Chapter 1 Introduction
1.4 Overview of the dissertation
The remainder of this paper is organized as follows. Section 2 introduces the formulation of the elastic potential energy and gravitational potential energy represented by the SBM. On the basis of the summation of the gravitational and elastic SBMs, the spring installation parameters are classified according to the role they play in the balancing equations. Section 3 describes the general criteria for the admissible spring configuration for an n-link VPM according to the number of required PDPs and balancing equations derived from the nonzero component matrices of the summation of the gravitational and elastic SBMs. The formulation of the PDPs and PIPs is then explored according to the spring configuration. In Section 4, because the displacement of the PDPs is expressed as an equation of the PIPs, the additional criteria for the installation of PIPs
are determined to reduce and equalize the nonzero displacement PDPs. Section 5 presents the derivation of a planar 2-, and 3-DOF VPM as design examples. The static equilibrium of quasistatic continuous motion is verified. Moreover, arrangement of standard springs, the method of reducing spring elongation, and the procedure of the adjustment of PDPs are explored for the engineering prototype of 2-DOF VPM. The energy consumption is estimated accordingly.
Chapter 2
Characteristics of SBM
2.1 Coordinate system of SBM
The coordinate system is defined in the Denavit-Hartenberg representation, shown in Fig. 6. The term rj denotes the direction vector of link j of an n-link manipulator, and xj and yj are unit orthonormal axis vectors, where j = 1, 2, …, n. rj, xj, and yj are defined to be 2 × 1 column matrices for 2D space. R(θj) represents the rotation matrix of the succeeding and preceding coordinate systems, where θj is the joint angle from the x(j − 1)
axis to the xj axis. Therefore, the change in the succeeding coordinate system can be determined according to the rotation of the preceding link.
Link i
Link 1
Link j
Link k rk
rj
ri
bik
aik
βik αik Sik
xik
pj
x1
xi-1
xi
xj
xk-1
xk
xn-1
xn
yn
yn-1
yj-1
yk
yk-1
yj
xj-1
yi
yi-1
y1 sj
g mp
pp
mj
rn
Joint 1
Joint i
Joint k
θj
Link n
Fig. 6 Gravity position of n-link articulated manipulator and changeable payload fitted at the end effector of link n and spring connected between link i and link k
2.2 Characteristics of gravitational SBM
The SBM method was proposed for analyzing a static balanced articulated manipulator with fixed potential energy in previous studies [16]. The term mj represents the mass of link j; pj represents the position vector of the mass center aligned along the line passing through the joints of links; g represents the gravitational acceleration; and I’
represents the rotation matrix that rotates 270°. The gravitational potential energy Ug is expressed as
j center of link j, and its orientation is the same as that of direction vector rj. In addition, rj
and sj are the magnitudes of rj and sj, respectively.
In this study, various payloads are embedded at the end effector of the articulated manipulator. Therefore, the gravitational potential energy consists of the gravity of links and various payloads. mp represents the mass of the payload, which may vary, and pp
represents the position vector of various payloads fitted at the end effector of link n. The gravitational potential energy Ug is rearranged as
j
According to Eqs. (1–4), the SBM representation for the gravitational potential energy can be obtained as
where [Guv] is called the gravitational SBM, and its elements Guv are called gravitational component matrices, which represent gravitational interacting potential energies between links u and v. Owing to its symmetry [18], Guv can be expressed as an SBM, as shown in Eq. (6), and only the upper triangular matrix is considered.
The gravitational component matrices G1v are expressed as
m v nr m s m g W
m g
n
v j
j v
v v p v
p
v 2,...,
1
1
I I
G (7)
where Wv is the sum of the mass of links from links v to n. Each component matrix Guv
in the gravitational SBM represents the quantity of the gravitational effect that acts between the ground (link 1) and link v. Therefore, Guv has a nonzero element in only the first row (i.e., u = 1) of off-diagonal matrices, as shown in Eq. (6). According to Eq. (7), Guv is a function of the payload and mass property of links.
2.3 Characteristics of elastic SBM
The spring configuration matrix [Sik] represents the configuration of fitted springs in an n-link SBM. Sik = 0 denotes that there is no spring installed between links i and k, and Sik = # denotes that at least one spring is installed between links i and k.
The expression of elastic potential energy is suggested by Lee et al.[15]. A spring with spring constant kik, fitted between links i and k of an n-link articulated manipulator, is shown in Fig. 6. Assume that the springs used are zero-free-length springs, and the distance between two attachment points |xik| can be considered the elongation of the spring. The elastic potential energy can be expressed as
ik T ik ik
ik k x x
2
U 1 (8)
where
position vectors extend from the joints of links i and k to the attachment points of the spring, respectively, and they can be expressed as
According to Eqs. (8–10), the SBM representation for the elastic potential energy can be obtained as
elastic component matrices, which represent elastic interacting potential energies between links u and v. Because [𝐊uvik] is a symmetric matrix, only the upper triangular matrix is
considered as represented in the following:
expressed as
Similarly, the diagonal terms of the elastic component matrices 𝐊uvik (for u = v) are expressed as
Each elastic component matrix 𝐊uvik shows the quantity of the elastic effect of the spring Sik between links u and v. Therefore, 𝐊uvik has nonzero terms in the diagonal part of rows u to v and columns u to v, as shown in Eq. (12). Because the elastic potential
energy of the spring affects only the links that it spans over, the nonzero elements of 𝐊uvik are within the range i ≤ u, v ≤ k. According to Eqs. (13–14), 𝐊uvik is a function of the spring constant, ratio of the distance between the joint and the attachment point of the spring to the length of the links to which the spring is attached, and attachment angles.
2.4 Static balance of total SBM
The gravitational and elastic potential energies change with the relative angular displacement between links u and link v, as shown in Eq. (15). To keep the total potential energy unchanged regardless of the configuration of the manipulator, all component matrices between every two distinct links (u ≠ v) must be zeros.
v T u uv cos1 r r (15)
Because the gravitational and elastic potential energies both have the same form, the total potential energy [Tuv] can be expressed as the sum of the gravity and elastic SBMs, and Tuv denotes the total component matrices, which represent total potential energies between links u and v.
The gravity potential energy induces nonzero elements in the first row of Tuv (for u
= 1) only. The associated spring Sik (for i = 1) with nonzero elements in the first row is required to balance Guv (for u = 1). Therefore, the first row of the total component matrices can be expressed as
n
The gravitational component matrix includes the gravity of various payloads. To calculate the balancing equation of T1v as scalar, the ground-attached end of spring S1k
must be installed at angle α1k = 90° or 270°, and the link-attached end must be installed at angle β1k = 0° or 180°. Therefore, the rotation matrix I’ can be ignored during the calculation. The elastic component matrices 𝐊uv1k of spring S1k can be rearranged as
expressed as
k
k
k
Considering the design parameters of spring S1k, there is one spring constant k1k and two installation parameters A1k and B1k. The installation parameter B1k is fitted in both first row and non-first-row component matrices, and it is attached to the link. Therefore, the adjustment of B1k would additionally influence the non-first-row component matrices and interfere with the motion of links. However, the installation parameter A1k is fitted in
first row component matrices only and is attached to the ground. Therefore, the adjustment of A1k can avoid these problems. Thus, the installation parameter A1k is determined to be adjustable for various payloads.
For the non-first-row elements of Tuv (for u ≠ 1), gravity has no influence but the ground-adjacent springs have elastic influence. Therefore, spring Sik (for i ≠ 1) must be installed to generate a counterbalancing force in the non-first-row parts to compensate for the influence of the non-zero elements of 𝐊uv1k (for u ≠ 1). defined as β1k = 0° or 180°, the attached end of the counterbalancing spring Sik should be installed at angles α1k = 0° or 180° and β1k = 0° or 180°, so the rotation matrix I can be ignored during the calculation. The elastic component matrices 𝐊uvik of the counterbalancing spring Sik can be rearranged as
where Aik and Bik are determined to represent dimensionless installation parameters as
ik
ikk ik
ik r
B b cos (21b)
In conclusion, the installation parameters are separated into three types on the basis of their balancing objectives and position. For the adjacent spring, the ground-attached installation parameter A1k is a PDP, and the link-attached installation parameter B1k is a distal PIP. For the non-ground-adjacent spring, the installation parameter Aik is a proximal PIP, and Bik is a distal PIP. The positive and negative directions of the PDPs and PIPs are determined as (90°, 270°) and (0°, 180°), respectively, as shown in Fig. 7. In addition, to enable adjustment, the PDPs must fit on PDP adjustment devices. PDP adjustment devices are used to adjust PDPs, and they can be locked at any position along the vertical axis, as shown in Fig. 7.
Positive direction (αik = 0° for i ≠ 1) Negative direction (αik = 180° for i ≠ 1)
Proximal PIP PDP adjustment
device
Positive direction (βik = 0°) Negative direction
(βik = 180° )
Distal PIP
Ground adjacent spring
Non-ground adjacent spring
Positive direction (αik = 90° for i = 1) Negative direction (αik = 270° for i = 1)
PDP Distal PIP
Fig. 7 Definition of the direction of PDPs and proximal and distal PIPs. Schematic of a PDP adjustment device.
Chapter 3
Determination PDP and PIP arrangements
3.1 Arrangement of PDPs
According to Section 2.2, only ground-adjacent springs contain PDPs. In this section, the characteristics of PDPs are obtained by analyzing the equations of the first row of the total SBM [Tuv]. To increase the efficiency of the adjustment, PDPs should be linear to the change of payload. For a set of balancing equations with linear solutions, the number of unknowns must be equal to the number of subsets of equations. To enable PDPs A1k to be adjustable for various payloads mp, A1k is determined to be unknown for the simultaneous equations of the entries in the first row.
In the balancing equations of the first row component matrices of Tuv, there are (n – 1) component matrices T12 to T1n. Therefore, (n – 1) equations are in the first row. For each fitted spring S1k, one distinct unknown of PDP A1k is added to the system. A total of (n – 1) springs should be connected on link 1; therefore, the total component matrices T1v
(for v = 2, …, n) can be rearranged as
All component matrices in Eq. (22) have the same orientation. Consequently, subset equations can be solved regardless of the orientation. The subset equations have the form of a first-order linear system. Therefore, Eq. (22) can be expressed as the matrix form of a linear system.
equations of payloads, which can be expressed as
n
where Ck is the coefficient of the payload, which represents the displacement of the PDP according to the payload, and Dk is the constant of the PDP, which represents the initial position of PDPs. In addition, Dk is determined by the constants of PDPs succeeding it.
Ck and Dk can be expressed as
According to Eqs. (25a–b), the PDPs are separated into two sets. The preceding set is used to balance the gravitational potential energy of various payloads mp. The latter set is used to balance the constant gravitational potential energy of the links. In conclusion, the configuration of ground-adjacent springs has the following characteristic.
CH1: For an n-link VPM, (n – 1) PDPs are used to compensate for the variations in
the potential energies owing to changes in the payload. These PDPs are linear functions of the payload and can be formed by installing (n – 1) ground-adjacent springs; that is, S1k (for k = 2, … , n) of an n × n spring configuration matrix is nonzero.
3.2 Arrangement of PIPs
The spring embedded between the ground and the end link produces nonzero terms in all components of the elastic SBM. However, only the terms in the first row are used to compensate for the gravitational potential energies; the other terms are excess elastic potential energies that are compensated by non-ground-adjacent springs.
Before installing non-ground-adjacent springs, it is necessary to identify the distribution of excess elastic potential energies in their elastic SBM. According to CH1, ground-adjacent springs S12 to S1n are installed, and the sum of their non-first-row elastic SBM has the same equations in the same column, as shown in Fig. 8 and Eq. (26). P (for
v = 3, …, n) represents the excess elastic potential energies of the ground-adjacent springs
Fig. 8 Distribution features of the summation of the elastic SBM of ground-adjacent springs.
According to Eq. (26), the excess elastic potential energies between column 3 and column (n – 1) can be balanced by setting the distal PIPs B1k (for k = 3, …, n) to a negative value that can be formed by their distal attachment angle β1k = 180°. However, owing to the constraint that B1n cannot equal zero, spring S2n must be installed to compensate for the excess elastic potential energies in column n. Therefore, the installation of spring S2n
entails the following characteristic.
CH2: For an n-link VPM, a spring must be connected between link 2 and link n to
compensate for the excess potential energies between link n and the links preceding it; that is, S2n of an n × n spring configuration matrix must be non-zero.
Therefore, after the installation of spring S2n, the final column of the non-first-row total SBM can be rearranged as
value (i.e., distal attachment angle β1q = 0°), spring S2q must be installed, and its distal PIP B2q must be set to a negative value (i.e., distal attachment angle β2q = 180°) to compensate for the excess elastic potential energy. Therefore, the distal PIP of springs S1qand S2q (for 3 ≤ q ≤ n) has the following characteristic.
CH3: For an n-link VPM, if the distal PIP of S1q is not attached in the negative direction of link q, a spring with a distal PIP attached in the negative direction is required to fit between links 2 and q (where q denotes a number between 3 and n).
Therefore, without the installation of spring S2q, column q of the non-first-row total SBM can be rearranged as
n SBM can be rearranged as
be considered in the balancing equations of T23.
According to Eqs. (27) and (29), after the installation of the springs connected between link 2 and the links succeeding link 2, the balancing equation in row 2 is different from those in the other rows. To solve the sequence of balancing equations without installing other springs Sik (for i ≠ 1, 2), each term in the same column should be the same
in its elastic SBM, and this can be achieved by setting its proximal PIP as A2q = –1 (i.e., position 𝑎2q = rq and attachment angle α2q = 180°). Therefore, the distribution features of
the nonzero terms are as shown in Fig. 9. Note the following exceptional case: when the elastic SBM of spring S23 is fitted in one field, it corresponds to one balancing equation only. Consequently, its proximal PIP A23 can be set at an arbitrary position.
Fig. 9 Distribution features of nonzero terms in the elastic SBM with attachment point ɑ2q = r2 and attachment angle α2q = 180° (for q > 3).
Therefore, the sequence of balancing equations in column q of the non-first-row total SBM can be rearranged as a single equation as follows:
characteristic.
CH4: For an n-link VPM, the proximal PIP of S2q must be attached to joint 1 to compensate for the excess elastic potential energy without fitting non-ground- and non-link-2 adjacent springs; that is, Sik (for i, k ≠ 1, 2) of an n × n spring configuration matrix must be zero.
3.3 Admissible spring configuration matrices for different DOF VPM
According to CH1–4, the spring configurations that can be applied for the VPM are
Base on Eq. (31), the admissible spring configurations of 2-, 3-, and 4-DOF VPMs are derived and shown in Table 1 (2-A), (3-A), (3-B), (4-A), (4-B), (4-C), and (4-D).
For a general arrangement with general displacement (i.e., the coefficients of PDPs are different), the number of required PDP adjustment devices is equal to the number of ground-adjacent springs, which is expressed as
1
n
Np (32)
3.4 Demonstration of 2, 3-DOF VPM
An example of a 2-DOF VPM with three springs is derived. The spring configuration matrix contributed by five springs is denoted as (2-A) in Table 1. The balancing equations of the off-diagonal upper triangular total SBM can be derived accordingly. T12, T13 are the balancing equations of the gravitational and elastic potential energies, and T23 is the
excess potential energies.
13 0
Similarly, an example of a 3-DOF VPM with five springs is derived. The spring configuration matrix contributed by five springs is denoted as (3-B) in Table 1. T12, T13, and T14 are the balancing equations of the gravitational and elastic potential energies, and
T23, T24, and T34 are those of the excess potential energies. manipulator can be expressed as
3
23 1
A (35d)
where B12 is free variable that can be predetermined arbitrarily.
According to Eqs. (34a–e) and CH1-4, PDPs and PIPs can be expressed as
4
The general arrangement of the VPM is based on the basic design equations of PDPs and PIPs. Because the number and displacement of PDPs are derived in general, (n – 1) PDP adjustment devices should adjust for A12 to A14 separately. However, some specific characteristics can reduce the number of PDP adjustment devices. This is discussed in the next section.
Chapter 4
Minimum number of PDP adjustment devices
For the design of an n-link VPM, (n – 1) PDP adjustment devices are considered to be adjustable while the payload changes. As the number of PDPs increases, the adjustment of associated springs becomes more difficult. To avoid this situation, the minimum number of PDP adjustment devices is discussed in this section.
4.1 Reduce the number of PDPs
According to Eq. (25a), (1 – 1 /B1q) is contained by a series of coefficients C1k (for k = 2, …, q – 1). When B1q = 1 (i.e., position b1q = rq and angle β1q = 0°), the coefficients C1k of the PDPs preceding link q equal 0, and (q – 2) PDPs are then eliminated. Therefore, the following necessary condition based on the reduced number of PDPs applies.
CH5: For an n-link VPM, (q – 2) PDPs are eliminated by setting the distal PIP of
S1q attached at joint q (where q denotes a number between 3 and n).
However, CH3 reveals that to set B1q at joint q, a spring S2q must be installed, and its distal PIP is set at angle β2q = 180°. The schematic of its arrangement is shown in Fig.
10(a).
For the general displacement, the number of PDP adjustment devices required is the same as the number of remaining PDPs, as expressed by
) 3
,
# (
1 If S2 q n
q n
Np q (37)
The number of nonzero terms in the second row of the spring configuration matrix indicates the number of different arrangements for the reduced number of PDPs. With respect to the general arrangement, the arrangement for the reduced number of PDPs is shown in Table 1 with the symbol (R), shown as (2-A-R), (3-A-R), (3-B-R1, R2), (4-A-R), (4-B-R1, R2), (4-C-R1, R2), and (4-D-R1, R2, R3).
4.2 Equivalent displacement of PDPs
For a VPM with more than one PDP, the equivalent displacement is designed to make the coefficients of the PDPs equal. Therefore, these PDPs can be set on the same PDP adjustment device. The equivalent equation of the coefficients of two adjacent PDPs is shown as
n q C
Cq q1 3 (38)
By rearranging Eq. (36), the relationship between two distal PIPs B1q and B1(q–1) is shown as
n q k B
B k q
q q
q 1( 1)1 3
1 ) 1 ( 1
1 (39)
Therefore, two adjacent PDPs can be fitted in the same PDP adjustment device to adjust them simultaneously; the schematic of this arrangement is shown in Fig. 10(b).
The following necessary condition for equalizing the displacement of adjacent PDPs applies.
CH6: For an n-link VPM, the displacement of every two adjacent PDPs can be
equalized by setting the distance between the distal PIP of S1q and joint q as the
product of the preceding distal PIP of S1q and spring constant ratio of S1(q – 1) and S1q
(for 3 ≤ q ≤ n).
Fig. 10 Arrangements for reducing the number of PDP adjustment devices
Because the displacement of every two adjacent PDPs can be equal, the displacement of all PDPs can be equalized by setting their distal PIPs in the relation as expressed by Eq. (37). With respect to the general displacement, the arrangement for the equivalent
Because the displacement of every two adjacent PDPs can be equal, the displacement of all PDPs can be equalized by setting their distal PIPs in the relation as expressed by Eq. (37). With respect to the general displacement, the arrangement for the equivalent