1. Introduction
With several recent developments in nanotechnology, there is considerable interest in exploring, manipulating, assembling, and fabrication of structures and systems at the micro and nanoscales. Nowadays, micromanipulators are being used for manipulation and motion generation in microscale environments such as in optical microscopes, electron microscopes and applications wherein 2D/3D micromanipulation of materials is carried out.
In the case of robotic manipulation, compliance is generally perceived as an undesirable but unavoidable characteristic. The external loading and self-weight lead to a nonnegligible deflection of the end effector. This deflection affects the accuracy of the operation and degrades the quality of the final product. While this is true for a large scale, high payload, and high range manipulation, there are situations wherein the external loads are extremely low, required motion is in micro/nano scales, and accuracy is of paramount importance. A flexure-based compliant mechanism such as the one discussed in this article is highly suitable in this context. Compliant mechanisms are a category of mechanisms that involve the deformation of some of its joints or members for creating movements [Reference Howell1]. This particular class of mechanisms purposefully incorporates compliance into the system to exploit certain benefits. For instance, the compliant flexure joints eliminate backlash and friction, thus, offering better accuracy. Furthermore, the low weight of these mechanisms ensures that they have negligible deformation/deflection due to self-weight. The flexure-based compliant mechanism has a limitation on the range of motion. However, as the required motions are in micro/nano scales, this shortcoming practically does not matter. To sum up, for specific applications involving low payload, precision micromanipulation, the benefits of inclusion of compliance outweigh its limitations; consequently, flexure-based compliant mechanisms are widely employed in such cases. Compliant mechanisms at microscales are increasingly being used for precise actuation or motion/force sensing mechanisms in micro electro mechanical systems.
Compliant mechanisms with serial structure offer large workspace and dexterous maneuverability, but as they have cantilever structures, they are more prone to deform/deflect under loading. Moreover, the serial arrangement of members leads to the accumulation of errors. In addition, at high speeds, vibration becomes a severe challenge in serial structured mechanisms. All these reasons make serial structure unsuitable for fast and precise micromanipulation. To conquer the limitations of serial structures, parallel structures have drawn extensive attention. Here, several independently actuated links connect the end effector to the base. The parallel structure ensures improved rigidity and higher load-carrying capacity. Moreover, this arrangement avoids error accumulation providing better positioning. Therefore, it can be concluded that a compliant mechanism with a parallel kinematic structure is the most preferred solution for micro-motion applications. These types of mechanisms are popularly known as compliant parallel micromanipulators (CPMs).
In the design process of a CPM, compliance analysis or stiffness analysis plays a significant role [Reference Pham and Chen2–Reference Wang, Zhang and Zhang4]. Analytical compliance modeling allows us to observe the effect of structural parameters of the CPM on performance. It also helps us to understand the static, modal, and dynamic behavior of the system whereby it is possible to provide in the design of a CPM with high stiffness that leads to higher natural frequency, good repeatability, dexterous, and precise manipulation [Reference Pham and Chen2].
A coverage of different methods of compliance modeling can be found in the work of Ling et al. [Reference Ling, Howell, Cao and Chen5]. The finite element analysis (FEA) has been often used to do compliance modeling and is considered most accurate. Alternatively, researchers use pseudo-rigid body (PRB) modeling [Reference Howell and Midha6–Reference Li and Wu9]. However, since the PRB model considers the compliance of the members in their working or on-axis direction only, it is not much useful for complete compliance analysis of a spatial mechanism. This method is mostly used for CPM with simpler designs or in cases where undesired or out of axis deformations are very low or negligible. A highly accurate analytical model can be formulated by considering the full nonlinear model of all the members of the mechanism [Reference Herpe, Walker, Dunnigan and Kong10, Reference Xu, Zhu, Dong and Liu11]. Unfortunately, such models become too involved for a spatial CPM with a large number of members. Further, the nonlinear method does not offer much benefit for a system undergoing small deformations. Therefore, it is suitable only where the system is subjected to large deformations like that of a compliant mechanism with distributed compliance. In the present context, with consideration of 6D compliance, a lumped parameter model of the members can be used as a compromise [Reference Pham and Chen2, Reference Li and Xu12, Reference Liang, Zhang, Chi, Song, Ge and Ge13]. With the assumption of Hooke’s law, this lumped model can be formulated very easily by using the matrix method.
Although an extensive amount of research has already been done in the area of compliance or stiffness analysis [Reference Hb, Hw, Xia, Ma and Zz3, Reference Wang, Zhang and Zhang4, Reference Klimchik, Pashkevich and Chablat14–Reference Al-Jodah, Shirinzadeh, Ghafarian, Das and Pinskier17], most of these study deal with the case of planar mechanisms or in applications involving planar motions. There is not much work available for spatial mechanisms having multiple degree-of-freedom (DOF). Some of the other recently reported works can be seen in refs. [Reference Chen, Dong, Yang, Sun and Du18–Reference Ling, Song, Zhang, He, Li and Wu22].
In a previous work of the authors, a compliant Stewart platform-based micromanipulator for high range 6-DOF motion applications was proposed [Reference Mishra and Kumar23]. It was shown that for its size, the design gives a better range of motion than other similar CPMs. The model has a full 6-DOF spatial motion capability that is achieved using six separate parallelogram linkages. The kinematic structure of these linkages so arranged offers a unique 6-DOF motion similar to that of a Stewart platform but at micro and nanoscales. However, carrying out a full compliance model of such a micromanipulator is not an easy task. The parallel paths of compliances in the combination of rigid and flexure linkages used along with a spatial disposition of each member needs a systematic process to build accurate models. In such a case, we consider a model for full 6-DOF motion with lumped compliance parameters in our paper.
The matrix method is used to derive the expressions for output compliance and input compliance. The proposed CPM takes into consideration parallelogram linkages in each limb to achieve a higher range of motions in a plane. The analytical expression for the amplification ratio (AR) for such parallelogram linkages has been derived in the article. Since the behavior of a compliant mechanism greatly depends on the compliance of the flexures incorporated in the design, factors like the type of joints, its geometrical parameters, and material are considered. In this article, we consider the spatial mechanism with two classes or types of flexure joints; and we allow different geometrical parameter for one of the types/classes. This is an important consideration in spatial CPM as compared to planar CPM that often assumes the same geometric features at flexure joints mostly due to manufacturing considerations. We determine the advantage of providing varied geometry and compliance in one type of flexure joints on the performance of the spatial mechanism.
The contributions of this article are as follows. We present a thorough approach to an analytical compliance analysis of a spatial 6-DOF micromanipulator, which has not been addressed much in the literature. For a spatial 6-DOF micromanipulator, the compliances are interwoven with the geometry of motion and the derivation of expression needs a systematic approach to represent the effective spatial compliance matrix. This procedure is highlighted and customized for a specific design of a system developed at Indian Institute of Technology (IIT) Kharagpur. The article also presents a procedure to obtain the analytical expression for the AR in terms of the compliance factors. The analytical expression of the AR is significant as it allows the designer to achieve a better amplification of motion by changing flexures’ geometry. Moreover, the availability of the analytical relations allows a designer to optimize the CPM’s geometry for the desired behavior. In most of the available work, authors do not consider the connecting links’ compliance and assume them as rigid members in their models. In our work, we have also incorporated the compliance of the connecting links along with that of the joints and have shown that the geometry of the connecting links has a significant effect on amplification of motion indeed. This is needed to access the cross-coupled effects of both joints and linkages in the distributed compliance system in all 6-DOF. Finally, the work establishes that the provision of flexures with different geometrical parameters can improve performance.
The organization of the rest of the article is as follows: In Section 2, the methodology used for compliance modeling has been discussed. A representation to define and to carry out transformation in the spatial domain is presented. Section 3 outlines the design of the micromanipulator briefly. The analytical model of output compliance, input compliance, and AR has been derived in Section 4. Validation of the derived mathematical model has been done in Section 5 with the help of Abaqus-6.14 simulation. Based on the analytical models, Section 6 presents the effect of dimensional parameters on the performance of the CPM. This section also presents the effects of dissimilar geometry. In the end, concluding remarks have been summarized in Section 7.
2. Compliance modeling of a CPM based on matrix method
Compliance modeling is carried out using the concept of a compliance matrix. For any flexure or compliant member, it can be shown that a compliance matrix maps the force vector and the displacement vector. Figure 1 shows a typical member in its frame
$i$
subjected to a generalized force at its end.
A flexure member subjected to a general force vector at its free end.
The
$6\times 6$
local compliance matrix for this member can be represented as
\begin{equation} C_i=\begin{bmatrix} C_{xFx} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & C_{yFy} & 0 & 0 & 0 & C_{yMz}\\[4pt] 0 & 0 & C_{zFz} & 0 & C_{zMy} & 0\\[4pt] 0 & 0 & 0 & C_{\theta xMx} & 0 & 0\\[4pt] 0 & 0 & C_{\theta yFz} & 0 & C_{\theta yMy} & 0\\[4pt] 0 & C_{\theta zFy} & 0 & 0 & 0 & C_{\theta zMz} \end{bmatrix} \end{equation}
where compliance factor
$C_{xFx}$
represents the displacement in
$x$
-direction due to a unit force in
$x$
-direction. Whereas
$C_{yMz}$
denotes the displacement in
$y$
-direction due to a unit moment
$Mz$
about
$z$
-axis. The other members of the matrix can be defined in a similar manner. The compliance factors for different types of flexures have been formulated in several refs. [Reference Paros and Weisbord24–Reference Smith26].
In Eq. (1),
$C_i$
denotes the local compliance at
$O_i$
. The absence of superscript in
$C_i$
means that the compliance is defined with respect to the fixed end or ground. Now, to transfer this compliance into any other coordinate system, let us say
$O_j$
, we can use the following expression as described in ref. [Reference Li and Xu12]
\begin{equation}{C_j} = T_i^j{C_i}{\left(T_i^j\right)^T} \end{equation}
where transformation matrix
$T_i^j$
is defined as
\begin{equation} T_i^j ={\begin{bmatrix}{R_i^j}\;\;\;&{S\left(r_i^j\right)R_i^j}\\[4pt] 0\;\;\;&{R_i^j} \end{bmatrix}} \end{equation}
where
$R_i^j$
is a rotation matrix which represents orientation of coordinate system
$O_i$
with respect to
$O_j$
,
$r_i^j$
represents the position vector of the point
$O_i$
with respect to
$O_j$
.
$S(r)$
is the skew-symmetric matrix, and for a position vector:
$r = [r_x\;\;\;r_y\;\;\;r_z]$
, it has the following form
\begin{equation} S(r) ={\begin{bmatrix} 0\;\;\;&{ -{r_z}}\;\;\;&{{r_y}}\\[4pt]{{r_z}}\;\;\;&0\;\;\;&{ -{r_x}}\\[4pt]{ -{r_y}}\;\;\;&{{r_x}}\;\;\;&0 \end{bmatrix}} \end{equation}
A flexure-based CPM may have members connected in a serial, a parallel, or in a compounded series–parallel configuration. The procedure to derive the compliance equation of a generalized system is presented in the following section.
2.1. Procedure for obtaining output compliance, input compliance, and amplification ratio for a generalized CPM with series–parallel configuration
This section discusses the procedures to formulate analytical models for output compliance, input compliance, and AR for a general CPM. The outlined methods can easily be applied to the other designs. Let us consider a typical simplified CPM as shown in Fig. 2. The CPM has
$N$
-limbs (
$S_1, S_2,\ldots, S_N$
) with
$m$
number of flexure members. The limb
$S_1$
comprises of a serial chain of joints and links;
$S_2$
comprises of series of sub chains
$SS_1$
and
$SS_2$
;
$SS_1$
is a parallel chain of joints and linkages;
$SS_2$
and
$SS_1$
are in series. The combined
$S_1, S_2,.., S_N$
are all in parallel chain configuration. Limb-
$S_N$
incorporates a motion amplification system wherein
$F$
is the input point and
$D$
is the output point.
Schematic diagram of a generalized compliant parallel manipulator.
2.1.1. Output compliance
The output compliance of this mechanism is the compliance of the whole system at output point
$O$
. Firstly, we should determine the compliance of each of the limbs. In the first limb
$S_1$
, flexure-1 and flexure-2 are in series, so their compliances can be added after transforming them to point
$A$
.
\begin{equation} C_A=T_1^A C_1 \left(T_1^A\right)^T+T_2^A C_2 \left(T_2^A\right)^T \end{equation}
The limb-
$S_2$
itself has a combination of series and parallel connections, therefore, it is further divided into subsystems
$SS_1$
and
$SS_2$
. Compliance of subsystem
$SS_2$
at
$B$
is
\begin{equation} C_{B:\;SS_2}=C_B=T_3^B C_3 \left(T_3^B\right)^T \end{equation}
The subsystem
$SS_1$
contains flexure-4 and flexure-5 in parallel. These compliances will first be transformed to the point
$C$
, and then, the inverse of these transformed compliances will be added. The desired compliance of the subsystem will be the inverse of the obtained sum. The equation for this process is presented below.
\begin{equation} C_{C:\;SS_1}^B=C_C^B= \left(\left(T_4^C C_4 \left(T_4^C\right)^T\right)^{-1}+\left(T_5^C C_5 \left(T_5^C\right)^T\right)^{-1}\right)^{-1} \end{equation}
The superscript
$B$
in the notation here indicates that point
$B$
is considered the reference or fixed point.
Subsystems
$SS_1$
and
$SS_2$
are in a serial chain. Therefore, the total compliance of the limb-
$S_2$
at
$C$
can be obtained by adding the transformed compliances of
$SS_1$
and
$SS_2$
as given below.
\begin{equation} C_C=T_B^C C_B \left(T_B^C\right)^T+ C_C^B \end{equation}
This process will be done for each of the limbs. Finally, the compliance for the last limb-
$S_N$
at point
$E$
is
\begin{equation} C_E=T_m^E C_m \left(T_m^E\right)^T+T_D^E C_{D} \left(T_D^E\right)^T \end{equation}
We have determined the compliance of all the limbs at their output points. Since all the limbs are in parallel, the output compliance of the full mechanism at
$O$
can be obtained by the following equation as done earlier.
\begin{equation} C_O=\left(\left(T_A^O C_A \left(T_A^O\right)^T\right)^{-1}+\left(T_C^O C_C \left(T_C^O\right)^T\right)^{-1}+\ldots..+\left(T_E^O C_E \left(T_E^O\right)^T\right)^{-1}\right)^{-1} \end{equation}
2.1.2. Input compliance
Input compliance is defined as the compliance of the actuated (or input) point of the CPM. If the CPM has a symmetric design, all the actuated points would yield identical input compliance. However, an asymmetric CPM may provide different values of input compliance. The generalized CPM in consideration here has an asymmetric design. In what follows, we shall try to determine one of the input compliances. The other input compliances can be obtained using a similar procedure.
Let us consider the input point
$B$
(Fig. 2) in limb-
$S_2$
for input compliance determination. The next step is to rearrange the system to make
$B$
the output point. Figure 3 shows the modified system after such rearrangement. The desired compliance is nothing but the output compliance of this modified system, which can be obtained by following the process discussed in the last section.
Schematic diagram of the modified CPM for input compliance determination.
2.1.3. Amplification ratio or displacement amplification
The current generalized CPM contains a motion amplifier in the limb
$S_N$
(Fig. 2). The AR for this system would be the ratio of displacement of its output point
$D$
to that of input point
$F$
. The determination of the AR typically involves the following steps.
-
1. Isolate the motion amplification system from the full mechanism as shown in Fig. 4.
-
2. Consider the mechanism left after removing the amplification system (Fig. 4a), and write force–displacement/(moment–rotation) equations at the bifurcation point (
$D$
). -
3. Write force–displacement/(moment-rotation) equations for the separated amplification system (Fig. 4b) at input point
$F$
and point
$D$
. -
4. Solve the algebraic equations generated from Step (2) and Step (3) to formulate the expression for AR.
Schematic diagram for amplification ratio determination.
3. Design of the 6-DOF flexure-based stewart platform-like CPM
Before carrying out the compliance analysis of the micromanipulator in consideration, we shall briefly discuss its design. The design of the mechanism is based on the Stewart platform [Reference Stewart27, Reference Gough28]. A compliant version of the mechanism developed at IIT Kharagpur is being considered [Reference Mishra and Kumar23]. In the design, right circular revolute and right circular spherical flexures (Figs. 5a and 5b, respectively) have been utilized in each limb of the mechanism. As mentioned earlier, the current design has a provision of parallelogram linkages in each limb. The kinematic diagram for a limb in Fig. 6 shows the locations wherein the flexure joints will be placed.
Design elements of the micromanipulator.
Kinematic diagram of a limb of the mechanism.
Figure 7 presents a detailed engineering description of the limb with flexure joints of the micromanipulator designed at IIT Kharagpur. All the revolute flexures in the parallelogram mechanism have been provided with identical dimensions. The full design of the Stewart CPM involves six limbs connected in a parallel configuration as shown in Fig. 8(b), which mimics the 6-DOF Stewart platform shown in Fig. 8(a). Figure 9 presents two views of the micromanipulator wherein the top view shows the relative description of the links and the side view/elevation is shown to represent the effect of motion/loads used in compliance analysis.
A limb of the micromanipulator with dimensions.
The general Stewart platform and compliant version of the micromanipulator.
Output compliance, input compliance, and amplification ratio.
4. Full 6-DOF spatial compliance modeling of the flexure-based stewart platform-like CPM
As per the objective of this work, for our micromanipulator, consider the following in our study: (a) output compliance of the entire mechanism, (b) input compliance at an input point of a limb, and (c) AR of the parallelogram linkage for a limb.
Output compliance of the present CPM can be defined as the motion of the end-effector point
$M$
(Fig. 9) when a unit external load is applied at the same point [Reference Yong and Lu29]. As per the compliance matrix defined in Eq. (1), output compliance has many components or factors. These factors describe the force/moment–displacement/rotation behavior of the CPM at the output point. The output compliance plays a very crucial role for accurate manipulation when the end effector is subjected to an external force or loading. A CPM with low output compliance (or high output stiffness) can perform its motion with precision without being influenced by external loads.
Input compliance for a CPM relates the input force and the resulting input motion [Reference Li, Huang and Tang30]. For the given Stewart platform-based CPM, input compliance is the compliance of input point
$A$
in
$y$
-direction with respect to link
$G$
of the parallelogram mechanism of the limb (Fig. 9). For safe operation, this input compliance should be more than the output compliance of the actuator.
The AR for the parallelogram linkage is defined as the ratio of output motion (motion of point
$D$
in
$y$
-direction in its local coordinate frame) to the input motion (motion of point
$A$
in
$y$
-direction in its local coordinate frame) of the parallelogram mechanism of the limb (Fig. 9). Again, the input and output motions are defined with respect to the link
$G$
, which is the actuator’s reference link. The compliance or stiffness of the actuators is not considered in the present study. The analytical expressions for compliances are generated by symbolic computation software Maple-18 [31], as these expressions can get cumbersome.
4.1. Modeling of output compliance
There are three different types of compliant members used in the design of the CPM in consideration: revolute flexures, spherical flexures, and deformable connecting links (modelled as Euler−Bernoulli beams as applicable). In compliance modeling, each of these members needs to be assigned coordinate frames. Figure 10 illustrates the scheme of coordinate system assignment followed in this article.
Coordinate system assignment scheme for compliant members.
With reference to Fig. 9, we can observe that the micromanipulator consists of six similar limbs with parallelogram linkages connected in a parallel configuration. In what follows, the expression of output compliance for one of the limbs will be derived.
4.1.1. Modeling the output compliance of a single limb
Referring to Fig. 7, it may be noted that the design of the limb involves several compliant members. For easier identification of these members, the key nodes and points are assigned numbers, as shown in Fig. 11. These comprise one of the three elements described in Fig. 10. The flexible members are shown in the legend table of Fig. 11; all other members in the system are considered as rigid. It can be seen that every member has been assigned two nodes. Out of these two nodes, one would be the output node based on how we proceed in the micromanipulator design for a particular calculation. The compliant member will be referred by this output node.
Numbering of key nodes and points in the limb-1 of the mechanism.
For the sake of convenience, a limb of the micromanipulator can further be divided into three subsystems (Fig. 11): (i) compliant universal joint (From base to
$U$
), (ii) compliant parallelogram mechanism (From
$U$
to
$D$
), and (iii) compliant spherical joint (from
$D$
to
$P1$
). The connectivity graph of all the members in the limb has been depicted in Fig. 12.
Connectivity graph of members in a limb for output compliance determination.
The first subsystem is the universal joint. It is constructed by the combination of three revolute flexures (joint-2, joint-4, and joint-6). For the universal joint, we can see that joint-2 and joint-4 are in parallel connection, and their end is in serial connection with joint-6 (Fig. 12). Hence, the compliance of the universal joint at point
$U$
can be written as
\begin{equation} C_{u} ={ \left({ \left(T_2^u{C_2}{\left(T_2^u\right)^T} \right)^{ - 1}} +{ \left(T_4^u{C_4}{\left(T_4^u\right)^T} \right)^{ - 1}} \right)^{ - 1}} + T_6^u{C_6}{\left(T_6^u\right)^T} \end{equation}
where
$C_2$
,
$C_4$
, and
$C_6$
are the compliance matrices for revolute flexure, and
$T_2^u$
,
$T_4^u$
, and
$T_6^u$
are corresponding transformation matrices. The detailed description of all the transformation and compliance matrices used in this article can be found in the appendices. The second subsystem is the parallelogram mechanism. Considering the series and parallel connection of different members (Fig. 12), the compliance of point
$D$
with respect to point
$U$
can be expressed as
\begin{equation} \begin{split} C_D^U & = \Big ({\big (T_8^D{C_8}{\left(T_8^D\right)^T} + T_9^D{C_9}{\left(T_9^D\right)^T} + T_{10}^D{C_{10}}{\left(T_{10}^D\right)^T}\big )^{ - 1}} + \big (T_{12}^D{C_{12}}{\left(T_{12}^D\right)^T} \\[4pt] & \quad + T_{16}^D{C_{16}}{\left(T_{16}^D\right)^T} + T_{17}^D{C_{17}}{\left(T_{17}^D\right)^T}{\big )^{ - 1}}{\Big )^{ - 1}} \end{split} \end{equation}
The third and final subsystem is the compliant spherical joint. For this joint, compliance at
$P1$
with respect to point
$D$
is
\begin{equation} C_{P1}^D = T_{19}^{P1}{C_{19}}{\left(T_{19}^{P1}\right)^T} \end{equation}
where
$C_{19}$
is the compliance matrix for spherical flexure joint. For the current design of the compliant spherical flexure, errors in the analytical relations (from ref. [Reference Lobontiu25]) were very high; hence for this flexure values of compliance factors has been taken from FEA simulations.
Figure 11 shows that the universal joint, parallelogram linkage, and spherical joint are serially connected. Therefore, from Eqs. (11) to (13), the final expression of compliance for full limb at
$P$
1 (with respect to ground) is
\begin{equation} C_{P1} = T_u^{P1}{C_u}{\left(T_u^{P1}\right)^T} + T_D^{P1}C_D^U{\left(T_D^{P1}\right)^T} + C_{P1}^D \end{equation}
The expression of output compliance for other five limbs can be formulated by following similar procedure.
4.1.2. Output compliance of the full model
The full compliant model of the current CPM is composed of six limbs connected in the parallel configuration, as shown in Fig. 9. Firstly, to derive the output compliance, the local compliances of these limbs will be transformed to the global coordinate frame
$M$
. Then, taking into account their parallel connection, the final output compliance at
$M$
can be expressed as
\begin{equation} \begin{split}{C_M} & = \Big ({\left(T_{P1}^MC_{P1}^{}{\left(T_{P1}^M\right)^T}\right)^{ - 1}} +{\left(T_{P2}^MC_{P2}^{}{\left(T_{P2}^M\right)^T}\right)^{ - 1}} +{\left(T_{P3}^MC_{P3}^{}{\left(T_{P3}^M\right)^T}\right)^{ - 1}} \\[4pt] &\quad +{\left(T_{P4}^MC_{P4}^{}{\left(T_{P4}^M\right)^T}\right)^{ - 1}} +{\left(T_{P5}^MC_{P5}^{}{\left(T_{P5}^M\right)^T}\right)^{ - 1}}+{\left(T_{P6}^MC_{P6}^{}{\left(T_{P6}^M\right)^T}\right)^{ - 1}}{\Big )^{ - 1}} \end{split} \end{equation}
Above is the equation for output compliance of the full micromanipulator.
4.2. Amplification ratio and input compliance
The provision of motion amplifier is common in flexure-based micromanipulators. The motion amplification in a compliant micromanipulator depends not only on its kinematics but also on the compliances of the members in it. Therefore, an analytical analysis in present context becomes essential as it can give the designer insights on the influence of design parameters on amplification. The analytical model can also be employed to optimize the geometry of mechanism for obtaining maximum possible amplification.
As defined before, the AR is the ratio of output motion to the input motion in the parallelogram mechanism of a limb. To formulate the expression for AR, the parallelogram mechanism of one of the limbs needs to be separated from the full mechanism. It should be noted that the input and output motions are defined with respect to the link
$G$
, and they do not involve the rigid-body motions of the parallelogram mechanism. Therefore, for the determination of input and output motion, the link
$G$
is assumed to be fixed in the present case. Figure 13(a) shows the separated parallelogram mechanism of limb-1 with suitable reactions. This separated mechanism will be referred to as parallelogram-1. Out of all six actuator locations, if actuation is only taking place at parallelogram-1, then the reaction forces at point
$D$
on the parallelogram-1 can be assumed to be in the plane of the mechanism. Even if any out-of-plane reaction forces occur, they will have a negligible effect on the amplification process of the mechanism. Moreover, since point
$D$
is situated at the output of the parallelogram mechanism, its planar rotation can be assumed to be zero. Let
$u_A$
be the input displacement, which is nothing but the displacement of point
$A$
in the
$y$
-direction (Fig. 13a). Similarly, let
$u_B$
be the output displacement, which is again nothing but the displacement of point
$D$
in
$y$
-direction (Fig. 13a). Additionally, the rotation of the point
$D$
has been denoted as
$\theta _D$
. In what follows, we shall write the force–displacement (and moment–rotation) equations for parallelogram-1 at point
$D$
(output point) and point
$A$
(input point). As the horizontal reaction at
$D$
(
$F_{D'}$
) does not have a significant contribution in the amplification process, it has been omitted in the calculations.
\begin{equation}{u_A} = C_A^G(2,2) \times{F_A} -{U_1} \times{F_D} -{U_2} \times{M_D} \end{equation}
\begin{equation}{u_D} = - C_D^G(2,2) \times{F_D} - C_D^G(2,6) \times{M_D} +{U_1} \times{F_A} \end{equation}
\begin{equation}{\theta _D} = - C_D^G(6,6) \times{M_D} +{U_2} \times{F_A}- C_D^G(6,2) \times F_D = 0 \end{equation}
Amplification ratio formulation.
Here
$C_A^G$
is the compliance at point
$A$
with respect to
$G$
, and
$C_A^G(2,2)$
is a compliance factor of the matrix
$C_A^G$
.
$C_D^G$
represents the compliance at the point
$D$
with respect to
$G$
.
$U_1$
relates the
$y$
-displacement at
$A$
(i.e.,
$u_A$
) due to force in the
$y$
-direction at location
$D$
(i.e.,
$F_D$
). From Maxwell’s reciprocal theorem,
$U_1$
should also relate the
$y$
-displacement at
$D$
(i.e.,
$u_D$
) as a result of force at
$A$
(i.e.,
$F_A$
). Similarly another parameter
$U_2$
relates
$M_D$
and
$u_A$
; therefore, it will relate
$F_A$
and
$\theta _D$
as well.
The mechanism that is obtained after removing the parallelogram-1 from the full mechanism will now be considered. This mechanism will also have point
$D$
. Figure 13(b) depicts this mechanism with reactions. So once again, the force–displacement (and moment/rotation) equations at point
$D$
can be written as
\begin{equation}{u_D} ={C_{D(2 - 6)}}(2,2) \times{F_D} +{C_{D(2 - 6)}}(2,6) \times{M_D} \end{equation}
\begin{equation}{\theta _D} ={C_{D(2 - 6)}}(6,6) \times{M_D} +{C_{D(2 - 6)}}(6,2) \times{F_D} = 0 \end{equation}
where
$C_{D(2-6)}$
is the compliance of the mechanism excluding limb-1 at point
$D$
.
For the original full mechanism, if input compliance is
$C_{in}^G$
, the following expression will also be valid at point
$A$
:
\begin{equation}{u_A} ={C_{in}^G}(2,2) \times{F_A} \end{equation}
We now have six force–displacement (and moment–rotation) equations (Eqs. 16−21). Eliminating
$F_D$
,
$F_A$
,
$M_D$
,
$U_1$
, and
$U_2$
from these equations, the expression for AR in terms of the compliance values is finally obtained as
\begin{equation} AR=\frac{u_D}{u_A}=\sqrt{\frac{Num}{Den}} \end{equation}
where,
\begin{align*} Num = &\! \left\{ C_A^G(2,2){\left({C_{D(2 - 6)}}(2,2)\right)^2}{\left({C_{D(2 - 6)}} (6,6)\right)^2} - 2C_A^G(2,2)\,{C_{D(2 - 6)}}(2,2)\,{C_{D(2 - 6)}}(2,6)\,{C_{D(2 - 6)}}(6,2) \,{C_{D(2 - 6)}}\right.\\& \quad (6,6)+ C_A^G(2,2)\,{\big({C_{D(2 - 6)}}(2,6)\big)^2}\,{\big({C_{D(2 - 6)}}(6,2)\big)^2} -{\big({C_{D(2 - 6)}}(2,2)\big)^2}\,{\big({C_{D(2 - 6)}}(6,6)\big)^2}\,{C_{in}^G}(2,2) \\& \quad +2C_{D(2-6)}(2, 2)\, C_{D(2-6)}(2, 6)\, C_{D(2-6)}(6, 2)\, C_{D(2-6)}(6, 6)\, C_{in}(2, 2) \\& \quad -\big(C_{D(2-6)}(2, 6)\big)^2 \, \big(C_{D(2-6)}(6, 2)\big)^2\,C_{in}(2,2)\Big\}\end{align*}
and
\begin{align*} Den= & \Big\{\big({C_{in}^G}(2,2)\big)^2\,\Big(C_D^G(2,2)\,{\big({C_{D(2 - 6)}}(6,6)\big)^2} - C_D^G(2,6)\,{C_{D(2 - 6)}}(6,2)\,{C_{D(2 - 6)}}(6,6) - C_D^G(6,2)\,{C_{D(2 - 6)}}(6,2)\,\\& \quad {C_{D(2 - 6)}}(6,6) + C_D^G (6,6)\,{\big({C_{D(2 - 6)}}(6,6)\big)^2} +{C_{D(2 - 6)}}(2,2)\,{\big({C_{D(2 - 6)}}(6,6)\big)^2} -{C_{D(2 - 6)}}(2,6)\,{C_{D(2 - 6)}}(6,2)\,\\& \quad {C_{D(2 - 6)}}(6,6)\Big)\Big\}.\end{align*}
From Eq. (22), additional information in terms of the expressions of
$C_A^G$
,
$C_D^G$
,
$C_{D(2-6)}$
, and
$C_{in}$
is needed. These expressions have been derived in the following sections.
4.2.1.
Expression for
$C_A^G$
With reference to Fig. 14, if we start from point
$G$
and proceed clockwise, we can observe that flexure member 8, 9, 10, 16, and 13’ are in serial connection. Hence, their resultant transformed compliance at
$A$
can be given as
Connectivity graph of members for calculation of
$C_A^G$
.
\begin{equation} C_{A:8,9,10,16,13^{\prime}}^G = T_8^A{C_8}{\left(T_8^A\right)^T} + T_9^A{C_9}{\left(T_9^A\right)^T} + T_{10}^A{C_{10}}{\left(T_{10}^A\right)^T} + T_{16}^A{C_{16}}{\left(T_{16}^A\right)^T} + T_{13^{\prime}}^A{C_{13^{\prime}}}{\left(T_{13^{\prime}}^A\right)^T} \end{equation}
Similarly, while proceeding anticlockwise from
$G$
, we can find joint-12 and 13” to be serially connected. Their resultant transformed compliance at
$A$
is
\begin{equation} C_{A:12,13^{\prime\prime}}^G = T_{12}^A{C_{12}}{\left(T_{12}^A\right)^T} + T_{13^{\prime\prime}}^A{C_{13^{\prime\prime}}}{\left(T_{13^{\prime\prime}}^A\right)^T} \end{equation}
The compliances
$C_{A:8,9,10,16,13^{\prime}}^G$
and
$C_{A:12,13^{\prime\prime}}^G$
are in parallel. Moreover, these two compliances together make a serial connection with the last flexure member, flexure-15, to reach the desired point of interest which is
$A$
. The final required expression can be calculated as
\begin{equation} C_A^G ={\big ({\left(C_{A:8,9,10,16,13^{\prime}}^G\right)^{ - 1}} +{\left(C_{A:12,13^{\prime\prime}}^G\right)^{ - 1}}\big )^{ - 1}} + T_{15}^A{C_{15}}{\left(T_{15}^A\right)^T} \end{equation}
4.2.2.
Expression for
$C_D^G$
The compliance
$C_D^G$
is same as
$C_D^U$
, which has already been obtained in Section 4.1.1 (Eq. 12). The same equation is presented below for
$C_D^G$
.
\begin{equation} \begin{split} C_D^G = &\left(\left(T_8^D C_8 \left(T_8^D\right)^T + T_9^D C_9 \left(T_9^D\right)^T + T_{10}^D C_{10} \left(T_{10}^D\right)^T\right)^{-1} + \left(T_{12}^D C_{12} \left(T_{12}^D\right)^T + T_{16}^D C_{16} \left(T_{16}^D\right)^T \right.\right.\\[4pt] &\left.\left. + T_{17}^D C_{17} \left(T_{17}^D\right)^T\right)^{-1} \right)^{-1} \end{split} \end{equation}
4.2.3.
Expression for
$C_{D(2-6)}$
In Section 4.1.2, the expression for output compliance of the current CPM was formulated (Eq. 15). The same equation is presented below.
\begin{equation} \begin{split}{C_M} &= \left({\left(T_{P1}^MC_{P1}^{}{\left(T_{P1}^M\right)^T}\right)^{ - 1}} +{\left(T_{P2}^MC_{P2}^{}{\left(T_{P2}^M\right)^T}\right)^{ - 1}} +{\left(T_{P3}^MC_{P3}^{}{\left(T_{P3}^M\right)^T}\right)^{ - 1}} \right.\\[4pt] & \left.\quad +{\left(T_{P4}^MC_{P4}^{}{\left(T_{P4}^M\right)^T}\right)^{ - 1}} +{\left(T_{P5}^MC_{P5}^{}{\left(T_{P5}^M\right)^T}\right)^{ - 1}} +{\left(T_{P6}^MC_{P6}^{}{\left(T_{P6}^M\right)^T}\right)^{ - 1}}\right)^{ - 1} \end{split} \end{equation}
In this equation, if we remove the first term in the right-hand side, the resulting expression would be the compliance at
$M$
for the mechanism without limb-1. Let us denote this compliance as
$C_{M(2-6)}$
.
$C_{M(2-6)}$
when connected serially with the spherical joint of limb-1 can give us the desired compliance
$(C_{D(2-6)})$
. Figure 15 outlines this process.
Simplified connectivity graph of steps involved in calculation of
$C_{D(2-6)}$
.
The final expression, obtained after following the steps mentioned above, is written below.
\begin{equation}{C_{D(2 - 6)}} = T_M^D{C_{M(2 - 6)}}{\left(T_M^D\right)^T} + T_{18}^D{C_{18}}{\left(T_{18}^D\right)^T} \end{equation}
4.2.4.
Expression for
$C_{in}^G$
The expression for
$C_{D(2-6)}$
was derived in the previous section. Connection of
$C_{D(2-6)}$
with the parallelogram-1 represents the full CPM (Fig. 16). In this context, the problem of determining the input compliance is nothing but finding out the compliance of point
$A$
with respect to
$G$
.
Simplified connectivity graph of members for calculation of input compliance.
As member 8, 9, and 10 are serially connected, their transformed compliance at A is
\begin{equation} C_{A\;:\;8,9,10}^G = T_8^A{C_8}{\left(T_8^A\right)^T} + T_9^A{C_9}{\left(T_9^A\right)^T} + T_{10}^A{C_{10}}{\left(T_{10}^A\right)^T} \end{equation}
This compliance is in parallel connection to
$C_{D(2-6)}$
as shown in Fig. 16. Therefore,
\begin{equation} C_{A\;:\;D}^G ={\Big ({\big (T_D^A{C_D}_{(2 - 6)}{(T_{D}^A)^T}\big )^{ - 1}} +{(C_{A\;:\;8,9,10}^G)^{ - 1}}\Big )^{ - 1}} \end{equation}
The rest of the process is similar to that followed in the derivation of compliance
$C_A^G$
. The expressions involved in the process are presented below.
\begin{equation} C_{A\;:\;(D,16,13^{\prime})}^G = C_{A\;:\;D}^G + T_{16}^A{C_{16}}{\left(T_{16}^A\right)^T} + T_{13^{\prime}}^A{C_{13^{\prime}}}{\left(T_{13^{\prime}}^A\right)^T} \end{equation}
\begin{equation} C_{A\;:\;12,13^{\prime\prime}}^G = T_{12}^A{C_{12}}{\left(T_{12}^A\right)^T} + T_{13^{\prime\prime}}^A{C_{13^{\prime\prime}}}{\left(T_{13^{\prime\prime}}^A\right)^T} \end{equation}
From Eqs. (31) and (32), the final expression can be written as
\begin{equation} C_{in}^G = \big ({({C_{A\;:\;(D,16,13^{\prime})}^G)^{ - 1}} +{(C_{A\;:\;12,13^{\prime\prime}}^G)^{ - 1}}\big )^{ - 1}} + T_{15}^A{C_{15}}{\left(T_{15}^A\right)^T} \end{equation}
This is the required expression for input compliance.
5. Comparison and validation of model using FEA
The analytical models for output compliance, input compliance, and AR of the CPM in consideration were obtained in the previous section. In this section, these models will be validated by FEA through Abaqus-6.14 simulations [32]. The various dimensional parameters (defined in Figs. 7 and 9) and simulation parameters for CPM are given in Table I. In the simulation, all the revolute flexures in the parallelogram linkages have been given identical dimensions.
Dimensional and simulation parameters.
Figure 17 represents the FEA model of the CPM prepared in Abaqus-6.14 environment. The 10 node tetrahedral element C3D10 has been used to create mesh of the CPM. As the deformations are mainly expected to be concentrated in the flexure joint locations, they have been provided with a fine mesh by setting curvature control to 0.01. A convergence analysis was done to determine the suitable mesh size. In the FEA model, a unit force in
$y$
-direction is applied at the input point (
$A$
) of the limb-1 (Fig. 18a) while keeping the link
$G$
fixed. The resulting displacements at point
$A$
(input displacement) and
$D$
(output displacement) in
$y$
-direction are observed. The input motion obtained in such a manner is nothing but the input compliance of the CPM in consideration. Whereas the ratio of the output and input displacement represents the AR. The output compliance in the FEA model can be determined by the application of a unit force/moment at the output point
$M$
of the CPM (Fig. 18b) and obtaining the corresponding displacement/rotation. Table II elaborates on the results obtained from the analytical (matrix) model and FEA simulations. Here, FEA results have been considered as the benchmark or true result for calculating the deviations.
FEA model of the CPM.
Input and output compliance determination using FEA simulations.
Comparison of results obtained from analytical and FEA model.
For output and input compliance, deviations were found to be from around 7% to 11%. These errors mainly originate from the inaccuracy of expressions of the adopted compliance factors for the flexure joints. Now let us look at the results for AR. From Eq. (22), we can observe that the derived expression for AR involves many compliance factors. Now, these factors themselves have errors ranging from 5% to 15%. This can be the reason for a high 24.3% deviation in AR result.
In order to reconcile the values closely, we assume that compliance values obtained by FEA are closer to reality than the analytical values and substitute them back into the expression for AR; it reduces the error from 24.3% to less than 15%. Finally, it can be established that the formulated expressions in Section 4 agree to the FEA results with acceptable deviations.
6. Influence of dimensional parameters on CPM’s compliances and AR
In Section 4, we formulated the analytical relationships for the given CPM. With the help of those expressions, we can observe the effect of CPM dimensions on its performance. We have seen that each limb of the current design has a parallelogram mechanism. This parallelogram mechanism incorporates four revolute flexures (8, 10, 12, and 17). As stated earlier, the dimensional parameters (
$r_r$
,
$b_r$
, and
$t_r$
(Fig. 5a)) of all these flexures are the same. The effect of dimensional parameters on output compliance, input compliance, and AR has been shown in Figs. 19(a), 19(b), and 20, respectively.
Effect of flexure dimensions on compliance.
Effect of flexure and beam parameters on AR.
6.1. Effect of dissimilarity in flexure parameter on the typical limb AR
The results in the earlier sections have shown that the performance of the CPM depends on geometrical parameters of the flexure. In general, most of the available CPMs in literature are designed with the assumption that a particular class of flexures have similar geometrical parameters when used repeatedly. It means that, for instance, if a CPM is made up of revolute flexures, then all of them would have the same radii, thicknesses, and widths. Manufacturing tolerances and variation can cause some deviation in the AR of the linkage. Furthermore, even additional coupled effects due to any dissimilarities can affect the AR. However, at the same time, one can introduce dissimilarity to alter the spatial compliance/performance of the micromanipulator. We shall consider the limb’s parallelogram mechanism (Fig. 13a) to explore this possibility in this section. In the parallelogram mechanism, the thickness of the four revolute flexures: flexure-8, 10, 12, and 17, will be changed one at a time, and its effect on the AR will be observed. Figure 21(a) represents an equivalent compliance scheme of the series–parallel chain of a typical limb shown. The input at a and the output of the link in the same plane is considered. The rest of the manipulator has an equivalent compliance
$C_{others}$
in the same plane. In this exercise, we change the compliance by adjusting thicknesses
$t_{r8}, t_{r10}, t_{r12},$
and
$t_{r17}$
independently, as mentioned. The result of this exploration is presented in Fig. 21(a). In this figure, point
$C$
(flexure thickness (
$t_r$
) = 1 mm) corresponds to the original design value. Furthermore, it will be a starting point for variation in this exploration. We reduce the thickness up to 50% and increase it to around 300%. The results presented in Fig. 21(b) reveal the following:
Effect of dissimilar flexures.
-
(i) Any increase in the flexure thickness increases the stiffness and reduces the AR significantly. Whereas a reduction in flexure thickness reduces the stiffness and increases the AR by a lesser amount. Further, the closer joints (one joint away) have less effect than the farther ones (more than one joint away) in the chain.
-
(ii) Thickness of flexure-12: This flexure is located on the input limb, nearest to the input point. For the higher thickness values, this flexure manifests the least reduction in AR compared to other flexures. While flexure-8, 10, and 17 yield almost the same result for a decrease in thickness, that is, a surge in the AR value, flexure-12 reveals different behavior for the same range of variation. During a decrease in the thickness of flexure-12, as we start approaching point
$C$
, we observe a drop in the rate of change of AR. Just beyond point
$C$
, AR stays constant for a while, and ultimately it starts decreasing. This observation can be explained as follows. The proper working of the parallelogram mechanism for amplification requires the input limb to rotate about flexure-12. For this to happen, flexure-12 should have a certain amount of flexibility as well as strength. As per on scheme, when the thickness of flexure-12 was decreased, the rest of the flexures’ thicknesses were kept constant. So, while this change does make the flexure-12 more flexible, it also renders the joint weak to resist the external load and moments applied by the rest of the mechanism. Therefore, beyond a limit, the decrease in thickness will only result in stretching of the joint and lead to a significant weakening of the mechanism.Let us understand this phenomenon through an FEA simulation. We shall reconsider the FEA model of the whole CPM as done previously in Section 5. In one of the parallelogram linkages of this model, flexure-
$12$
thickness has been reduced to 0.2
$mm$
. All other dimensions are the same as before (Table I). Material, element type, and curvature control settings are also unchanged. However, the element size is kept smaller (0.5
$mm$
) due to the presence of a thin flexure. We subject the parallelogram (with thin flexure) to an input force of 4
$N$
. The system’s response in Fig. 22 shows the flexure stretching that was predicted earlier.Figure 22.Stretching of a thin flexure.
As a consequence of the joint stretching, some part of the input motion gets lost. For the reasons mentioned above, we observe a reduction in AR for flexure-12 for low thickness values in Fig. 21(b). Thus, to obtain a high AR, as a guideline, we can state that the flexure nearest to the input point should not be designed weaker than other flexures and can be used as a baseline constant.
-
(iii) Thickness of flexure-17 : As shown in Fig. 21(a), flexure-17 is on the left side of the input limb. This flexure also exhibits a reduction in AR for an increase in thickness. However, in this case, the rate of decline is more than that of flexure-12. On the other hand, if we decrease the thickness, then unlike flexure-12, we observe an increase in AR.
-
(iv) Thickness of flexure-8 and flexure-10 : Two flexures are more than one joint away from input actuation point. Flexure-8 lies on the right side of the bottom limb in the parallelogram mechanism. As we increase its thickness, it leads to a reduction in AR. Moreover, a decrease in the thickness causes growth in AR. The change in AR for flexure-8 and flexure-10 (the other flexure in the same limb) is almost coincident. It is also important to note that in the opposing parallel paths, the AR reduces very sharply by increasing the thickness of the flexure. In fact, for higher thickness values, these two flexures produce more reduction in AR than the other two remaining flexures (flexure-12 and 17).
The effect of change in thickness for each joint in the parallelogram linkage has been understood from previous discussions. Now, if the aim is to improve the AR, then to achieve that, we can fix the geometry of flexure-12 and decrease the thickness of any of the other three joints. It should be appreciated that this is, in fact, one of the many ways to improve AR. In the present case, at a time, the thickness of only one flexure was changed, the rest being the same. A more detailed study of the combined effects of the stiffness variation can give a better design choice. An optimization study can also be carried out in this context.
7. Conclusions
The compliance modeling of a full 6-DOF series–parallel flexure-based Stewart platform-like micromanipulator was presented in this article. The matrix method has been considered to carry out the mathematical modeling. The compliance matrix, which depends on the dimensions and material of the member, was utilized in the matrix method. The analytical expression for output compliance was formulated for the current design by dividing the full CPM into multiple subsystems. Furthermore, the AR relation was also obtained for the parallelogram linkage of a limb. Finally, the expression for input compliance of the CPM has been derived. The validation of the analytical model was performed using FEA in the Abaqus-6.14 environment. Simulation results revealed that analytical models have less than 15% deviations from the FEA model. Then with the help of formulated analytical models, the effect of some major dimensional parameters on the performance of the CPM was presented in various plots. In the end, it has been indicated that dissimilarity of geometrical parameters of the flexures may lead to better performance in terms of displacement amplification.
We should keep in mind that limits on output and input compliance constrain the geometry of any CPM. Furthermore, being the thinnest part of the mechanism, the flexures are most prone to material failures. There would also be upper limits on flexure sizes based on space constraints. All these factors should be considered while finalizing the flexure geometry. Such models, along with the analytical models derived in this paper, can be extremely helpful in discovering optimal configurations in future.
Conflicts of interest
The design of the micromanipulator is subject to a patent filed by the institute where the authors are carrying out their research.
Financial support
None.
Ethical considerations
None.
Authors’ contributions
None.
A. Compliance matrices of flexures
-
1. Compliance matrix for revolute flexures with geometry
$(r_r, b_r, t_r)$
[Reference Koseki, Tanikawa, Koyachi and Arai33]:(A1)where,
\begin{align} C_r=\begin{bmatrix} c_5 \;\;\;\;\; & 0 \;\;\;\;\; & 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0 \;\;\;\;\; & c_1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &c_3\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &c_2\;\;\;\;\; &0\;\;\;\;\; &-c_4\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &c_8\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &-c_4\;\;\;\;\; &0\;\;\;\;\; &c_6\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &c_3\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &c_7 \end{bmatrix} && \end{align}
$c_1=\tfrac{9\, \pi \, r_r^{5/2}}{2 \, E \, b_r \, t_r^{5/2} }+ \tfrac{3\, \pi \, r_r^{3/2}\, }{2\, E\, b_r\, t_r^{5/2}};\; {\it c_2}={\tfrac{12\, \pi \,{{\it r_r}}^{2}}{E\,{{\it b_r}}^{3}} ( \sqrt{{\tfrac{{\it r_r}}{{\it t_r}}}}-\tfrac{1}{4} ) };\; {\it c_3}= \tfrac{9}{2}\,{\tfrac{\pi \,{{\it r_r}}^{3/2}}{E\,{\it b_r}\,{{\it t_r}}^{5/2}}};$
${\it c_4}={\tfrac{{12\,\it r_r}}{E\,{{\it b_r}}^{3}}}{ ( \pi \,\sqrt{{ \tfrac{{\it r_r}}{{\it t_r}}}}-1-\tfrac{\pi}{2} ) }; \:$
${\it c_5}={\tfrac{1}{E\,{\it b_r}} ( \pi \,\sqrt{{\tfrac{{\it r_r}}{{ \it t_r}}}}-\tfrac{\pi}{2} )};\,{\it c_6}=12\,{\tfrac{1}{E\,{{\it b_r}}^{3}} ( \pi \,\sqrt{{\tfrac{{ \it r_r}}{{\it t_r}}}}-1-\tfrac{\pi}{2} ) };\!\!$
${\it c_7}=\tfrac{9}{2}\,{\tfrac{\pi \,\sqrt{{\it r_r}}}{E\,{\it b_r}\,{{\it t_r}}^{5/2}}} ;\:$
${\it c_8}=\tfrac{9}{4}\,{\tfrac{\pi \,\sqrt{{\it r_r}}}{G\,{\it b_r}\,{{\it t_r}}^{5/2}}};$
$G=\tfrac{E}{2\,(1+\nu )}.$
-
2. Compliance matrix for connecting links (modelled as Euler-Bernoulli beams) with geometry (depth-
$a$
, width-
$w$
, length-
$l$
) [Reference Koseki, Tanikawa, Koyachi and Arai33]:(A2)where,
\begin{align} C_b=\begin{bmatrix} b_5 \;\;\;\;\; & 0 \;\;\;\;\; & 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0 \;\;\;\;\; & b_1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &b_3\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &b_2\;\;\;\;\; &0\;\;\;\;\; &-b_4\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &b_8\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &-b_4\;\;\;\;\; &0\;\;\;\;\; &b_6\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &b_3\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &b_7 \end{bmatrix} & & \end{align}
${\it b_1}={\dfrac{4\,{l}^{3}}{E\,{a}^{3}\,w}};\:$
${\it b_2}={\dfrac{6\,{l}^{2}}{E\,{a}^{3}\,w}};\:$
${\it b_3}={\dfrac{4\,{l}^{3}}{E\,a\,{w}^{3}}};\:$
${\it b_4}={\dfrac{6\,{l}^{2}}{E\,a\,{w}^{3}}}; \:$
${\it b_5}={\dfrac{l}{E\,a\,w}};\:$
${\it b_6}={\dfrac{12\,l}{E\,a\,{w}^{3}}};\:$
${\it b_7}={\dfrac{12\,l}{E\,{a}^{3}\,w}};\:$
${\it b_8}={\dfrac{10\,l}{G\,{a}^{3}w}}.\:$
-
3. Compliance matrix for spherical flexure with geometry (
$r_s$
= 4 mm,
$t_s$
= 1.5 mm):(A3)where,
\begin{align} C_s=\begin{bmatrix} s_5 \;\;\;\;\; & 0 \;\;\;\;\; & 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0 \;\;\;\;\; & s_1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &s_3\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &s_1\;\;\;\;\; &0\;\;\;\;\; &-s_4\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &s_8\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &-s_4\;\;\;\;\; &0\;\;\;\;\; &s_6\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &s_3\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &s_6 \end{bmatrix} && \end{align}
$s_5=10.490\times 10^{-6}$
mm/N;
$s_1=875.000\times 10^{-6}$
mm/N;
$s_3=s_4=199.000\times 10^{-6}$
mm/N-mm;
$s_6=49.760\times 10^{-6}$
rad/N-mm;
$s_8=63.930\times 10^{-6}$
rad/N-mm. For the current design of the compliant spherical flexure, errors in the the analytical relations (from [Reference Lobontiu25]) were very high. Moreover, any other accurate formulation could not be found in the literature for the concerned flexure. Therefore, for this flexure, values of compliance factors (presented above) has been taken from FEA simulations.
B. Transformation matrices
\begin{align} T_2^u= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &b/2+g+{\it r_c}+{ \it t_c}/2\;\;\;\;\; &-{\it l_{2n}}-{\it r_c}\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &-b/2-g-{\it r_c}-{\it t_c}/2\;\;\;\;\; &0\;\;\;\;\; &-r\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{2n}}+{\it r_c}\;\;\;\;\; &r\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{4}^{u}=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-b/2-g-{\it r_c}-{\it t_c}/2\;\;\;\;\; &-{\it l_{2n}}-{\it r_c}\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &b/2+g+{\it r_c}+{\it t_c}/2\;\;\;\;\; &0\;\;\;\;\; &-r\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{2n}}+{\it r_c}\;\;\;\;\; &r\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{6}^{u}= \begin{bmatrix} 0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &{\it l_{2n}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{2n}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0 \end{bmatrix} && \end{align}
\begin{align} T_{8}^{D}=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_6}+{\it r_r}+{\it t_r}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2+{\it l_5}-2\,{\it r_r}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_6}-{\it r_r}-{\it t_r}/2\;\;\;\;\; &-{\it l_{11}}/2-{\it l_5}+2\,{\it r_r}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{9}^{D}= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_6}+{\it r_r}+{\it t_r}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2+2\,{\it r_r}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_6}-{\it r_r}-{\it t_r}/2\;\;\;\;\; &-{\it l_{11}}/2-2\,{\it r_r}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{10}^{D}= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_6}+{r_r}+{\it t_r}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_6}-{\it r_r}-{\it t_r}/2\;\;\;\;\; &-{\it l_{11}}/2\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} {T_{12}^D}= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2+{\it l_5}-2\,{\it r_r}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{10}}/2\;\;\;\;\; &-{\it l_{11}}/2-{\it l_5}+2\,{r_r}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{16}^D= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2+2\,{\it r_r}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{10}}/2\;\;\;\;\; &-{\it l_{11}}/2-2\,{\it r_r}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{17}^D= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{11}}/2\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{ \it l_{10}}/2\;\;\;\;\; &-{\it l_{11}}/2\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{19}^P=\begin{bmatrix} 0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_9}\\[4pt] 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &-{l_9}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_u^{P1}=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-(l_9+2\,r_s+l_8+l_7+l_3)\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &(l_{11}/2\,+l_5+l_4/2)\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &(l_9+2\,r_s+l_8+l_7+l_3)\;\;\;\;\; &-(l_{11}/2\,+l_5+l_4/2)\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_D^{P1}=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-(l_9+2\,r_s+l_8+l_7+l_3)\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &(l_9+2\,r_s+l_8+l_7+l_3)\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{Pi}^M ={\begin{bmatrix}{R_{Pi}^M}\;\;\;\;\; &{S(r_{Pi}^M)R_{Pi}^M}\\[4pt] 0\;\;\;\;\; &{R_{Pi}^M} \end{bmatrix}} & & \end{align}
where,
\begin{align*} & R_{Pi}^M = \\ & \!\!\left [\! \begin {array}{ccc} \cos (\psi _i) \cos (\theta _i)\;\;\; &\cos (\psi _i) \sin (\theta _i) \sin (\phi _i) -\sin (\psi _i) \cos (\phi _i)\;\;\; &\cos (\psi _i) \sin (\theta _i) \cos (\phi _i) +\sin (\psi _i) \sin (\phi _i) \\[4pt] \sin (\psi _i) \cos (\theta _i)\;\;\; &\sin (\psi _i) \sin (\theta _i) \sin (\phi _i) +\cos (\psi _i) \cos (\phi _i)\;\;\; &\sin (\psi _i) \sin (\theta _i) \cos (\phi _i) -\cos (\psi _ i) \sin (\phi _i) \\[4pt] -\sin (\theta _i)\;\;\; &\cos (\theta _i) \sin (\phi _i)\;\;\; &\cos (\theta _i) \cos (\phi _i) \end {array} \!\right ] \end{align*}
where
$i$
varies from 1 to 6 for all limbs. Angles (in degrees) for the corresponding rotation matrices are as follows:
$\psi _1=187.825, \theta _1=12, \phi _1=90;\; \psi _2=172.175, \theta _2=12, \phi _2=90;\; \psi _3=307.875, \theta _3=12,$
$\phi _3=90;\; \psi _4=292.175, \theta _4=12, \phi _4=90;\; \psi _5=67.825, \theta _5=12, \phi _5=90; \psi _6=52.175, \theta _6=12, \phi _6=90.$
Also,
$S_{P1}^M= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&-{\it r_p}\,\sin \left ({ \frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it l_{14}}&0&-{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt]{ \it r_p}\,\sin \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180 }} \right ) &{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) &0\end{array} \right ];$
${\it S_{P2}^M}= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&-{\it r_p}\,\sin \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it l_{14}}&0&-{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt]{ \it r_p}\,\sin \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180 }} \right ) &{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) &0\end{array} \right ];$
${\it S_{P3}^M}= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&{\it r_p}\,\sin \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it l_{14}}&0&-{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it r_p}\,\sin \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{ 180}} \right ) &{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60+{\it \alpha _p}/2 \right ) }{180}} \right ) &0\end{array} \right ] ;$
${\it S_{P4}^M}= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&{\it r_p}\,\sin \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it l_{14}}&0&-{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it r_p}\,\sin \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{ 180}} \right ) &{\it r_p}\,\cos \left ({\frac{\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) &0\end{array} \right ] ;$
${\it S_{P5}^M}= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&-{\it r_p}\,\sin \left ({\frac{\pi \,{\it \alpha _p}}{360}} \right ) \\[4pt] -{ \it l_{14}}&0&{\it r_p}\,\cos \left ({\frac{\pi \,{\it \alpha _p}}{360}} \right ) \\[4pt]{\it r_p}\,\sin \left ({\frac{\pi \,{ \it \alpha _p}}{360}} \right ) &-{\it r_p}\,\cos \left ({\frac{\pi \,{\it \alpha _p} }{360}} \right ) &0\end{array} \right ] ;$
${\it S_{P6}^M}= \left [ \begin{array}{c@{\quad}c@{\quad}c} 0&{\it l_{14}}&{\it r_p}\,\sin \left ({\frac{\pi \,{\it ap}}{360}} \right ) \\[4pt] -{ \it l_{14}}&0&{\it r_p}\,\cos \left ({\frac{\pi \,{\it \alpha _p}}{360}} \right ) \\[4pt] -{\it r_p}\,\sin \left ({\frac{\pi \,{ \it \alpha _p}}{360}} \right ) &-{\it r_p}\,\cos \left ({\frac{\pi \,{\it \alpha _p} }{360}} \right ) &0\end{array} \right ] .$
\begin{align} T_8^A=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{14}}-{\it p_l}-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{13}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{14}}+{\it p_l}+{\it l_{10}}/2\;\;\;\;\; &-{\it l_{13}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} {\it T_9^A}= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{14}}-{\it p_l}-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_5}+4\,{\it r_r}+{\it l_{13}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{14}}+{\it p_l}+{\it l_{10}}/2\;\;\;\;\; &{\it l_5}-4\,{\it r_r}-{\it l_{13}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} {\it T_{10}^A}=\begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{14}}-{\it p_l}-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_5}+2\,{\it r_r}+{\it l_{13}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &{\it l_{14}}+{\it p_l}+{\it l_{10}}/2\;\;\;\;\; &{ \it l_5}-2\,{\it r_r}-{\it l_{13}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} {\it T_{16}^A}=\begin{bmatrix} -1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{12}}-2\,{ \it r_r}-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_5}+4\,{\it r_r}+{\it l_{13}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &{\it l_{12}}+2\,{\it r_r}+{\it l_{10}}/2\;\;\;\;\; &{\it l_5}-4\,{\it r_r}-{\it l_{13}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1 \end{bmatrix} \;\;\;\;\; &\;\;\;\;\; & \end{align}
\begin{align} {\it T_{13^{\prime}}^A}=\begin{bmatrix} -1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_{12}}-2\,{ \it r_r}-{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &{\it l_{12}}+2\,{\it r_r}+{\it l_{10}}/2\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1 \end{bmatrix} && \end{align}
\begin{align} T_{13"}^A= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{12}}+2\,{ \it rr}+{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &-{\it l_{12}}-2\,{\it r_r}-{\it l_{10}}/2\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{12}^A= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{12}}+2\,{\it r_r}+{\it l_{10}}/2\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{13}}\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &-{\it l_{12}}-2\,{\it r_r}-{\it l_{10}}/2\;\;\;\;\; &-{\it l_{13}}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{15}^A= \begin{bmatrix} 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{l_{12}}+2\,{r_r}\\[4pt] -1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &-{ l_{12}}-2\,{r_r}\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_D^A= \begin{bmatrix} 1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &{\it l_{12}}+2\,{\it r_r}+{\it l_{10}}\\[4pt] 0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &-{\it l_5}+2\,{\it r_r}+{ \it l_{13}}-{\it l_{11}}/2\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &-{\it l_{12}}-2\,{\it r_r}-{\it l_{10}}\;\;\;\;\; &{\it l_5}-2\,{\it r_r}-{\it l_{13}}+{\it l_{11}}/2\;\;\;\;\; &0 \\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1\;\;\;\;\; &0\\[4pt] 0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &0\;\;\;\;\; &1 \end{bmatrix} & & \end{align}
\begin{align} T_{M}^D ={\begin{bmatrix}{R_{M}^D}\;\;\;\; &{S(r_{M}^D)R_{M}^D}\\[4pt] 0\;\;\;\; &{R_{M}^D} \end{bmatrix}} && \end{align}
where,
$R_M^D=(R_{P1}^M)^{-1}$
;
and
\begin{align*}& S(r_M^D)=\\ &\!\!\left [\! \begin {array}{ccc} 0\;\;\;&-{\it K_1} \,\sin \left ( {\dfrac {\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right )\;\;\; &{\it K_2}+{\dfrac {{\it l_{14}}}{\cos \left ( \pi/15 \right ) }}+{ \it r_p}-{\dfrac {{\it l_{14}}}{\sin \left ( \pi/15 \right ) }} \\[4pt] {\it K_1}\,\sin \left ( {\dfrac {\pi \, \left ( 60-{ \it \alpha _p}/2 \right ) }{180}} \right )\;\;\; &0\;\;\;&-{\it K_1}\,\sin \left ( {\dfrac { \pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{180}} \right ) \\[4pt] -{\it K_2}-{\dfrac {{\it l_{14}}}{\cos \left ( \pi/15 \right ) }}-{\it r_p}+{\dfrac {{\it l_{14}}}{\sin \left ( \pi/15 \right ) }}\;\;\; &{\it K_1}\,\sin \left ( {\dfrac {\pi \, \left ( 60-{\it \alpha _p}/2 \right ) }{ 180}} \right )\;\;\; &0\end {array}\! \right ] \end{align*}
Again, where,
$K_1= r_p+(l_9+2\,r_s+l_8+l_7)\,\sin \left (\dfrac{\pi }{15}\right )$
;
$K_2=l_7+l_8+2\,r_s+l_9.$
