CHAPTER 3 Mounting region : practical spring configurations on manipulators
3.3 Cable-driven mechanism of spring on the basis of the reels
The reels can be utilized for cable driven mechanisms. By using a reel which has two different sizes (e.g., 𝑅𝑤 and 𝑅𝑠 for the wire and spring, respectively; Fig. 5), the elongation of the springs can be reduced through another shortening performance, 𝑠𝑟, as described in Eq. (8a). The reels are attached to the links. To ensure consistent performance of the pull, the torque on the reels should be kept as usual. Therefore, the spring coefficient is enlarged as the square of 𝑠𝑟, that is, 𝑘 becomes 𝑠𝑟2𝑘. With the cable driven mechanism on the basis of the reels, the mounting length of the springs is shortened as expressed in Eq. (8b). To quantify the minimum performance of 𝑠𝑟, the maximum elongation, pseudo-mounting length, and spring free length are utilized as described in Eq. (8c).
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Fig. 5 Cable driven mechanism with the reel for the spring
The benefit of using the reels is that only one reel is required for one spring;
furthermore, the reels are attached to the links, leading to higher stability of the pulley systems. Moreover, 𝑠𝑟 can be any positive number; thus, the spring coefficient will not become unmanageable. The string length considering the remaining space of the
mounting region is derived as Eq. (8d).
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CHAPTER 4
Placing region : Spring configurations for minimizing the
placing length
4.1 Determination of pseudo-mounting length
In Eqs. (6), (7) and (8), the string length is proportional to the placing length.
Intentionally minimized the placing length helps reduce the string length. The placing length can be shortened by arranging the spring on the proximal, distal, or adjacent links. Moreover, the placing length can be further minimized by arranging the spring in the mounting region in reverse.
Fig. 6 Mounting length and pseudo-mounting length of springs on the adjacent links
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The spring configurations for the minimized placing length are determined by three parameters, namely the length of the spring attachment vectors 𝑎, 𝑏 and the pseudo-mounting length 𝑚′ of the spring configurations. The pseudo-mounting length is the maximum length available for the springs to be arranged with a minimal placing length and is theoretically limited by the link configurations, because the springs are arranged on the links. The magnitude of the pseudo-mounting length is based on the residual link length defined away from the link joints to link’s proximal or distal ends (e.g., 𝑟𝑖𝑝 and 𝑟𝑖𝑑 in Fig. 1(a)). An appropriate link configuration has a long pseudo-mounting length,
and candidates for the pseudo-mounting length can be described using Eqs. (9) for
scenarios where the springs are arranged on adjacent links, as shown in Fig. 6
𝑚𝑖−1′ = max(𝑟(𝑖−1)𝑝+ 𝑟𝑖−1, 𝑟(𝑖−1)𝑑) (9a) 𝑚𝑖+1′ = max(𝑟(𝑖+1)𝑝, 𝑟𝑖+1+ 𝑟(𝑖+1)𝑑) (9b)
The pseudo-mounting length for the springs on proximal (distal) links can be realized by giving a vertical distance from the springs to the links that shortens the placing length such that 𝑝 equals 𝑎 sin 𝛼 (𝑏 sin 𝛽). To give a vertical distance to the placing length, the springs should be arranged from the point of vertical intersection on the links. The pseudo-mounting length is therefore solved as the longest length
measured away from the point of vertical intersection.
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Fig. 7 Mounting length and pseudo-mounting length of springs on the proximal link
However, this leads to various placing length scenarios, making the equations more varied and complex because the rotation angle can be any value, as shown in Fig. 7.
Two situations of pseudo-mounting length for arranging springs on proximal and distal links is as expressed in Eqs. (10a) and (10b), respectively. If the point of vertical intersection is located on the links, the pseudo-mounting length can be determined as the longest length of the two sections separated by the point. For scenarios where the point of vertical intersection of the spring is not located on the links, the total link length can be utilized fully as the pseudo-mounting length. Depending on the rotation angle of the springs, the pseudo-mounting length with the aligned or unaligned springs can also
be classified. The pseudo-mounting length of the springs is listed in Table 1.
max( cos cos ) or
i ip i id ip i id
m' r r a , r a r r r (10a)
max( cos cos ) or
k kp k kd kp k kd
m' r b , r r b r r r (10b)
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Table 1 The pseudo-mounting length of springs on the basis of the mounting sides and the attachment links with the springs aligned or not aligned
4.2 Minimized placing length
According to the spring configurations on the proximal link, distal link, or adjacent links, the minimization of the placing length on the basis of the pseudo-mounting length can be described for each configuration. Moreover, two scenarios—where the spring attachment vectors are either aligned or not aligned on the links—are discussed separately. Table 2 is a reference table for all cases of placing length equations and
classifications. Gray table cells are cases with an overextending mounting length, i.e., 𝑚 > 𝑚′, whereas the other cells are cases where 𝑚 < 𝑚′.
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A function is utilized for simplifying the law of cosines such that 𝐿𝑜𝑐(𝑎, 𝑟, 𝛼) represents √𝑎2+ 𝑟2− 2𝑎𝑟 cos 𝛼, as shown in the Table 2.
Table 2 Placing length equations of springs on the basis of the mounting sides and the attachment links
For cases where the mounting length is longer than the link length and pseudo-mounting length, the links should be extended so the springs can be arranged on the links. Multiple sets of springs can be arranged on one link. Therefore, the links may be further extended for arranging other springs.
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As an extreme case, extending the links for elongating the pseudo-mounting length
such that 𝑚′> 𝑚 always holds can be utilized. Therefore, the placing length on the proximal and distal link is a vertical distance (𝑝 = |𝑎 sin α| and |𝑏 sin 𝛽|, respectively),
which is the length measured from the point of vertical intersection to the spring
attachment point. This placing length for arranging springs on adjacent links means that the term 𝑚 − 𝑚′ can be eliminated when the springs are arranged starting from the point of vertical intersection. The equations in Table 2 can then be simplified.
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CHAPTER 5
Extension region : Routing configuration synthesis
5.1 Superposition property of articulated springs
Vectors possess superposition property. The routing configuration synthesis of the springs is a technique on the basis of the superposition property. The attachment vectors of springs are 𝒂, 𝒃 and 𝒗 defined from the aforementioned spring configures. By the superposition property of vectors, the attachment vectors are divided into 𝑛 sub vectors that 𝑛 sub loops are given herein as shown in Fig. 8, called “routing configurations” of the springs. The number of 𝑖 represents the 𝑖 sub loop of the springs, which is
composited by the sub vectors 𝒂𝒊, 𝒗𝒊 and 𝒃𝒊, where 𝑖 = 1~𝑛. The sub vectors are divided from the spring attachment vectors, i.e. 𝑛 ≥ 2. And the vectors are limited such as 0 < 𝒂𝒊 < 𝒂. The sub vectors are summed to be original magnitude as Eqs. (11) described that the superposition property of vectors is followed.
The routing configurations enable springs to be arranged with smaller sub
attachment vectors that occupies less space.
(𝒂𝟏… + 𝒂𝒊… + 𝒂𝒏) = 𝒂 (11a)
(𝒗𝟏… + 𝒗𝒊… + 𝒗𝐧) = 𝒗 (11b)
(𝒃𝟏… + 𝒃𝒊… + 𝒃𝒏) = 𝒃 (11c)
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Fig. 8 The illustrated model for the superposition property of spring attachment vectors
The superposition in virtual link vector can be realized by the utilization of
auxiliary links as Table 3 presents. Parallelogram linkages provide a copying property of the vectors to form the sub loops of the springs. Because the number of auxiliary links increases with the more joints that a spring spans over, the number of spanned links less than 4 is suggested. Moreover, the minimum auxiliary links are utilized for each spring configurations so the springs are arranged neither on proximal link nor distal link.
Table 3 The number and configurations of auxiliary links for virtual link
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To reduce the variables of the vector equations, scale ratio of the sub vectors w.r.t.
the attachment vectors are used such that 𝑁𝑎𝑖 = 𝑎𝑎𝑖, 𝑁𝑣𝑖 = 𝑣𝑣𝑖 and 𝑁𝑏𝑖 =𝑏𝑏𝑖, so
normalized vectors can be shown, i.e. 0 < 𝑁𝑎𝑖 < 1. With the normalization of vectors, following by the superposition property of vectors, the initial constraints of the sub
vectors in Eqs. (11) can be described as Eqs. (12) shown:
∑𝑛𝑖=1𝑁𝑎𝑖 = 1 (12a)
∑𝑛𝑖=1𝑁𝑣𝑖 = 1 (12b)
∑𝑛𝑖=1𝑁𝑏𝑖 = 1 (12c)
5.2 Routing configuration synthesis on the basis of equivalent articulated springs
While designing the routing configurations of the articular springs, to maintain the consistent spring capability in joint torques as expressed in Eqs. (3), a concept of conservation law is followed that the summation of the spring capability of sub-loops are supposed to be as usual as the original spring configuration described in Eqs. (13),
where the spring coefficients of sub-loops are normalized by the origin springs, i.e.
𝑁𝑘𝑖 =𝑘𝑘𝑖> 0, so the variables can be further reduced such as Eqs. (12), and the routing
configuration synthesis will be more general for different design cases.
∑𝑛𝑖=1𝑁𝑘𝑖𝑁𝑎𝑖𝑁𝑏𝑖 = 1 (13a)
∑𝑛𝑖=1𝑁𝑘𝑖𝑁𝑎𝑖𝑁𝑣𝑖 = 1 (13b)
∑𝑛𝑖=1𝑁𝑘𝑖𝑁𝑏𝑖𝑁𝑣𝑖 = 1 (13c)
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The physical meaning of Eqs. (13) is that a spring can be substituted by the other springs with their spring capability in joint torque are kept consistent, where the springs are call equivalent springs. To find the spring coefficients of the equivalent springs, the value of sub attachment vectors can be served as known parameters to solve the spring coefficients such that the 𝑁𝑘1, 𝑁𝑘2 … and 𝑁𝑘𝑛 are variables to be determined.
Simplifications can be utilized, for instance, the same spring capability in each sub
loop, 𝑁𝑘1𝑁𝑎1𝑁𝑏1 = 𝑁𝑘𝑖𝑁𝑎𝑖𝑁𝑏𝑖 can be assumed, or using the same spring coefficients, 𝑁𝑘𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 to derive available solutions. In order to compressively discuss the
general solution, no simplifications will be applied in the derivation, and the derivation of the routing configuration synthesis with two steps, 𝑛 = 2, is introduced in this paper as Fig. 9 describes.
Fig. 9 The routing configuration synthesis of the springs by two steps
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Two steps of the routing configurations are assumed, and 𝑁𝑘1 and 𝑁𝑘2 are the spring coefficients of the two equivalent springs to be solved shown in Eqs. (14). The simultaneous equations can be realized by graphical method that the coordinates of 𝑁𝑘1
and 𝑁𝑘2 are used and 𝑁𝑎1𝑁𝑏1 or 𝑁𝑎2𝑁𝑏2, for instance, are therefore the coefficient of 𝑁𝑘1 and 𝑁𝑘2 respectively.
𝑁𝑘1𝑁𝑎1𝑁𝑏1+ 𝑁𝑘2𝑁𝑎2𝑁𝑏2=1 (14a)
𝑁𝑘1𝑁𝑎1𝑁𝑣1+ 𝑁𝑘2𝑁𝑎2𝑁𝑣2=1 (14b)
𝑁𝑘1𝑁𝑏1𝑁𝑣1+ 𝑁𝑘2𝑁𝑏2𝑁𝑣2=1 (14c)
The sub vectors are realized as the intercept of each equation. As Fig. 10(a) presents, a dependent solution of Eqs. (14) can be solved if the three lines are collinear to each other. The intercepts are equal as Eq. (15a) shows, and the relationship between each sub vector can be derived in Eq. (15b) while the spring coefficients can then be
solved as Eq. (15c). A linear dependence solution of the equivalent springs exists.
⇒ 𝑁𝑎1𝑁𝑏1 = 𝑁𝑎1𝑁𝑣1= 𝑁𝑏1𝑁𝑣1 (15a)
⇒ 𝑁𝑎1 = 𝑁𝑏1 = 𝑁𝑣1 (15b)
⇒ 𝑁𝑘1 = 𝑁1
𝑎12 −𝑁𝑁𝑎22
𝑎12 𝑁𝑘2 (15c)
On the other hand, a unique solution of the spring coefficients of Eqs. (14) can be solved once these three lines intersect each other. The situation can be distinguished into two cases: an intersection of the lines with and without overlapping shown in Fig. 10(b).
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For the case of the lines with overlapping, because two lines overlap to each other, coefficients are same and equations are simplified. By the simplified two equations, an intercept of the two lines can be found and the value of the spring coefficients can be calculated as a function inversely proportional to the parameters. That is to say, if the Eqs. (14a) and (14b) overlap to each other, a solution of the spring coefficient can be found in the first part of Eqs. (16). If the Eqs. (14a) and (14c) are chosen, the solution will be the second part of Eqs. (16), while the third part is derived for the Eqs. (14b) and
(14c).
For the case of the lines without overlapping, one intercept of the three lines can be given as shown in Fig. 10(b). However, considering the constraints of the routing
configurations that the superposition property of the vectors should be followed, the unique solution that none of these three lines are parallel to each other are unable to be derived. A contradiction solution will be given such as the lines are overlapping if none overlapping case are assumed. These solutions show that the form 𝑁𝑎1 ≠ 𝑁𝑣1≠ 𝑁𝑏1 is invalid and not available for the routing configurations as a result.
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(a) Collinear case (b) An intersection with and without overlapping Fig. 10 The possible solutions of spring configurations on the basis of graphical method
With the previous derivations, the general solutions of the routing configurations can be found. Two cases of routing configurations show that the number of diverging parameters in the sub vectors 𝑁𝑎, 𝑁𝑣, 𝑁𝑏 should be 0 or 1. The first situation is that the sub vectors are equal to each other, and are resulted in a geometric similarity such as each sub loop is a smaller version of original loop.
Another situation is that one sub vector owns different value with respect to the
other two vectors, so called diverging parameters. The diverging parameter can locate in 𝑁𝑎, 𝑁𝑣 and 𝑁𝑏. If the diverging parameter locates in 𝑁𝑣, two ways to reach the
situation of scaled 𝑁𝑣 can be used. One is by scaling it’s all compositions such that a
geometric similarity is utilized for each sub vector, i.e. (𝑁𝑟(𝑖+1)… = 𝑁𝑟(𝑗)… = 𝑁𝑟(𝑘−1)) = 𝑁𝑣. However, this gives a variable length of the vector 𝑁𝑣 that makes it
impractical for applications since the length of the auxiliary links are constant but the length of the virtual link is variable. So another way to reach the situation of scaled 𝑁𝑣
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by just scaling one arbitrary composition of the virtual link is introduced. Only one link is chosen as a substitute of the diverging. In this way, the auxiliary links can be actually applied on the manipulators rather than be applied on the variable length of the virtual link. Two criteria for briefly describing the routing configuration synthesis are
concluded and established in Table 4.
Table 4 Criteria of the routing configuration synthesis for articulated manipulator
For the status that the virtual link does not exist, i.e. 𝑣 = 0, mono-articular springs are therefore exist. The derivation of the routing configuration synthesis for the mono-articular springs with two steps, 𝑛 = 2, can then be calculated. Only Eq. (14a) is
considered, and the result is shown that the 𝑁𝑘1 can be a function of 𝑁𝑎1, 𝑁𝑏1 and 𝑁𝑘2. If 𝑁𝑎1, 𝑁𝑏1 are determined, a linear dependent equation between 𝑁𝑘1 and 𝑁𝑘2
can be followed such as Eq. (17) describes.
1 1
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5.3 Routing configuration synthesis of articulated springs by wire winding
With the aforementioned routing configurations, the springs can be substituted by two or more equivalent springs that performs consistent spring torque to the
manipulator. However the utilization of multiple springs increases the difficulty of using routing configurations because the more springs used, the more interference may occur.
One solution to make it easier to arranging springs is by the technique of “wire winding”. The wire winding is that the redundant springs are removed and only one
equivalent sole spring is utilized for the routing configurations as shown in Fig. 11. The sub loops are connected. On this condition, the spring elongations of each sub loop are shared to each other and the spring coefficient are unique, that is, 𝑁𝑘1= 𝑁𝑘2 = 𝑁𝑘 if the two steps routing configuration are assumed. Therefore, elastic elements with similar characteristics can be utilized, e.g. a rubber band or an elastic Band.
Fig. 11 The routing configuration synthesis of the springs with the wire winding technique by an equivalent sole spring.
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Because the spring elongations are summed, by distributive law, two separated elongations should have the same term for combination. A scale parameter of the elongation is set as 𝑠𝑐 so Eq. (18a) can be realized as Eqs. (18b) and (18c) for the wire winding.
𝛿 = ∑2𝑖=1√(𝒗𝒊+ 𝒃𝒊− 𝒂𝒊) ∙ (𝒗𝒊+ 𝒃𝒊− 𝒂𝒊) (18a) 𝛿 = (1 + 𝑠𝑐)√(𝒗𝟏+ 𝒃𝟏− 𝒂𝟏) ∙ (𝒗𝟏+ 𝒃𝟏− 𝒂𝟏) (18b) 𝛿 = (1 + 𝑠𝑐)√𝑣12+ 𝑏12+ 𝑎12+ 2(𝑣1𝑏1𝑐𝑜𝑠𝜃𝑣𝑏− 𝑣1𝑎1𝑐𝑜𝑠𝜃𝑎𝑣− 𝑎1𝑏1𝑐𝑜𝑠𝜃𝑎𝑏) (18c)
For multi-articular springs, because each joint angle is the independent variable of the springs, one way to make the spring elongation reasonable is by ensuring the ratio of each coefficient to be a constant as described in Eq. (19a). The normalization of sub vectors are then used and the results of the routing configurations with wire winding are
shown in Eq. (19b).
The wire winding technique for the routing configurations give a symbiosis relationship to the springs since the elongation are shared. The symbiosis relationship can be realized by the extra elongation given from other sub loops as Eqs. (20a) to (20c) shown. So, the spring coefficient of the wire winding in multi-articular springs is
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therefore solved as Eq. (20d) on the basis of Eqs. (19), which is further reduced with
respect to the routing configurations without wire winding since 𝑠𝑐 > 0.
(1 + 𝑠𝑐)𝑁𝑘𝑁𝑎1𝑁𝑏1+ (1 +𝑠1
If the virtual link does not exist, i.e. 𝑣 = 0, mono-articular springs are therefore exist. The derivation of the routing configurations with the wire winding in mono-articular spring by two steps, 𝑛 = 2, is shown in Eq. (21a) and calculated in Eq. (21b).
However, the term, 𝑏𝑎, is waited to be defined. A shape parameter of the attachment vectors are therefore utilized for the term and notified as 𝑠ℎ so Eq. (21c) can be derived and the results can be shown in Eq. (21d). Two cases of the routing
configurations with the wire winding for mono-articular springs are presented. The wire winding technique can be applied once the shape parameter follows Eq. (21d). Or the shape parameter can be any once two sub vectors are equal to each other. With Eqs.
(21), the spring coefficient of mono-articular springs can be therefore solved as Eqs.
(20a) and (20d).
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The wire winding for multi-articular springs can be realized by the geometric
similarity, which means that the two sub loops have the similar shape. The difference in size can be represented by the elongation as a scale parameter 𝑠𝑐, which is one edge of
the sub loops. While the wire winding for mono-articular springs are not, and can be diversity.
5.4 Classifications and types of routing configurations
The routing configurations with and without wire winding are derived. The routing configurations can be designed with one or no diverging parameter in it. Moreover, classifications and types of the routing configurations can be presented on the basis of the correlation between attachment vectors. A simplified model for temporarily deriving the classifications and types of the routing configurations is shown in Fig. 12. A multi-articular spring can be distinguished into multi-part and mono-part by shifting the
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springs to the proximal or distal link. Because the geometric similarity in multi-part gives a proportional ratio to the vectors, the connection vector of the sub vectors introduce the mono-part to be discussed. The connection vectors are then redefined as multi-part vectors and mono-part vectors for analysis.
Fig. 12 The exploded view of the articular springs with multi-part and mono-part
The equivalence of multi-part vectors and mono-part vectors is identified as the similar classification that 𝑁𝑚𝑢𝑖 = 𝑁𝑚𝑜𝑖. In this classifications, the shape parameter can
be any, and the routing configurations with wire winding can also be applied. There are two type in this classification that the first one is “equal type”: all sub-loops have
equal size of the vectors which the loops will overlap as Table 5 (a) shows. The second one is “unequal type” with respect to the equal type that at least one of sub vector is
different from the others, not all sub-loops overlaps as Table 5 (b) shows.
Contrary to the similar classifications, inequivalence of the multi- and mono-part vectors is identified as dissimilar classifications that 𝑁𝑚𝑢𝑖 ≠ 𝑁𝑚𝑜𝑖. In this
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classification, the shape parameter is constrained so that the routing configurations with wire winding cannot be applied to arbitrary cases of the mono-articular springs, neither all of the multi-articular springs. Only specific case of the mono-articular springs enable
to use wire winding. There are also two type in the classification. The first one is
to use wire winding. There are also two type in the classification. The first one is