Design of planar static balancer with associated linkage

Mechanism and Machine Theory 81 (2014) 79–93

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Design of planar static balancer with associated linkage Sang-Hyung Kim a, Chang-Hyun Cho b,⁎ a b

Chosun University, Dept. of Mechanical Engineering, College of Engineering, Engineering Building 1, 309 Pilmun-daero Dong-gu, Gwangju 501-759, Republic of Korea Chosun University, Dept. of Mechanism & Systems, College of Engineering, Engineering Building 1, 309 Pilmun-daero Dong-gu Gwangju 501-759, Republic of Korea

a r t i c l e

i n f o

Article history: Received 18 March 2014 Received in revised form 24 June 2014 Accepted 24 June 2014 Available online xxxx Keywords: Associated linkage Gravity compensation Static balancing Space mapping Design equation

a b s t r a c t This paper presents a design method for a static balancer with associated linkage. Various mechanisms can be obtained with modifications to the associated linkage. Gravity compensators for various mechanisms can be achieved similarly from a gravity compensator for the associated linkage. The space mapping method is adopted to design a gravity compensator for the associated linkage. Conversion rules are derived by investigating the variances of a mechanism from the associated linkage and are applied to the design equation for the associated linkage generated by the space mapping method. Rows and columns of the design equation are deleted by conversion rules, leading to deletion rules. A new gravity compensator for the mechanism derived from the associated linkage is obtained by applying the deletion rules to the design equation (i.e., gravity compensator) for the associated linkage. The four-bar mechanism is adopted as the associated linkage, and various gravity compensators for planar mechanisms are derived from the gravity compensator of the four-bar linkage. Simulations are conducted, and the results show that complete gravity compensation is possible for various planar mechanisms. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Gravity compensators with springs have been proposed for several decades. Nathan has proposed a 1-DOF (degree of freedom) gravity compensator in which one end of a spring is fixed at the base and the other end is attached at a moving link [1]. Ulrich and Kumar have suggested a 1-DOF gravity compensator that uses wire and a pulley [2]. An internal cam mechanism has been designed by Koser and applied to a five-bar mechanism [3]. Herder has presented a static balancer for a variable load equipped with a storage spring [4]. A hybrid design has been suggested in which a parallel linkage is adopted to indicate the COM (center of mass) [5]. A 2-DOF gravity compensator for roll-pitch rotations has been proposed [6]. The roll-pitch rotations are decoupled with bevel gears, and two 1-DOF gravity compensators are installed at the rotating bevel gears. Static balancers of a parallel mechanism have been studied using counter masses and springs [7,8]. A gravity compensator for a service robot has been developed [9]. A gravity compensator using a hemispherical magnet has been proposed [10]. Relations between static balancing parameters of the cognates of a four-bar linkage have been studied and a static balancing method for a general n-DOF revolute and spherical jointed rigid-body linkages has been developed [11]. This paper proposes a design method of a static balancer by which gravity compensators of various mechanisms are obtained from a gravity compensator of the associated linkage. Gravity compensators of various mechanisms are designed with intensive analyses of various mechanisms (e.g., computation of the potential energy) in general. In the proposed method, however, gravity compensators of various mechanisms are simply obtained with modifications to the gravity compensator for the associated linkage.

⁎ Corresponding author. Tel.: +82 62 230 7389. E-mail addresses: [email protected] (S.-H. Kim), [email protected] (C.-H. Cho).

http://dx.doi.org/10.1016/j.mechmachtheory.2014.06.012 0094-114X/© 2014 Elsevier Ltd. All rights reserved.

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Various mechanisms can be obtained with modifications to the associated linkage. For example, the slider crank is derived from the four-bar linkage by replacing a revolute joint with a prismatic joint. When the associated linkage is converted into a new mechanism, some joints are ignored, and some masses of linkages are merged into adjacent links. Therefore, conversion rules are derived by investigating the variances of the mechanism derived from the associated linkage. The space mapping method [12] is adopted to design a gravity compensator for associated linkage. The mapping between joints and unit gravity compensators is determined with the mapping matrix, and the design equation is derived. The mapping matrix indicates both the number of unit gravity compensators and their locations (i.e., their kinematic constraints). The conversion rules are applied to the design equation for the associated linkage generated by the space mapping method. Rows and columns of the design equation are deleted by using the conversion rules, resulting in a new design equation and a new mapping matrix for the derived mechanism. The so-called deletion rules are obtained. Hence, gravity compensators for mechanisms derived from the associate linkage can be simply obtained from the gravity compensator for the associate linkage by applying the deletion rules. In this paper, the four-bar mechanism is adopted as the associated linkage, and various gravity compensators for planar mechanisms are derived from the gravity compensator of the four-bar linkage. Simulations are conducted, and the results show that complete gravity compensation is possible for various planar mechanisms. 2. The space mapping method The design of a spring balancer is considered as a mapping between two spaces (i.e., the joint space for gravitational torques and the gravity compensator space for compensating torques). The mapping matrix indicates the mechanical connections of unit gravity compensators with respect to a target mechanism. The design method with space mapping is briefly summarized in this paper. Please refer to [12] for more details. 2.1. Space mapping The joint space is predetermined as θ = [θ1, θ2,…, θn]T ∈ Rn × 1, where θi denotes the rotation angle of the i-th joint, and n represents the number of joints. For simple analysis, we assume that the gravity compensator space consists of only a 1-DOF gravity compensator. In this case, the gravity compensator space can be determined as θg = [θg1, θg2,…, θgm]T ∈ Rm × 1, where θgi denotes the rotation angle of the i-th gravity compensator, and m represents the number of 1-DOF gravity compensators. The rotation angles of the gravity compensators (i.e., θg) are passively determined by the pose of the mechanism (i.e., θ). Thus, functions or relationships exist between the joint space and the gravity compensator space. Suppose that θg is computed with θ as follows: θg ¼ Jθ þ ϕ

ð1Þ

where J ∈ Rm × n and ϕ ∈ Rm × 1. J denotes a mapping matrix between the joint space θ and the gravity compensator space θg, and ϕ represents a vector of constant phase angles. 2.2. Potential energy in both spaces Let 0Pi be the position of the COM of link i with respect to the {0} frame. Then, the potential energy of mass mi is obtained by Vm ¼ −

n X

0

ð2Þ

mi g Pi

i¼1

where g represents the gravitational vector. 0Pi is computed from [0Pi;1] = 0 Ti[iPi;1], where 0 Ti ¼0 T1 1 T2 …i−1 Ti denotes the transformation matrix. Because 0 Ti is determined by θ, Eq. (2) represents the potential energy in the joint space.

A h

k b

O

x

l

m

B

θ

Fig. 1. A 1-DOF gravity compensator [1].

g y

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81

The potential energy in gravity compensator space is derived. The 1-DOF gravity compensator in [1] is used in this study and is shown in Fig. 1. One end of a spring is attached at point A, which is fixed in the ground, and the other end is attached to point B, which is located at the link. Point A (or B) is located at a distance h (or b) from the origin O. A zero-length spring, which has zero length at zero deflection (e.g., the initial state), is adopted. For practical implementation of the zero-length spring, the section between points A and B can be interconnected with a wire, and the spring is installed at the base or the arm. Refer to [4] for practical implementations of the zero-length spring. The potential energy of the spring is computed as Vk(θ) = kS2(θ)/2 = C + kbhcosθ, where C = k(h2 + b2)/2; k and S denote a spring constant and the deflection of a spring, respectively. The potential energy in the gravity compensator space is derived as Vk ¼

m X

  C j þ k j b j h j cos θgj

ð3Þ

j¼1

where Cj = kj(h 2j + b 2j )/2; and hj, bj, and kj are parameters of the j-th gravity compensator. 2.3. Design equation Complete gravity compensation is achieved when the total potential energy in the joint space and the gravity compensator space is invariant for all θ and θg; i.e., V(θ, θg) = Vm(θ) + Vk(θg) = constant. Because θg = Jθ + ϕ, V(θ, θg) becomes V(θ) = Vm(θ) + Vk(Jθ +ϕ). The partial differential of V(θ) by θi is written in matrix form: f ðθÞVm

max −J

T

MK ¼ 0

ð4Þ

where f(θ) ∈ Rn × n and Vm_max = [Vm1, Vm2,…, Vmn]T ∈ Rn × 1 ⋅ Vmj represents the maximum potential energy in the joint space measured at joint j, and fij(θ) denotes the ratio of the variance of Vmj by changing the pose of a given manipulator. (Vmj is constant because the link parameters and masses are constant in most cases.) Thus, we have M = diag[sin(J1θ + ϕ1), sin(J2θ + ϕ2),…, sin(Jmθ + ϕm)] = diag[sin(θg1), sin(θg2),…, sin(θgm)] ∈ Rm × m; and K = [k1b1h1, k2b2h2,…, kmbmhm]T ∈ Rm × 1, where diag[x1,…, xn] denotes an n × n diagonal matrix, and JT denotes the transpose of J (i.e., JT ∈ Rn × m). Note that the sine terms in M originate from the partial differential of the potential energy in the gravity compensator space of Eq. (3). 2.4. Determination of the mapping matrix The goal of the design is to determine the mapping matrix J. The number of row vectors of J represents the number of unit gravity compensators. The row vectors of J indicate the locations (or the kinematic constraints) of the unit gravity compensators. Thus, determination of J means that both the number of unit gravity compensators and their locations are obtained simultaneously. The characteristics of a given mechanism are investigated from the viewpoint of energy. Recall that f(θ) denotes ratios of the variance of potential energy in the joint space by changing the pose of a given mechanism. Eigenvalues of f(θ) are investigated; these represent characteristics of f(θ). Assume that the eigenvalues of f(θ) are λ = [λ1, λ2,…, λn]T ∈ Rn × 1, and λi can be decomposed with some basis functions. Considering M, the gravity compensator space is described with only simple sine functions because it consists of only 1-DOF gravity compensators in this study. In this situation, the basis function is chosen as a sine function that is a component of the gravity compensator space. λi can be decomposed with sine functions (i.e., basis functions):     m m λi ðθÞ ¼ Σ j¼1 dij sin θgj ¼ Σ j¼1 dij sin J j θ þ ϕ j

ð5Þ

where dij are any real values; m is the number of unit gravity compensators; and Jj = [ Jj1, Jj2,…, Jjn] ∈ R1 × n represents the j-th row vector of J. Thus, the mapping matrix J is simply obtained by decomposing the eigenvalues of f(θ) with components in the gravity compensator space. Suppose, for example, that λ1 = sin(θ1)cos(θ2). λ1 is decomposed with sine functions as λ1 = (sin(θ1 + θ2) + sin(θ1 − θ2))/2. In this situation, m = 2, J1 = [1, 1], J2 = [1, −1], and ϕj = 0 for j = 1, 2. Note that the row vector of J, Jj, indicates the kinematic constraints between joints and a unit gravity compensator. That is, the exact locations of springs are not determined with this method. Various designs can be achieved by considering the kinematic inversion for practical implementations. 3. Associated linkage 3.1. Concept of the associated linkage Type and dimensional syntheses are often used to design an appropriate mechanism. Type synthesis is applied to determine the kinematic structure of a mechanism or linkage in that the concept of associated linkage for type synthesis is introduced [13,14]. For the concept of associated linkage, a basic kinematic chain is selected and modified to obtain the desired mechanism. In this paper, the four-bar linkage is chosen as a basic kinematic chain consisting of only binary links and revolute joints. Note that the coupler

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point is not considered in this paper. The graph representation is used. For the graph representation, the vortex denotes a link, and the edge represents a joint. Numbers at the vortices indicate the i-th link; and P and R at the lines denote prismatic and revolute joints, respectively.

Fig. 2. Four-bar linkages.

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83

Fig. 2 shows mechanisms derived from the four-bar linkage, when a revolute joint is (or joints are) replaced by the prismatic joint (or joints) [15]. The associated linkage (i.e., the basic kinematic chain) is presented in Fig. 2(a). li and lig denote the length of link i and the distance of the COM of link i, respectively. θi represents the rotation angle of link i, and mi denotes the mass of link i. When a mechanism possesses one prismatic joint, the turning-block linkage, swinging-block linkage, and slider crank are obtained. When a mechanism possesses two prismatic joints, the cardanic motion linkage is obtained. Fig. 2(c), (e), and (g) represents mirror images of Fig. 2(b), (d), and (f) in the graph representation, respectively. Therefore, the five mechanisms of Fig. 2(a), (b), (d), (f), and (h) are considered in this paper. 3.2. The conversion rules We investigated the relationships between the basic kinematic chain (i.e., an associated linkage) and the derived mechanisms. When the revolute joint of the associated linkage is replaced by the prismatic joint, the parameters (e.g., li, lig, and mi) and rotation angles (e.g., θi) of the links to be considered vary during the computation of gravitational torques. We investigated variances of the parameters and rotation angles between the derived mechanisms (i.e., Fig. 2(b), (d), (f), and (h)) and the associated linkage (i.e., Fig. 2(a)) from the viewpoint of the gravitational torque. For easy analysis, link 4 is set as the fixed (or reference) link, and the indices of the links of all mechanisms are set to be identical. Table 1 lists the parameters and rotation angles eliminated from the associated linkage during computation of the gravitational torques of the derived mechanisms. For the derived mechanisms, the mass of link i of the associated linkage is merged into an adjacent link, when link i becomes the slider. A link into which a mass is merged is represented as ■ in Fig. 2. Conversion rules between the derived mechanism and associated linkage are summarized from the viewpoint of the gravitational torques as follows: 1. 2. 3. 4.

The mass of the fixed link is not considered. When link i of the associated linkage becomes a slider, li and lig are set to zero. mi of slider i is merged into a link connected slider i with a revolute joint. When mi is merged into the fixed link, mi is set to zero. All unnecessary joint angles for computation of positions of all mi's are ignored. When revolute joint i is replaced by a prismatic joint and is required for computation of positions of all mi's, the rotation angle of revolute joint i (θi) is set to a constant angle to indicate the direction of movement of a slider.

For example, the conversion rules are explained for the cases of Fig. 2(b) and (f). For Fig. 2(b), l1, l2, l1g, l2g, m1, m2, and m3 are used to compute gravitational torques. In this situation, m4 is ignored because link 4 is set as the fixed link. Link 3 becomes a slider, so l3 and l3g are set to zero. The mass of slider 3 (m3) is merged into link 2, where slider 3 is connected with a revolute joint. The positions of all m1's can be determined with θ1 and θ2, so θ3 and θ4 can be ignored. A joint space for Fig. 2(b) is determined as θ = [θ1, θ2]T. For Fig. 2(f), the mass of the fixed link (m4) is not considered. Because link 2 becomes a slider, l2 and l2g are set to zero. The mass of slider 2 (m2) is merged into link 1. θ2 and θ3 are ignored because the positions of all mi's can be determined with θ1 and θ4. Therefore, the joint space in the case of Fig. 2(f) becomes θ = [θ1, θ4]T. l1, l3, l1g, l3g, m1, m2, and m3 are considered in the computation of the gravitational torques of Fig. 2(f). 4. The deletion rules of the design equation When a prismatic joint is applied to an associated linkage, several parameters and joint angles are eliminated or ignored during the computation of gravitational torques as indicated in the conversion rules. The joint space of the derived mechanism (e.g., θ = [θ1, θ2]T for Fig. 2(b)) occurs as a subspace of the joint space of the associated linkage (e.g., θ = [θ1, θ2, θ3]T for Fig. 2(a)). Gravitational torques (or potential energy) of the derived mechanism may be obtained from those of the associated linkage by applying the conversion rules. Therefore, a gravity compensator for the derived mechanism can be obtained from that of the associated linkage by applying the conversion rules. Assume that an associated linkage is given with a joint space of θ = [θ1, θ2,…, θn]T ∈ Rn × 1, and a gravity compensator of the associated linkage satisfying the design equation exists. That is, a mapping matrix of J is determined. When reduction of the joint space occurs (some θi's are ignored), the i-th column vector of J and the i-th row vector of f(θ) for the associated linkage can be eliminated. Note that the i-th row vector of f(θ) represents partial differentials of the potential energy by θi. Link parameters of slider i are set to zero (i.e., li = lig = 0), and the mass of slider i is merged into an adjacent link, so Vmi becomes zero. When Vmi is set to zero, the i-th column vector of f(θ) can be ignored. The elimination of the i-th column vector of J is identical to the deletion of the i-th row Table 1 Eliminated parameters and joint angles.

Fig. 2(b) Fig. 2(c) Fig. 2(d) Fig. 2(e) Fig. 2(f) Fig. 2(g) Fig. 2(h)

li

lig

mi

θi

l3 l1 l3 l1 l2 l2 l1, l3

l3g l1g l3g l1g l2g l2g l1g, l3g

m4 m4 m4, m3 m4, m1 m4 m4 m4

θ3, θ4 θ1, θ2 θ3, θ4 θ1, θ2 θ2, θ3 θ2, θ3 θ1 = const., θ3, θ4

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vector of JT. When deletion of the i-th row vector of JT occurs, the i-th column vector of M and Ki are ignored. Because M is a diagonal matrix, the i-th row vector of M can also be eliminated by deletion of the i-th column vector of M. Elimination of the i-th row vector of M results in deletion of the i-th column vector of JT. Deletion of row and column vectors is presented in Fig. 3, and the numbers in Fig. 3 denote the sequence of deletion. The deletion rules are summarized as follows: Deleting the i-th row vector of JT by reduction of the joint space. Eliminating the i-th column and row vectors of M by procedure 1. Deleting Ki by procedure 2. Deleting the i-th column vector of JT by procedure 2. Deleting the i-th row vector of f(θ) by reduction of the joint space. Eliminating Vmi because the link parameters of slider i are set to zero (i.e., li = lig = 0) and the mass of slider i is merged into an adjacent link. 7. Deleting the i-th column vector of f(θ) by procedure 6. 1. 2. 3. 4. 5. 6.

When the conversion rules are applied to the design equation of an associated linkage, several row and column vectors are eliminated, resulting in a new design equation for the derived mechanism. Thus, a design equation for the derived mechanism can be easily obtained by deleting the row and column vectors of the design equation for an associated linkage. When slider i is located remotely from the fixed link, two open kinematic chains occur that consist only of revolute joints (see Fig. 2(f) and (g)). That is, deletion of rows occurs at the mid as shown in Fig. 3. In this situation, the masses of one open kinematic chain are eliminated mutually from the other chain during the computation of gravitational torques. Thus, masses of m1,…, mi − 1 (or mi + 1,…, mn) should be eliminated from Vm(i + 1),…, Vmn (or V m1 ; …; V mði−1Þ ). 5. Gravity compensators of the associated linkage We designed gravity compensators of the four-bar linkage (i.e., the associated linkage) with the space mapping method. The joint spaces can be chosen arbitrarily to represent the motions of all links. Positions of all mi's for the four-bar linkage can be determined with various joint spaces (e.g., [θ1, θ2, θ3]T, [θ1, θ2, θ4]T, [θ1, θ3, θ4]T, and [θ2, θ3, θ4]T). Two joint spaces of the four-bar linkage are considered in this paper: θb = [θ1, θ2, θ3]T and θa = [θ1, θ2, θ4]T. θb and θa denote the base joint space and alternative joint space, respectively. Therefore, two design equations (or gravity compensators) are obtained for the four-bar linkage. 5.1. A gravity compensator for the base joint space When joint 4 is eliminated, the four-bar linkage in Fig. 2(a) becomes a three-link open loop chain. The base joint space of θb = [θ1, θ2, θ3]T is adopted, and the position vectors of all mi's are determined as P1 ¼ l1g ½c1 ; s1 

T

ð6Þ

 T T P2 ¼ l1 ½c1 ; s1  þ l2g c1þ2 ; s1þ2

ð7Þ

 T  T T P3 ¼ l1 ½c1 ; s1  þ l2 c1þ2 ; s1þ2 þ l3g c1þ2þ3 ; s1þ2þ3

ð8Þ

where c1 (or s1) and c1 + 2 (or s1 + 2) denote cos(θ1) (or sin(θ1)) and cos(θ1 + θ2) (or sin(θ1 + θ2)), respectively. Suppose that the gravity exerts in the positive direction of the x-axis (i.e., g = [g, 0]T ∈ R2 × 1). Potential energy of the masses (or potential energy in the joint space) is computed by

Vm ¼ −

n¼3 X

      mi g  Pi ¼ −g m1 l1g þ ðm2 þ m3 Þl1 c1 −g m2 l2g þ m3 l2 c1þ2 −g m3 l3g c1þ2þ3 :

i¼1

Fig. 3. A new design equation from the design equation of an associated linkage.

ð9Þ

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85

Partial differentials of Eq. (9) are written in the matrix form 9 2 8 s1 < ∂V m =∂θ1 = ∂V =∂θ2 ¼ 4 0 ; : m 0 ∂V m =∂θ3

s1þ2 s1þ2 0

9 38 s1þ2þ3 < V m1 = s1þ2þ3 5 V m2 ¼ f ðθÞVm ; : s1þ2þ3 V m3

max

ð10Þ

where Vm1 = g(m1l1g + (m2 + m3)l1), Vm2 = g(m2l2g + m3l2) and Vm3 = gm3l3g. Eigenvalues of f(θ) are computed as  T λ ¼ s1 ; s1þ2 ; s1þ2þ3 :

ð11Þ

λ takes the form of a simple sine function that is a component of M in Eq. (4). Referring to Eq. (5), m ¼ 3; J 11 ¼ 1; J 12 ¼ J 13 ¼ 0; J 21 ¼ J 22 ¼ 1; J 23 ¼ 0; J 31 ¼ J 32 ¼ J 33 ¼ 1; d11 ¼ d22 ¼ d33 ¼ 1; d12 ¼ d13 ¼ d21 ¼ d23 ¼ d31 ¼ d32 ¼ 0, and all ϕi = 0. The mapping matrix of the four-bar mechanism is obtained by 2

1 J ¼ 41 1

0 1 1

3 0 33 05 ∈ R : 1

ð12Þ

M and K become M = diag[s1, s1 + 2, s1 + 2 + 3] ∈ R3 × 3 and K = [k1b1h1, k2b2h2, k3b3h3]T ∈ R3 × 1. Substituting Eqs. (12), (10), M, and K intoEq. (4) yields 2

s1 40 0

s1þ2 s1þ2 0

9 2 38 s1þ2þ3 < V m1 = 1 5 s1þ2þ3 V m2 −4 0 ; : s1þ2þ3 0 V m3

32 s1 1 1 1 1 54 0 0 0 1

0 s1þ2 0

9 38 < k1 b1 h1 = 5 k2 b2 h2 ¼ 0: ; : s1þ2þ3 k3 b3 h3 0 0

ð13Þ

Spring constants k1, k2, and k3 are computed with Eq. (13):  . k1 ¼ g m1 l1g þ ðm2 þ m3 Þl1 ðb1 h1 Þ

ð14Þ

 . k2 ¼ g m2 l2g þ m3 l2 ðb2 h2 Þ

ð15Þ

k3 ¼ gm3 l3g =ðb3 h3 Þ:

ð16Þ

Note that θ2, θ3, and θ4 can be determined with θ1 in general, because the four-bar mechanism has 1 DOF. However, nonlinear relationships between θ2, θ3, θ4 and θ1 are not considered in this paper. 5.2. A gravity compensator with the alternative joint space For Fig. 2(a), θ4 can be determined as θ4 ¼ θ1 þ θ2 þ θ3 :

ð17Þ

Eq. (17) indicates a kinematic constraint of the four-bar mechanism. Applying Eq. (17) to the base joint space of θb = [θ1, θ2, θ3]T yields a new joint space of θa = [θ1, θ2, θ4]T, which denotes an alternative joint space. Assume that θb ∈ Rn × 1 and θa ∈ Rp × 1. Here, n and p represent the number of joints in the base joint space and alternative joint space, respectively. Mapping between the base and alternative joint spaces is determined as θb ¼ Jba θa

ð18Þ

where Jba ∈ Rn × p indicates constraints between the base and alternative joint spaces. Assume that J and ki for θb exist satisfying Eq. (4). Applying Eq. (18) to the design equation yields f(Jbaθa)Vm_max − JTM(JJbaθa + ϕ)K = 0, and multiplying f(Jbaθa)Vm_max − JTM(JJbaθa + ϕ)K = 0 by J Tba results in T

Jba f ð Jba θa ÞVm

T max −ðJJ; baÞ Mð J Jba θa

þ ϕÞK ¼ 0:

ð19Þ

Eq. (19) is satisfied regardless of θa because the design equation with θb is already satisfied. Thus, JJba represents a new mapping matrix for θa. Relationships between the base and alternative joint spaces and the gravity compensator space are depicted in Fig. 4. Therefore, the mapping between the joint space of the four-bar mechanism and the gravity compensator space can be described with both θg = Jθb + ϕ and θg = JJbaθa + ϕ.

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Fig. 4. Relationships between the base and alternative joint spaces and gravity compensator space.

With θb, θa, and Eq. (18), Jba is determined as Jba = [1, 0, 0; 0, 1, 0; − 1, − 1, 1] ∈ R3 computed as 2

J Jba

1 ¼ 41 0

0 1 0

3 0 33 0 5∈R : 1

× 3

. The new mapping of JJba is

ð20Þ

Substitutions of Jba and Eq. (20) into Eq. (19) result in a new design equation with θa: 2

s1 40 0

s1þ2 s1þ2 0

9 2 38 1 0 < V m1 = 5 0 V m2 −4 0 ; : 0 s4 V m3

1 1 0

32 0 s1 0 54 0 1 0

0 s1þ2 0

9 38 0 < k1 b1 h1 = 5 ¼ 0: 0 k b h : 2 2 2; s4 k3 b3 h3

ð21Þ

Spring constants of Eq. (21) are identical to those of Eq. (13). 5.3. Practical implementations of mapping matrices Two gravity compensators for the joint spaces of θb and θa are depicted in Fig. 5. Considering the mapping matrix of Eq. (12) for θb, 1-DOF gravity compensators are located at θg1 = θ1, θg2 = θ1 + θ2, and θg3 = θ1 + θ2 + θ3. In this situation, θ1 + θ2 and θ1 + θ2 + θ3 represent the parallel constraint, so two parallelograms are equipped as shown in Fig. 5(a). Thus, three 1-DOF gravity compensators are employed at θ1, parallelogram I, and parallelogram II, respectively. Considering the mapping matrix of Eq. (20) for θa, 1-DOF gravity compensators should be installed at θg1 = θ1, θg2 = θ1 + θ2, and θg3 = θ4. Therefore, three 1-DOF gravity compensators are

Fig. 5. Gravity compensators of the four-bar link.

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87

Fig. 6. Derived design equations.

employed at θ1, parallelogram I, and θ4, respectively. θg3 for θb is identical to θ4 referring to Eq. (17). Thus, θ4 is an alternative position of θg3 for θb. 6. Design examples We designed gravity compensators for the cases of Fig. 2(b), (d), (f), and (h) by applying the deletion rules in Section 4. Note that analyses (e.g., computation of the potential energy) are necessary for all specific cases without application of the deletion rules. With application of the deletion rules, however, gravity compensators for Fig. 2(b), (d), (f), and (h) are simply obtained from the design equation for the associated linkage (i.e., Eq. (13) or (21)).

Fig. 7. Simulation models.

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Table 2 Parameters and spring constants for case 1.

li lig = bi mi hi ki

Link 1

Link 2

Link 3

0.82 m 0.41 m 20 kg 0.6 m 1144.10 N/m

1.04 m 0.52 m 15 kg 0.7 m 490.33 N/m

0.54 m 0.27 m 10 kg 0.3 m 326.89 N/m

For Fig. 2(b), the base joint space and the base design Eq. (13) are used. Note that selection of the base or alternative joint space is dependent on the joint configuration of the derived mechanism. For example, θa can be adopted for Fig. 2(f) and (g), and both θa and θb can be adopted for Fig. 2(b), (d), and (h). According to the conversion rules, θ3 and θ4 are ignored, so the joint space of Fig. 2(b) becomes θ = [θ1, θ2]T by eliminating θ3 from θb = [θ1, θ2, θ3]T. Because link 3 becomes a slider, l3 and l3g are set to zero. The mass of the slider, m3, is merged to the adjacent link connected with the revolute joint (i.e., link 2). The deletion rules are applied to Eq. (13), and the deletion procedures are presented in Fig. 6(a). A new design equation derived from the design equation of the associated linkage is obtained by 

s1 0

s1þ2 s1þ2



  V m1 1 1 s1 − 0 V m2 0 1



0 s1þ2

k1 b1 h1 k2 b2 h2

¼ 0:

ð22Þ

Spring constants are computed by k1 = g(m1l1g + (m2 + m3)l1)/(b1h1) and k2 = g(m2l2g + m3l2)/(b2h2) and are identical to Eqs. (14) and (15). For Fig. 2(d), the same joint space for Fig. 2(b) can be used. However, the mass of the slider, m3, is merged into link 4 (i.e., the fixed link), so m3 is set to zero according to the conversion rules. Therefore, the design equation for Fig. 2(d) has the same form of Eq. (22), except that Vm1 and Vm2 in Eq. (22) have different values because m3 is set to zero. Vm1 and Vm2 are recomputed as Vm1 = g(m1l1g + m2l1) and Vm2 = gm2l2g, respectively. Spring constants are obtained by k1 = Vm1/(b1h1) and k2 = Vm2/(b2h2) and can be also computed with Eqs. (14) and (15) by setting m3 to zero. For Fig. 2(f), link 2 becomes a slider, so l2 and l2g are set to zero. The mass of the slider, m2, is merged into link 1. θ1 and θ4 are used to determine the poses of all links with only revolute joints. Thus, the alternative joint space of θa = [θ1, θ2, θ4]T is adopted, and θ2 is ignored. The reduced joint space from the alternative joint space is achieved by θ = [θ1, θ4]T for Fig. 2(f). The deletion rules are applied to Eq. (21), and the deletion procedures are presented in Fig. 6(b). A new design equation is obtained by 

s1 0

0 s4



 V m1 1 − V m3 0

0 1



s1 0

0 s4



k1 b1 h1 k3 b3 h3

¼ 0:

ð23Þ

Because slider 2 is located remotely from the fixed link, m3 (or m1 and m2) is eliminated from Vm1 (or Vm3), according to Section 4. Therefore, from Eq. (10), Vm1 and Vm3 are recomputed as Vm1 = g(m1l1g + m2l1) and Vm3 = gm3l3g, respectively. Spring constants are obtained by k1 = Vm1/(b1h1) and k3 = Vm3/(b3h3) from Eq. (23). For Fig. 2(h), links 1 and 3 become sliders, so l1, l1g, l3, and l3g are set to zero according to the conversion rules. Because θ1 and θ3 are ignored, the joint space is obtained as θ = [θ2] ∈ R1 × 1. The base joint space θb and Eq. (13) are adopted. The deletion rules are applied to Eq. (13), and the deletion procedures are presented in Fig. 6(c). A new design equation is obtained by V m2 s1þ2 −k2 b2 h2 s1þ2 ¼ 0:

ð24Þ

In this situation, θ1 of Eq. (24) has the constant value of π/2 from Fig. 2(h). (θ1 = π/2 indicates the direction of movement of slider 1.) The spring constant k2 in Eq. (24) is identical to that in Eq. (15).

Table 3 Parameters and spring constants for case 2.

li lig = bi mi hi ki

Link 1

Link 2

Slider 3

0.82 m 0.41 m 20 kg 0.6 m 1144.10 N/m

1.04 m 0.52 m 15 kg 0.7 m 490.33 N/m

– – 10 kg – –

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89

Table 4 Parameters and spring constants for case 3.

li lig = bi mi hi ki

Link 1

Link 2

Slider 3

0.47 m 0.24 m 20 kg 0.3 m 1634.44 N/m

0.60 m 0.30 m 15 kg 0.6 m 245.17 N/m

– – 10 kg – –

7. Simulations Dynamic simulations are conducted for the gravity compensators of Fig. 2(a), (b), (d), (f), and (h) with Recurdyn. Simulation models are presented in Fig. 7. Cases 1, 2, 3, 4, and 5 represent models for Fig. 2(a), (b), (d), (f), and (h), respectively. The design Eqs. (13), (22), (23), and (24) are used to design the gravity compensators. Note that the design equation for Fig. 7(c) (i.e., Fig. 2(d)) takes the same form as that of Eq. (22) (i.e., the design equation of Fig. 7(b) or Fig. 2(b)). Comparisons of Fig. 7(b) and (c) with Fig. 7(a) show that the parallelogram II in Fig. 7(a) is eliminated by the conversion rules. Note that parallelogram II is employed for θ3 (or link 3), and θ 3 is eliminated by the conversion rules. Similar situations occur for Figs. 7(d) and (e). Parameters and spring constants are presented in Tables 2–6. For all cases, the gravity exerts in the positive direction of the x-axis (i.e., g = [g, 0]T ∈ R2 × 1), and g is set to 9.81 m/s2. θ1 is actuated with virtual motors: θ1(t) = 0.1 ⋅ sin(2πt/5) + 1.82 (radian) for case 1; θ1(t) = 0.0962 ⋅ sin(2πt/5) + 2.21 for case 2; θ1(t) = 0.1 ⋅ sin(2πt/5) + 2.21 for case 3; and θ1(t) = 0.15 ⋅ sin(2πt/5) + 2.29 for case 4, respectively, where t denotes time. For case 5 θ2 is actuated as θ2(t) = −0.1017 ⋅ sin(2πt/5) + 1.86. Results of the simulations are presented in Fig. 8. Potential energy of the masses and the springs is measured. Vm and Vk denote the potential energy of the masses and that of the springs, respectively. The total potential energy is computed with Vm + Vk. The total potential energy is invariant for all cases, whereas the poses of the mechanisms vary. Thus, the results of the simulations indicate that complete gravity compensation is achieved for all cases, verifying that the proposed design method is applicable. Control responses are presented in Fig. 8(f) for the slider-crank in Fig. 7(b). A PD controller is equipped at θ1 and the reference motion of θ1 is given as θr1(t) = 0.1 ⋅ sin(2πt/5) + 2.21 (radian). The same parameters in Table 3 are utilized. The input and output in Fig. 8(f) denote θr1(t) and measured angles of θ1, respectively. The error is computed by θr1(t) — measured angle. The proportional and derivative gains KP and KD are set to KP = 200 and KD = 50, respectively. Oscillatory motions are observed at 0 ≤ t ≤ 1 due to an abrupt change in the acceleration at t = 0. Errors are bounded within ± 0.01 ranges. 8. Discussions 8.1. Variance of direction of gravity Suppose that the gravity is applied with an angle of α with respect to the x-axis as shown in Fig. 9 (i.e., g = g[cα, sα]T). The position vectors of Eqs. (6), (7), and (8) are used to compute the potential energy in the joint space. With Eq. (2), the potential energy is computed as     V m ¼ −g m1 l1g þ ðm2 þ m3 Þl1 c1−α −g m2 l2g þ m3 l2 c1þ2−α −gm3 l3g c1þ2þ3−α :

ð25Þ

Partial differentials of Eq. (25) are written in the matrix form 9 2 8 s1−α < ∂V m =∂θ1 = ∂V m =∂θ2 ¼ 4 0 ; : 0 ∂V m =∂θ3

s1þ2−α s1þ2−α 0

9 38 s1þ2þ3−α < V m1 = s1þ2þ3−α 5 V m2 ¼ f ðθÞVm ; : s1þ2þ3−α V m3

ð26Þ

max

Table 5 Parameters and spring constants for case 4.

li lig = bi mi hi ki

Link 1

Slider 2

Link 3

1.07 m 0.53 m 20 kg 0.6 m 817.22 N/m

– – 15 kg – –

0.50 m 0.25 m 10 kg 0.4 m 245.17 N/m

90

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Table 6 Parameters and spring constants for case 5.

li lig = bi mi hi ki

Link 1

Slider 2

Link 3

– – 20 kg – –

1.04 m 0.52 m 15 kg 0.7 m 490.33 N/m

– – 10 kg – –

where Vm1, Vm2, and Vm3 have the same values of Eq. (10). According to the mapping matrix of Eq. (12) and f(θ)Vm_max of Eq. (26), a mapping relationship considering the offset angle a is obtained by 2

1 θg ¼ 4 1 1

0 1 1

38 9 8 9 0 < θ1 = < α = 0 5 θ2 − α ¼ Jθ þ ϕ: : ; : ; α 1 θ3

ð27Þ

Thus, the offset angle in the gravity is represented as a phase angle in the space mapping. Because no variances in the mapping matrix occur except the phase angle, the spring constants of Fig. 9 are identical to Eqs. (14), (15), and (16) (i.e., spring constants for α = 0). The phase angle of Eq. (1) is implemented by rotating a unit gravity compensator with the phase angle as in Fig. 9. Simulation results are presented in Fig. 10. θ1 is actuated as θ1(t) = 0.1 ⋅ sin(2πt/5) + 1.82 (radian). Parameters in Table 2 are utilized. Experimental results show that the total potential energy is invariant for all times for α = −π/4.

Vm + Vk Energy (J)

200 0

Energy (J)

Vk

Vm

0

1

2

3 Time (s) a) Four-bar link

4

400 Vk

Vm

Vm + Vk

200

Vm + Vk

400

Vm 200 0

5

Energy (J)

Energy (J)

400

0

1

2

3 Time (s) b) Slider Crank

Vk

4

5

Vm + Vk

400

Vm

200 Vk

0

0

1

2

3 Time (s) c) Swing block linkage

4

0

5

0

1

2

3 Time (s) d) Turning block linkage

4

5

rad

400

200 Vm 0

0

1

2

3 Time (s) e) Cardanic motion

Output Input

2.2 2 0.01

V m + Vk rad

Energy (J)

2.4

Vk 4

5

Error

0

- 0.01 0

Fig. 8. Results of simulations.

1

2

3 Time (s) f) Control responses

4

5

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8.2. Consideration of the prismatic joint The prismatic joint must be considered for expressing the pose of a mechanism during computing the potential energy (e.g., the cardanic motion in Fig. 2(h)) in some cases. The potential energy may vary according to the displacement of the prismatic joint in this situation. In our design method, however, the prismatic joint is not considered (or eliminated) because of the conversion rules. At least one revolute joint connected to the reference frame is necessary to determine the poses of links. That is, the potential energy should be recomputed inevitably, leading to the design equation being rebuilt, when the prismatic joint is considered. The cardanic motion of Fig. 2(h) is considered as an example. Assume that the gravity is given as g = g[cα, sα]T. When we consider the four-bar mechanism in Fig. 2(a), the poses of the cardanic motion mechanism can be determined by substituting θ1 = π/2, l1 = −l2s1 + 2, and l1g = −βl2s1 + 2 into position vectors of Eqs. (6), (7), and (8), where β = l1g/l1. Substituting θ1 = π/2, l1 = −l2s1 + 2, and l1g = −βl2s1 + 2 into Eq. (25) yields the potential energy of the masses for the cardanic motion mechanism:     V m ¼ g ðβm1 þ m2 Þl2 −m2 l2g sα s2þπ=2 −g m2 l2g þ m3 l2 cα c2þπ=2 −gm3 l3g c2þ3þπ=2−α :

ð28Þ

Applying the conversion rules (i.e., l3g = 0; θ3 is eliminated; and m1 and m3 are merged into link 2 (thus, β = 1, because l1 = l1g)) to Eq. (28) yields V m ¼ −V m

prismatic c2 −V m revolute c2þπ=2

ð29Þ

where Vm_prismatic = − g((m1 + m2)l2 − m2l2g)sα and Vm_revolute = g(m2l2g + m3l2)cα. A design equation for Eq. (29) is obtained as 

s2

s2þπ=2

  V m prismatic −½ 1 V m revolute

 1

s2 0

0 s2þπ=2



k1 b1 h1 k2 b2 h2

¼ 0:

ð30Þ

The signs of Vm_prismatic and Vm_revolute vary according to α. The phase angle of the mapping relationship should be adjusted to maintain positive Vm_prismatic and Vm_revolute according to the variances of sα and cα. Supposing, for example, that α = π/6, cα N 0 and sα N 0, and, therefore, Vm_prismatic b 0 and Vm_revolute N 0. In this situation, c2 is changed to − c2 ± π to set Vm_prismatic to a positive value.

Fig. 9. Gravity compensator of the four-bar linkage with the offset angle α in the gravity.

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Fig. 10. Results of simulation with the offset angle α = −π/4.

Because − c2± π is identical to c2, no variances occur in Eqs. (29) or (30), and only the phase angle of π is implemented. Spring constants are computed from Eq. (30):   .



k1 ¼ g m2 l2g þ m3 l2 cα ðb1 h1 Þ

ð31Þ

  .



k2 ¼ g ðm1 þ m2 Þl2 þ m2 l2g sα ðb2 h2 Þ:

ð32Þ

Gravity compensators of Eq. (30) are presented in Fig. 11. Simulation results are presented in Fig. 12. θ2 is actuated as θ2(t) = − 0.1191 ⋅ sin(2πt/5) + 2.42 (radian). Parameters in Table 6 are utilized. Experimental results show that the complete gravity compensation is achieved for α = π/6. Note that the conversion rules are applied after rewriting the total potential energy with displacement of the prismatic joint (i.e., l1 = −l2s1 + 2 in this situation). When the conversion rules are applied prior to rewriting the total potential energy, l1 and l1g are eliminated, and there is no opportunity to use the displacement of the prismatic joint (i.e., l1 in this situation). Supposing that α = 0 as a special case, Vm_prismatic and Vm_revolute become Vm_prismatic = 0 and Vm_revolute = g(m2l2g + m3l2), respectively. The design Eq. (30) becomes identical to Eq. (24). Thus, the gravity compensator of Fig. 7(e) represents the special case of gravity compensators for the cardanic motion mechanism. 9. Conclusions This paper presents a design method for a gravity compensator with associated linkage (i.e., the basic kinematic chain). Because the derived mechanisms originate from the associated linkage, a gravity compensator for the derived mechanism is also obtained from the gravity compensator of the associated linkage. The space mapping method is adopted, and conversion and deletion rules are developed. Based on various examples and the results of simulations, the following conclusions are derived: 1. When a mechanism is derived from an associated linkage (i.e., a revolute joint of an associated linkage becomes a prismatic joint), conversion rules can be obtained from the viewpoint of the gravitational torque.

Fig. 11. Gravity compensators of the cardanic motion with the offset angle of a in the gravity.

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Fig. 12. Results of simulation of the cardanic motion with the offset angle α = π/6.

2. When the conversion rules are applied to the design equation, eliminations of row and column vectors or components of the design equation occur. Thus, deletion rules can be obtained from the conversion rules. 3. A new design equation (or gravity compensator) for the derived mechanism from the associate linkage can be simply achieved by applying the deletion rules to the design equation for the associated linkage. 4. By considering the joint configuration of the derived mechanism, the joint space can be chosen accordingly. 5. When only the revolute joints are considered so as to compute the poses of a kinematic chain, the mapping matrix is invariant, although there is an offset angle in the gravity. The offset angle in the gravity is shown in only the phase angle of the mapping relationship. 6. When a prismatic joint (or joints) is considered so as to compute the poses of a kinematic chain, the potential energy should be recomputed and the design equation should also be rebuilt to implement the displacement of the prismatic joint. Even in this situation, the conversion rules can be applied. For further research, the proposed method will be evaluated with mechanisms possessing multi-loops. Because the four-bar linkage is adopted as an associated linkage, only the binary link and single loop are considered in this paper. When a ternary link is used, several loops are achieved, and sophisticated mass distribution occurs. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2009007) in study design. This study was supported by research funds from Chosun University, 2013 in the writing of the report. References [1] R.M. Nathan, A constant force generation mechanism, ASME J. Mech. Transm. Autom. Des. 107 (1985) 508–512. [2] N. Ulrich, V. Kumar, Passive mechanical gravity compensation for robot manipulator, Proc. of the 1991 IEEE Int. Conf. on Robotics and Automation, Sacramento, 1991, pp. 1536–1541. [3] K. Koser, A cam mechanism for gravity-balancing, Mech. Res. Commun. 36 (2009) 523–530. [4] J.L. Herder, R. Barents, M. Schenk, W.D. van Dorsser, B.M. Wisse, Spring-to-spring balancing as energy-free adjustment method in gravity equilibrators, Proc. of the ASME 2009 Int. Eng. Technical Conf. and Computer and Inf. in Eng. Conf, 2009, pp. 1–12. [5] A. Agrawal, S.K. Agrawal, Design of gravity balancing leg orthosis using non-zero free length springs, Mech. Mach. Theory 40 (2005) 693–709. [6] C.H. Cho, W.S. Lee, J.Y. Lee, S.C. Kang, A 2-dof gravity compensator with Bevel gears, J. Mech. Sci. Technol. 26 (2012) 2913–2919. [7] C.M. Gosselin, J. Wang, On the design of gravity-compensated six-degree-of-freedom parallel mechanisms, Proc. of the 1998 IEEE Int. Conf. on Robotics and Automation, 1998, pp. 2287–2294. [8] A. Russo, R. Sinatra, F. Xi, Static balancing of parallel robots, Mech. Mach. Theory 40 (2005) 191–202. [9] K.A. Wyrobek, E.H. Berger, H.F.M.V. Loos, J.K. Salisbury, Towards a personal robotics development platform: rationale and design of an intrinsically safe personal robot, Proc. of the 2009 IEEE Int. Conf. on Robotics and Automation, 2009, pp. 2165–2170. [10] B. van Ninhuijs, B.L.J. Gysen, J.W. Jansen, E.A. Lomonova, Multi-degree-of-freedom spherical permanent magnet gravity compensator for mobile arm support systems, Proc. 2013 IEEE Int. Electr. Mach. and Drives Conf. (IEMDC) (2013) 1443–1449. [11] S. Deepak, Static Balancing of Rigid-Body Linkages and Compliant Mechanisms, (PhD. dissertation) 2012. [12] C.H. Cho, S.C. Kang, Design of a static balancing mechanism for a serial manipulator with an unconstrained joint space using one-DOF gravity compensators, IEEE Trans. Robot. 30 (2014) 421–431. [13] A.G. Erdman, G.N. Sandor, S. Kota, Mechanism design analysis and synthesis, vol. 1, Prentice Hall, 2001, pp. 526–530. [14] R.C. Johnson, Mechanical Design Synthesis: Creative Design and Optimization, RE Krieger Publishing Company, Malabar, 1971, pp. 35–128. [15] L.-W. Tsai, Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press, 2010, pp. 112–139.