Chapter 3 Spring installation configurations
3.3 Determination of spring attachment angles
The determination of spring attachment angles following the rules described in the
aforementioned section is demonstrated in this section.
First of all, rule 1 is always valid for any admissible spring configuration, because the spring attached between ground and end link (S1n) is the only spring able to neutralize
the gravitational energy contribution in entry 1n. It is determined that the attachment
angle β1n− α1n is equal to 270° according to rule 1. Then, based on the requirement
described in rule 2, if no spring is attached between link 3 and end link, for the spring attached between link 2 and end link (S2n), it is determined that the attachment angle α2n
is equal to 180° according to rule 2. Take the spring configuration matrix 5-4 as an
example, it is determined that the attachment angle β15− α15 is equal to 270° for S15
according to rule 1, and that the attachment angle α25 is equal to 180° for S25
according to rule 2.
Based on rule 5, for each spring attached between ground and link k (S1k) where
2 ≤ k ≤ n, if no spring attached between links 2 and v for k ≤ v ≤ n contributes an
energy component with 180° direction to entry 2k, the sufficient condition of rule 5 is
satisfied for the spring S1k. Therefore, the spring S1k is in charge of contributing an
energy component with 180° direction to entry 2k and the attachment angle β1k is
determined to be 180° according to rule 5. Take the spring configuration matrix 5-4 as
an example, because no spring is attached between links 2 and 4 and the attachment angle α25 is 180° for the spring attached between links 2 and 5 according to rule 2. It is
determined that the attachment angle β14 is equal to 180° for S14 according to rule 5.
In accordance with rule 3, if S1k = # and S1(k+1) = 0 in the first row of spring
configuration matrix, it is determined that the attachment angle β1k− α1k is equal to 270° for S1k. After the attachment angles of spring S1k is determined, in the case that
S1(k−1) = #, it is determined that the attachment angle β1(k−1)− α1(k−1) is equal to
270° for S1(k−1) if the direction of energy components 𝐊𝐏𝟏𝐤− 𝐊𝐋𝟏𝐤 is equal to 270°
for S1k according to rule 3. Likewise, in the case that S1(k−2)= #, S1(k−3) = #, and so
according to rule 3. Take the spring configuration matrix 5-4 as an example, because no
spring is attached between ground and link 3, it is determined that attachment angle β12− α12 is equal to 270° for S12 according to rule 3.
On the basis of rule 4, if S1k = # and S1(k−1) = 0 in the first row of spring
configuration matrix, it is determined that the direction of energy components 𝐊𝐏𝟏𝐤− 𝐊𝐋𝟏𝐤 must be 90° for S1k. After the attachment angles of S1k is determined, in the case
that S1(k+1) = #, it is determined that the direction of energy components 𝐊𝐏𝟏(𝐤+𝟏)− 𝐊𝐋𝟏(𝐤+𝟏) must be 90° for S1(k+1) if the attachment angle β1k− α1k is equal to 90°
for S1k according to rule 4. Likewise, in the case that S1(k+2)= #, S1(k+3) = #, and so
on, attachment angles can be determined iteratively if the requirements are achieved
according to rule 4. Take the spring configuration matrix 5-4 as an example, because no
spring is attached between ground and link 3, it is determined that the direction of energy
components 𝐊𝐏𝟏𝟒− 𝐊𝐋𝟏𝟒 must be 90° for S14 according to rule 4. Combined with the
fact that the attachment angle β14 is equal to 180° for S14 according to rule 5, the
attachment angle α14 is determined to be 90° accordingly. Iteratively, it is determined
that the direction of energy components 𝐊𝐏𝟏𝟓− 𝐊𝐋𝟏𝟓 must be 90° for S15 according to
rule 4 because the attachment angle β14− α14 is equal to 90° for S14. Combined with
the fact that the attachment angle β15− α15 is equal to 270° for S15 according to rule
1, the attachment angle α15 is determined to be 90° accordingly.
To ensure that the neutralization is accomplished for every entry in the non-first rows
of the total stiffness block matrix, at least one energy component with 180° direction
must be contributed to each entry in the non-first rows. Hence, for every the non-zero
element in the spring configuration matrix, the requirements according to rules 5, 6 and
7 have to be examined, starting from springs with larger span over number k − i to those
with smaller span over number.
If a spring attached between links i and k (Sik) is the only spring able to contribute
an energy component with 180° direction to entry (i+1)k, it is determined that the
attachment angle βik is equal to 180° for Sik according to rule 5. If a spring attached between links i and k (Sik) is the only spring able to contribute an energy component with
180° direction to entry i(k-1), it is determined that the attachment angle αik is equal to
0° for Sik according to rule 6. If a spring attached between links i and k (Sik) is the only
spring able to contribute an energy component with 180° direction to entry ik, it is
determined that the attachment angle βik− αik is equal to 0° for Sik according to rule
7. Take the spring configuration matrix 5-4 as an example, it is determined that the
β 180° for S
attachment angle β25− α25 has to be 0° for S25 according to rule 7, and that the
attachment angle β23− α23 has to be 0° for S23 according to rule 7.
Chapter 4
Qualitative efficiency index
4.1 Identification of energy contributions by spring installation configurations
Attachment angles of installed springs are determined dependent on the spring
configuration matrix. Moreover, for a spring configuration under the objective of
minimum span over number of springs [25], all spring attachment angles must be
determined according to the described rules because the design parameters of springs are
just enough to satisfy the constraint equations derived from the total stiffness block matrix.
Therefore, the spring attachment angles are not arbitrary choices but determined values
when the spring configuration matrix is selected. That is to say, we can select an
admissible spring configuration matrix and determine the attachment angles of installed
springs by examining the counteraction between gravitational and elastic potential energy
expressed in the total stiffness block matrix.
The direction properties of elastic energy components are also determined
simultaneously. For the elastic energy contributions in the first row, contributions with 270° direction are providing effects aligned with gravity and identified as negative
contributions which are harmful for static balance; elastic energy contributions with 90°
which are beneficial for static balance. For the elastic energy contributions in the
non-first rows, contributions with 0° direction are redundant spring side effects which are
undesired for static balance and identified as negative contributions; elastic energy
contributions with 180° direction are counteracting against spring side effects and
identified as positive contributions.
Take the articulated manipulator with spring configuration matrix 5-4 as the example,
attachment angles of spring S14 are α14= 90° and β14 = 180° as described in Table
3. The direction of the proximal elastic energy component 𝐊𝐏𝟏𝟒 is 90° according to Eq.
(15a); therefore, 𝐊𝐏𝟏𝟒 is a positive contribution. Likewise, the distal elastic energy
component 𝐊𝐃𝟏𝟒 is a positive contribution with 180° direction according to Eq. (15b);
the leading elastic energy component 𝐊𝐋𝟏𝟒 is a negative contribution with 270°
direction according to Eq. (15c); the side effect elastic energy component 𝐊𝐒𝟏𝟒 is a
negative contribution with 0° direction according to Eq. (15d).
The distribution of total stiffness block matrix can be rearranged as the form
expressed below in Eq. (21) where the positive contributions are placed in the right side
and the negative contributions are placed in the left side.
N
4.2 Derivation of qualitative efficiency index
The negative elastic energy contributions caused by installed springs are required to
be neutralized by additional springs to achieve static balance. In other words, the negative
contributions cause a waste of elastic potential energy. The ratio of positive elastic energy
contributions ought to be as high as possible. Therefore, the qualitative efficiency index
is defined as the number ratio of positive elastic energy contributions to total elastic
energy contributions represented in Eq. (22). Statically balanced articulated manipulators
with higher qualitative efficiency indexes are assessed to be more energy efficient.
Number of positive elastic energy contributions Qualitative efficiency index =
Number of total elastic energy contributions (22)
For example, the number of positive elastic energy contributions for the articulated
distribution expressed in Eq. (21). The qualitative efficiency index is obtained accordingly.
The number of positive elastic energy contributions and the qualitative efficiency index
for four-link and five-link articulated manipulator with admissible spring configurations
are listed in Tables 2 and 3 respectively.
Table 2
Spring installation configurations and qualitative efficiency indexes for four-link articulated manipulators
0 0 # #
Table 3
Spring installation configurations and qualitative efficiency indexes for five-link articulated manipulators
0 0 # 0 #
4.3 Comparison of energy efficiency of each spring
Energy efficiency of each spring is listed in Tables 2 and 3. The qualitative efficiency
index of each spring is the number ratio of positive elastic energy contributions to total
elastic energy contributions from each spring respectively.
A mono-articular spring is a spring that only spans over one joint, i.e. a spring attached between links i and k (Sik) with k − i = 1. The qualitative efficiency index of a
mono-articular spring is always 1.000 because mono-articular springs always provide
exactly one positive energy contribution to the stiffness block matrix. Moreover, it is generally observed that the greater the span over number k − i of a spring is, the lower
the qualitative efficiency index of the spring is. The main reason is that the side effect
energy components effect more entries as the span over number increases, leading the
number of negative contributions from the spring to be larger. Physically, the larger the
span over number of a spring is, the more complicated the form of the elastic energy
performance is, leading the possible energy waste more likely to occur. Therefore,
replacing larger span over number springs with smaller span over number ones could be
an intuitive way to increase the energy efficiency if possible.
4.4 Comparison of energy efficiency by qualitative efficiency index
Among the four-link statically balanced articulated manipulators, the articulated
manipulators with spring configuration matrices 4-1 and 4-2 have higher qualitative
efficiency indexes than those with spring configuration matrix 4-3; thus, articulated
manipulators with spring configuration matrices 4-1 and 4-2 are more energy efficient
than those with spring configuration matrix 4-3.
Among the five-link statically balanced articulated manipulators, the articulated
manipulator with spring configuration matrix 5-4 have the highest qualitative efficiency
index among the articulated manipulators with spring configuration matrices 5-1 to 5-5;
thus, articulated manipulator with spring configuration matrix 5-4 is the most energy
efficient among those with spring configuration matrices 5-1 to 5-5.
Notice that qualitative efficiency index of articulated manipulators with spring
configuration matrix 4-1 and 4-2 are the same, and qualitative efficiency index of
articulated manipulators with spring configuration matrix 5-1, 5-2 and 5-3 are the same.
The reason is that the difference between spring configuration matrices 4-1 and 4-2 is just
a substitution of the equivalent spring configuration installation for three adjacent links
[25]. Specifically, the springs S23 and S24 in the spring configuration matrix 4-1 is
S S
acquire the spring configuration matrix 4-2. The summation of elastic energy
contributions from springs S23 and S24 in the spring configuration matrix 4-1 is equal
to the summation of elastic energy contributions from springs S24 and S34 in the spring
configuration matrix 4-2. Although the distributions of energy contributions are not
identical to each other for spring configuration matrices 4-1 and 4-2, the number ratio of
positive contributions to total contributions are the same. The same rationale can be
applied for the spring configuration matrices 5-1 and 5-2 as well as the spring
configuration matrices 5-2 and 5-3.
Chapter 5
Quantitative efficiency index
5.1 Derivation of quantitative efficiency index
The qualitative efficiency indexes for energy efficiency assessment are applied in
the case that the magnitudes of gravitational energy contributions are not taken into
consideration since the specific parameters of articulated manipulators are unknown.
Nevertheless, for statically balanced articulated manipulators in practical, it is necessary
to take the magnitudes of gravitational energy contributions into consideration to obtain
a more accurate assessment.
If the specific parameters of articulated manipulators are given, magnitudes of
gravitational energy contributions can be obtained by the equations derived from the
counteraction between gravitational and elastic potential energy in the total stiffness block
matrix. For a general case, under the assumption that all link masses are equal to each
other as displayed in Eq. (23a), all link lengths are equal to each other as displayed in Eq.
(23b) and all mass centers are located in the middle point of corresponding links as
displayed in Eq. (23c), the magnitudes of gravitational contributions can be determined.
r = r for j = 1, 2, j , n (23b)
According to Eq. (10b), the magnitudes of gravitational contributions are obtained
and listed in Eq. (24a-d):
The ratio of the magnitudes of gravitational energy contributions from G12 to G15
is 7: 5: 3: 1. The magnitudes of elastic energy contributions can be obtained relative to
gravitational energy contributions considering the counteraction between gravitational
and elastic potential energy in the total stiffness block matrix.
Following the distribution of total stiffness block matrix in Eq. (21) based on the
spring configuration matrix 5-4 as the example, the distribution of magnitudes of
corresponding elastic energy contributions are determined and demonstrated in Eq. (25):
14
25 14 25
elastic energy contributions to total elastic energy contributions as represented in Eq. (26).
Statically balanced articulated manipulators with higher quantitative efficiency indexes
are assessed to be more energy efficient.
Magnitude of positive elastic energy contributions Quantitative efficiency index =
Magnitude of total elastic energy contributions (26)
For example, sum up the magnitudes of the terms in the right side in Eq. (25) to
obtain the magnitude of positive elastic energy contributions. Sum up the magnitudes of
all terms except gravitational energy contributions in Eq. (25) to obtain the magnitude of
total elastic energy contributions. The quantitative efficiency index of articulated
manipulator with spring configuration matrix 5-4 is acquired accordingly as shown in Eq.
(27):
2
Quantitative efficiency index with spring configuration matrix 5 - 4
b b b
The symbolic representation of quantitative efficiency index is a function of some
spring attachment length ratios. For the example shown in Eq. (27), the quantitative
efficiency index is dependent on spring attachment length ratios b14
r4, a15
r1, and br25
5. These spring attachment length ratios and spring constants are dependent to each other, and can
be determined through trial-and-error to make sure the maximum elongation of each
spring is appropriate [25]. Proper spring attachment length ratios are suggested to be less
than 1. Assume that the attachment length ratios are equal to each other denoted by ρ.
Thus, the quantitative efficiency index can be simplified as a function of ρ. Eq. (27) can
be reformulated as Eq. (28) shown below:
4 3 2
4 3 2
Quantitative efficiency index with spring configuration matrix 5 - 4
18 34 20 7 1
= 20 52 40 14 2
(28)
Following the process mentioned above, the quantitative efficiency index for
four-link and five-four-link articulated manipulators with admissible spring configuration matrices
are obtained as functions of ρ and listed in Tables 4 and 5 respectively.
Table 4
Quantitative efficiency index of four-link articulated manipulators
0 0 # #
Table 5
Quantitative efficiency index of five-link articulated manipulators
5.2 Comparison of energy efficiency by quantitative efficiency index
The quantitative efficiency index indicates the proportion of elastic energy
contributions that does assist in achieving static balance. Therefore, the higher the
quantitative efficiency index is, the better the energy efficiency the mechanism is. The
quantitative efficiency index approaches the minimum value as the attachment length ratio ρ approaches zero. The minimum value is 0.5 for every case as the spring constants
0 0 # 0 #
become infinite large leading both negative and positive elastic energy contributions
infinite large. The quantitative efficiency index approaches the maximum value as the attachment length ratio ρ approaches infinity. The maximum values for each case are
different as listed in Tables 4 and 5.
Since proper spring attachment length ratio ρ is suggested to be less than 1,
quantitative efficiency indexes for ρ = 0.2, 0.5, 1 are listed in Tables 4 and 5 for
comparisons. The difference among quantitative efficiency indexes becomes larger when
the spring attachment length ratio ρ increases.
Furthermore, the quantitative efficiency index of four-link and five-link articulated
manipulators with admissible spring configuration matrices are illustrated and compared
in Figs. 7 and 8 respectively.
Fig. 7. Quantitative efficiency index of four-link articulated manipulators Configuration 4-3
Configuration 4-1, 4-2
Fig. 8. Quantitative efficiency index of five-link articulated manipulators
In theory, it is expected to maximize the quantitative efficiency index; nevertheless,
higher quantitative efficiency indexes requires higher spring attachment length ratios,
leading the installed springs to occupy larger space. The trade-off between energy
efficiency and space saving has to be properly considered according to requirements of
each individual design. The comparisons in this study are made when the spring
attachment length ratio are equivalent to each other.
Among the four-link statically balanced articulated manipulators, those with spring
configuration matrices 4-1 and 4-2 have higher quantitative efficiency indexes than those
Configurations 5-1, 5-2, 5-3
Configuration 5-5 Configuration 5-4
configuration matrices 4-1 and 4-2 are more energy efficient than those with spring
configuration matrix 4-3.
Among the five-link statically balanced articulated manipulators, the articulated
manipulators with spring configuration matrices 5-1, 5-2 and 5-3 have the highest
qualitative efficiency indexes among the articulated manipulators with spring
configuration matrices 5-1 to 5-5; thus, articulated manipulators with spring configuration
matrices 5-1, 5-2 and 5-3 are the most energy efficient among those with spring
configuration matrices 5-1 to 5-5. Notice that the assessment results for five-link
articulated manipulators are different between using qualitative efficiency index and
quantitative efficiency index.
Chapter 6
Design example
6.1 Articulated manipulator with equal link mass
A five-link statically spring-balanced articulated manipulator is considered as a
design example. In the case that a designer intends to design a five-link
(four-degree-of-freedom) spring-balanced articulated manipulator, the admissible spring configurations
under the objective of minimum span over number of spring are listed in Table 3. In the
preliminary stage of the mechanism design, the size and mass parameters of the
mechanism might remain unknown. However, the qualitative efficiency index still can be
obtained following the aforementioned descriptions since it is known that the designated
mechanism is a five-link articulated manipulator.
As shown in Table 3, the articulated manipulator with spring configuration matrix
5-4 have the highest qualitative efficiency index among the five-link articulated
manipulators. Thus, the articulated manipulator with spring configuration matrix 5-4 is
considered the most energy efficient design in the preliminary stage and selected as the
fundamental form for further development of the specific mechanism.
In the subsequent stages, the energy efficiency assessment with quantitative
and link mass are specifically determined. If the specific parameters are determined that
all link mass are equal to each other as displayed, and that all link lengths are equal to
each other, and that all mass centers are located in the middle point of corresponding links,
as shown in Eqs. (23a-c).
The quantitative efficiency index is obtained and shown in Table 5 and Fig. 8. The
quantitative efficiency indexes of articulated manipulators with spring configuration
matrices 5-1, 5-2 and 5-3 are the highest. The quantitative efficiency index indicates the
proportion of elastic energy contributions that does assist in achieving static balance. In
the case that ρ = 1. The quantitative efficiency indexes could vary from 0.613 to 0.638
as shown in Table 5. Therefore, 2.5% energy waste could be prevented is the quantitative
efficiency index method is adopted during the mechanism design processes.
6.2 Articulated manipulator with descending link mass
In the subsequent stages, if the specific parameters are determined that the mass of
each link is half as heavy as the preceding link owing to the strength requirement of the
manipulator as displayed in Eq. (29a), and that all link lengths are equal to each other as
shown in Eq. (29b), and that all mass centers are located in the middle point of
corresponding links as shown in Eq. (29c).
j j-1
m = m for j = 1, 2, , n
2 (29a)
r = r for j = 1, 2, j , n (29b)
j j
s 1
= for j = 1, 2, , n
r 2 (29c)
The quantitative efficiency index can be obtained following the aforementioned
processes. The magnitudes of gravitational energy contributions can be obtained, and the
magnitudes of elastic energy contributions is determined accordingly. The quantitative
magnitudes of elastic energy contributions is determined accordingly. The quantitative