CHAPTER 1 Intorduction
1.3 Motivation and preview
Traditionally, springs are arranged through trial and error or excessive mounting space for inducing ZFL characteristics to the springs. In this paper, a methodology for exactly inducing ZFL characteristics in conventional springs to serial manipulator is presented, and the compressive arrangements for mounting spring on manipulators with unrestricted motion is discussed. Which are the first issue of the paper. For the second issue, methods to reduce the spring space in the manipulators are neglected. To enhance the advantage of the serial manipulators with respect to the previous researches, a routing configuration synthesis for reorganizing the springs with less spring space and aligning on the manipulators is given. So, a spring can be arranged on the manipulators from theorem to practical.
In Chapter 2, a spring–string arrangement is employed and is divided into three regions for mounting, tensioning, and placing springs. And the joint torque provided by the spring can be derived.
In Chapter 3, the derivation of the spring configurations for developing ZFL characteristics as well as the applications of pulley and cable driven mechanisms are discussed. ZFL characteristics can be induced if adequate length can be ensured for mounting the spring according to the spring configuration. Although the spring configurations for ZFL characteristics are presented, feasible configurations and
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unlimited mounting length for the springs with respect to the links are yet to be achieved because the overextended springs and links caused by the long mounting length shroud the workspace of the other links. Because the mounting length is solved as a function of spring elongation, by the use of pulleys and reels, the elongation and mounting length can be shortened. The springs can then be arranged in alignment on the links. The spring configuration by this additional arrangement shows that a
comprehensive spring configuration exists. Theoretical springs can then be arranged practically on the manipulators.
In Chapter 4, the length for placing springs is determined on the basis of the spring configuration. To optimize the placing length, a reference length acquired from the link configurations is utilized. The springs are assumed to be arranged on the specific links;
hence, the placing lengths can be controlled, and minimization of the placing length can therefore be described clearly.
In Chapter 5, to further adjust the configuration of springs in the regions for tensioning springs, a general approach for re-utilizing the spring space on manipulators, so called routing configuration synthesis, is discussed. And the assessment of the routing configurations by shrinkage performance of the spring space is provided. The proposed routing configuration synthesis can provide consistent performance of spring torque but occupy less space of the springs during motion. And the springs are thereby
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replaced by equivalent springs derived from the routing configuration synthesis. The method of the configurations is based on the superposition approach of vectors, and the conservation law of spring capability in joint torques. Constraints for deriving the routing configurations are developed with the foregoing two concepts. By incorporating the constraints of vectors and energy, general solutions of the routing configurations of the springs can be proposed. And design equations of the spring coefficients or
attachment vectors of the equivalent springs can be derived. To further reduce the use of the equivalent springs in the routing configurations, wire winding technique of the springs is then applied by using one equivalent sole springs. Following by the
distributive law of spring elongation, the wire winding for the routing configurations is then established. The optimized routing configuration is observed when the springs occupy least space, and align on the linkage, from which, the spring configuration can be readily obtained.
In Chapter 6, an example illustrating the routing configuration synthesis of the springs of a serial three-articular manipulator is discussed. Thereafter, the derivations of the spring configurations for developing ZFL characteristics as well as the applications of pulley and reels are discussed.
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CHAPTER 2
Parameters of link and spring configurations
2.1 Link and spring configurations
A serial-type planar articulated manipulator with rigid multiple links is shown in
Fig. 1(a). Each succeeding link 𝑖+1 is connected to its preceding link 𝑖 by joint 𝐽𝑖. Link 𝑘 is the distal link and link 𝑖 is the proximal link closer to the ground link.
Configuration parameters are defined in this section, and each link vector
𝒓𝒊, 𝒓𝒋, … and 𝒓𝒌 of the articulated manipulator is placed along the line passing through
centers of the joints of links 𝑖, 𝑗, and 𝑘, respectively, and link vectors are represented by 2 × 1 matrices. Residual link length is the length from the link joints to the proximal and distal ends of the link and is represented by 𝑟𝑖𝑝 and 𝑟𝑖𝑑, respectively. The
succeeding link 𝑖+1 with respect to its preceding link 𝑖 can be represented using a 2 × 2
rotation matrix with joint angle 𝜃𝑖+1, where 𝜃𝑖+1 is measured counterclockwise from 𝒓𝒊 to 𝒓𝒊+𝟏. One mean is used to describe the links between link 𝑖 and 𝑘 as a virtual
link 𝑣, which is variable in length, with link vector 𝒗 from 𝐽𝑖 to 𝐽𝑘−1, that is, 𝒗 = 𝒓𝒊+𝟏+ ⋯ + 𝒓𝒋+ ⋯ + 𝒓𝒌−𝟏.
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(a) Link configurations
(b) Spring configurations
Fig. 1 Spring attached between links 𝑖 and 𝑘 of the kinematic chain
In Fig. 1(b), a tension spring is labeled 𝑆𝑖𝑘, free length as 𝑙𝑓, and string length as 𝑙𝑠. The two ends are hinged to links 𝑖 and 𝑘, and the spring coefficient is 𝑘𝑖𝑘. Springs
are distinguished by the number of joints that a spring spans over. If links 𝑖 and 𝑘 are contiguous, the spring is a mono-articular spring; otherwise, it is a multi-articular spring.
In the spring–string arrangement shown in Fig. 1(b), the point 𝐻 is fixed on the
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spring for arranging strings or wires; points 𝐴, 𝐴′, and 𝐵 are fixed on the links. The point 𝐴 is a turning point, and 𝐴′ is a pseudo-turning point dividing the spring into three regions, as shown in Fig. 1(b). The first region is called the “mounting region”
with mounting length 𝑚; it contains the spring and a part of the string. The second is the “extension region” between separate links in a serial-type manipulator with
extension 𝑥 during motion, which includes the remainder of the string. The turning points are not immutable. The mounting region can be arranged on the distal side such as link 𝑘 to obtain the turning point and the pseudo-turning as 𝐵 and 𝐵′, respectively.
Furthermore, the mounting region can be arranged elastically on the other links. The third region is the “placing region,” represented by 𝑝, is the distance between the two
turning points or the two aforementioned regions. The placing length of the spring can be theoretically expressed using Eqs. (1a) and (1b) when the spring is arranged on the
proximal and distal side of the manipulator, respectively.
𝑝 = 𝐴′𝐴̅̅̅̅̅ (1a)
𝑝 = 𝐵′𝐵̅̅̅̅̅ (1b)
Under tension, the length of a tension spring equals the sum of its free length and
elongation. Therefore, the length of the spring–string arrangement can be calculated, i.e.
𝛿 + 𝑙𝑓+ 𝑙𝑠. By the sum of regions’ lengths and the placing length, that is, 𝑥 + 𝑚 + 𝑝,
the elongation equation for a tension springs is derived and expressed in Eq. (2).
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𝛿 = 𝑥 + 𝑚 + 𝑝 − 𝑙𝑓− 𝑙𝑠 (2)
The extension 𝑥 is determined by the link kinematics and described as a function of link vectors 𝒗 and attachment vectors 𝒂, 𝒃. Attachment vectors are used to describe
the positions of points 𝐴 and 𝐵 with respect to the link joints 𝐽𝑖, and 𝐽𝑘−1. α and 𝛽 are the rotation angles of the attachment vectors measured counterclockwise from 𝒓𝒊, 𝒓𝒌 to 𝒂, 𝒃, respectively. 𝒂 and 𝒃 can be represented by the rotation matrices such
as 𝐑(α) = [cosα −sinαsinα cosα], 𝐑(𝛽) = [cos𝛽 −sin𝛽
sin𝛽 cos𝛽 ] and the unit vectors of links, that is, 𝒂 = 𝑎𝐑(α)𝒓̂𝒊, 𝒃 = 𝑏𝐑(𝛽)𝒓̂𝒌. Joint angle 𝜃𝑎𝑣, 𝜃𝑣𝑏 and 𝜃𝑎𝑏 are measured
counterclockwise from 𝒂 to 𝒗, 𝒗 to 𝒃 and 𝒂 to 𝒃 respectively.
2.2 Joint torques of springs on manipulators
By tensioning the springs, a force based on the spring coefficient and the spring elongation will be provide, i.e. 𝑘𝛿. And then, the spring force introduce torque to each spanned joint in Fig. 2. The joint torque can be presented by the function of the force and a perpendicular distance of the force to joints 𝑃 as Eq. (3a) shows. With the link kinematics are determined, the perpendicular distance can therefore be calculated as a component of the attachment vectors, and the joint torques 𝜏𝑎𝑣 and 𝜏𝑏𝑣 on the virtual link are herein derived in Eqs. (3b) and (3c), respectively. The negative value represents the clockwise direction.
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Fig. 2 The spring configuration of the serial manipulator
In Eqs (3), the joint torques are shown as the function of spring coefficients, attachment vectors and joint angles. Because sine terms are the variables during link
motion, consistent torque performance of the springs on each joint will be shown if 𝑘𝑎𝑏, 𝑘𝑎𝑣, 𝑘𝑏𝑣 are constant. These terms are the maximum joint torques that a spring can provide to each joints, which is defined as the “spring capability” of articulated
springs.
𝜏 = 𝑃 ∙ 𝑘𝛿 (3a)
𝜏𝑎𝑣 = −𝑎𝑘𝛿 𝑠𝑖𝑛 𝜑𝑎 = 𝑎𝑘 (𝑏 + 𝑣− 𝑠𝑖𝑛 𝜃− 𝑠𝑖𝑛 𝜃𝑎𝑣
𝑎𝑏) 𝑠𝑖𝑛 𝜃𝑎𝑏 = 𝑘𝑎𝑏 𝑠𝑖𝑛 𝜃𝑎𝑏+ 𝑘𝑎𝑣 𝑠𝑖𝑛 𝜃𝑎𝑣 (3b) and
𝜏𝑏𝑣 = 𝑏𝑘𝛿 𝑠𝑖𝑛 𝜑𝑏 = −𝑏𝑘 (𝑎 − 𝑣𝑠𝑖𝑛 𝜃𝑠𝑖𝑛 𝜃𝑣𝑏
𝑎𝑏) 𝑠𝑖𝑛 𝜃𝑎𝑏 = −𝑘𝑎𝑏 𝑠𝑖𝑛 𝜃𝑎𝑏+ 𝑘𝑏𝑣 𝑠𝑖𝑛 𝜃𝑣𝑏 (3c)
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CHAPTER 3
Mounting region : practical spring configurations on
manipulators
3.1 Methodology of spring configurations for ZFL characteristics
A spring is called a ZFL spring if 𝛿 equals extension 𝑥. The configurations of non-ZFL springs for ZFL characteristics can then be derived by eliminating the nonzero term in Eq. (2). The necessary condition for ZFL characteristics, Eq. (4), is a function of the mounting length, free length, string length, and placing length. The condition reveals that the necessary length can be determined by two conditions: if the string length 𝑙𝑠 is given at first, 𝑐 is the mounting length to be solved, that is, 𝑚 = 𝑙𝑓+ 𝑙𝑠− 𝑝, and vice
versa.
𝑚 + 𝑝 − 𝑙𝑓− 𝑙𝑠 = 0 (4)
With the spring configurations for ZFL characteristics built, the elongation of springs in Eq. (2) can be represented as Eq. (5a) and then derived as Eq. (5b). ZFL characteristics can be achieved even if the spring is installed at a rotation angle with respect to the line passing through joint centers. For full rotation, the maximum elongation can be calculated from the links’ total length.
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𝛿 = 𝑥 = √(𝒗 + 𝒃 − 𝒂) ∙ (𝒗 + 𝒃 − 𝒂) (5a)
v b a2 2 2 2vb vTR rk 2va vTR ri 2ab riR rk
(5b) However, the necessary condition for the springs to have ZFL characteristics, Eq.
(4), is unconstrained, which may yield infeasible solutions for practical applications, for example, the problem of springs bent at turning points such that the mounting length is less than the sum of the elongation and the free length of springs, that is, 𝑚 < 𝛿 + 𝑙𝑓.
Because a spring should provide linear force without being bent, to ensure non-limitation of spring motion, the constraint in the spring–string arrangements shown in Fig. 1(b) is that the mounting length of a spring should be longer than the sum of the elongation and the free length at any time, that is, 𝑚 ≥ 𝛿 + 𝑙𝑓, so that an adequate workspace for springs is available. Because the elongation varies, one conservative design is set with the constraint of the maximum elongation 𝛿𝑚𝑎𝑥 rather than tracing the variation of elongations, as expressed by Eq. (6a). With the constraint for springs presented, the string length 𝑙𝑠 is given by Eq. (6b), obtained by substituting Eq. (6a) into Eq. (4). Two available choices for designers to arrange non-ZFL springs for ZFL characteristics with the spring motion unlimited are shown. By following either Eq. (6a)
or (6b), the minimum string length or mounting length can then be determined.
𝑚 ≥ 𝛿max+ 𝑙𝑓 (6a)
𝑙𝑠 ≥ 𝛿max+ 𝑝 (6b)
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3.2 Cable-driven mechanism of spring on the basis of pulley systems
The aforementioned example shows the practical arrangements of springs.
However, the spring mounting region is infeasible on account of the overextended mounting region with respect to the links. The overextended regions hinder the motion of the manipulators because the workspace of the distal link may be shrouded.
Therefore, a proper mounting length should be determined. The mounting length is a function of the spring elongation. To make the mounting region feasible for the given configurations, a pretension spring can be used to shorten the spring elongation while maintaining the spring force, as shown in Fig. 3(b). The pretension force can be achieved using winding springs [23] or a tension-compression set of springs that provides the initial force.
(a) Curve of original spring (b) Curve of pretension spring (c) Curve of stronger spring Fig. 3 Spring elongation and its workspace (in gray blocks) in three cases
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However, the springs in this paper are not preloaded. Therefore, another method is to adjust the elongation by scaling it so that the elongation of the springs in the
mounting regions is shorter than that in the extension regions. Springs provide the elongation while the linkage get times of the elongation simultaneously. For the mounting regions, the spring curve can be realized as shown in Fig. 3(c) while that for the extension regions can be realized as shown in Fig. 3(a). The mounting length is then
shortened on the basis of the decrease in elongation because ∆𝛿′ is shorter than ∆𝛿.
The approach of scaling the elongation is called “cable-driven.” The spring
configurations without scaling the elongation are shown in Fig. 4(a), where a spring is arranged in the mounting region. To make the mounting region feasible for
configurations such that its length is controllable and proper for manipulators, spring elongation should be decreased. This can be achieved by a cable-driven mechanism with the use of pulley systems. Cable-driven for the mounting region can be represented by the shortening performance 𝑠𝑝, which is a natural number. The case without using pulley, that is, 𝑠𝑝 = 1 is shown in Fig. 4(a). The other case is the multiple use of cable driven mechanism. For example, the spring elongation is halved, that is, 𝑠𝑝 = 2, while the extension is maintained because a set of movable pulleys is used, as shown in Fig.
4(b). For the case of higher shortening performance, that is, 𝑠𝑝 = 3, two sets of movable pulleys are used as shown in Fig. 4(c). By this method, spring elongation is
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physically minimized that the 2 pulleys are enough for all cases. The multiple loops of pulley systems decrease the size of the spring mounting region, as expressed by Eq. (7a)
that the springs mounting length is smaller than the most available mounting length 𝑚ava such that 𝑚ava ≥ 𝑚 always holds.
(a) Shortening performance 𝑠𝑝 = 1 (b) 𝑠𝑝 = 2 (c) 𝑠𝑝 = 3 Fig. 4 Cable driven mechanisms with pulley systems for the springs
By this method, the spring coefficient should be squared to ensure the consistent spring force, i.e. 𝑘 becomes 𝑠𝑝2𝑘, which can be achieved by using parallel sets of the
springs. Because the scaling of the elongation might cause the spring coefficient to become unmanageable, arbitrarily shrinking the elongation is not desired. An
assessment to determining the shortening performance can be derived as Eq. (7b). By determining the maximum elongation, pseudo-mounting lengths, and spring free
lengths, the minimum shortening performance can be determined, so proper adjustments
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can be made. The string length considering the remaining space of the mounting region is derived as Eq. (7c) if the springs are not arranged with the most available mounting
length.
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).
w
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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
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