eqs. (15) and (9) into Eq. (14) and inserting the result thus obtained into Eq. (13) and then substituting this into Eq. (12), we obtain 1 V s ¼ k½ðd1 dn1 Þ2 þ ðd2 dn2 Þ2 þ ðd3 dn3 Þ2 : ð16Þ 2 Upon substitution of Eqs. (11) and (9) into Eq. (10) and expanding the result thus obtained, we get Vg as V g ¼ Mgðd1 n1 þ d2 n2 þ d3 n3 Þ:
ð17Þ
Therefore, the total potential energy of the system is the sum of potential energies due to gravity and the spring 1 V ¼ V g þ V s ¼ ðMg kdÞd1 n1 þ ðMg kdÞd2 n2 þ ðMg kdÞd3 n3 þ kðn21 þ n22 þ n23 Þd 2 2 1 þ kðd21 þ d22 þ d23 Þ: ð18Þ 2 The total potential energy is invariant with configuration if the coefficients of configuration variable terms vanish. Hence, the stiffness of spring k is determined by setting the coefficients of dini, i = 1, 2, 3 to zero, i.e., Mg : ð19Þ k¼ d The only configuration varying term remaining in the total potential energy is 1 kðd21 þ d22 þ d23 Þ or simply 12 krOC rOC . It may be noted that this term is function of all joint 2 angles, i.e., hi, i = 1, . . ., n. The strategy is to add appropriate springs to the system such that the total potential energy of the system becomes configuration invariant. To this end, we should select springs such that the configuration dependent terms in potential energy vanish.
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First, we derive and classify these terms in terms of variable joint angles and then we try to attach springs to appropriate points such that they can cancel out the variable terms in total potential energy. The number of extra springs depends on the number of variable terms in total potential energy expression. If we change the base of the manipulator with respect to the gravity vector, the unit gravity vector g can be re-expressed in the inertial frame as g ¼ n01 x0 þ n02 y0 þ n03 z0 ;
ð20Þ
n0i ,
i = 1, 2, 3 are constants. This is the only change that we should make when the base where all has a different orientations with respect to the gravity vector. It is clearly seen from the procedure mentioned in this section that the gravity balancing of the manipulator does not depend on the coefficients n0i , i = 1, 2, 3. Therefore, this method results in gravity balancing for all directions of the gravity vector. In other words, the base of machine can be installed in any direction with respect to gravity vector. However, the point P in Fig. 2 remains always on the axis of the gravity vector. In other words, OP is kept in the direction of the gravity when the base of the platform is rotated. In this work, it is assumed that the undeformed length of the spring is zero. In other words, the spring force is zero when the deformation of the spring is also zero. In the physical implementation of zero free length, non-zero free length spring can be used behind the pulley where the spring force can be transmitted through the wire [6]. The method of gravity balancing mentioned in this section is applied to n-link spatial serial manipulators. However, to show some examples, we apply the method to gravity balancing of two and three degrees-of-freedom (DOF) spatial manipulators in the next section. The method can be extended to consider example of more dof serial manipulators.
3. Examples 3.1. Two DOF robotic manipulator Consider a two DOF spatial manipulator, i.e., n = 2 in Fig. 2. Here, the axis of joint 1 (z0) is not parallel to the axis of joint 2 (z1), i.e., a1 5 0. However, we assume z1 is parallel to the axis of joint 3 (z2), i.e., a2 = 0. The location of the system center of mass of two DOF manipulator can be derived using n = 2 in Eq. (8) as follows: rOC
2 X ¼ ðcix xi þ ciy yi þ ciz zi Þ;
ð21Þ
i¼1
where the coefficients are constants and can be computed inserting n = 2 into Eq. (7). The resulting coefficients are m1 ð22Þ c1x ¼ a1 þ b1x ; M c1y ¼
m1 b ; M 1y
ð23Þ
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c1z ¼
m1 m2 b1z þ d 2 ; M M
ð24Þ
c2x ¼
m2 ða2 þ b2x Þ; M
ð25Þ
c2y ¼
m2 b ; M 2y
ð26Þ
m2 b : M 2z The expression 12 krOC rOC for two-DOF manipulator is c2z ¼
ð27Þ
1 1 krOC rOC ¼ kðc21x þ 2c1x c2x x1 x2 þ 2c1x c2y x1 y2 þ 2c1x c2z x1 z2 þ c21y þ 2c1y c2x y1 2 2 x2 þ 2c1y c2y y1 y2 þ 2c1y c2z y1 z2 þ c21z þ 2c1z c2x z1 x2 þ 2c1z c2y z1 y2 þ 2c1z c2z z1 z2 þ c22x þ c22y þ c22z Þ:
ð28Þ
Upon substitution of i = 2, a2 = 0 in the transformation matrix given by Eq. (1), we obtain the angle between the axes of the local frames of two successive links 1 and 2 shown in Table 1. Then, inserting the results from Table 1 into Eq. (28), one obtains the non-zero configuration varying terms in 12 krOC rOC . These terms are kðc1x c2x þ c1y c2y Þ cos h2 and kðc1x c2y þ c1y c2x Þ sin h2 . We add two springs to compensate for the two configuration varying terms: a spring with stiffness k1 between the axes x1 and x2 and a second spring k2 between the axes y1 and x2. The connection points d 0i , i = 1, . . ., 4 are shown in Fig. 3. The expression for the potential energy of the two springs and configuration varying terms remaining in the total potential energy is written as
Table 1 The angle between the axes of the local frames at links 1 and 2 x2 y2 z2
x1
y1
z1
cos h2 sin h2 0
sin h2 cos h2 0
0 0 1
Fig. 3. Spring connections for 2-link manipulator.
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1 02 0 0 V r ¼ kðc1x c2x þ c1y c2y Þ cos h2 þ kðc1x c2y þ c1y c2x Þ sin h2 þ k 1 ðd 02 1 þ d 2 2d 1 d 2 Þ cos h2 2 1 02 0 0 þ k 2 ðd 02 ð29Þ 3 þ d 4 2d 3 d 4 Þ sin h2 : 2 The values of k1 and k2 are selected such that Eq. (29) becomes configuration invariant. The expressions for these two stiffnesses are k1 ¼
kðc1x c2x þ c1y c2y Þ ; d 01 d 02
ð30Þ
k2 ¼
kðc1x c2y þ c1y c2x Þ : d 03 d 04
ð31Þ
Therefore, the two DOF spatial manipulator is completely gravity balanced by using three from the center springs shown in Fig. 4. These three springs are: (i) a spring with stiffness k ¼ Mg d of mass of the manipulator to a point on a line from the origin of the inertially fixed frame parallel to the gravity vector, (ii) two more springs with stiffnesses k1 and k2. Please note that the location of the center of mass of the manipulator can be identified using auxiliary parallelograms based on Eq. (21). However, the location of the center of mass has a simpler expression when described with respect to point O1 instead of O and is given as rO1 C ¼ rOC rOO1 ¼ rOC a1 x1 ;
ð32Þ
where rOC was defined in Eq. (21). 3.1.1. Two DOF manipulator: special case We obtain the conditions for gravity balancing of a 2-link spatial manipulator when the center of mass of the links are located on xizi plane, i.e., ciy = 0 in Eq. (21). Using these simplifications in Eqs. (30) and (31), one gets
Fig. 4. 2-link gravity balanced spatial manipulator with spring connections.
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Fig. 5. 2-link gravity balanced spatial manipulator with spring connections: special case.
k1 ¼
kc1x c2x ; d 01 d 02
k 2 ¼ 0:
ð33Þ ð34Þ
Therefore, the 2-link spatial manipulator is gravity balanced using only two springs—one from the center of mass of the manipulator to an inertially fixed point along the gravity direction passand the second spring of stiffness k1 between the axis ing through the origin, with stiffness k ¼ Mg d x1 and x2, shown in Fig. 5. 3.1.2. Prototype of two DOF manipulator An engineering prototype based on the design explained in Section 3.1.1 was fabricated at University of Delaware and is shown in Fig. 6. This prototype is made of wooden parts and it has the following DH and inertia parameters. a1 = 0.5 m, d1 = 0, a1 = p/6 rad, d2 = 0.3 m, a2 = 0, a2 = 0.4 m, m1 = 4.188 kg and m2 = 3.903 kg. It verifies the conceptual idea of spatial gravity balancing of two DOF manipulator and was tested on different configurations of end-effector and also different orientations of the base (see http://mechsys4.me.udel.edu/movies/rehrob/spabalmov/). Adding auxiliary links results in some limitation for the motion of the prototype. As an example, the second link of the manipulator cannot be fully extended because of the auxiliary links. However, this limitation can be alleviated by changing the shape of these links. Also the manipulator cannot be rotated fully (360°) about the base because of the spring which is connected to the system center of mass. This is not a problem since the usual range of the motion for the joints are much smaller than 360°. 3.2. Three DOF spatial manipulator Consider a three DOF spatial manipulator, i.e., n = 3, as shown in Fig. 7. Here, the three joint axes 1, 2, and 3 axes are not parallel to each other. However, we assume that joint axis 3 (z2) is
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Fig. 6. Photographs of prototype of the two DOF spatial gravity balanced manipulator at different orientations of the base.
Fig. 7. A 3-link spatial manipulator.
parallel to joint axis 4 (z3), i.e., a3 = 0. The location of the system center of mass of three DOF manipulator can be derived using n = 3 in Eq. (8) as follows: rOC ¼
3 X ðcix xi þ ciy yi þ ciz zi Þ;
ð35Þ
i¼1
where the coefficients are constants and can be computed by inserting n = 3 into Eq. (7). Here, we assume the center of mass of the links are located on xizi plane, i.e., ciy = 0 in Eq. (21). The nonzero terms of expression 12 krOC rOC for the three-DOF manipulator are
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1 1 krOC rOC ¼ kðc21x þ c1x c2x x1 x2 þ c1x c2z x1 z2 þ c1x c3x x1 x3 þ c1x c3z x1 z3 þ c21z 2 2 þ c1z c2x z1 x2 þ c1z c2z z1 z2 þ c1z c3x z1 x3 þ kc1z c3z z1 z3 þ c22x þ c2x c3x x2 x3 þ c2x c3z x2 z3 þ c22z þ c2z c3x z2 x3 þ c2z c3z z2 z3 þ c23x þ c23z Þ:
ð36Þ
Upon substituting i = 2, 3 and a3 = 0 in transformation matrix given by Eq. (1), we obtain the angles between the different axes of the coordinate frames as shown in Table 2. Inserting the results from Table 2 into the non-zero terms of Eq. (36) and noting that c3z = 0 because the center of mass of link 3 is located on the z2x3 plane, one obtains the remaining non-zero terms to be as kc1x c2x cos h2 þ kc1x c2z sin a2 sin h2 þ kc1x c3x x1 x3 þ kc1z c3x sin a2 sin h3 þ kc2x c3x cos h3 . We add four springs to compensate for these five terms: (i) a spring k1 between axis x1 and x3, (ii) a spring k2 between x2 and x3, (iii) a spring k3 between y2 and x3, and (iv) a spring k4 between axis x1 and x2 with connection points d 0i , i = 1, . . ., 8, respectively, as shown in Fig. 8.
Table 2 The dot products of the axes of the local coordinate frames x2 y2 z2
x3 y3 z3
x1
y1
z1
cos h2 sin h2cos a2 sinh2sin a2
sin h2 cos h2cos a2 cos h2sin a2
0 sin a2 cos a2
x2
y2
z2
cos h3 sin h3 0
sin h3 cos h3 0
0 0 1
Fig. 8. 3-link gravity balanced spatial manipulator with springs connection.
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The expression for the potential energy of the four springs can be written as 1
!
! V k 1 ¼ k 1 b1 b2 b1 b2 ; 2
ð37Þ
1 02 0 0 V k 2 ¼ k 2 ðd 02 3 þ d 4 d 3 d 4 cos h3 Þ; 2
ð38Þ
1 02 0 0 V k 3 ¼ k 3 ðd 02 5 þ d 6 d 5 d 6 sin h3 Þ; 2
ð39Þ
1 02 0 0 V k 4 ¼ k 4 ðd 02 ð40Þ 7 þ d 8 d 7 d 8 cos h2 Þ; 2
! where b1 b2 is given by
! ð41Þ b1 b2 ¼ d 01 x1 þ d 2 z1 þ a2 x2 þ d 3 z2 þ d 02 x3 :
! Upon substitution of b1 b2 into Eq. (37) and inserting the angles between the different axes from Table 2 into the result thus obtained, one can obtain V k1 as 1 1 2 2 02 0 0 2 V k 1 ¼ k 1 ðd 02 1 þ d 2 þ a2 þ d 3 þ d 2 þ 2d 2 d 3 cos a2 Þ þ k 1 ð2d 1 a2 cos h2 2d 1 d 3 sin h2 sin a2 2 2 2d 01 d 02 x1 x3 þ 2d 2 d 02 sin h3 sin a2 þ 2a2 d 02 cos h3 Þ: ð42Þ Finally, the expression for the potential energy of the four springs and configuration varying terms remaining in the total potential energy is written as V r3 ¼ kc1x c2x cos h2 þ kc1x c2z sin a2 sin h2 þ kc1x c3x x1 x3 þ kc1z c3x sin a2 sin h3 þ kc2x c3x cos h3 1 þ C þ k 1 ð2d 01 a2 cos h2 2d 01 d 3 sin h2 sin a2 2d 01 d 02 x1 x3 þ 2d 2 d 02 sin h3 sin a2 2 þ 2a2 d 02 cos h3 Þ k 2 d 03 d 04 cos h3 k 3 d 05 d 06 sin h3 k 4 d 07 d 08 cos h2 ; ð43Þ where C is the constant. This expression can be classified in terms of variable joint angles as V r3 ¼ ðkc1x c2x k 1 d 01 a2 k 4 d 07 d 08 Þ cos h2 þ ðkc1x c2z k 1 d 01 d 3 Þ sin h2 sin a2 þ ðkc1x c3x k 1 d 01 d 02 Þx1 x3 þ ðkc2x c3x þ k 1 a2 d 02 k 2 d 03 d 04 Þ cos h3 þ ðkc1z c3x sin a2 þ k 1 d 2 d 02 sin a2 k 3 d 05 d 06 Þ sin h3 :
ð44Þ
The values of ki for i = 1, . . ., 4 are derived to make the potential energy invariant with configuration. The expressions for the spring stiffnesses are kc c ð45Þ k 1 ¼ 1x 02z ; d 3d 1 d 02 ¼
kc1x c3x ; k 1 d 01
ð46Þ
k2 ¼
kc2x c3x þ k 1 a2 d 02 ; d 03 d 04
ð47Þ
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Fig. 9. Fractional mechanism to locate system center of mass.
k3 ¼
kc1z c3x sin a2 þ k 1 d 02 d 2 sin a2 ; d 05 d 06
ð48Þ
k4 ¼
kc1x c2x k 1 d 01 a2 s: d 07 d 08
ð49Þ
Note that all d 0i , i = 1,. . .,8, are arbitrary except d 02 , specified by Eq. (47). The location of the system center of mass can be realized using the following procedure: (i) locate the center of mass of the first two links C12 using the method shown in Section 3.1; (ii) locate the center of mass of the third link C3; (iii) locate the system center of mass C using a fractional mechanism shown in Fig. 9. A fractional mechanism works in the following way: If the masses m1 + m2 and m3 are located at points C12 and C3, respectively, we use two links of arbitrary 1 þm2 l1 lengths l1 and l2 connected by a revolute joint. Two auxiliary links of lengths g1 l1 ¼ m1mþm 2 þm3 m3 and g2 l2 ¼ m1 þm2 þm3 l2 are attached to the parent links to form a parallelogram with revolute joints so that all joints of the fractional mechanism are parallel to each other. g1l1 and g2l2 form a parallelogram and locate the system center of mass C if this mechanism is attached to the manipulator at points C12 and C3 using two spherical joints. This method can be extended to include more links within the spatial manipulator.
4. Conclusion The paper described the underlying theory of gravity balanced for spatial robotic manipulators through a hybrid strategy which uses springs in addition to identifying the center of mass using parallelograms and auxiliary links. In the literature, it has been proved that springs alone cannot gravity balance a spatial multi-degree-of-freedom mechanism unless the attachment points of the springs are actively changed during motion of the manipulator. This paper showed that springs with ends fixed on the mechanism are sufficient to gravity balance a spatial mechanism if a hybrid method for gravity balancing is used, where the center of mass is identified first through auxiliary parallelograms. Also, the system remains gravity balanced even if the orientation of the base is
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changed, i.e., the direction of the gravity is changed with respect to the base. The method for gravity compensation was illustrated by two and three degrees-of-freedom (DOF) spatial manipulators. Currently, a prototype with the underlying principles of this paper is under fabrication.
Acknowledgments The support of NIH grant # 1 RO1 HD38582-01A2 is gratefully acknowledged. The authors acknowledge the help of Mark Deaver in the construction of the demonstration prototype.
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