Singularity analysis and dexterity performance on a novel parallel mechanism with kinematic redundancy

Abstract

This article presents a novel spatial parallel mechanism with kinematic redundancy. The design strategy and evolution of the proposed mechanism is introduced, and kinematic model of the mechanism is established. To simplify singularity analysis of this kind of mechanism, the virtual plane method which can separate the whole parallel mechanism into two parts is presented. The relative Jacobian matrices are established and illustrated with singularity configurations of three types. Kinematic performance is obtained to see redundancy effects on the mechanism. The orientational workspace is obtained by the regions of orientational angles with varied platform position. It shows the orientational workspace of the redundant mechanism is significantly larger. Evaluation of condition number demonstrates the proposed mechanism can clearly stay away from singularities with a large range of rotational angles. A trajectory example is conducted to further prove the proposed mechanism can produce a large range of rotational angles without meeting with singularities.

Keywords

Parallel mechanism kinematic redundancy kinematic model singularity dexterity performance

Introduction

It is widely known that spatial parallel mechanisms (PMs) possess many advantages compared to serial mechanisms in the areas include manufacturing, motion simulation, and pick and place manipulation. Such benefits include higher stiffness, higher payload ability, higher acceleration, the solution to inverse kinematics problem is simpler, the position of actuators located in or near to the rack.¹ However, the aforementioned advantages often suffer lower dexterity performance, finite smaller workspace, more complex kinematic models, and dynamic models. The additional complexity of these kinematically constrained architectures is more than that of serial mechanisms.²

So far, most of the proposed PMs in the literature have focused on architecture with nonredundant. In a bid to overcome the aforementioned drawbacks of PMs, the redundant PMs have been proposed.³ Totally, only three redundancy types were defined by Lee and Kim⁴ and can be separated into two types. One is kinematic redundancy and the other is actuation redundancy, which can include all three redundancy types.⁴ Many researchers proposed the actuation redundancy PM to alleviate the disadvantage such as limited rotational capability^5

–11 . Generally, the majority of actuation redundancy studies have focused on replacing some or all existing passive joints by active joints or adding additional limbs to connect the fixed base with moving platform. The studies of adding additional limbs are more probably to avoid singularities. However, actuation redundancy mechanism does not alter mobility. Besides, adding actuation redundancy may alleviate or eliminate its singularities but not change the mechanism’s workspace. Unfortunately, this method can generate internal forces between links and moving platform. Therefore, controlling of such mechanisms requires the advanced controllers and becomes more challenging.^12
–14

In recent decades, other researchers have proposed the PM with kinematic redundancy to eliminate or reduce singularities.^2,3,15,16 This method is introduced by adding extra active joints into the limbs to improve the capacity to modulate the limbs or avoid singularities. The kinematic redundancy research on high applicable nonredundant PMs such as 3-RPR and Gough–Stewart was proposed by Gosselin and colleagues,^16,17 where an underlined letter represents an actuated joint. However, the architecture of mechanism is also an influence of performance. Other considerations such as actuators are located at fixed platform should be design guidance. Sagi and Bandyopadhyay¹⁸ present a comparison between a hybrid and a parallel architecture. The actuator arrangement based on the architecture design is very important to produce a distinction of performance. According to the successful design strategy in DELTA structure, the links are not burdened with the load formed by prismatic actuators. Therefore, Srivatsan and Bandyopadhyay¹⁹ introduced a four-bar linkage into each limb, which generates a better weight distribution among limbs and reduces limb loads burdened with prismatic actuators. Unfortunately, the workspace of this mechanism is not very large because of the limitation from the four-bar linkage structure of each limb.

To make the mechanism structure overcome the above disadvantages, a novel PM with kinematic redundancy is proposed by introducing five-bar structure into each limb. The evolution of the novel redundant PM is briefly introduced as shown in the “Architecture of the kinematically redundant mechanism” section. In the “Kinematic model” section, the kinematic model of the redundant PM is provided. Based on the proposed virtual plane (VP) method for decreasing difficulty of singularity analysis, the configurations of three singularity types are discussed and some of the configurations are illustrated in the “Singularity analysis” section. Then, the orientational workspace of the proposed 3-RRRRRS mechanism is established by the region of orientational angles with varied platform position (z) as shown in the “Workspace analysis” section. To get the effect of adding redundancy into mechanism, evaluation of condition number and an example circle trajectory are performed in the “Dexterity analysis” and “Trajectory simulation” sections, respectively.

Architecture of the kinematically redundant mechanism

Evolution of 3-RRRRRS mechanism from the 3-RPS mechanism

The architecture of MaPaMan¹⁹ was introduced by the evolution of the 3-RPS. And the 3-RPS mechanism was introduced in 1978 by Hunt and studied by Lee and Shah in 1988.^20,21 Each limb of the 3-RPS mechanism has a structure of RPS. The passive revolute joints (R) are located at the base of limbs and connect with fixed platform. The spherical joints (S) followed by the prismatic actuators (P) are connected with moving platform. The redundant architecture is essentially formed by a series of modifications from the nonredundant limbs architecture, while reserving the overall connectivity between the limbs and moving platform. However, the motion capacity of MaPaMan or 3-RPS was completely limited by the limb’s architecture. Assuming that each limb can be added by a redundant joint, thereby the kinematically redundant mechanism will be formed. The key purpose of these modifications of architecture is to propose a mechanism without any prismatic type of actuator or passive joint. Because the impact of inertial loads on actuators is much higher than the inertial loads in the links, the actuators need to be arranged on the ground. Through doing this, some of other improvements may be also achieved, such as motion capacity, singularity avoidance, and versatility. Consequently, the five-bar linkage is an appropriate unit for limb architecture.

The evolution is shown in Figure 1 and described as follows: (1) Prismatic actuator (P) is replaced by the passive revolute joint (R₂). As a result, the limb architecture becomes an RRS which can be seen in MaPaMan.¹⁹ (2) In the same vertical plane, a redundant actuator (P₁) is added to connect moving platform with fixed platform through revolute joint (R₃) and spherical joint (S). (3) Finally, the revolute joint (R₄) at the bottom of P₁ is acted as an output point of the aforementioned five-bar linkage. Redundant actuator (P₁) is replaced by a passive revolute joint (R₅) based on the same method. Therefore, a novel redundant PM, namely 3-RRRRRS mechanism, is obtained by following these three steps. It will be demonstrated that the novel mechanism is a PM with kinematic redundancy in the “DOF and RDOF of 3-RRRRRS mechanism” section.

Figure 1.

The evolution of the limb architecture. (a) RPS architecture; (b) RRS architecture; (c) RRRPS architecture; (d) RRRRRS architecture.

Architecture of 3-RRRRRS mechanism

Three identical limbs connect fixed platform with moving platform to form the mechanism. The moving platform is designed as a rigid equilateral triangle form, which acts as the output manipulator of the mechanism. Architecture of the mechanism and details of the first limb are shown in Figure 2. The limb consists of a five-bar linkage and a strut link. The five-bar linkage is actuated by two actuators located at A_i and E_i , respectively. The motion is produced by the strut link attached to the output rotational joint (C_i ) of five-bar linkage. C_i is in the strut base and transmits motion to moving platform via the strut link. Then, the coordinate frames O-xyz and p-uvw are set up in the center of fixed platform and moving platform, respectively.

Figure 2.

Architecture of 3-RRRRRS mechanism and kinematic diagram of limb 1. (a) The 3-RRRRRS mechanism Architecture and (b) Kinematic diagram of limb 1.

DOF and RDOF of 3-RRRRRS mechanism

The degree of freedom (DOF) of the mechanism is the number of independent actuators that needed to define a unique configuration. The rotational degree of freedom (RDOF) of the mechanism is equal to the number of independent motions of moving platform. In the kinematically redundant mechanism, the DOF of the whole mechanism is not equal to the RDOF between the moving platform and the fixed base.²² In this section, the RDOF and DOF of the mechanism are calculated to show the proposed mechanism has a structure with kinematic redundancy. As shown in Figure 2, the coordinates of kinematic joints center (A_i , B_i , C_i , D_i , E_i , S_i ) can be noted as $a_{i} = {[x_{A i}, y_{A i}, z_{A i}]}^{T}$ , $b_{i} = {[x_{B i}, y_{B i}, z_{B i}]}^{T}$ , $c_{i} = {[x_{C i}, y_{C i}, z_{C i}]}^{T}$ , $d_{i} = {[x_{D i}, y_{D i}, z_{D i}]}^{T}$ , $e_{i} = {[x_{E i}, y_{E i}, z_{E i}]}^{T}$ , and $s_{i} = {[x_{S i}, y_{S i}, z_{S i}]}^{T}$ . The axes of revolute joints A ₁, B ₁, C ₁, D ₁, E ₁ are parallel to the y-axis. Based on the screw theory, the screws of revolute joints in limb 1 can be written as

{\begin{matrix} $_{1} = {[0 1 0; 0 0 x_{A 1}]}^{T} \\ $_{2} = {[0 1 0; - z_{B 1} 0 x_{B 1}]}^{T} \\ $_{3} = {[0 1 0; - z_{C 1} 0 x_{C 1}]}^{T} \\ $_{4} = {[0 1 0; - z_{D 1} 0 x_{D 1}]}^{T} \\ $_{5} = {[0 1 0; 0 0 x_{E 1}]}^{T} \end{matrix}

Assuming the direction of three equivalent rotation axes of joint S ₁ respectively along the x-, y- and z-axes, screws of joint S ₁ can be written as

{\begin{matrix} $_{6} = {[1 0 0; 0 z_{S 1} 0]}^{T} \\ $_{7} = {[0 1 0; - z_{S 1} 0 x_{S 1}]}^{T} \\ $_{8} = {[0 0 1; 0 - x_{S 1} 0]}^{T} \end{matrix}

Based on equations (1) and (2), the screw system of limb 1 can be obtained as

$_{11} = ($_{1} $_{2} $_{3} $_{4} $_{5} $_{6} $_{7} $_{8})

Then, the reciprocal screw of screw system of limb 1 $$_{11}$ can be deduced as

$_{1}^{r} = {[0 1 0; - z_{S 1} 0 x_{S 1}]}^{T}

where $$_{1}^{r}$ represents a restraint force vector along the y-axis direction passing through the center of joint S ₁.

Due to the symmetrical properties of this PM, there are three restraint force vectors focus on the moving platform. The vectors are parallel to the revolute joints axes in each limb respectively. Assuming that the initial position is when moving platform is parallel to fixed platform, the three restraint force vectors are coplanar consequently. Then one can be obtained is that the three restraint force vectors are linearly independent. This means that the movement of the moving platform along the x- and y-axes and the rotation around the z-axis are restricted. Therefore, the RDOF of the moving platform is the movement along the z-axis and the rotation around the x- and y-axes.

In this section, the modified Grübler–Kutzbach equation (5) is used to find the DOF of this mechanism. Due to the 3-RRRRRS mechanism is kinematically redundant, the main work is getting v from the structure of this mechanism.

M = d (n - g - 1) + \sum_{i = 1}^{g} f_{i} + v

where M is the DOF of the mechanism, d is the order of mechanism, d = 6 − λ, λ is the number of common constraints on the mechanism, n is the total number of links of the mechanism, g is the total number of the joints of the mechanism, f_i is the number of degrees of freedom of the ith joint, and v is the number of redundant constraints of the mechanism.

To get the number of redundant constraints v, the reciprocal screw of $$_{1}$ and $$_{2}$ can be written as

{\begin{array}{l} $_{11}^{r} = {[- \frac{1}{\sqrt{3}} 1 0; 0 0 0]}^{T} \\ $_{12}^{r} = {[0 0 0; \sqrt{3} 1 0]}^{T} \\ $_{13}^{r} = {[0 0 1; 0 0 0]}^{T} \\ $_{14}^{r} = {[\frac{- x_{A 3} + \sqrt{3} y_{A 3}}{\sqrt{3}} 0 0; - \frac{x_{A 3} - x_{D 3} - \sqrt{3} y_{A 3} + \sqrt{3} y_{D 3}}{z_{D 3}} 0 1]}^{T} \end{array}

Then, the reciprocal screws of $$_{3}$ and $$_{4}$ can be written in the same manner

{\begin{array}{l} $_{21}^{r} = {[- \frac{1}{\sqrt{3}} 1 0; 0 0 0]}^{T} \\ $_{22}^{r} = {[0 0 0; \sqrt{3} 1 0]}^{T} \\ $_{23}^{r} = {[0 0 1; 0 0 0]}^{T} \\ $_{24}^{r} = {[\frac{- x_{E 3} + \sqrt{3} y_{E 3}}{\sqrt{3}} 0 0; - \frac{x_{B 3} - x_{E 3} - \sqrt{3} y_{B 3} + \sqrt{3} y_{E 3}}{z_{D 3}} 0 1]}^{T} \end{array}

Comparing equation (6) with equation (7), three groups of reciprocal screws are the same ( $$_{11}^{r}$ and $$_{21}^{r}$ , $$_{12}^{r}$ and $$_{22}^{r}$ , $$_{13}^{r}$ and $$_{23}^{r}$ ). It means there are three redundant constraints that exist in limb 1 consequently. According to the symmetrical structure of the mechanism, there are three redundant constraints in limb 2 and limb 3 respectively. So, the total redundant constraint number leads to v = 9.

The DOF of the mechanism can be calculated by equation (5)

M = 6 (17 - 21 - 1) + 27 + 9 = 6

Therefore, the calculation results of DOF and RDOF demonstrate the mechanism is a kinematically redundant mechanism. In this article, E_i is selected as the redundant actuation.

Kinematic model

Referring to Figure 2, the limb architecture of the redundant mechanism contains an actuation unit (five-bar linkage) and a strut link (l _s). The design purpose of the redundant mechanism is to furthest reducing the passive joints number to simplify the calculation of kinematic problem. It is obvious to observe that the actuation unit of each limb contains mostly all passive joints.

Therefore, to simplify symbolic expressions involved the active and passive variables, the positions of C_i are selected as output of actuation units and expressed by active variables θ _1i, θ _5i. Where θ_ji is the joint θ_j in the ith limb, θ _3i is the function involved active variables θ _1i, θ _5i and the geometric link parameters. The position of C_i is noted by a vector c _i as mentioned in the “DOF and RDOF of 3-RRRRRS mechanism” section. In the five-bar linkage (A ₁ B ₁ C ₁ D ₁ E ₁) of limb 1, the constraint equations of C ₁ can be written as

{(b_{1} - c_{1})}^{T} (b_{1} - c_{1}) = l_{4}^{2}

{(d_{1} - c_{1})}^{T} (d_{1} - c_{1}) = l_{3}^{2}

where $b_{1} = {[x_{A 1} - l_{5} cos θ_{1}, 0, l_{5} sin θ_{1}]}^{T}$ , $d_{1} = {[x_{E 1} - l_{2} cos θ_{5}, 0, l_{2} sin θ_{5}]}^{T}$ .

By solving equations (9) and (10), the position of C ₁ can be obtained. Here, the result is two sets of solutions for C ₁, which corresponds to the two locations of C ₁ as shown in Figure 2. The solution corresponding to the protruding C ₁ is chosen.

The coordinate of point S ₁ can be expressed by the active variables θ ₁₁, θ ₅₁ and passive variable θ ₃₁

s_{1} = (\begin{matrix} x_{C_{1}} + l_{s} cos θ_{31} \\ 0 \\ z_{C_{1}} + l_{s} sin θ_{31} \end{matrix})

where s _i is the function involved active variables θ _1i, θ _5i and the geometric link parameters.

The symmetric architecture makes it easy to obtain the coordinates of points S ₂ and S ₃ in the other two limbs. Assuming that limb 2 and limb 3 are in the x–z plane as the same case as limb 1, coordinates of S ₂ and S ₃ can be obtained through replacement of subscript “1” by “2” in equation (11). Then the actual coordinates of S ₂ are easy to be found by rotating this vector through 120° about the z-axis. The actual coordinates of S ₃ can be obtained in a similar method

s_{2} = R_{Z} (2 π / 3) s_{2 o}

s_{3} = R_{Z} (4 π / 3) s_{3 o}

In equations (12) and (13), R _Z (*) represents the rotation matrix about z-axis, $s_{2 o} = {[x_{C_{2}} + l_{s} cos θ_{32} 0 z_{C_{2}} + l_{s} sin θ_{32}]}^{T}$ , $s_{3 o} = {[x_{C_{3}} + l_{s} cos θ_{33} 0 z_{C_{3}} + l_{s} sin θ_{33}]}^{T}$ . The coordinates of points C ₂ and C ₃ in limb 2 and limb 3 can be obtained by using the same method. If c ₂ and c ₃ represent the coordinates of points C ₂ and C ₃, then we have $c_{2} = R_{Z} (2 π / 3) c_{1}$ and $c_{3} = R_{Z} (4 π / 3) c_{1}$ .

Forward kinematic

Due to the moving platform is designed as a rigid equilateral triangle form, the geometric constraint is that the points S ₁, S ₂, and S ₃ can form an equilateral triangle architecture, which can result in the constraint equations. Hence, the constraint equations can be formulated as

μ_{1} ≜ {(s_{1} - s_{2})}^{T} (s_{1} - s_{2}) - h_{t} ​^{2} = 0

μ_{2} ≜ {(s_{2} - s_{3})}^{T} (s_{2} - s_{3}) - h_{t} ​^{2} = 0

μ_{3} ≜ {(s_{1} - s_{3})}^{T} (s_{1} - s_{3}) - h_{t} ​^{2} = 0

where h_t is the side of the equilateral triangle formed by the points S ₁, S ₂, and S ₃.

Put the above three equations in a vector

μ = {(μ_{1}, μ_{2}, μ_{3})}^{T} = 0

At this stage, it only remains three unknown variables (θ ₃₁, θ ₃₂, and θ ₃₃) to solve in equations (14) to (16). Computations of these variables lead to the derivation of a 16-degree univariate polynomial equation.²³ The method of getting these unknown variables can be seen in Sagi and Bandyopadhyay¹⁸. After calculation of θ _3i, the coordinates of points S ₁, S ₂, and S ₃ can be obtained to find the position and orientation of moving platform.

Inverse kinematic

Kinematically redundant mechanisms have an infinite number of possible solutions of inverse kinematic. Therefore, if there are any, there may be one or some loci of solutions instead of the finite solutions. Figure 2 describes a case about the locus of solutions. The search region in Figure 2 is an assuming area that point C ₁ of five-bar linkage can reach based on the maximum and minimum actuation angles of θ ₁, θ ₅. In the inverse kinematic solution, if the moving platform location is given, then all possible locations of point C ₁ are the points on the circle of radius l _s centered at S ₁. Therefore, all possible solutions of θ ₁ and θ ₅ are the intersection of the aforementioned region and location, which are located on the MC ₁ N arc.

In the inverse kinematic solution, the moving platform Cartesian coordinates are obtained by the vector of the central point p, noted by p . The moving platform orientation is expressed by rotational matrix Q . It stands for the rotation of the moving coordinate frame p-uvw from the fixed coordinate frame O-xyz. To express the orientational capability of the mechanism more conveniently, the Tilt-and-Torsion (T–T) angles convention²⁴ is adopted here. Where α represents the torsion angle, β represents the tilt angle. The rotation matrix for T–T angle convention can be written as

\begin{array}{l} Q =^{TT} R ​_{p}^{O} (α, β, 0) = R_{z} (α) R_{y} (β) R_{z} (- α) R_{z} (0) \\ = [\begin{matrix} c^{2} α c β + s^{2} α & s α c α (c β - 1) & c α s β \\ s α c α (c β - 1) & s^{2} α c β + c^{2} α & s α s β \\ - c α s β & - s α s β & c β \end{matrix}] \end{array}

Hence, the center position vector of the ith spherical joint, which connects limb i with the moving platform can be obtained as

s_{i} = p + Q s_{i o} = p + v_{i}, i = 1, 2, 3

where s _io represents the position vector of S_i , expressed in the moving coordinate frame p-uvw. s _i is the vector of point S_i , expressed in the fixed coordinate frame.

As shown in Figure 2, the constraint equation in terms of link l _s length connecting points S_i with C_i can be written as

{(s_{i} - c_{i})}^{T} (s_{i} - c_{i}) = l_{s}^{2}

When the orientation and position of the moving platform are given, the point C_i of every limb has a possible locus of location for the inverse kinematic solution as shown in Figure 2. Since each constraint equation contains two unknown variables (θ _1i and θ _5i), there is an infinite solution for θ _1i and θ _5i to generate C_i located on the MC ₁ N arc. If θ _1i and θ _5i are determined as actuated angles, one of the angles θ _1i and θ _5i should be given first.

Jacobian matrix of the 3-RRRRRS mechanism

The output parameters of the mechanism can be noted by $x = {[α, β, z]}^{T}$ , and the input parameter is noted by $q = {[θ_{11}, θ_{51}, θ_{12}, θ_{52}, θ_{13}, θ_{53}]}^{T}$ . Therefore, the vector s _i includes the output parameters x, and the input parameter q is included in the vector c _i . Referring to Figure 2, equation (20) can be rewritten as

l_{s}^{2} = {(x_{S_{1}} - x_{E_{1}} + l_{2} c θ_{51} + l_{3} c θ_{41})}^{2} + {(z_{S_{1}} - l_{2} s θ_{51} - l_{3} s θ_{41})}^{2}

l_{s}^{2} = {(x_{S_{2}} + 1 / 2 (x_{E_{2}} - l_{2} c θ_{52} + l_{3} c θ_{42}))}^{2} + {(y_{S_{2}} - \sqrt{3} / 2 (x_{E_{2}} - l_{2} c θ_{52} + l_{3} c θ_{42}))}^{2} + {(z_{S_{2}} - l_{2} s θ_{52} - l_{3} s θ_{42})}^{2}

l_{s}^{2} = {(x_{S_{3}} + 1 / 2 (x_{E_{3}} - l_{2} c θ_{53} + l_{3} c θ_{43}))}^{2} + {(y_{S_{3}} + \sqrt{3} / 2 (x_{E_{3}} - l_{2} c θ_{53} + l_{3} c θ_{43}))}^{2} + {(z_{S_{3}} - l_{2} s θ_{53} - l_{3} s θ_{43})}^{2}

where θ _4i is the function of the two actuation angles (θ _1i and θ _5i) and link parameters of the five five-bar linkages. The symbols x_Si , y_Si , and z_Si are the coordinates of sphere joints S_i , which include the output parameters of the moving platform.

Assuming $\dot{x} = {[\dot{α}, \dot{β}, \dot{z}]}^{T}$ and $\dot{q} = {[{\dot{θ}}_{11}, {\dot{θ}}_{51}, {\dot{θ}}_{12}, {\dot{θ}}_{52}, {\dot{θ}}_{13}, {\dot{θ}}_{53}]}^{T}$ as the vectors of the moving platform Cartesian coordinates and active joints coordinates, respectively, the differentiations of equation (21) to equation (23) with respect to time can be deduced and written in the form of $J_{x} \dot{x} = J_{q} \dot{q}$ .

The direct and inverse matrices ( J _x , J _q ) can be obtained as

J_{x} = {[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]}_{3 \times 3}

J_{q} = {[\begin{matrix} u_{1} & v_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & u_{2} & v_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & u_{3} & v_{3} \end{matrix}]}_{3 \times 6}

Hence, the overall Jacobian matrix J can be written as

J = J_{x}^{- 1} J_{q}

where the a_ij , u_i , and v_i are the functions of α, β, z, θ _1i, θ _5i and link parameters, i, j=1, 2, 3.

Singularity analysis

Currently, three types of singularity of closed-loop kinematic chains are presented based on the inverse Jacobian matrix ( J _q ) and direct Jacobian matrix ( J _x ).²⁵ The conditions for singularity are (1) | J _x | = 0, direct kinematic singularity occurs. (2) | J _q | = 0, inverse kinematic singularity occurs. (3) | J _x | = 0 and | J _q | = 0, combined singularity occurs. Hence, the Jacobian matrices of the 3-RRRRRS can be deduced according to the kinematic model. Due to the complicated expressions of the Jacobian matrix elements, however, it is difficult to obtain the analytical structure parameters when mechanism in singularities. In this section, a method of virtual platform construction is proposed. The mechanism can be divided into two parts, the actuation unit and a virtual PM.

In the “DOF and RDOF of 3-RRRRRS mechanism” section, it is shown that the translation perpendicular to plane A_iB_iC_iD_iE_i is restrained by the five-bar linkage in each limb. That is, the points C_i and S_i are always located in the plane A_iB_iC_iD_iE_i . Here, we assume that the initial position of the mechanism is the configuration when moving platform is parallel to fixed platform. Because of the symmetrical mechanism with identical geometric parameters, an equilateral triangle VP can be formed through points C ₁ C ₂ C ₃ as shown in Figure 3. Figure 4 is the initial position of parallel structure formed by links C_iS_i and moving platform.

Figure 3.

The illustration of virtual platform formed by C ₁ C ₂ C ₃.

Figure 4.

The configuration of virtual parallel mechanism in the initial position.

In Figure 4, the symbol r is defined as circumradius of the equilateral triangle C ₁ C ₂ C ₃ in initial position. It is obvious that locations of C_i will be changeable with the motion of five-bar linkage actuated by θ _1i and θ _5i. The new locations of C_i are noted by ${C^{'}}_{i}$ . Therefore, a new triangle plane will be formed by the points ${C^{'}}_{1}, {C^{'}}_{2}$ , and ${C^{'}}_{3}$ during the motions of points C_i . As shown in Figure 5, the purple triangle plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ is an arbitrary configuration in the moving process of mechanism. The new triangle plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ cannot always be equilateral. However, the centroid of the new triangle noted by $P^{'}$ can be calculated by the points ${C^{'}}_{i}$ . Then, a circle can be plotted in the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ in which radius is r and the center is the point $P^{'}$ . As a result, the circle will intersect with $P^{'} {C^{'}}_{i}$ at the points noted by C_i as shown in Figure 6. The symbol d_i is equal to subtraction of ${C^{'}}_{i}$ and C_i .

Figure 5.

The illustration of initial configuration and the arbitrary configuration in the moving process.

Figure 6.

Top view of the virtual plane $C_{1}^{'} C_{2}^{'} C_{3}^{'}$ .

As r is the circumradius of the triangle C ₁ C ₂ C ₃ when moving platform is in the initial position, d_i can be regarded as the distance produced by the initial point C_i that moves to point ${C^{'}}_{i}$ in the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ . The initial position of ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ can be determined by plotting a circle with radius r in the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ . If we define a virtual PM which is formed by the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ , strut links C_iS_i , and moving platform, the distant of d_i can be regarded as the distant of actuation P of the virtual PM. The VP ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ can be treated as fixed platform of the virtual mechanism. Thus, the link C_iS_i , platform S ₁ S ₂ S ₃ and the VP can be regarded as a virtual PM 3-PRS.

Consequently, the initial mechanism 3-RRRRRS is separated into actuation units (five-bar linkage under the VP) and virtual 3-PRS mechanism (upon the VP). Hence, the singularity types of 3-RRRRRS mechanism can be concluded as follows: (1) singularity occurs in actuation units; (2) singularity occurs in virtual 3-PRS mechanism; and (3) combined singularity when both of actuation units and virtual 3-PRS mechanism in singularity configurations.

Singularity analysis of the actuation units

The actuation units are composed by five-bar linkage A_iB_iC_iD_iE_i (I = 1, 2, 3) of three limbs. Whatever the mechanism configuration is, the output points C_i of limbs can always construct a plane C ₁ C ₂ C ₃. The plane C ₁ C ₂ C ₃ is named as virtual platform. Generally, singularity occurs in any of the possible conditions: (1) singularity occurs in one or more limbs of five-bar linkage; (2) singularity occurs in PM constructed by virtual platform; and (3) singularity occurs when both conditions (1) and (2) are satisfied.

The singularity configurations of plane five-bar linkage can be obtained by Jacobian matrices. As shown in Figure 2, two constraint equations can be obtained to solve the kinematic problem

{\begin{matrix} Δ_{1} : l_{5} cos θ_{1} + l_{4} cos θ_{2} - l_{2} cos θ_{5} - l_{3} cos θ_{4} - l_{1} = 0 \\ Δ_{2} : l_{5} sin θ_{1} + l_{4} sin θ_{2} - l_{1} sin θ_{5} - l_{3} sin θ_{4} = 0 \end{matrix}

The Jacobian matrix J can be obtained by the differentiation of equation (27) with respect to time and expressed in the form of equation (28)

J = [\begin{matrix} - l_{2} sin θ_{5} - \frac{l_{2} sin θ_{4} A_{1}}{A_{4}} & - \frac{l_{5} A_{2}}{A_{4}} \\ l_{2} cos θ_{5} + \frac{l_{2} cos θ_{4} A_{1}}{A_{5}} & \frac{l_{5} A_{3}}{A_{5}} \end{matrix}]

where $A_{1} = sin (θ_{2} - θ_{5})$ , $A_{2} = sin (θ_{1} - θ_{2})$ , $A_{3} = cos (θ_{1} - θ_{2})$ , $A_{4} = sin (θ_{4} - θ_{2})$ , $A_{5} = cos (θ_{4} - θ_{2})$ .

The determinant of J can be written as equation (29)

Det (J) = \frac{l_{2} l_{5} sin (θ_{4} - θ_{5}) sin (θ_{1} - θ_{2})}{sin (θ_{4} - θ_{2})}

When singularity occurs in five-bar linkage, the condition is Det( J ) = 0 or Det( J ) = ∞. If Det( J ) = 0, the relations between joint angles can be written as $θ_{4} = θ_{5}$ or $θ_{4} = θ_{5} + π$ ; $θ_{1} = θ_{2}$ or $θ_{1} = θ_{2} + π$ . If Det( J ) = ∞, the relation is $θ_{4} = θ_{2}$ or $θ_{4} = θ_{2} - π$ . Figure 7 shows singularity configurations that the joint angles of five-bar mechanism are satisfied in different conditions. Parallel lines in red represent two links are stretched-out or folded-back.

Figure 7.

Singularity configurations in different conditions.

It is easy to obtain that there are only nine singularity configurations of five-bar linkage in each limb as shown in Figure 7. Figure 8 is the topological diagram of the proposed mechanism. S_i represents the strut link, and F_i stands for five-bar linkage in each limb. The red area is the moving platform. In Figure 8, C_i stands for case i of singularity configuration and 0 stands for non-singularity configuration. These nine singularity configuration cases are noted by C_i in each limb as shown in Table 1. If the five-bar linkage of each limb meets singularity, it means inverse singularity or boundary singularity occurs in the 3-RRRRRS mechanism.

Figure 8.

Topological diagram of the proposed 3-RRRRRS mechanism.

Table 1.

Singularity configurations in each limb.

Limb number	Singularity configurations in different cases (C_i , i = 1, 2…, 9)
L ₁	C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ C ₈ C ₉
L ₂	C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ C ₈ C ₉
L ₃	C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ C ₈ C ₉

Considering there are nine singularity configuration cases in each limb, the singularity types of three limbs can be concluded in Table 2. As shown in Table 2, singularity type is the number of limbs that is in singularity configurations. The combinations when occurring singularity cases can be calculated through combinations of these nine cases among three limbs. If only one limb and no matter which one occurs singularity, it belongs to singularity type 1 case in combinations. Hence, the combination number in three singularity types can be written in Table 2 by means of permutations and combinations. Singularity types 1, 2, and 3 in case 3 configuration are illustrated in Figure 9.

Table 2.

The types of singularity configurations.

Singularity Type	Configurations in different cases (C_i ) Number	Combination Number
1	C_i 0 0 (i = 1, 2…, 9) 0 C_j 0 (j = 1, 2…, 9) 0 0 C_k (k = 1, 2…, 9)	9
2	C_i C_j 0 (i = 1, 2…, 9) C_i 0 C_k (j = 1, 2…, 9) 0 C_j C_k (k = 1, 2…, 9)	45
3	(i = 1, 2…, 9) C_i C_j C_k (j = 1, 2…, 9) (k = 1, 2…, 9)	129

Figure 9.

Singularity types 1, 2, and 3 in case 3 configuration.

Singularity of the virtual PM

The other one of the singularities of 3-RRRRRS mechanism occurs when the virtual PM 3-PRS meets a singular configuration. Since the active revolute joints are added to 3-RPS, it is still possible to occur singularity. As mentioned in the “Singularity analysis” section, the links C_iS_i , platform S ₁ S ₂ S ₃, and VP C ₁ C ₂ C ₃ can be regarded as a virtual PM 3-PRS. The actuation distance of P in the virtual PM 3-PRS is noted by d_i , which d_i is equal to the distance that the initial point C_i moves to point ${C^{'}}_{i}$ in the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ . Assuming the coordinate frame p′-u′v′w′ is set up in the plane ${C^{'}}_{1} {C^{'}}_{2} {C^{'}}_{3}$ as shown in Figure 6, the coordinate frame p′-u′v′w′ corresponding to O-xyz can be obtained through the three points ${C^{'}}_{i}$ . The coordinate frame corresponding to the coordinates of ${C^{'}}_{i}$ in p′-u′v′w′ can be expressed as

{C^{'}}_{1} = {[d_{1} + r, 0, 0]}^{T}, {C^{'}}_{2} = {[- 1 / 2 (d_{2} + r), - \sqrt{3} / 2 (d_{2} + r), 0]}^{T}, {C^{'}}_{3} = {[- 1 / 2 (d_{3} + r), - \sqrt{3} / 2 (d_{3} + r), 0]}^{T}

In the virtual PM 3-PRS mechanism, the constraint equation in terms of link l _s length connecting points S_i with ${C^{'}}_{i}$ can be written as

{(s_{i} - {c^{'}}_{i})}^{T} (s_{i} - {c^{'}}_{i}) = l_{s}^{2}

where ${c^{'}}_{i}$ is a function in terms of d_i and s _i include the output parameters α, β, z.

Using the same method as in the “Jacobian matrix of the 3-RRRRRS mechanism” section, we can obtain the Jacobian matrix of the virtual PM 3-PRS by taking the partial derivative of equation (31) with respect to time on both sides. The result can be arranged in matrix form as $J_{x}^{3 - p r s} \dot{x} = J_{q}^{3 - p r s} \dot{q}$

[\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}] [\begin{matrix} \dot{α} \\ \dot{β} \\ \dot{z} \end{matrix}] = [\begin{matrix} c_{1} & 0 & 0 \\ 0 & c_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix}] [\begin{matrix} {\dot{d}}_{1} \\ {\dot{d}}_{2} \\ {\dot{d}}_{3} \end{matrix}]

where the b_ij and c_i are the functions of α, β, z, d_i and link parameters, i, j = 1, 2, 3.

When the determinant of $J_{x}^{3 - p r s}$ or $J_{q}^{3 - p r s}$ goes to zero, the singularity of the virtual PM 3-PRS will occur. A combined singularity of the virtual PM 3-PRS occurs when the determinant of $J_{x}^{3 - p r s}$ or $J_{q}^{3 - p r s}$ are both zero. The singular characteristic of 3-PRS similar to 3-RPS has been completely researched.^26,27 The typical configuration of this singularity is the main factor to limit the orientation workspace. It occurs when all links C_iS_i are coplanar, any of the links C_iS_i (i = 1, 2, 3) are coplanar with the platform S ₁ S ₂ S ₃. The cases of the virtual PM singularity configuration are illustrated in Figure 10. Parallel lines in blue represent any of the links C_iS_i are coplanar with platform S ₁ S ₂ S ₃ or links C_iS_i are coplanar. Here, we take the singularity that all links C_iS_i are coplanar as an example to illustrate the singularity analysis. The configuration that all links C_iS_i are coplanar is the third configuration in Figure 10. In this configuration, the platform S ₁ S ₂ S ₃ is coplanar with VP C ₁ C ₂ C ₃, which leads to α, β, z equal to zero. If α, β, and z are equal to zero simultaneously, the determinant of $J_{x}^{3 - p r s}$ and $J_{q}^{3 - p r s}$ will go to zero. The configuration that α, β, and z are equal to zero in the virtual PM 3-PRS corresponding to the configuration that α and β are equal to zero and the platform S ₁ S ₂ S ₃ is coplanar with VP C ₁ C ₂ C ₃ in the whole mechanism. By substituting the values of α, β, z corresponding to this configuration into the Jacobian matrices of the whole mechanism in the “Jacobian matrix of the 3-RRRRRS mechanism” section, the determinant of J _x goes to zero. Therefore, the singularity of 3-RRRRRS mechanism will occur when the singularity of the virtual 3-PRS PM occurs.

Figure 10.

Configurations of the singularity caused by the virtual PM.

Combined singularity of the 3-RRRRRS5

As previously described, the combined singularity occurs when both the five-bar linkage and the virtual 3-PRS mechanism are in singularity configurations. That is, if the determinant of $J_{x}^{3 - p r s}$ or $J_{q}^{3 - p r s}$ goes to zero and simultaneously the determinant of J in five-bar linkage goes to zero, then combined singularity occurs in the mechanism. Figure 11 illustrates some special configuration cases when the mechanism is in combined singularity configurations.

Figure 11.

Combined singularity configurations of the 3-RRRRRS.

However, adding redundant actuation joints in the other two limbs can help to avoid singularity configurations by changing the direction of link C_iS_i . As a result, this would be prevented that any of links C_iS_i are parallel to the platform S ₁ S ₂ S ₃ or all the links C_iS_i intersect at a common plane S ₁ S ₂ S ₃. The configurations of the third case are shown in Figure 11. Reminding the initial purpose, one of the purposes is adding these redundant revolute joints to avoid singularities. The avoidance of the typical singularity can be realized by using the five-bar linkage to control the direction of link C_iS_i . The controlling strategy is not developed in this article, and future work will focus on the control strategies. Another purpose is workspace will be increased by adding the redundant joints. The workspace analysis of 3-RRRRRS will be described in the next section.

Performance analysis

Workspace analysis

In the traditional PM, there are many factors to limit the region of workspace. Whereas the main factor is the singularity configurations occurring, especially the configuration that link C_iS_i is parallel to the moving platform. The purpose of adding redundancy is to adjust the direction of the link C_iS_i through redundancy actuation to avoid singularity, so that the moving platform of the mechanism can generate large rotational angle motions. As shown in the “Singularity of the virtual PM” section, all Cartesian positions and poses of the moving platform can be met with one non-singularity configuration at least. Such as link C_iS_i is not coplanar with plane S ₁ S ₂ S ₃ of the proposed mechanism. In this section, the workspace is used to demonstrate that the redundancy can reduce the limitation of motion ability. For the purpose of providing a visualized workspace of the mechanism, the necessary evaluation of the orientational workspace is more effective than the evaluation of the translational workspace. The orientational workspace is mapped by the ranges of orientational angles (α, β) with varied moving platform position (z). The procedure was completed in this section. For a given position z incremented by steps of one unit (mm), the angle (β) will be a traversal of 360° when incremented by a step size of 1°. Then, to every values of angle β, the moving platform is pitched by torsion angle (α) incremented by a step size of 1° When actuator limits or singularity configuration is determined, then incrementations of z and α will stop. In this section, dimension parameters of this mechanism and actuator limits are presented in Table 3. The orientational workspace is mapped as shown in Figure 12 with the regions of α, β, and z. In this section, passive joint limits and limbs interfere with the moving platform are not considered. Thus, the obtained result can be regarded as an upper boundary of the workspace.

Table 3.

Geometric parameters of the mechanism (all lengths in mm).

Parameters	Symbols	Values
Crank length (θ _1i)	l ₅	200
Crank length (θ _5i)	l ₂	200
Coupler length (B_iC_i )	l ₃	300
Coupler length (C_iD_i )	l ₄	300
Strut length	l _s	200
Platform side	h_t	100
Actuator limit	θ _1i	[30, 150]
Actuator limit	θ _5i	[30, 150]
Crank base position (A_i )	[l+l ₁, 0, 0]^T	[450, 0, 0]^T
Crank base position (E_i )	[l ₁, 0, 0]^T	[250, 0, 0]^T

Figure 12.

Orientational workspace of the architecture of Table 3.

The results are shown in Figures 12 to 14. Cylindrical coordinates system is used in this demonstration. In Figure 12, the z-axis measures the platform position (z), and the color varied with the z scale that goes from −60 mm to 60 mm. Figure 12 shows the workspace is projected to the z = 0 mm plane. Here, the hollow dots are invalid region that they don’t satisfy the limits in Table 3, and the red dots are valid region. As shown in Figure 12, the yellow line represents the minimum boundary of the valid region, where α = 40°. In Figure 14, the angle (α) is measured by the radius direction, and the angle (β) is measured by the peripheral direction. It can be observed that the maximum pitched angle (α) can reach 90° in a certain direction, while the pitched angle will yet reach about 50° in the worst direction. Therefore, a large orientational workspace is obtained with the proposed structure, even the combination of rotations exists. Figure 15 shows two special cross-section view of the workspace, the orientation workspace is symmetric with respect to A–A cross section. The results are consistent with the structural symmetry of the mechanism.

Figure 13.

Projected orientational workspace.

Figure 14.

Top view of the orientational workspace.

Figure 15.

Special cross-section view of the orientational workspace: (a) A–A cross-section view and (b) B–B cross-section view.

Dexterity analysis

In this section, the condition number noted by κ is chosen as the performance index to demonstrate the mechanism can stay away from singularities. Thus, the value of κ can be obtained through the matrix J in equation (26). If the condition number is infinite in the measurement range, it means the mechanism will meet singularity configuration. κ can be expressed by the spectrum norm of the matrix

κ = \frac{σ_{max}}{σ_{min}}

where $σ_{max}$ and $σ_{min}$ are the maximum and minimum eigenvalue of the matrix J , respectively. $σ_{max}$ and $σ_{min}$ are equal to the norm of J and the reciprocal of the norm of $J_{}^{- 1}$ .

Figure 16 shows the performance index κ varies with orientational angles α and β. Figure 16 shows the evaluation of index κ varies with α under the conditions of β = 0, P_i /6, and P_i /3. Figure 16 shows the evaluation of index κ varies with β under the conditions of α = 0, P_i /6, and P_i /3. The vertical axis represents variation of index κ. The abscissa axis represents variations of α and β.

Figure 16.

Evaluation of κ varies with orientational angles α and β: (a) the relationship between κ and α and (b) the relationship between κ and β.

It can be observed that the performance index never exceeds 8 under different conditions of β in Figure 16. Referring to Figure 16, the performance index with α varied in different values of β is slightly lower than that of the former case. All the condition numbers are a constant less than about 8, and the values near the center are close to 1 in the evaluation. It should be noted that the interference of limbs and moving platform is not considered in this case. This cause the large angles range in abscissa axis. Nevertheless, the results show condition numbers are constant, not infinity, namely the mechanism can clearly stay away from singularity.

Trajectory simulation

A circle trajectory is performed by simultaneously giving the moving platform a distance from 0 to x _max along the u-axis and another distance from 0 to y _max along the v-axis, which can be seen in Figure 17. The composite motion of two direction displacements can make the center point p of the moving platform perform a circle trajectory.²⁸ To demonstrate, the mechanism can produce the large rotational angles without occurring singularities, the range from 0 to x _max and y _max should correspond to or contain large angle values of α and β. In addition, the obtained variation curves of actuators should be continuous because the actuators have no values when the mechanism is in singularity configuration.

Figure 17.

Trajectory simulation: (a) 3-D model of trajectory simulation, (b) top view of trajectory simulation, and (c) geometry trajectory of the point p.

The relationship between the positions (x, y) and the rotational angles (α and β) can be obtained by the structural characteristics of the mechanism. Referring to Figure 17, three limbs maintain an angular separation of 120° symmetry in the architecture of the mechanism, which are caused by the equilateral triangle structure of the moving platform.

The five-bar linkage restrains the joints S_i will always be located in the plane A_iB_iC_iD_iE_i . Hence, the relationship between coordinates (x, y) of S_i maintains the expressions $y = 0$ , $y = \sqrt{3} x$ , and $y = - \sqrt{3} x$ , respectively, which can be rewritten as

s_{1} (2) = 0, s_{2} (2) = \sqrt{3} s_{2} (1), s_{3} (2) = - \sqrt{3} s_{3} (1)

Here, s _i (j) denotes the jth row of the position vector s _i in the ith limb. The values of s _i can be obtained by equations (11) to (13). Based on equation (34), the relations of the moving platform positions (x, y) and the rotational angles α and β can be deduced as

{\begin{matrix} x = r (cos 2 α) (cos β - 1) / 2 \\ y = r (sin 2 α) (1 - cos β) / 2 \end{matrix}

Hence, according to equation (35), when we determine the value of x and y, then the value of α and β will be obtained. If α and β are in the range of values in the workspace, then the given values of x and y are feasible. Here, we give the value of x _max and y _max is 10, so that the value range of x and y can demonstrate the moving platform will produce a larger angle when it performs the circle trajectory. In addition, to get a valid set of solutions from an infinite number of inverse kinematics, the given constraint condition is linked to C_iS_i which is not coplanar with S ₁ S ₂ S ₃ platform in the simulation.

Figure 18 shows the variation curves of actuators (θ _1i and θ _5i) in the process of moving platform performs the circle trajectory, where the initial position is defined as the fixed platform is parallel to the moving platform. In Figure 18, “theta i” represents “θ _1i” and “elta i” represents “θ _5i”, i = 1, 2, 3. It can be observed that the cosinoidal and sinusoidal displacements of moving platform in the x-axis direction and the y-axis direction are varying from 0 mm to 10.0 mm, which can produce a circle trajectory. The variation is measured by the right vertical axis. According to equation (35), the range of displacement values from 0 to 10 can cause a large range of rotational angles. It can also be obtained that the variation curves of actuator angles (θ_i and θ _5i) are continuous, which demonstrate the mechanism can accomplish the circle trajectory motion without occurring singularities.

Figure 18.

The actuator curves under the moving platform performs a circle trajectory.

Conclusions

To improve the small workspace of traditional PM, a novel kinematically redundant PM with symmetric limbs is proposed, which can maintain the characteristics of weight distribution before and after adding redundant actuation. The Jacobian matrix of the mechanism is derived using the constraint derivation and further simplified by proposing a VP method. The method can separate the whole mechanism into two sub-mechanisms and is confirmed by the resultant Jacobian matrices used for the singularity analysis. Meanwhile, the orientational workspace and the condition number as performance index are analyzed, the results demonstrate that the designed mechanism can efficiently stay away from singularities in large-range rotation. A trajectory example is carried out to verify the large rotation ability of the proposed mechanism. The proposed method can provide a reference for the design of redundant PM. In the field of machine tool processing and other areas which require large angle output, the mechanism has certain application prospects.

Footnotes

Acknowledgment

The authors would like to thank the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. 51875033) and the Fundamental Research Funds for the Central Universities (grant no. 2018YJS140).

ORCID iD

Fuqun Zhao

References

Zhang

Wang

. New indices for optimal design of redundantly actuated parallel manipulators. J Mech Robot 2017; 9: 1–15.

Ebrahimi

Carretero

Boudreau

. 3-PRRR redundant planar parallel manipulator: inverse displacement, workspace and singularity analyses. Mech Mach Theory 2007; 42(8): 1007–1016.

Wang

Gosselin

. Kinematic analysis and design of kinematically redundant parallel mechanisms. J Mech Des 2004; 126(1): 109–118.

Lee

Kim

. Kinematic analysis of generalized parallel manipulator systems. In: IEEE Conference on Decision and Control 2, San Antonio, TX, USA, 18–22 December 1993, pp. 1097–1102. Piscataway, NJ: IEEE.

Cheng

Yiu

. Dynamics and control of redundantly actuated parallel manipulators. IEEE/ASME Trans Mech 2004; 8(4): 483–491.

Dasgupta

Mruthyunjaya

. Force redundancy in parallel manipulators: theoretical and practical issues. Mech Mach Theory 1998; 33(6): 727–742.

Firmani

Podhorodeski

. Force-unconstrained poses for a redundantly-actuated planar parallel manipulator. Mech Mach Theory 2004; 39(5): 459–476.

Firmani

Podhorodeski

. Force-unconstrained poses for parallel manipulators with redundant actuated branches. Trans Can Soc Mech Eng 2005; 29(3): 343–356.

Nokleby

Firmani

Zibil

, et al. An analysis of the force-moment capabilities of branch-redundant planar-parallel manipulators. In: ASME 2007 international design engineering technical conferences and computers and information in engineering conference, Las Vegas, Nevada, USA, 4–7 September 2007, pp. 1013–1020.

10.

Taghirad

Nahon

. Jacobian analysis of a macro-micro parallel manipulator. In: Proceeding of IEEE/ASME international conference on advanced intelligent mechatronics, Zurich, Switzerland, 4–7 September 2007, pp. 1–6. Piscataway, NJ: IEEE.

11.

Zhang

, et al. Mobility, kinematic analysis, and dimensional optimization of new three-degrees-of-freedom parallel manipulator with actuation redundancy. J Mech Robot 2017; 9(4): 041008.

12.

Marquet

Krut

Company

, et al. ARCHI: a new redundant parallel mechanism—Modeling, control and first results. In: Proceeding of IEEE/RSJ international conference on intelligent robots and system. Maui, HI, USA, 29 October–3 November 2001, pp. 183–188. Piscataway, NJ: IEEE.

13.

Marquet

Company

Krut

, et al. Control of a 3-dof over-actuated parallel mechanism. In Proceeding of ASME design English technology conference. Montreal, Quebec, Canada, 29 September–2 October 2002, pp. 1185–1191.

14.

Han

. Kinematically redundant parallel haptic device with large workspace. Int J Adv Robot Syst 2012; 9: 1–9.

15.

Cha

Lasky

Velinsky

. Kinematically-redundant variations of the 3-RRR mechanism and local optimization-based singularity avoidance. Mech Based Des Struc Mach 2007; 35(1): 15–38.

16.

Gosselin

Laliberte

Veillette

. Singularity-free kinematically redundant planar parallel mechanisms with unlimited rotational capability. IEEE Trans Robot 2015; 31(2): 457–467.

17.

Gosselin

Schreiber

. Kinematically redundant spatial parallel mechanisms for singularity avoidance and large orientational workspace. IEEE Trans Robot 2016; 32(2): 286–300.

18.

Sagi

Bandyopadhyay

. Comparison of hybrid and parallel architectures for two-degrees-of-freedom planar robot legs. In: The Third international and 18th national conference on machines and mechanisms, Mumbai, India, December 2017, pp. 83–93.

19.

Srivatsan

Bandyopadhyay

. On the position kinematic analysis of MaPaMan: a reconfigurable three-degrees-of-freedom spatial parallel manipulator. Mech Mach Theory 2013; 62(4): 150–165.

20.

Hunt

. Kinematic geometry of mechanisms. Oxford: Clarendon Press, 1978.

21.

Lee

Shah

. Kinematic analysis of a three degrees of freedom in-parallel actuated manipulator. In: Proceedings of IEEE international conference on robotics and automation, Raleigh, NC, USA, 31 March–3 April 1987, pp. 345–350.

22.

Guo

. Kinematics analysis of a novel planar parallel manipulator with kinematic redundancy. J Mech Sci Technol 2017; 31(4): 1927–1935.

23.

Kumar

Bandyopadhyay

. Forward kinematics of the 3-RPRS parallel manipulator using a geometric approach. In: The Third international and 18th national conference on machines and mechanisms, Mumbai, India, December 2017, pp. 61–69.

24.

Wang

Guan

, et al. A novel 3-PUU parallel mechanism and its kinematic issues. Robot Cim-Int Manuf 2016; 42: 86–102.

25.

Gosselin

Angeles

. Singularity analysis of closed-loop kinematic chains. IEEE Trans Robot Autom 1990; 6(3): 281–290.

26.

Basu

Ghosal

. Singularity analysis of platform-type multi-loop spatial mechanisms. Mech Mach Theory 1997; 32(3): 375–389.

27.

Joshi

Tsai

. Jacobian analysis of limited-DOF parallel manipulators. J Mech Des 2002; 124(2): 254–258.

28.

Bozek

Pokorny

Svetlík

, et al. The calculations of Jordan curves trajectory of the robot movement. Int J Adv Robot Syst 2016; 13(5): 1–7.