1. Introduction
In recent years, there has been an increasing interest in parallel manipulators(PMs). Numerous PMs have been proposed, and kinematics and dynamics methods are suggested and applied [Reference Merlet1]–[Reference Gao, Li, Zhao, Jin and Zhao5]. There has been a particular interest in lower mobility parallel manipulators (LMPMs) due to their advantages as having simpler mechanical designs, larger workspace, error tolerance, agility and robustness. Overconstrained parallel manipulators (OPMs) are subclass for LMPMs. The overconstrained manipulators are arranged due to their subspaces. Moreover, the overconstrained manipulators are perfect candidates for the job done by LMPMs because the desired motion of the end effector is in a subspace.
Research has been done for the overconstrained manipulators in the areas such as structural synthesis, type synthesis, kinematic analysis, dynamic analysis, stiffness, force analysis and dynamic analysis. In the structural synthesis, the needed number of joints and their configurations are found. Intuition, screw theory, or group theory are the most common methods. Gogu [Reference Gogu6] Suggested a methodology for the structural synthesis of fully isotropic OPMs. In their research, Kong and Gosselin [Reference Kong and Gosselin7] proposed a screw theory-based virtual chain approach for the type synthesis of PMs where some are overconstrained. Dai et al. [Reference Dai, Huang and Lipkin8] prompted a screw theory based mobility formulation for overconstrained parallel mechanisms. Lee and Herve [Reference Lee and Hervé9] created a lie group approach for the structural synthesis of 4 DoF (3T1R) OPMs with uncoupled actuation. Hu [Reference Hu10] investigated the Exechon PM, expressed kinematically identical overconstrained and non-overconstrained manipulators and compared them with the Exechon PM. Structural synthesis of overconstrained manipulators with partial subspaces is described with a methodology based on unit screws is described in the work of Selvi [Reference Selvi11].
Some overconstrained manipulators are suggested by researchers [Reference Yan, Li, Li and Zhao12]–[Reference Arian, Isaksson and Gosselin15], and kinematic analysis is done by analytical and geometric methods. Yan et al. [Reference Yan, Li, Li and Zhao12] suggested two overconstrained 2-RPU&SPR PMs and compared their kinematic behaviors, workspace and dexterity. They also made a comparative stiffness analysis for the same two manipulators [Reference Yan, Li, Zhao and Li16]. Yan et al. [Reference Li, Li and Zhao13] proposed a 3 DoF overconstrained manipulator. Using an analytical method, it analyses the manipulator’s kinematics, recommends an algorithm for the manipulator’s workspace analysis, then finds the reachable workspace. Li et al. [Reference Xu, Chen, Ye and Li14] proposed a 2R1T OPM, namely Hex4. After the kinematic analysis is done, the link parameters are optimized for a better transmission workspace. Arian et al. [Reference Arian, Isaksson and Gosselin15] provides a Schönflies motion generator with infinite tool rotation in their research. The system is overconstrained, and the kinematic problem is investigated for the position, velocity and acceleration of the mechanisms. Singularity analysis and dynamic models are also developed for the system. Kinematic analysis of a 5 DoF overconstrained manipulator for rehabilitation is described in the work of Selvi and AL-Dulaimi [Reference Selvi and Al-Dulaimi17].
Also, dynamic analysis of OPMs is done by several researchers. Sharifzadeh et al. [Reference Sharifzadeh, Arian, Salimi, Tale Masouleh and Kalhor18] claim to have obtained a closely realistic dynamic model of 3 DoF translational OPM Tripteron using the white box and black box model and genetic algorithm. For the overconstrained 2PUR–PSR PM, researchers [Reference Chen, Xu, Zhang and Li19] described two dynamic models with and without constrained forces/moments using the Newton–Euler approach and natural orthogonal complement method. Arian et al. [Reference Arian, Isaksson and Gosselin15] carried out the kinematic and dynamic analysis of overconstrained manipulator Tripteron through Newton–Euler approach. Research also has been done related to the force and stiffness relations of the overconstrained manipulators. Xu et al. [Reference Xu, Liu, Yao and Zhao20] carried out a screw theory based method considering the link elastic deformations and stiffness matrix for analyzing the force relation of overconstrained LMPMs. Liu et al. [Reference Liu, Xu, Yao and Zhao21] reviewed the methods for the force analysis of OPMs. They discussed the problem of passive and active OPMs’ statically indeterminacy. They also prompted a universal method for those two kinds based on the screw theory. Dynamic analysis of a 5 DoF overconstrained manipulator is done with the decomposition of the partial subspaces is described in the work of Selvi and Yilmaz [Reference Selvi and Yilmaz22].
Based on linear algebra and screw theory, Liu et al. [Reference Liu, Huang, Kecskeméthy, Chetwynd and Li23] presented a systematic approach for force/motion transmissibility of redundantly actuated and OPMs. Comparing both types, they found out that the effect of being overconstrained has little concerning having actuation redundancy. To overcome the statically indeterminate problem in the force analysis of OPM’s, Xu et al. [Reference Xu, Lu, Liu, Guo, Yao and Zhao24] investigates the presence of the linearly dependent overconstrained wrenches. They reformulate the deformation compatibility equations between the overconstrained wrenches and describe the principle force model for OPMs and analyze these on a 2RPU–SPR OPM.
Hu and Huang [Reference Hu and Huang25] provided a kinetostatic model for an overconstrained LMPM with 2-RPU+UPR joint leg configuration. Using this kinetostatic model, the stiffness and the deformations of links are exemplified. Yang et al. [Reference Yang, Li, Chen and Xu26] proposed a modeling method for the elastostatic stiffness of OPMs using screw theory and applying the method to 2UPR–RPU PMs. Zhang and Fang [Reference Zhang and Fang27] proposed a 1T2R OPM with 2RPU–2SPR joint configuration and they calculated the linear and angular stiffness of the PM to find distributions law of the performance indices of redundantly actuated and overconstrained. Li et al. [Reference Li, Xu, Chen and Chai28] proposed a method for the analytical elastostatic stiffness modeling of OPMs. They elaborated geometric algebra along with strain energy. They also showed proof of concept comparison with finite element methods. In the study of Zhao et al. [Reference Zhao, Yan, Li and Xie29], the reachable workspace of an OPM with 2RPU&SPR joint configuration is investigated by using an analytical approach. Decomposition is used to divide a closed chain system into serial chains for the direct task and path planning of manipulators in the work of Han and Amato [Reference Li and Nanc30]. A random loop generator is presented and used for the kinematic analysis of closed-loop chains by Cortes et al. [Reference Cortés, Siméon and Laumond31]. The selection of active and passive links is essential and is mainly used for path planning for probabilistic road mapping.
Overconstrained manipulators for lower subspaces can be created and analyzed for different tasks with unique motions. However, far too little attention has been paid to creating a generic method for OPM’s kinematic analysis. This study aimed to evaluate a generic methodology for kinematic analysis of overconstrained parallel manipulators with partial subspaces (OPM-PS) and validate the method with examples. The essay has been organized in the following way. It begins by laying out the theoretical dimensions of the research and looks at how the partial subspaces can be used to describe an overconstrained manipulator. The new method that decomposes the mechanism into multiple lower subspace manipulator loops is presented. The methodology is then exemplified by designing and evaluating an overconstrained manipulator with a 5 DoF overconstrained manipulator. The inverse kinematic analysis for position analysis is shown. The Jacobian is derived for the system, and the inverse velocity task is described. The workspace calculation for the methodology is also described, and some numerical examples are depicted. Additionally, the procedure is applied for a 3 DoF (2T1R) PM. Finally, the methodology and examples are discussed.
2. Decomposition Method in OPMs
Mechanisms that belong to a subspace are described to be overconstrained. Depending on the geometry of links, these subspaces can have partial subspaces, which can be helpful in the kinematic analysis of overconstrained manipulators of this type.
Bennett [Reference Bennett32] proposed a method for creating an overconstrained mechanism by combining two mechanisms by using an intersecting joint. Removing the joint is possible due to the overconstrained subspace property (Fig. 1). These mechanisms behave in how the motion has both independent parts at each subspace and intersecting motion. The resulting 6R mechanism can be defined with two partial spherical subspaces with an imaginary joint, as shown in Fig. 1.
Bennet double spherical overconstrained mechanism.
In Selvi’s (2012) work, this method was generalized and used for the structural design of PMs. Several manipulators are generated using the method. Kinematic analysis and synthesis of the resulting mechanisms are done by reversing the idea and decompose the mechanism with an imaginary joint. The output of the first mechanism with a lower subspace will be input for the second one with another lower subspace. The partial subspaces for overconstrained manipulators are listed as spherical, planar, cylindrical, line and plane with just translation. An overconstrained manipulator generated using the combination method (Selvi 2012) will include subspaces with lower subspace degrees. In that case, the decomposition method can be used to separate the manipulator into two subspaces by adding imaginary joints. Imaginary joints are selected according to the intersection motion of the partial subspaces. The overconstrained manipulator is simulated as two manipulators where one’s input is the other manipulator’s output. One of the manipulators will include passive joints, and the other will include active joints. Also, end effector motion will belong to these two subspaces and their intersection motion.
The manipulator to be analyzed is an N DoF manipulator (Fig. 2(a)) and should be dividable into two partial subspaces (Fig. 2(b)). J imaginary joints are added to the manipulator (Fig. 2(c)). J is determined by investigating each leg of the manipulator, in each leg, the added joint should not crate any redundancy on the newly created manipulator in the upper subspace. The type and direction of the imaginary joints should be selected according to the intersection motion of the subspaces. The first section below will be defined with an N DoF manipulator that consists of the real base and a virtual platform (Fig. 2(e)). N DoF controls the motion of the moving platform and controls accompanying links to the platform that transmits motion to the second section (upper part) of the manipulator. The motion of these links will be the input for the second section. The second section is described with a manipulator with J DoF (Fig. 2(d)) and a virtual base. This J DoF generates a motion about the geometry of the second section. The input of the manipulator in the second section comes from a part of the output of the first section manipulator. To formulate the number of motions that describe the platform motion, Eq. (1) is suggested. Where the number of motions belongs to the first subspace (M) is found with the difference between the DoF of whole manipulator and second section manipulator.
\begin{align}M = N - J \end{align}
3. Overconstrained Manipulator Analysis with Decomposition Method
Two OPMs with partial subspaces are newly created and analyzed to demonstrate the methodology’s aspects. The first one is a 5 DoF manipulator in subspace
$\lambda=5$
and the second example is a 3 DoF manipulator in
$\lambda=3$
subspace.
(a) Overconstrained manipulator with N DoF, (b) joints relation to two partial subspaces, (c) imaginary joints added to the manipulator, (d) second section parallel manipulator with J DoF, (e) first section parallel manipulator with N DoF.
3.1. 5 DoF PM in subspace
$\lambda=5$
The manipulator is designed with four legs, where three legs have PRR(RR) joint combination and one leg has RRS combination, as shown in Figs. 3(a) and 4(a). The joints in the planar region shown in Fig. 3(a) will be used to create the active planar manipulator. The joints shown in the spherical region will be used to create the passive spherical manipulator. For the manipulator, J is found as three because adding joints to three legs with PRR(RR) do not create redundancy, and adding a joint to RRS leg will create redundancy if connected with the spherical joint. In Fig. 3(b), three revolute joints are shown as imaginary joints. Revolute joints are selected due to the intersection motion between the planar and spherical subspaces is rotation, and the direction of the joints should be perpendicular to the planar subspace because the intersecting motion of two partial subspaces is in that direction. Figure 3(c) 3 DoF passive spherical manipulator and Fig. 3(d) 5 DoF active planar PM configuration are shown. The actuated joints in the system are shown as underlined in Fig. 3(d). The imaginary actuators of the spherical manipulator are imaginary joints. Platform translates along the plane in two directions and rotates around three directions. According to the Eq. (1), motion of the end effector that belongs to the lower platform (M) is found as two, which is the two translations of the lower platform.
Decomposition method applied to 5 DoF parallel manipulator (a) OPM with subspace regions, (b) OPM with redundant imaginary joints, (c) 3 DoF passive spherical manipulator and (d) 5 DoF active planar manipulator.
Plano spherical
$\lambda=5$
manipulator (a) actuators configuration, (b) passive spherical manipulator and (c) active planar manipulator.
Inverse Kinematic Analysis
Imaginary Primary Manipulator (Spherical Subspace)
The upper part is a 3RRR spherical manipulator shown in Fig. 4(b), where three input revolute joints are inline. The kinematic scheme for the identical leg of the imaginary manipulator is shown in Fig. 5. End-effector orientation of the manipulator is defined by ρ. It will be determined for a specific motion by using Euler angles in the R(z’-y’-z’)(
$\zeta ,\xi ,\psi $
) configuration, as
${\boldsymbol{\rho }} = Ro{t_z}\left( \zeta \right).Ro{t_y}\left( \xi \right).Ro{t_z}\left( \psi \right)\!.$
Kinematic scheme of the imaginary spherical manipulator.
Kinematic scheme for the planar part of 5 DoF manipulator.
The vectors w i = [wx,i wy,i wz,i]T, end effector joint positions for the upper manipulator will be found from Eqs. (2) and (3).
\begin{align}{\textbf{w}_i} = {\boldsymbol{\rho }}\,{.}\,Ro{t_z}( {{\eta _i}} )\,{.}\,Ro{t_y}( {{\beta _i}} )\,{.}\,{\textbf{s}}\end{align}
The use of rotation matrices from a forward kinematics view describes the position of the vectors w i in terms of the orientation of the input joint u = [0,0,Reference Merlet1]T and kinematic parameters of the upper manipulator as shown in Eq. (3).
\begin{align}{\textbf{w}_i} = Ro{t_z}( {{\theta _{1i}}})\,{.}\,Ro{t_x}( {{\alpha _1}})\,{.}\,Ro{t_z}( {{\theta _{2i}}})\,{.}\,Ro{t_x}( {{\alpha _2}})\,{.}\,\textbf{u}\end{align}
The closure formed by Eqs. (2) and (3) gives three separate closure equations. The first two equations of Eq. (3) are used to find
$\theta$
1i
values for each leg of the mechanism.
$\theta$
1i values will be used as output values for the secondary manipulator.
Velocity analysis
Let the angular velocity of the platform be
$\boldsymbol\omega $
= [
$\omega $
x
$\omega $
y
$\omega $
z]T and the angular velocities of imaginary joints as
${\dot{\boldsymbol\theta }} = {\left[ {{{\dot{\boldsymbol\theta} }_\textbf{11}}{{\dot{\boldsymbol\theta} }_\textbf{12}}{{\dot{\boldsymbol\theta} }_\textbf{13}}} \right]^\textbf{T}}$
.
$\boldsymbol\Omega $
is related to the orientation velocity matrix of the platform as
\begin{align}{\boldsymbol{\Omega }} = {\dot{\textbf{s}}}{{\textbf{s}}^{ - 1}} = \left( {\begin{array}{c@{\quad}c@{\quad}c}\textbf{0}&{ - {\omega _z}}&{{\omega _y}}\\{{\omega _z}}&\textbf{0}&{ - {\omega _x}}\\{ - {\omega _y}}&{{\omega _x}}&\textbf{0}\end{array}} \right)\end{align}
Angular velocity of the rigid body is defined with the linear equations as,
\begin{align}\boldsymbol\omega = {\textbf{u}_i}{\dot{\boldsymbol\theta} _{\textbf{1i}}} + {\textbf{v}_i}{\dot{\boldsymbol\theta}_{\textbf{2i}}} + {\textbf{w}_i}{\dot{\boldsymbol\theta} _{\textbf{3i}}}\,\,\,\,\, i = 1,2,3\end{align}
Where
${\textbf{v}_i} = Ro{t_z}\left( {{\theta _{1i}}} \right) \cdot Ro{t_x}\left( {{\alpha _1}} \right) \cdot{\textbf{u}}$
Multiplying both sides of the Eq. (5) with
$\left( {{\textbf{v}_i} \times {\textbf{w}_i}} \right)$
will eliminate
${\dot {\boldsymbol\theta} _{\textbf{2i}}}$
and
${\dot{\boldsymbol\theta}_{\textbf{3i}}}$
from the constraint equation
\begin{align}{\boldsymbol{\omega }} \cdot \left( {{\textbf{v}_i} \times {\textbf{w}_i}} \right) = {\textbf{u}_i} \cdot \left( {{\textbf{v}_i} \times {\textbf{w}_i}} \right){\dot{\boldsymbol\theta} _{\textbf{i}}}\,\,\,\,\,i = 1,2,3\end{align}
Eq. (6) results in three linear equations that we can form a Jacobian matrix for the spherical manipulator.
\begin{align}{\textbf{J}_\textbf{x,s}}\boldsymbol\omega = {\textbf{J}_{\textbf{q,s}}}{\dot{\boldsymbol\theta }}\end{align}
\begin{align}{\textbf{J}_\textbf{s}}\boldsymbol\omega = {\dot{\boldsymbol\theta }}\end{align}
\begin{align}{\textbf{J}_\textbf{s}} = \textbf{J}_{\textbf{q,s}}^{-\textbf{1}}{\textbf{J}_{\textbf{x,s}}}\end{align}
Where
${\textbf{J}_{\textbf{q,s}}} = \left( {\begin{array}{c@{\quad}c@{\quad}c}{{v_{1,x}}{w_{1,y}} - {v_{1,y}}{w_{1,x}}}&0&0\\0&{{v_{2,x}}{w_{2,y}} - {v_{2,y}}{w_{2,x}}}&0\\0&0&{{v_{3,x}}{w_{3,y}} - {v_{3,y}}{w_{3,x}}}\end{array}} \right)$
,
\begin{align*}{\textbf{J}_{\textbf{x,s}}} = \left( \begin{array}{c@{\quad}c@{\quad}c} {{v_{1,y}}{w_{1,z}} - {v_{1,z}}{w_{1,y}}} & { - {v_{1,x}}{w_{1,z}} + {v_{1,z}}{w_{1,x}}} & {{v_{1,x}}{w_{1,y}} - {v_{1,y}}{w_{1,x}}} \\ \\[-7pt] {{v_{2,y}}{w_{2,z}} - {v_{2,z}}{w_{2,y}}} & { - {v_{2,x}}{w_{2,z}} + {v_{2,z}}{w_{2,x}}} & {{v_{2,x}}{w_{2,y}} - {v_{2,y}}{w_{2,x}}} \\ \\[-7pt] {{v_{3,y}}{w_{3,z}} - {v_{3,z}}{w_{3,y}}} & { - {v_{3,x}}{w_{3,z}} + {v_{3,z}}{w_{3,x}}} & {{v_{3,x}}{w_{3,y}} - {v_{3,y}}{w_{3,x}}} \\ \end{array} \right)\end{align*}
Imaginary secondary manipulator (planar subspace)
Planar manipulator
The imaginary secondary manipulator will be a planar 5 DoF redundant manipulator shown in Fig. 4(c). The manipulator’s output is the coordinates of point P and orientations of links joined at point P.
For the inverse kinematics of point P and joints at E and D loop closure equation will be,
\begin{align}\overline {OP} = \overline {OD} + \overline {DE} + \overline {EP} \end{align}
It will result in two equations as
\begin{align}Px = Dx + {b_1}\, cos\left( {{\phi _{14}}} \right) + {b_2}\, cos\left( {{\phi _{14}} + {\phi _{24}}} \right)\end{align}
\begin{align}Py = Dy + {b_1}\sin \left( {{\phi _{14}}} \right) + {b_2}\sin \left( {{\phi _{14}} + {\phi _{24}}} \right)\end{align}
Solving these two equations will result in
\begin{align*}{\phi _{14}} = Atan2\left( {Dy - Py,Dx - Px} \right) \pm co{s^{ - 1}}\left( {\frac{{{b_1}^2 + {b_2}^2 + {{\left( {Dx - Px} \right)}^2} + {{\left( {Dy - Py} \right)}^2}}}{{ - 2{b_1}\sqrt {{{\left( {Dx - Px} \right)}^2} + {{\left( {Dy - Py} \right)}^2}} }}} \right)\\[3pt] {\phi _{24}} = - {\phi _{14}} + Atan2\left( {Py - Dy - {b_1}sin\left( {{\phi _{14}}} \right)\!,Px - Dx - {b_1}cos\left( {{\phi _{14}}} \right)} \right)\qquad\qquad\qquad\end{align*}
the second loop closure equation (Eq. (13)) will be used to find actuator values for legs 1,2,3
\begin{align}\overline {OP} + \overline {PC} = \overline {OA} + \overline {AB} + \overline {BC} \end{align}
\begin{align}Px + {a_{2i}}cos\left( {{\theta _{1i}}} \right) = {h_i}cos\left( {{\alpha _i}} \right) + {S_i}cos\left( {{\alpha _i} + \frac{\pi }{2}} \right) + {a_{1i}}cos\left( {{\alpha _i} + \frac{\pi }{2} + {\phi _{1i}}} \right)\end{align}
\begin{align}Py + {a_{2i}}sin\left( {{\theta _{1i}}} \right) = {h_i}sin\left( {{\alpha _i}} \right) + {S_i}sin\left( {{\alpha _i} + \frac{\pi }{2}} \right) + {a_{1i}}sin\left( {{\alpha _i} + \frac{\pi }{2} + {\phi _{1i}}} \right)\end{align}
Linear actuator values can be found as
\begin{align}{S_i} = - b \pm \sqrt {{b^2} - 4c} \end{align}
Where
\begin{align*}b &= \left( { - 2{\textrm{Py}}cos\left( {{\alpha _i}} \right) + 2\left( {{\textrm{Px}}sin\left( {{\alpha _i}} \right) + {a_{2i}}sin\left( {{\alpha _i} - {\theta _1}} \right)} \right)} \right)\\[3pt] c &= - {a_{1i}}^2 + {a_{2i}}^2 + {h_i}^2 + {\textrm{P}}{{\textrm{x}}^2} + {\textrm{P}}{{\textrm{y}}^2} - 2{h_i}{\textrm{Px}}cos\left( {{\alpha _i}} \right) - 2{a_{2i}}{h_i}cos\left( {{\alpha _i} - {\theta _{1i}}} \right) - 2{h_i}{\textrm{Py}}sin\left( {{\alpha _i}} \right)\\[3pt] &\quad + 2{a_{2i}}\left( {\textrm{Px}}cos\left( {{\theta _{1i}}} \right) + {\textrm{Py}}sin\left( {{\theta _{1i}}} \right) \right) \end{align*}
Velocity analysis
For the planar manipulator, output velocities are stated with
${\dot{\textbf{X}}} = {\left[ \dot{\theta}_{11}\ \dot{\theta}_{12}\ \dot{\theta}_{13}\ \dot{\textrm{P}}\textrm{x}\ \dot{\textrm{P}}\textrm{y} \right]^T}$
and the input velocities are shown with vector
${\dot{\textbf{q}}} = {\left[ {{{\dot S}_1}\ {{\dot S}_2}\ {{\dot S}_3}\ {{\dot \phi }_{14}}\ {{\dot \phi }_{24}}} \right]^T}$
.
\begin{align}Px + {a_{2i}}cos\left( {{\theta _{1i}}} \right) = {h_i}cos\left( {{\alpha _i}} \right) + {S_i}cos\left( {{\alpha _i} + \frac{\pi }{2}} \right) + {a_{1i}}cos\left( {{\alpha _i} + \frac{\pi }{2} + {\phi _{1i}}} \right)\end{align}
\begin{align}Py + {a_{2i}}sin\left( {{\theta _{1i}}} \right) = {h_i}sin\left( {{\alpha _i}} \right) + {S_i}sin\left( {{\alpha _i} + \frac{\pi }{2}} \right) + {a_{1i}}sin\left( {{\alpha _i} + \frac{\pi }{2} + {\phi _{1i}}} \right)\end{align}
From Eqs. (14) and (15),
${\phi _{1i}}$
is eliminated to create a constraint equation
\begin{align}{a_{1i}}^2 &= h_i^2 + S_i^2 + P{x^2} + P{y^2} + {a_{2i}}^2 - 2Px{h_i}cos\left( {{\alpha _i}} \right) - 2{h_i}cos\left( {{\alpha _i} - {\theta _{1i}}} \right)a2i - 2Px{S_i}cos\left( {{\alpha _i} + {\pi \over 2}} \right)\nonumber\\ &\quad + 2Pxa2icos\left( {{\theta _{1i}}} \right) - 2Py{h_i}sin\left( {{\alpha _i}} \right) - 2Py{S_i}sin\left( {{\alpha _i} + {\pi \over 2}} \right) + 2Py{a_{2i}}sin\left( {{\theta _{1i}}} \right)\end{align}
Moreover, it was derived by time for the velocity relations.
$$\matrix{
{} \hfill & {\left[ { - {\rm{Px}} - cos\left( {{\theta _{1i}}} \right){\rm{a}}{2_i} + cos\left( {{\alpha _i}} \right){h_i} - sin\left( {{\alpha _i}} \right){S_i}} \right]\mathop {\rm{P}}\limits^ {{\rm{x}}^ + }\left[ { - {\rm{Py}} - sin\left( {{\theta _{1i}}} \right){\rm{a}}{2_i} + sin\left( {{\alpha _i}} \right){h_i} + cos\left( {{\alpha _i}} \right){S_i}} \right]} \hfill \cr
{} \hfill & {\quad \times \mathop {\mathop {\rm{P}}\limits^. }\limits^ {{\rm{y}}^ + }{\rm{a}}{2_i}\left[ { - {\rm{Py}}cos\left( {{\theta _{1i}}} \right) + {\rm{Px}}sin\left( {{\theta _{1i}}} \right) + sin\left( {{\alpha _i} - {\theta _{1i}}} \right){h_i} + cos\left( {{\alpha _i} - {\theta _{1i}}} \right){S_i}} \right]\mathop {\rm{\theta }}\limits^. {1_i}} \hfill \cr
{} \hfill & {\quad = [ - {\rm{Py}}cos({\alpha _i}) + {\rm{Px}}sin({\alpha _i}) + sin({\alpha _i} - {\theta _{1i}}){\rm{a}}{2_i} + {S_i}]{{\dot S}_i}} \hfill \cr
} $$
Eqs. (10) and (11) is derived by time for further velocity relations
\begin{align}{\dot{\textrm{P}}\textrm{x}} = \left( { - {b_1}sin\left( {{\phi _{14}}} \right) - {b_2}sin\left( {{\phi _{14}} + {\phi _{24}}} \right)} \right){\dot \phi _{14}} - {b_2}sin\left( {{\phi _{14}} + {\phi _{24}}} \right){\dot \phi _{24}}\end{align}
\begin{align}{\dot{\textrm{P}}\textrm{y}} = \left( {{b_1}\cos \left( {{\phi _{14}}} \right) + {b_2}\cos \left( {{\phi _{14}} + {\phi _{24}}} \right)} \right){\dot \phi _{14}} + {b_2}\cos \left( {{\phi _{14}} + {\phi _{24}}} \right){\dot \phi _{24}}\end{align}
\begin{align}{J_{x,p}}\dot{\textbf{x}} = {J_{q,p}}\dot{\textbf{q}}\end{align}
\begin{align*}{J_p}\dot {\textbf{x}} = \dot {\textbf{q}}\end{align*}
\begin{align*}{J_p} = J_{q,p}^{ - 1}{J_{x,p}}\end{align*}
\begin{align*}{J_{{\textbf{q}},{\textbf{p}}}} = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{J_{q,p}^{1,1}} & 0& 0& 0 &0 \\0&{J_{q,p}^{22}}&0&0&0\\0 & 0& {J_{q,p}^{3,3}}& 0 &0 \\0 & 0& 0& {J_{q,p}^{4,4}} & {J_{q,p}^{4,5}}\\ \\[-6pt]0 & 0& 0& {J_{q,p}^{5,4}} & {J_{q,p}^{5,5}}\\\end{array} \right)\nonumber\end{align*}
Where,
\begin{align*}J_{q,p}^{1,1} &= - {\textrm{Py}}cos\left( {{{{\alpha }}_1}} \right) + {\textrm{Px}}sin\left( {{{{\alpha }}_1}} \right) + sin\left( {{{{\alpha }}_1} - {{{\theta }}_{11}}} \right){{\textrm{a}}_{21}} + {S_1}\\[3pt] J_{q,p}^{2,2} &= - {\textrm{Py}}cos\left( {{{{\alpha }}_2}} \right) + {\textrm{Px}}sin\left( {{{{\alpha }}_2}} \right) + sin\left( {{{{\alpha }}_2} - {{{\theta }}_{12}}} \right){{\textrm{a}}_{22}} + {S_2}\\[3pt]J_{q,p}^{3,3} &= - {\textrm{Py}}cos\left( {{{{\alpha }}_3}} \right) + {\textrm{Px}}sin\left( {{{{\alpha }}_3}} \right) + sin\left( {{{{\alpha }}_3} - {{{\theta }}_{13}}} \right){{\textrm{a}}_{23}} + {S_3}\\[3pt]J_{q,p}^{4,4} &= \left( { - {b_1}sin\left( {{\phi _{14}}} \right) - {b_2}sin\left( {{\phi _{14}} + {\phi _{24}}} \right)} \right)\\[3pt]J_{q,p}^{4,5} &= - {b_2}sin\left( {{\phi _{14}} + {\phi _{24}}} \right)\\[3pt]J_{q,p}^{5,4} &= \left( {{b_1}cos\left( {{\phi _{14}}} \right) + {b_2}cos\left( {{\phi _{14}} + {\phi _{24}}} \right)} \right)\\[3pt]J_{q,p}^{5,5} &= {b_2}cos\left( {{\phi _{14}} + {\phi _{24}}} \right)\end{align*}
\begin{align*}{J_{{\textbf{x}},{\textbf{p}}}} = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{J_{x,p}^{1,1}}&0&0&{J_{x,p}^{1,4}}&{J_{x,p}^{1,5}}\\ \\[-7pt]0&{J_{x,p}^{2,2}}&0&{J_{x,p}^{2,4}}&{J_{x,p}^{2,5}}\\ \\[-7pt]0&0&{J_{x,p}^{3,3}}&{J_{x,p}^{3,4}}&{J_{x,p}^{3,5}}\\ \\[-7pt]0&0&0&1&0\\ \\[-7pt]0&0&0&0&1\\\end{array} \right)\end{align*}
Where,
\begin{align*}J_{x,p}^{1,1} &= {{\textrm{a}}_{21}}\left[ { - {\textrm{Py}}cos\left( {{{{\theta }}_{11}}} \right) + {\textrm{Px}}sin\left( {{{{\theta }}_{11}}} \right) + sin\left( {{{{\alpha }}_1} - {{{\theta }}_{11}}} \right){h_1} + cos\left( {{{{\alpha }}_1} - {{{\theta }}_{11}}} \right){S_1}} \right]\\[3pt] J_{x,p}^{2,2} &= {{\textrm{a}}_{22}}\left[ { - {\textrm{Py}}cos\left( {{{{\theta }}_{12}}} \right) + {\textrm{Px}}sin\left( {{{{\theta }}_{12}}} \right) + sin\left( {{{{\alpha }}_2} - {{{\theta }}_{12}}} \right){h_2} + cos\left( {{{{\alpha }}_2} - {{{\theta }}_{12}}} \right){S_2}} \right]\\[5pt]J_{x,p}^{3,3} &= {{\textrm{a}}_{23}}\left[ { - {\textrm{Py}}cos\left( {{{{\theta }}_{13}}} \right) + {\textrm{Px}}sin\left( {{{{\theta }}_{13}}} \right) + sin\left( {{{{\alpha }}_3} - {{{\theta }}_{13}}} \right){h_3} + cos\left( {{{{\alpha }}_3} - {{{\theta }}_{13}}} \right){S_3}} \right]\\[5pt]J_{x,p}^{1,4} &= - {\textrm{Px}} - cos\left( {{{{\theta }}_{11}}} \right){{\textrm{a}}_{21}} + cos\left( {{{{\alpha }}_1}} \right){h_1} - sin\left( {{{{\alpha }}_1}} \right){S_1}\\[5pt]J_{x,p}^{1,5} &= - {\textrm{Py}} - sin\left( {{{{\theta }}_{11}}} \right){{\textrm{a}}_{21}} + sin\left( {{{{\alpha }}_1}} \right){h_1} + {\textrm{Cos}}\left( {{{{\alpha }}_1}} \right){S_1}\\[5pt]J_{x,p}^{2,4} &= - {\textrm{Px}} - cos\left( {{{{\theta }}_{12}}} \right){{\textrm{a}}_{22}} + cos\left( {{{{\alpha }}_2}} \right){h_2} - sin\left( {{{{\alpha }}_2}} \right){S_2}\\[5pt]J_{x,p}^{2,5} &= - {\textrm{Py}} - sin\left( {{{{\theta }}_{12}}} \right){{\textrm{a}}_{22}} + sin\left( {{{{\alpha }}_2}} \right){h_2} + {\textrm{Cos}}\left( {{{{\alpha }}_2}} \right){S_2}\\[5pt]J_{x,p}^{3,4} &= - {\textrm{Px}} - cos\left( {{{{\theta }}_{13}}} \right){{\textrm{a}}_{23}} + cos\left( {{{{\alpha }}_3}} \right){h_3} - sin\left( {{{{\alpha }}_3}} \right){S_3}\\[5pt]J_{x,p}^{3,5} &= - {\textrm{Py}} - sin\left( {{{{\theta }}_{13}}} \right){{\textrm{a}}_{23}} + sin\left( {{{{\alpha }}_3}} \right){h_3} + cos\left( {{{{\alpha }}_3}} \right){S_3}\end{align*}
Using Eqs. (7) and (22) recurrently, the velocity analysis of the whole manipulator can be done.
Results of kinematic calculations
The parameters for the 1st manipulator is selected with trial and error as follows,
\begin{align*}{\eta _{\textrm{1}}} &= 0^\circ ,\,{\eta _{\textrm{2}}} = {\textrm{12}}0^\circ ,\,{\eta _{\textrm{3}}} = {\textrm{24}}0^\circ ,\,\beta_{\textrm{i}} = {\textrm{6}}0^\circ ;{\textrm{ i}} = {\textrm{1}},{\textrm{2}},{\textrm{3}}\\[4pt] \alpha 1 &= \alpha 2 = 75^\circ ,\,Dx = 0,\,Dy = 75,\,b1 = 100,\,b2 = 100\end{align*}
To test the inverse kinematic solution for the manipulator and an objective motion is defined that has sinusoidal characteristic with different phase angles and frequencies. The objective motion of the orientation and translation of the end effector is shown in Fig. 7(a) and (b), respectively.
Using Eq. (3), the imaginary joint values are found for the objective orientation data and presented in Fig. 8.
Objective values for 5 DoF manipulator (a) orientation and (b) translation.
Calculated orientation values of imaginary joints.
These imaginary joint values are used as an objective function for the bottom manipulator. The actuated joints of the manipulator’s position and orientation are shown in Fig. 9(a) and (b), respectively.
Calculated position and orientation values of actuated joints: (a) prismatic and (b) revolute.
The velocity relations for imaginary joints are tested by describing angular velocity at the end effector. The related imaginary joint velocities are found in Fig. 10(a), which will be used along with the sinusoidal translational velocity defined for the end effector to calculate the actuator velocities. The translational and angular velocities are found as presented in Fig. 10(b).
(a) Imaginary joints velocity, (b) linear actuator velocities and (c) angular actuator velocities.
The decomposition method is also used to determine or design the workspace of the overconstrained manipulator. The determined workspace for the upper part of the 5 DoF manipulator is shown in Fig. 11(a), where the orientation is dexterous around ζ. The range of the imaginary joints is calculated for ±20° in all directions as,
$\theta_{11}$
: −0.55 to +0.55 radians,
$\theta_{12}$
from 1.4 to 3.2 radians,
$\theta_{13}$
from 3.5 to 4.9 radians. The workspace for the whole manipulator is then calculated using these ranges and presented in Fig. 11(b) and (c). It is seen that the
$\theta_{13}$
direction is dexterous.
(a) Individual workspace of the upper part of 5 DoF manipulator at ζ = 0, (b) workspace of the manipulator for xy motion and (c) workspace for the imaginary joints.
For a design case of the manipulator for a better workspace, number of parameters in design should be 25. The use of imaginary joints helps divide the system into two parts and split the number of design parameters by 9 and 16. Thus, the design space is decreased from the order of 25 to 16 and 9. Also, the workspace of the imaginary joints shown in Fig. 11(c) can be used to set constraint equations for an optimization problem.
3.2. 3 DoF manipulator in subspace
$\lambda=3$
An overconstrained manipulator with 3 DoF is shown in Figs. 12(a) and 13(a). Manipulator has three legs, where two legs have PPR joint combination and one leg has PPH combination as shown in Fig. 12. The translational region’s joints belong to the translational subspace, and joints in the cylindrical region belong to the cylindrical subspace. For the proposed manipulator, J is found as one because only one joint added to the PPH leg will not create redundancy for the upper passive manipulator. It is selected to be a prismatic joint due to the intersection motion of the planar with translation only space and cylindrical subspace is translational motion, and the direction of the prismatic joint should be coaxial with the intersection of these subspaces. Figure 12(c) 1 DoF cylindrical mechanism is seen, and Fig. 12(d) 3 DoF translational PM is seen. The actuated joints in the system are shown as underlined in Fig. 12(d). Using Eq. (1), motion of the end effector that belongs to the lower platform (M) is found as two related to the lower platform’s translations.
Decomposition method applied to 3 DoF parallel manipulator (a) 3 DoF overconstrained manipulator, (b) 3 DoF overconstrained manipulator with redundant imaginary joints, (c) 1 DoF passive cylindrical manipulator and (d) 3 DoF active translational manipulator.
3 DoF overconstrained manipulator (a) actuators configuration, (b) kinematics of the 1 DoF cylindrical mechanism and (c) kinematics of the 3 DoF planar manipulator.
The imaginary actuator of the cylindrical manipulator is an imaginary joint. Platform translates along the plane in two directions and rotates around one of the directions.
Three parameters define the manipulator’s output—two translations along x and y axes as Px and Py and rotation around the y-axis as
$\phi$
. The upper part of the system is a one DoF mechanism with PHR joint configuration in cylindrical subspace. The output of the mechanism is
$\phi$
, and the input is S as shown in Fig. 13(b). The closure equation of the mechanism can be given as follows
\begin{align}\phi = \theta = {\textrm{s/p}}\end{align}
where p is the pitch value for the helical joint.
From Eq. (24), the imaginary input value S can be calculated and deriver by time to get the velocity equation
\begin{align}{\dot{\textbf{S}}} = p*\dot \phi \end{align}
The secondary imaginary manipulator is a 3 DoF planar translational manipulator as shown in Fig. 13(c).
With the leg configuration as 2PP+PPP
\begin{align}{\textrm{OA}} + {\textrm{AB}} + {\textrm{BC}} + {\textrm{CP}} = {\textrm{OP}}\end{align}
\begin{align}\left( {{h_i} - {a_i}} \right)cos\left( {{\alpha _i}} \right) - {S_i}sin\left( {{\alpha _i}} \right) - {k_i}cos\left( {{\beta _i} + {\alpha _i}} \right) = {\textrm{Px}}\end{align}
\begin{align}\left( {{h_i} - {a_i}} \right)sin\left( {{\alpha _i}} \right) - {S_i}cos\left( {{\alpha _i}} \right) - {k_i}sin\left( {{\beta _i} + {\alpha _i}} \right) = {\textrm{Py}}\end{align}
For link 1 ki = k1+S
\begin{align*}{S_1} &= \left[ {{\textrm{Py}}\,cos\left( {{\alpha _1}} \right) - {\textrm{Px}}\,sin\left( {{\alpha _1}} \right) + sin\left( {{\beta _1}} \right)({k_1} + S)} \right]/cos\left( {2{\alpha _1}} \right)\\[3pt] {S_2} &= \left[ {{\textrm{Py}}cos\left( {{\alpha _2}} \right) - {\textrm{Px}}sin\left( {{\alpha _2}} \right) + sin\left( {{\beta _2}} \right)({k_2})} \right]/cos\left( {2{\alpha _2}} \right)\\[3pt]{S_3} &= \left[ {{\textrm{Py}}cos\left( {{\alpha _3}} \right) - {\textrm{Px}}sin\left( {{\alpha _3}} \right) + sin\left( {{\beta _3}} \right)({k_3})} \right]/cos\left( {2{\alpha _3}} \right) \end{align*}
Velocity analysis can be done by deriving the position equation by time. The velocity relation of the second example manipulator is found in Eq. (29).
\begin{align}\left[ {\begin{array}{*{20}{c}}{{{\dot S}_1}}\\ \\[-7pt]{{{\dot S}_2}}\\ \\[-7pt]{{{\dot S}_3}}\end{array}} \right] = \left( {\begin{array}{c@{\quad}c@{\quad}c}{\dfrac{{ - sin\left( {{\alpha _1}} \right)}}{{cos\left( {2{\alpha _1}} \right)}}}&{\dfrac{{cos\left( {{\alpha _1}} \right)}}{{cos\left( {2{\alpha _1}} \right)}}}&{\dfrac{{sin\left( {{\beta _1}} \right)}}{{cos\left( {2{\alpha _1}} \right)}}}\\ \\[-7pt]{\dfrac{{ - sin\left( {{\alpha _2}} \right)}}{{cos\left( {2{\alpha _2}} \right)}}}& {\dfrac{{cos\left( {{\alpha _2}} \right)}}{{cos\left( {2{\alpha _2}} \right)}}}&0\\ \\[-7pt]{\dfrac{{ - sin\left( {{\alpha _3}} \right)}}{{cos\left( {2{\alpha _3}} \right)}}}&{\dfrac{{cos\left( {{\alpha _3}} \right)}}{{cos\left( {2{\alpha _3}} \right)}}}&0\end{array}} \right) \left[ {\begin{array}{*{20}{c}}{{{{\dot{\textrm{P}}\textrm{x}}}}}\\ \\[-7pt]{{{\dot{\textrm{P}}\textrm{y}}}}\\ \\[-7pt]{\dot{\textbf{S}}}\end{array}} \right]\end{align}
The construction parameters for the second example manipulator is selected as, p = 20 (constant pitch of the screw joint), α1 = 210°, α2 = 330°, α3 = 90°, β1 = 45°, β2 = 285°, β3 = 165°, k1 = 10 mm, k2 = 15 mm, k3 = 25 mm with a ±400 mm range of sliders.
Again a sinusoidal objective motion with different phase angles and frequencies was determined for the position and orientation of the 3 DoF manipulator, as shown in Fig. 14(a) and (b), respectively. The imaginary joint position S is found in Fig. 15(a), and the three slider values are calculated as shown in Fig. 15(b).
Displacement and orientation values of decided objective (a) translation and (b) orientation.
(a) Imaginary joint position changes of 3 DoF manipulator and (b) inverse kinematic position analysis of 3 DoF manipulator.
The velocity is determined and shown in Fig. 16(a), and corresponding velocity values are calculated as shown in Fig. 16(b).
(a) Objective velocity of the platform and (b) calculated velocity of the actuator.
The sliders’ limits are selected as ±400 mm, and the resulting workspace for
$\phi$
= 0, 120, 240, 360, 480, 600 are presented in Fig. 17. It is seen that the workspace is shifting due to the rotation in the helical joint.
Workspace of the second example manipulator.
5. Discussion
The decomposition method is applied by dividing the system into lower subspace close loops rather than open loops. This unique approach shows several advantages in kinematic calculation simplicity, understanding of the motion of the PM, workspace and singularity determination. With the help of decomposition, the manipulator is divided into two parts. Lower dimensional subsets can be investigated separately. Then singularities can be determined regarding the lower subspace and related motion. The use of the decomposition method is also helpful in means of calculations. Solving the kinematics for the manipulator commonly, they are needed to be dissected into arms and then for each arm, kinematic calculations should be done. Assuming the use of homogeneous transformation matrices for single arms of the system, some complex calculations should have been handled for spatial kinematics. For example, if this method is applied, the first example multiplication of five six by six homogeneous matrices should be solved. Also, dividing the motion into lower subspaces helps to identify the motion quickly. The use of the decomposition method shows that the workspace for the manipulator can be divided concerning the partial subspaces of the manipulator. Then individual parts can be calculated and investigated in means of workspace. The intersection of the workspace of the imaginary joints and the sub manipulator with the actuators will result in the actual workspace. The visualization of the separate workspace also helps in the interpretation in means of design. Among the two examples shown in the text, additional four manipulators are described in the appendix without detailed kinematics but just the use of the decomposition method.
6. Conclusion
The Decomposition method is defined and applied for two example manipulators. Results show that using the method and adding imaginary joints to the inverse position and velocity analysis with Jacobian can be achieved. Furthermore, it is also shown that the decomposition method will be a convenient tool for the design process of these types of manipulators for desired workspaces. Unlike most OPMs, the OPMs-PS have a clear definition for the subspace itself. Applicability of the method will open a way of research in this area of OPMs.
Conflicts of Interest
The author declares none.
Financial Support
This research received no specific grant from any funding agency, commercial or not-for-profit sectors.
Supplementary Material
To view supplementary material for this article, please visit https://doi.org/10.1017/S0263574721001351.
