Motivation and preview

^Tuv ^Guv



^Kikuv ⁽²⁾

Considering the properties of stiffness block matrices, admissible spring

configurations were derived under objectives of minimum number of springs and

minimum span over number of springs [25].

1.3 Motivation and preview

Based on the stiffness block matrix approach, spring attachment angles were

predetermined and discussed before the discussion about criteria of admissible spring

configurations in previous study [25]. However, previous research focused on the location

and configuration of installed springs, valuable insight from the determination of spring

attachment angles for understanding the function and efficiency of elastic energy was not

well described. The properties of stiffness block matrix can be further discussed to

understand the energy efficiency of statically spring-balanced articulated manipulators.

In this study, the spring installation configurations such as the spring attachment

angles and direction properties of elastic energy contributions are discussed based on the

Rules for determination of spring installation configurations are elaborated; therefore, the

spring attachment angles and direction properties of elastic energy contributions can be

found for articulated manipulators with different spring configuration matrices. Elastic

energy contributions are therefore identified to be positive contributions or negative

contributions by their direction properties. Elastic energy contributions providing effects

aligned with gravity are redundant for static balance and regarded as negative

contributions. Elastic energy contributions counteracting against gravity and redundant

elastic effects are regarded as positive contributions. A qualitative efficiency index is

proposed as the number ratio of positive elastic energy contributions to total elastic energy

contributions. Furthermore, in the case that the magnitudes of gravitational energy

contributions are taken into consideration, a quantitative efficiency index is proposed as

the magnitude ratio of positive elastic energy contributions to total elastic energy

contributions. Energy efficiency of an n-link statically spring-balanced articulated

manipulators can be assessed according to the proposed efficiency indexes. Comparisons

among four-link and five-link statically balanced articulated manipulators with different

admissible spring configuration matrices are illustrated respectively. A design example is

demonstrated to highlight the practical uses of the efficiency indexes. The energy

efficiency of statically spring-balanced articulated manipulators can be assessed and

compared during the design processes to help the designer obtained the articulated

manipulator design with minimum energy wastes without trails and errors.

Chapter 2

Stiffness block matrix representation

2.1 Gravitational stiffness block matrix

On the basis of Denavit-Hartenberg representation [26], the coordinate system of the

n-link planar articulated manipulator is constructed as shown in Fig. 5. The Cartesian

planar coordinate system is applied on each link. Unit axes x_𝐣 and y_𝐣 are orthonormal

vectors and axis x_𝐣 is aligned on link vector r_𝐣. Each succeeding coordinate system with

respect to its preceding coordinate system can be represented by a rotation matrix 𝐑(θ_j)

[25]. θ_j is the joint angle from the (j − 1)^th coordinate to the j^th coordinate frame.

Moreover, θ_ik is defined as the relative angle from the i^th coordinate frame to the k^th

coordinate frame.

Fig. 5. Coordinate system of an n-link articulated manipulator

Referring to the n-link articulated manipulator as shown in Fig. 5, the position vector

of mass center of link j denoted by 𝐩_𝐣 is expressed as Eq. (3):

matrix times a direction vector [25], the gravitational potential energy denoted by U_G is

derived as Eq. (4):

n

G j

j=2

U = -



m ^{g p T j (4)}

where m_j represents the mass of link j, and g represents the gravitational acceleration

vector. The gravitational potential energy in Eq. (4) can be arranged as the form of

stiffness block matrix representation [23,25], as shown in Eq. (5):

T

where [𝐆_𝐮𝐯] is named the gravitational stiffness block matrix [23,25]. The terms 𝐆_𝐮𝐯

in the gravitational stiffness block matrix are named the gravitational energy components,

where subscribes indicating the term is located in the u^th row of v^th column. Values of

gravitational energy components are expressed as Eq. (6):

 

v ^v

 

v ⁿ w gravitational stiffness block matrix [𝐆_𝐮𝐯] is expressed as a symmetric matrix in which

only upper triangular matrix is considered [23,25], the distribution of gravitational

stiffness block matrix is shown in Eq. (7):

0

2.2 Property of gravitational stiffness block matrix

Each non-zero gravitational energy component 𝐆_𝟏𝐯 represents the gravitational

energy contribution that acts between ground and link v. Thus, non-zero gravitational

energy components are located only in the first row of the gravitational stiffness block

matrix as expressed in Eq. (7).

Because energy components are mathematically representing planar vector space,

we can rearrange and divide each gravitational energy component into a direction and a

magnitude as expressed in Eq. (8):

     

mag dir for v = 2, 3, ..., n

1v  1v 1v

G G R G (8)

where dir(𝐆_𝟏𝐯) is the direction and mag(𝐆_𝟏𝐯) is the magnitude of the gravitational

energy component 𝐆_𝟏𝐯. Moreover, the direction and magnitude can be expressed as Eqs.

(9a) and (9b):

The directions of gravitational energy components are the gravitational direction

270° plus an arctangent angle denoted as φ’_v as represented in Eq. (9a). In the case that

the mass center deflection angle φ_v is not equal to zero, the angle φ’_v will not be equal

to zero correspondingly. To be more specifically, the range of φ’_v can be discussed

according to some basic assumptions to obtain a general understanding. If all magnitudes of mass center deflection angles |φ_v| are not greater than 45°, all mass m_v are equal

to each other, and all mass center length ratio ^s_r^v

v are not greater than 0.5, the angle ranges of all φ’_v are determined to be between −22.5° and 22.5°. Since the angle ranges of φ’_v are not considerable in magnitude, it is found that the directions of all gravitational

energy components dir(𝐆_𝟏𝐯) are approximately 270°. On the other hand, the direction

of gravitational energy component dir(𝐆_𝟏𝐯) becomes far from 270° only when the link

shape is extremely strange, designed deliberately to make φ’_v considerably great in

magnitude.

In this paper, all mass center deflection angles φ_v are assumed to be zero. In other

words, all mass centers of links are assumed to be exactly on the line through the joints

of links. Under the zero φ_v assumption, the direction and magnitude of each

gravitational energy component can be simplified as Eqs. (10a) and (10b):

 

The direction of every gravitational energy component dir(𝐆_𝟏𝐯) is exactly 270°,

as expressed in Eq. (10a), making the discussion in the following sections more clearly

to understand. However, the findings in this study are also valid for the cases with

reasonable mass center deflection angle φ_v.

The mass center is approximately located in the centroid of each link so that the mass center length ratio ^s_r^v

v is generally suggested to be less than 1 intuitively. Under the circumstances, the magnitudes of gravitational energy components mag(𝐆_𝟏𝐯) are

distinct to each other in a descending order from v = 2 to v = n according to Eq. (9b).

2.3 Elastic stiffness block matrix

For a statically balanced articulated manipulator, a spring configuration matrix [S_uv], an n × n matrix, is defined to represent the configuration of installed springs

[24,25]. Only one spring at most is allowed to be attached between two links. The

non-zero element of S_uv = # which lies in the u^th row of v^th column stands for a spring

attached between links u and v. Only the upper triangular spring configuration matrix is

considered because of the symmetry and redundancy [24,25].

As illustrated in Fig. 6, a spring with spring constant k_ik attached between links i

and k of an n-link articulated manipulator is denoted by S_ik= # in the spring

configuration matrix. Furthermore, since link i is the proximal attached link and link k is

the distal attached link, i is defined to be less than k.

Fig. 6. Spring attached between links i and k of an n-link articulated manipulator

Because the zero-free-length springs are applied, the distance between two attachment points |𝐱_𝐢𝐤| is regarded as the elongation of the spring, as expressed in Eq.

(11):

of a transformation matrix times a direction vector [25], the elastic potential energy from

Notice that U_E shown in Eq. (1) denotes the elastic potential energy of the whole

system, while U_E^ik represents the elastic potential energy from spring S_ik. The elastic

potential energy in Eq. (12) can be arranged as the form of stiffness block matrix

representation as shown in Eq. (13) [23,25]:

T

superscripts indicate that the term is induced by the spring attached between links i and

k, and the subscripts indicate that the term is located in the u^th row of v^th column.

Diagonal elastic energy components 𝐊_𝐮𝐯^𝐢𝐤 with u = v are expressed as Eqs.

(14a-c):

2

r_k are the proximal and distal attachment length ratios respectively.

2.4 Property of elastic stiffness block matrix

Energy components are mathematically representing planar vector space, and can

therefore be divided into a direction and a magnitude. Notice that the representation of

elastic energy components expressed in Eqs. (14a-c) and (15a-d) are already divided into

a direction and a magnitude as the form shown in Eq. (16):

     

contribution that acts between links u and v caused by the spring attached between links i and k (S_ik).

The change of the relative angle θ_uv causes an energy variation of 𝐫_𝐮^𝐓𝐊_𝐮𝐯^𝐢𝐤𝐫_𝐯.

Therefore, the elastic energy contribution from the diagonal elastic energy component 𝐊_𝐮𝐯^𝐢𝐤 with u = v does not vary when the relative position of links change, since the

relative angle θ_uv with u = v is always zero. That is to say, the diagonal energy

components in Eq. (14a-c) are irrelevant to statically balanced condition; therefore, only

off-diagonal energy components are considered during the discussion about static balance.

The total number of off-diagonal elastic energy components for the spring attached

between links i and k is determined to be (k−i+1)(k−i)

2 , and is regarded as the number of total elastic energy contributions. The number of total elastic energy contributions are

listed in Tables 2 and 3 respectively for the four-link and five-link articulated

manipulators with the spring configuration matrices assessed in this study. Intuitively, the

number of total elastic energy contributions should be as little as possible for the reason

that springs with less number of total contributions are less complicated in the form of

energy performance, leading the possible energy waste less likely to occur. Thus, the

examples demonstrated in this study shown in Tables 2 to 5 are ordered ascendingly by

the number of total elastic energy contributions as a brief guess that the less the number

of total contributions is, the more efficient the articulated manipulator is.

Off-diagonal elastic energy components are denoted by the symbol 𝐊_𝐮𝐯^𝐢𝐤 with

different subscripts according to the locations. However, only four distinct values exist

among them. It is confusing when symbols with different subscripts are representing the

same value. Therefore, the denotations of off-diagonal elastic energy components are

further simplified as 𝐊_𝐏^𝐢𝐤, 𝐊_𝐃^𝐢𝐤, 𝐊_𝐋^𝐢𝐤, 𝐊_𝐒^𝐢𝐤.

Elastic energy components dominated by the proximal installation configurations

a_ik

r_i and α_ik of the spring are called the proximal elastic energy components, denoted as 𝐊_𝐏^𝐢𝐤 shown in Eq. (15a). All the proximal elastic energy components 𝐊_𝐏^𝐢𝐤 are located in

the i^th row. Elastic energy components dominated by the distal installation

configurations ^b_r^ik

k and β_ik of the spring are called the distal elastic energy components, denoted as 𝐊_𝐃^𝐢𝐤 shown in Eq. (15b). All the distal elastic energy components 𝐊_𝐃^𝐢𝐤 are

located in the k^th column. Elastic energy component dominated by both proximal and

distal installation configurations ^a_r^ik

i, α_ik, ^b_r^ik

k and β_ik of the spring is called the leading elastic energy component, denoted as 𝐊_𝐋^𝐢𝐤 shown in Eq. (15c). The leading elastic energy

𝐢𝐤 th th

with value k_ik𝐑(0°) are called the side effect elastic energy components, denoted as 𝐊_𝐒^𝐢𝐤 expressed in Eq. (15d). All the side effect elastic energy components 𝐊_𝐒^𝐢𝐤 are

located in the (i + 1)^th to (k − 1)^th rows of (i + 1)^th to (k − 1)^th columns,

directions of which are all 0°. Side effect elastic energy components 𝐊_𝐒^𝐢𝐤 are considered

troublesome and needed to be balanced by other springs.

The distribution of elastic stiffness matrix [𝐊_𝐮𝐯^𝐢𝐤] is expressed in Eq. (17):

0 0 0 0 0 0 0

Chapter 3

Spring installation configurations

3.1 Examination of energy counteraction from vector perspective

For four-link and five-link articulated manipulators, admissible spring configuration

matrices under the objective of minimum span over number of springs [25] are listed in

Tables 2 and 3 respectively. Starting from an admissible spring configuration matrix, the

energy contributions in the total stiffness block matrix of a specific spring configuration

can be presented according to the definition of gravitational and elastic stiffness block

matrix shown in Eqs. (7) and (17).

For example, starting from the spring configuration matrix 5-4 shown in Table 3,

following the definition of gravitational and elastic stiffness block matrix shown in Eqs.

(7) and (17), the energy contributions in the total stiffness block matrix can be expressed

as shown below in Eq. (18):

Once the distribution of the total stiffness block matrix is acquired, equations derived

from the entries of total stiffness block matrix can be obtained according to the fact that

summation of energy components in the same entry equal to zero.

Energy components are 2 × 2 matrices which consist of four terms for each of them,

so the fact that summation of energy components in the same entry equal to zero ought to

provide four linear equations. However, only two out of the four linear equations are

independent because both gravitational and elastic energy components are

mathematically representing planar vector space which consist of only two independent

variables, directions and magnitudes. By dividing each energy component into a direction

and a magnitude as expressed in Eqs. (8) and (16), the general knowledge of vector

operations can be applied throughout the study.

Thus, it is much easier to understand the philosophy of the neutralization among

energy components for static balancing if each energy component is considered as a

vector; while it is more compact to present the whole system in matrix form. By

examination of the total stiffness block matrix, which expresses the counteraction

between gravitational and elastic potential energy, the spring attachment angles are found

to be determined according to some specific rules.

3.2 Characteristics for determination of spring installation configurations

First of all, only energy components 𝐆_𝟏𝐧 and 𝐊_𝐋^𝟏𝐧 exist in entry 1n for every

admissible spring configuration. Since the sum of 𝐆_𝟏𝐧 and 𝐊_𝐋^𝟏𝐧 is equal to zero to

achieve static balance, 𝐊_𝐋^𝟏𝐧 must have the opposite direction and the equal magnitude

compared with 𝐆_𝟏𝐧. On account of the direction property of gravitational energy

components, rule 1 can be derived as shown in Table 1: for the spring attached between ground and end link (S_1n), the attachment angle β_1n− α_1n must be 270°.

If no spring is attached between link 3 and end link, Eqs. (19a) and (19b) can be

obtained considering the neutralization within entry 2n and entry 3n. Therefore, for the spring attached between link 2 and end link (S_2n), 𝐊_𝐋^𝟐𝐧 must be equal to 𝐊_𝐃^𝟐𝐧 to keep

both Eqs. (19a) and (19b) satisfied; hence, the attachment angle α_2n must be 180° and

attachment length ratios ^a_r²ⁿ

2 must be 1 to keep 𝐊_𝐋^𝟐𝐧 and 𝐊_𝐃^𝟐𝐧 equal. Therefore, rule 2 can be obtained as shown in Table 1.

 0

General equations based on the neutralization within entries 1(k+1), 1k and 1(k-1)

derived by subtracting Eq. (20a) from Eq. (20b). For a spring attached between ground and link k (S_1k), if no spring or only spring whose direction of energy components

𝐊_𝐏^{𝟏(𝐤+𝟏)}− 𝐊_𝐋^{𝟏(𝐤+𝟏)} equals to 270° is attached between ground and link (k+1),

considering Eq. (20d) on account of the direction property and magnitude property of

gravitational energy components, it is required that the direction of energy component 𝐊_𝐋^𝟏𝐤 must be 90° to keep entry 1k balanced because the direction of the summation of

all the other energy components in Eq. (20d) is 270°. Therefore, the attachment angle β_1k− α_1k for S_1k must be 270°. Rule 3 shown in Table 1 is obtained accordingly. attached between ground and link k (S_1k), if no spring or only spring with attachment

angle β_1(k−1)− α_1(k−1) = 90° is attached between ground and link (k-1), considering

Eq. (20e) on account of the direction property and magnitude property of gravitational

energy components, the direction of energy components 𝐊_𝐏^𝟏𝐤− 𝐊_𝐋^𝟏𝐤 must be 90° to

keep entry 1k balanced because the direction of the summation of all the other energy

components in Eq. (20e) is 270°. Therefore, rule 4 can be obtained as shown in Table 1.

It is noticed that rule 4 does not provide direct constraints on spring attachment angles

and only gives a relationship between elastic energy components. Nevertheless this

relationship is useful for the determination of spring attachment angles, if other

constraints are considered at the same time. For example, if rules 3 and 4 are applied to a

spring S_1k at the same time, the attachment angle β_1k− α_1k must be 270° according

to rule 3; the attachment angle α_1k must be 90° according to the constraints provided

by both rules 3 and 4. Another example is that if rules 4 and 5 are applied to a spring S_1k

at the same time, the attachment angle β_1k must be 180° according to rule 5; the

attachment angle α_1k must be 90° according to the constraints provided by both rules

4 and 5.

Considering the neutralization within an arbitrary entry ik with i ≠ 1 and k ≠ n, all springs attached between links u and v (S_uv) for 1 ≤ u ≤ i − 1 and k + 1 ≤ v ≤ n

elastic energy components are inevitably to be 0° as mentioned in the aforementioned

section shown in Eq. (15d). Hence, to neutralize these 0° direction energy components

in entry ik, the direction of the summation of all the following elastic energy components

contributed to entry ik must be 180°: the leading elastic energy components from springs

attached between links i and k, the proximal elastic energy components from springs

attached between links i and v for each k + 1 ≤ v ≤ n, and the distal elastic energy

components from springs attached between links u and k for each 1 ≤ u ≤ i − 1. Thus,

some of the spring attachment angles can be determined in order to neutralize the 0°

direction energy components caused by other springs.

A spring attached between links i and k (S_ik) with k − i ≥ 2 must contribute its

distal elastic energy component as a 180° direction energy component to balance the

entry (i+1)k, if no spring or only springs with attachment angle β_uk = 0° are attached

between links u and k for each 1 ≤ u ≤ i − 1, and if no spring or only springs with

attachment angle α_(i+1)v = 180° are attached between links (i+1) and v for each k + 1 ≤ v ≤ n, and if no spring or only spring with attachment angle β_(i+1)k− α_(i+1)k =

180° is attached between links (i+1) and k. Therefore, the attachment angle β_ik for S_ik

must be 180° and rule 5 is obtained in Table 1 accordingly.

A spring attached between links i and k (S_ik) with k − i ≥ 2 and i ≠ 1 must

contribute its proximal elastic energy component as a 180° direction energy component

to balance the entry i(k-1), if no spring or only springs with attachment angle α_iv= 180°

are attached between links i and v for each k + 1 ≤ v ≤ n, and if no spring or only

springs with attachment angle β_u(k−1) = 0° are attached between links u and (k-1) for

each 1 ≤ u ≤ i − 1, and if no spring or only spring with attachment angle β_i(k−1)− α_i(k−1) = 180° is attached between links i and (k-1). Hence, the attachment angle α_ik

for S_ik must be 0° as rule 6 describes shown in Table 1.

A spring attached between links i and k (S_ik) with i ≠ 1 must contribute its leading

elastic energy component as a 180° direction energy component to balance the entry ik,

if no spring or only springs with attachment angle α_iv= 180° are attached between links

i and v for each k + 1 ≤ v ≤ n, and if no spring or only springs with attachment angle β_uk = 0° are attached between links u and k for each 1 ≤ u ≤ i − 1 .Thus, the

attachment angle β_ik− α_ik for S_ik must be 0°, and rule 7 can be obtained accordingly

as described in Table 1.

Table 1

Rules for determination of spring attachment angles from admissible spring configuration matrices

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 (S_1n) is the only spring able to neutralize

the gravitational energy contribution in entry 1n. It is determined that the attachment

angle β_1n− α_1n 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 (S_2n), it is determined that the attachment angle α_2n

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 (S_1k) where

Based on rule 5, for each spring attached between ground and link k (S_1k) where

在文檔中 彈簧靜平衡機構之能量效率分析 (頁 13-0)