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Abstract

An analytical elastostatic stiffness modeling approach for kinematically redundant parallel mechanisms (KR-PMs) according to screw theory and the second theorem of Castigliano is proposed. The 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs are used as two examples of KR-PMs to validate the precision of the proposed approach. The relative error is less than 1.55% when the theoretical model is compared to the FEA model. The PM’s stiffness performance is assessed using the minimum linear displacement stiffness performance index, which is based on the theoretical model. By integrating Particle Swarm Optimization (PSO) with polynomial fitting technology, an analytical mapping model of the response surface between the optimal actuating distances and the moving platform (MP)’s given position and orientation is established. It is demonstrated that 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs can significantly enhance the stiffness performance when compared to the stiffness performance of 2-UPR/RPU PM.

Keywords

Parallel mechanism kinematically redundant elastostatic stiffness stiffness performance

Introduction

Compared with serial mechanisms, PMs have obviously advantages such as rapid speed, high rigidity, big bearing capacity and excellent dynamic performances. PMs have recently become widely used in industrial application, such as parts sorting, motion simulation, medical surgical robots, and complex parts processing.^1–4 However, PMs may encounter problems such as small workspaces and singularities in industrial applications.^5–7 Therefore, it is an important issue to increase the workspaces and avoid singularities with redundancy.

The redundancy of PMs is mainly divided into two types: kinematic redundancy and actuation redundancy. Kinematic redundancy means that the overall DOF of the mechanism is greater than the number of independent motions of the moving platform. Actuation redundancy means that the number of independent actuators of the mechanism is greater than the overall DOF of the mechanism. To increase workspaces and avoid singularities, researchers have proposed some configurations of KR-PMs. In 2004, Wang and Gosselin⁸ designed three new types of KR-PMs: planar 4-DOF mechanism, spherical parallel mechanism and spatial 7-DOF Stewart platform. Ebrahimi et al.^9–11 proposed a 3-PRRR planar KR-PM according to 3-RRR PM. After that, Gosselin and Schreiber^12–14 proposed a variety of KR-PMs, and the kinematic modeling, workspace and singularity have been analyzed. The results show that the KR-PMs have significant advantages of avoiding singularities, increasing workspaces and improving the kinematic performances of the mechanisms. Isaksson¹⁵ proposed three types of kinematically redundant planar PMs, which can avoid singularities effectively by selecting the appropriate actuators. In 2017, Qu and Guo¹⁶ proposed a new kinematically redundant planar PM.

The elastostatic stiffness performance is one of the most important performances of PMs. Since the elastostatic stiffness modeling is the basis of stiffness performance analysis, it is necessary to carry out the PM’s elastostatic stiffness modeling during the design stage. The elastostatic stiffness modeling methods of PMs contained these types: finite element method and assume mode method,^17–19 matrix analysis method,^20,21 virtual joint method,^22,23 screw¹ theory method,^24,25 and strain energy method.^26,27 The precision of the FEM is high, but the model needs to be reconstructed and divided with high calculation cost for different configurations, which is inadequate for performance analysis. The method of matrix structure analysis does not require grid division, but it involves high order matrix operations and several deformation compatibility equations, making it improper for analytical elastostatic stiffness modeling. The virtual joint method needs to deal with redundant spring restrictions, which makes the stiffness model more difficult. The screw theory method reduces the dimension of the limb constraint screws system and improves the calculation efficiency. The strain energy method can analyze the PM’s stored strain energy under external wrenches with definite physical meaning. However, fewer scholars research the elastostatic stiffness modeling of KR-PMs. Shin and Lee²⁸ proposed an elastostatic stiffness modeling method of planar KR-PMs. Jamshidifar et al.²⁹ established an elastostatic stiffness model of the kinematically redundant constrained cable parallel robot. Wang et al.³⁰ deduced a KR-PM’s stiffness model and natural frequency using the matrix structure analysis method. Zhang et al.³¹ discussed the stiffness matrix for a 3-DOF planar KR-PM.

To estimate the PM’s stiffness performance, researchers have defined a few stiffness indices, such as eigenvalue index, determinant value index, main diagonal index of stiffness matrix, trace of matrix, virtual work index, etc.^32–35 However, the above stiffness indices except virtual work index are based on algebraic properties with unclear physical meanings, and the dimensions of the stiffness matrix are not uniform, which may lead to interpretation errors. The virtual work index describes the stiffness performance using energy. Although it can avoid the dimensions of the stiffness matrix are not uniform, it is related to the external wrenches of the mechanism.

An analytical elastostatic stiffness modeling approach of KR-PMs according to the Castigliano’s second theorem and screw theory has been put forward. The approach has been validated by 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs. Combining PSO and polynomial fitting technology, the mapping model of the response surface between the optimal actuating distances and the position and orientation of the MP is established. According to the obtained theoretical stiffness model of 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs, the minimum linear displacement stiffness performance index³⁶ is used to estimate the stiffness performance. Compared with the stiffness performance of 2-UPR/RPU PM, it is shown that 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs can significantly enhance the stiffness performance.

Analytical elastostatic stiffness modeling of KR-PMs

Modeling process

Figure 1 depicts the general model of the KR-PMs. The KR-PM contains n limbs connected to the MP. The kinematically redundant limb contains multiple actuating joints, while the non-redundant limb only contains one actuating joint. To facilitate the establishment of analytical elastostatic stiffness model, the fixed coordinate system O-xyz, the moving coordinate system o-uvw and the limb coordinate system B_i-x_iy_iz_i were set as indicated in Figure 1. The external wrenches imposed on the MP can be simplified as a screw W = ( F M )^T acting on the moving coordinate system. To build an analytical elastostatic stiffness model of a KR-PM, this paper considers tension/compression, shear, bending and torsion deformation of the links on the limb. Without taking into account the effects of joint clearance and deformation, gravity and friction, it is assumed that the MP, fixed base, and all joints are rigid.

Figure 1.

The general model of KR-PMs.

The modeling process of the analytical elastostatic stiffness of KR-PMs is as indicated in Figure 2, and the detailed steps are outlined as follows:

Step 1: The i-th limb twist system is established based on the screw theory and the actuating force screw $S_{i}^{T}$ of kinematically redundant limb can be calculated by locking all actuating joints. The constraint wrench system amplitude W _i = ( f _i m _i)^T is obtained using the reciprocal screw theory, f _i represents the force, m _i represents the couple;

Step 2: Exploiting strain energy method, the i-th limb analytical expression of strain energy U_i is obtained;

Step 3: The elastic deformation $δ_{i}$ associated constraint wrench system is obtained utilizing the constraint wrench system’s amplitude W _i to calculate the partial derivative of the strain energy U_i;

Step 4: According to the connection among the elastic deformation $δ_{i}$ and the amplitude of the constraint wrench system W _i, the flexibility matrix C _i and stiffness matrix K _i of the i-th limb are generated;

Step 5: The deformation compatibility equation and the balancing equation of the MP are created based on the virtual work principle;

Step 6: In accordance with the stiffness matrix and deformation compatibility equation of each limb, the overall stiffness matrix and the constraint wrench system amplitude of the KR-PM are derived.

Step 7: The elastostatic stiffness performance of the KR-PM is related to the configuration. Thus, it can be improved by adjusting the actuating distances given position and orientation of the MP.

Figure 2.

Flow chart for elastostatic stiffness modeling of KR-PMs.

Limb stiffness matrix

The mapping relationship between the constraint wrench system amplitude and the associated elastic deformation is represented by the limb stiffness matrix. The limb stiffness matrix shares the same dimension as the constraint wrench system. For a lower-DOF KR-PM, as long as there are exist passive joints, the limb stiffness matrix is a compact stiffness matrix. Assume that the i-th limb contains (6-m_i + 1) constraint force/couple, that includes a force screw f_ij f _ij (j = 1,2,…,a) and b couple screw m_ij m _ij (j = 1,2,…,b), f_ij and m_ij are the amplitude of the other. In this paper, a limb coordinate system B_i-x_iy_iz_i is established at the B_i point. The cross section’s neutral axis direction is where the x- and y-axes are positioned, whereas the z-axis runs along the direction of the limb. Any cross section of the i-th limb’s internal force/moment can be stated as

\begin{matrix} f_{ix} = \sum_{j = 1}^{a} f_{ij} f_{ij} \cdot e_{1} \\ f_{iy} = \sum_{j = 1}^{a} f_{ij} f_{ij} \cdot e_{2} \\ f_{iz} = \sum_{j = 1}^{a} f_{ij} f_{ij} \cdot e_{3} \\ m_{ix} = \sum_{j = 1}^{a} v_{i} e_{3} \times f_{ij} f_{ij} \cdot e_{1} + \sum_{j = 1}^{b} m_{ij} m_{ij} \cdot e_{1} \\ m_{iy} = \sum_{j = 1}^{a} v_{i} e_{3} \times f_{ij} f_{ij} \cdot e_{2} + \sum_{j = 1}^{b} m_{ij} m_{ij} \cdot e_{2} \\ m_{iz} = \sum_{j = 1}^{a} v_{i} e_{3} \times f_{ij} f_{ij} \cdot e_{3} + \sum_{j = 1}^{b} m_{ij} m_{ij} \cdot e_{3} \end{matrix}

(1)

where f _ij and m _ij stand for the unit vectors of constraint force and couple, respectively, and e ₁, e ₂, and e ₃ stand for the unit vectors along the three axes of the limb coordinate system. The cross section shear force magnitudes are represented by f_ix and f_iy, whereas the axial forces magnitude is represented by f_iz. Two bending moments are m_ix and m_iy, while the torque at the cross section is m_iz. v_i is the measurement of the separation from the cross section to the limb’s end.

The i-th limb strain energy is determined based on material mechanics,

U_{i} = \int_{0}^{l_{i}} (\frac{f_{ix}^{2}}{2 G_{i} A_{ix}} + \frac{f_{iy}^{2}}{2 G_{i} A_{iy}} + \frac{f_{iz}^{2}}{2 E_{i} A_{i}} + \frac{m_{ix}^{2}}{2 E_{i} I_{ix}} + \frac{m_{iy}^{2}}{2 E_{i} I_{iy}} + \frac{m_{iz}^{2}}{2 G_{i} J_{i}}) d v_{i}

(2)

where l_i stands for the i-th limb length, E_i and G_i stand for the elastic and shear moduli, respectively; A_ix and A_iy stand for the cross section’s effective shear area along the x_i- and y_i-axes, respectively; A_i represents the cross section area; I_ix and I_iy stand for the cross section inertia moment beside the x_i- and y_i-axes, and J_i represents the polar inertia moment.

As stated in the second theorem of Castigliano, the deformation vector $δ_{i}$ along the direction of the constraint wrench system W _i is equal to the strain energy’s partial derivative of to the constraint wrench system’s amplitude vector,

δ_{i} = \frac{\partial U_{i}}{\partial W_{i}}

(3)

where $δ_{i} = {(\begin{matrix} \begin{matrix} \begin{matrix} δ_{i 1} & \dots \end{matrix} & δ_{ia} \end{matrix} & \dots & δ_{i (6 - m_{i} + 1)} \end{matrix})}^{T}$ and $W_{i} = {(\begin{matrix} \begin{matrix} f_{i 1} & \dots \end{matrix} & f_{ia} & \dots & m_{i (6 - m_{i} + 1)} \end{matrix})}^{T}$ are (6-m_i + 1) × 1 vector.

Equation (3) can be expanded as

{\begin{matrix} δ_{i 1} = \frac{\partial U_{i}}{\partial f_{i 1}} = c_{11}^{i} f_{i 1} + \dots + c_{1 a}^{i} f_{ia} + \dots + c_{1 (6 - m_{i} + 1)}^{i} m_{ib} \\ ⋮ \\ δ_{ia} = \frac{\partial U_{i}}{\partial f_{ia}} = c_{a 1}^{i} f_{i 1} + \dots + c_{aa}^{i} f_{ia} + \dots + c_{a (6 - m_{i} + 1)}^{i} m_{ib} \\ ⋮ \\ δ_{i (6 - m_{i} + 1)} = \frac{\partial U_{i}}{\partial m_{i 1}} = c_{(6 - m_{i} + 1) 1}^{i} f_{i 1} + \dots + c_{(6 - m_{i} + 1) a}^{i} f_{ia} \\ + \dots + c_{(6 - m_{i} + 1) (6 - m_{i} + 1)}^{i} m_{ib} \end{matrix}

(4)

In a matrix, equation (4) is written as

δ_{i} = C_{i} W_{i}

(5)

C_{i} = (\begin{matrix} c_{11}^{i} & \dots c_{1 a}^{i} & \dots c_{1 (6 - m_{i} + 1)}^{i} \\ ⋮ ⋮ ⋮ \\ c_{a 1}^{i} & \dots c_{aa}^{i} & \dots c_{a (6 - m_{i} + 1)}^{i} \\ ⋮ & ⋮ & ⋮ \\ c_{(6 - m_{i} + 1) 1}^{i} & \dots c_{(6 - m_{i} + 1) a}^{i} & \dots c_{(6 - m_{i} + 1) (6 - m_{i} + 1)}^{i} \end{matrix})

(6)

where C _i is the flexibility matrix for the i-th limb and c_ij is its internal elements. c_ij represents the strain increment component in the i direction when the unit stress in the j direction only increases on the micro cell and there is no stress increment in other directions.

Thus, by flipping the flexibility matrix, it is possible to get the limb stiffness matrix as

K_{i} = C_{i}^{- 1}

(7)

As a result, the computations above can be used to generate the analytical expressions of the flexibility matrix and stiffness matrix of each limb.

Overall stiffness matrix and amplitude of constraint wrench system

By assembling all of the limb stiffness matrices together, the overall KR-PM’s stiffness matrix may be found. Non-overconstrained PMs and overconstrained PMs are the two categories into which KR-PMs fall. The overall stiffness matrix can be determined directly from the balance equations of the MP if the KR-PM is a non-overconstrained PM, the quantity of constrained responses and the quantity of balancing equations are equal. If the KR-PM is an overconstrained PM and there are more constrained reactions than equilibrium equations, then there must be an equal number of deformation compatibility equations inserted in order to derive the overall KR-PM’s stiffness matrix. In the fixed coordinate system, the balancing equation for the MP can be expressed as follows

W = G_{f}^{F} f = (J_{1} J_{2} \dots J_{n}) f

(8)

where J _i is a $6 \times (6 - m_{i} + 1)$ matrix, which represents the i-th limb constraint wrench system; $G_{f}^{F}$ is a $6 \times ((6 - m_{1} + 1) + 6 \times (6 - m_{2} + 1) + . . . + 6 \times (6 - m_{n} + 1))$ matrix, which represents the combination of the constraint wrench system in each limb; f _i is a $((6 - m_{1} + 1) + 6 \times (6 - m_{2} + 1) + . . . + 6 \times (6 - m_{n} + 1)) \times 1$ vector, which represents the combination of the force/couple amplitude vector in each limb.

The deformation vector $Δ$ at the MP’s center point and the deformation vector $δ_{i}$ at each of its limbs should match the following mapping relationship in accordance with the virtual work principle.

\begin{matrix} W^{T} Δ = {(\begin{matrix} W_{1} & W_{2} & \dots & W_{n} \end{matrix})}^{T} (\begin{matrix} δ_{1} & δ_{2} & \dots & δ_{n} \end{matrix}) \\ = W_{1}^{T} δ_{1} + W_{2}^{T} δ_{2} + \dots + W_{n}^{T} δ_{n} \end{matrix}

(9)

where $Δ = {(\begin{matrix} δ x & δ y & δ z & δ θ & δ φ & δ ψ \end{matrix})}^{T}$ represents the MP’s center minuscule movement under the influence of external wrenches.

Combining equations (8) and (9), the deformation compatibility equation between $δ_{i}$ and $Δ$ can be expressed as

δ_{i} = {J_{i}}^{T} Δ

(10)

Equation (5) can be rewritten as

W_{i} = K_{i} δ_{i}

(11)

Substituting equations (10) and (11) into equation (8), one obtains

W = \sum_{i = 1}^{n} J_{i} K_{i} {J_{i}}^{T} Δ = K Δ

(12)

Therefore, the overall elastostatic stiffness matrix of a KR-PM is

K = \sum_{i = 1}^{n} J_{i} K_{i} {J_{i}}^{T}

(13)

Besides, the amplitude of the constraint wrench system in each limb can thus be obtained by combing equations (10)–(13).

W_{i} = K_{i} {J_{i}}^{T} K^{- 1} W

(14)

Example 1: 2-UPR/PRPU KR-PM

Modeling of analytical elastostatic stiffness

To validate the analytical elastostatic stiffness approach of KR-PMs, 2-UPR/PRPU KR-PM has been taken to set up the theoretical model in this section. The 3-DOF non-redundant 2-UPR/RPU PM is proposed by Li and Herve.³⁷ According to 2-UPR/RPU PM, 2-UPR/PRPU KR-PM has been designed based on Lie group theory (for more details, see Wu et al.³⁸). Hence, the 2-UPR/RPU PM has been taken as a reference group to show the improvement of stiffness performance of 2-UPR/PRPU KR-PM. For the stiffness performance analysis of this PM, we can obtain the actuating distance q₄ using the optimization algorithm according to the best stiffness performance.

Structure description

The 2-UPR/PRPU KR-PM has three limbs, limb 1 and limb 2 are two non-redundant limbs, and limb 3 is a kinematically redundant limb as shown in Figure 3. The first and second limbs are fixedly connected with the framework through the U joints, and the third limb is fixedly connected with the framework through the guide. The first and second limbs take the only P joint as the actuator, and the third limb takes two P joints as the actuators to update the MP’s position and orientation.

Figure 3.

Schematic diagram of 2-UPR/PRPU KR-PM.

A₁, A₂ represent the U joint’s rotation center in the first and second limbs, respectively, and A₃ represents the R joint’s rotation center in the third limb. B₁, B₂ represent the R joint’s rotation center in the first and second limbs, respectively, and B₃ represents the U joint’s rotation center in the third limb. At the midway of A₁A₂, the fixed coordinate frame O-xyz is fixed. The x- and y-axes are vectors along OA₃ and OA₂, and the right-hand rule can be used to get the z-axis. Besides, the u- and v-axes of the moving coordinate frame o-uvw are vectors along oB₃ and oB₂, and the w-axis is vertical to the MP. Additionally, the 2-UPR/PRPU KR-PM’s dimension parameters are outlined below: OA₁ = OA₂ = l_b, OA₃ = q₄, oB₁ = oB₂ = oB₃ = l_a.

Inverse kinematics

For modeling and performance analysis of elastostatic stiffness, the inverse kinematics is required. The 2-UPR/PRPU KR-PM’s inverse kinematics can be established using the relationship between the input variables q_i (i = 1,2,3,4) and the output parameters ( $α, β, z$ ) of the MP. Figure 3 provides for the derivation of the closed loop vector equation for this PM as

P_{m} + a_{i} = q_{i} + b_{i} (i = 1, 2, 3)

(15)

where P_m represents the position vector Oo; b _i represents the position vector OA_i; a _i represents the position vector oB_i; q _i represents the position vector A_iB_i. P_m = (x y z)^T, b_i = (−1)ⁱl_be_i(i = 1,2), b₃ = q₄e₃, a_i = (−1) ⁱ R _p e _i(i = 1,2), a₃ = R_p e ₃, where e₁ = e₂ = (0 1 0)^T, e ₃ = (1 0 0)^T, q_i = q_i e _li (i = 1,2,3).

The rotation matrix R _p can be stated on the basis of the 2-UPR/PRPU KR-PM’s rotation characteristics, which state that the PM rotates first around the y-axis of the fixed coordinate frame before rotating around the u-axis of the moving coordinate frame.

R_{p} = R_{y, β} R_{u, α} = (\begin{matrix} \cos β & \sin α \sin β & \cos α \sin β \\ 0 & \cos α & - \sin α \\ - \sin β & \sin α \cos β & \cos α \cos β \end{matrix})

(16)

The following equation yields the coordinate of point o in the fixed coordinate frame.

P_{m} = {(\begin{matrix} z \tan β & 0 & z \end{matrix})}^{T}

(17)

Combining equations (15)–(17), the inverse position solution of 2-UPR/PRPU KR-PM is derived as

{\begin{matrix} q_{1} = \pm \sqrt{{(z \tan β - l_{a} \sin β \sin α)}^{2} + {(l_{b} - l_{a} \cos α)}^{2} + {(z - l_{a} \cos β \sin α)}^{2}} \\ q_{2} = \pm \sqrt{{(z \tan β + l_{a} \sin β \sin α)}^{2} + {(- l_{b} + l_{a} \cos α)}^{2} + {(z + l_{a} \cos β \sin α)}^{2}} \\ q_{3} = \pm \sqrt{{(z \tan β - q_{4} + l_{a} \cos β)}^{2} + {(z - l_{a} \sin β)}^{2}} \end{matrix}

(18)

Through equation (18), we can easily obtain the input variables q_i (i = 1,2) of first and second limbs given the output parameters ( $α, β, z$ ). However, the input variables q_i (i = 3,4) of third limb are not unique. Hence, there are infinity input variables, and an optimal input variable can be chosen according to the best stiffness performance.

Stiffness matrix of UPR limb and PRPU limb

According to the limb stiffness modeling approach, the UPR limb’s actuating force screw $S_{i}^{T}$ can be obtained by locking one actuating joint. As shown in Figure 4, besides, the constraint wrench system of UPR limb using the screw theory is deduced as

{\begin{matrix} {S^{T}}_{i} = (\begin{matrix} q_{i} / | q_{i} |; & b_{i} \times q_{i} / | q_{i} | \end{matrix}) \\ {S^{r}}_{i 1} = (\begin{matrix} \cos β & 0 & \sin β; \begin{matrix} 0 & 0 & 0 \end{matrix} \end{matrix}) \\ {S^{r}}_{i 2} = (\begin{matrix} 0 & 0 & 0; \begin{matrix} - \sin β & 0 & \cos β \end{matrix} \end{matrix}) \end{matrix} i = (1, 2)

(19)

where ${S^{T}}_{i}$ denotes an actuating force screw passing through point A_i and along the direction A_iB_i; ${S^{r}}_{i 1}$ denotes a constraint force screw passing through point A_i and in the direction of x-axis; ${S^{r}}_{i 2}$ denotes a constraint couple screw which is vertical to the U joint.

Figure 4.

The constraint wrench system of 2-UPR/PRPU KR-PM.

Figure 5(a) shows the constraint wrench system on the UPR limb. f_ai, f_i₁, and m_i₁ represent the amplitudes of the actuating force screw ${S^{T}}_{i}$ , the constraint force screw ${S^{r}}_{i 1}$ and the constraint couple screw ${S^{r}}_{i 2}$ , respectively. The UPR limb’s coordinate system B_i-x_iy_iz_i is constructed as demonstrated in Figure 5(b). The x_i-axis runs parallel to the axis of the R joint, the z_i-axis along the vector A_iB_i, and the y_i-axis is obtained using the right-hand rule.

Figure 5.

The UPR limb’s constraint wrench system: (a) force/moment state and (b) projection state.

Any cross section of the UPR limb’s internal force/moment can be stated as

{\begin{matrix} f_{ix} = f_{i 1} \\ f_{iy} = 0 \\ f_{iz} = f_{ai} \\ M_{ix} = 0 \\ M_{iy} = m_{i} τ_{i} \cdot e_{2} + (v_{i} (- e_{3}) \times f_{i 1} e_{1}) \cdot e_{2} \\ M_{iz} = m_{i} τ_{i} \cdot e_{3} \end{matrix} (i = 1, 2)

(20)

where e ₁, e ₂, and e ₃ represent the unit vectors along the three axes of the limb coordinate system; τ _i is the unit vector of the constraint couple m _i.

Therefore, the UPR limb’s strain energy can be represented as

\begin{matrix} U_{i} = \int_{0}^{L_{i}} (\frac{f_{ix}^{2}}{2 G_{i} A_{ix}} + \frac{f_{iz}^{2}}{2 E_{i} A_{i}} + \frac{M_{iy}^{2}}{2 E_{i} I_{iy}} + \frac{M_{iz}^{2}}{2 G_{i} I_{i}}) d v_{i} \\ = \frac{L_{i}}{2 E_{i} A_{i}} f_{ai}^{2} + (\frac{L_{i}}{2 G_{i} A_{ix}} + \frac{L_{i}^{3}}{6 E_{i} I_{iy}}) f_{i 1}^{2} - \frac{τ_{i} \cdot e_{2} L_{i}^{2}}{2 E_{i} I_{iy}} f_{i 1} m_{i} \\ + (\frac{{(τ_{i} \cdot e_{2})}^{2} L_{i}}{2 E_{i} I_{iy}} + \frac{{(τ_{i} \cdot e_{3})}^{2} L_{i}}{2 G_{i} I_{i}}) m_{i}^{2} \end{matrix}

(21)

The second theorem of Castigliano states that the limb’s infinitesimal elastic deformation under the constraint wrench system is

{\begin{matrix} δ_{ia} = \frac{\partial U_{i}}{\partial f_{ai}} = \frac{L_{i}}{E_{i} A_{i}} f_{ai} \\ δ_{if} = \frac{\partial U_{i}}{\partial f_{i 1}} = (\frac{L_{i}}{G_{i} A_{ix}} + \frac{L_{i}^{3}}{3 E_{i} I_{iy}}) f_{i 1} - \frac{τ_{i} \cdot e_{2} L_{i}^{2}}{2 E_{i} I_{iy}} m_{i} \\ φ_{i} = \frac{\partial U_{i}}{\partial m_{i}} = - \frac{τ_{i} \cdot e_{2} L_{i}^{2}}{2 E_{i} I_{iy}} f_{i 1} + [\frac{{(τ_{i} \cdot e_{2})}^{2} L_{i}}{E_{i} I_{iy}} + \frac{{(τ_{i} \cdot e_{3})}^{2} L_{i}}{G_{i} I_{i}}] m_{i} \end{matrix}

(22)

Equation (22) is expressed in matrix form as

δ_{i} = C_{i} f_{i}

(23)

C_{i} = (\begin{matrix} \frac{L_{i}}{E_{i} A_{i}} & 0 & 0 \\ 0 & \frac{L_{i}}{G_{i} A_{ix}} + \frac{L_{i}^{3}}{3 E_{i} I_{iy}} & - \frac{τ_{i} \cdot e_{2} L_{i}^{2}}{2 E_{i} I_{iy}} \\ 0 & - \frac{τ_{i} \cdot e_{2} L_{i}^{2}}{2 E_{i} I_{iy}} & \frac{{(τ_{i} \cdot e_{2})}^{2} L_{i}}{E_{i} I_{iy}} + \frac{{(τ_{i} \cdot e_{3})}^{2} L_{i}}{G_{i} I_{i}} \end{matrix})

(24)

where δ _i= (δ_ia, δ_if, φ_i)^T stands for the UPR limb’s deformation vector; f _i= (f_ai, f_i₁, m_i) ^T stands for the constraint wrench system’s amplitude vector. The flexibility matrix produced from equation (24) is a 3 × 3 compact matrix, which can increase the stiffness performance’s computation efficiency.

Therefore, according to equation (24), the UPR limb’s stiffness matrix is

K_{i} = C_{i}^{- 1}

(25)

The PRPU limb’s actuating force screw ${S^{T}}_{3}$ can be obtained by locking two actuating joints based on the screw theory. In a fixed coordinate system, the constraint wrench system of this limb is

{\begin{matrix} {S^{T}}_{3} = (\begin{matrix} q_{3} / | q_{3} |; & b_{3} \times q_{3} / | q_{3} | \end{matrix}) \\ {S^{r}}_{31} = (\begin{matrix} 0 & \cos β & 0; x_{B_{3}} \begin{matrix} \sin β - z_{B_{3}} \cos β & 0 & 0 \end{matrix} \end{matrix}) \\ {S^{r}}_{32} = (\begin{matrix} 0 & 0 & 0; \begin{matrix} - \sin β & 0 & \cos β \end{matrix} \end{matrix}) \end{matrix}

(26)

where ${S^{T}}_{3}$ stands for a actuating force screw, which passes point A₃ and along the direction of A₃B₃; ${S^{r}}_{31}$ stands for a constraint force screw, which passes point B₃ and along the direction of B₁B₂; ${S^{r}}_{32}$ stands for a constraint couple screw which is vertical to the U joint.

The constraint wrench system and limb coordinate system of the PRPU limb are indicated in Figure 6. The y₃-axis runs parallel to the axis of the R joint, z₃-axis runs along the vector A₃B₃, and the x₃-axis on the basis of the right-hand rule. The f_a₃, f₃₁, and m₃₁ denote the amplitudes of the actuating force screw ${S^{T}}_{3}$ , the constraint force screw ${S^{r}}_{31}$ and the constraint couple screw ${S^{r}}_{32}$ , respectively.

Figure 6.

The PRPU limb’s constraint wrench system: (a) force/moment state and (b) projection state.

In the limb coordinate system, the force/moment in any section of PRPU limb is

{\begin{matrix} f_{3 x} = 0 \\ f_{3 y} = f_{31} \\ f_{3 z} = f_{a 3} \\ M_{3 x} = m_{3} τ_{3} \cdot e_{1} + [v_{3} e_{3} \times f_{31} e_{2}] \cdot e_{1} \\ M_{3 y} = 0 \\ M_{3 z} = m_{3} τ_{3} \cdot e_{3} \end{matrix}

(27)

The PRPU limb’s strain energy on the basis of the constraint wrench system can be stated as

\begin{matrix} U_{3} = \int_{0}^{L_{3}} (\frac{f_{3 y}^{2}}{2 G_{3} A_{3 y}} + \frac{f_{3 z}^{2}}{2 E_{3} A_{3}} + \frac{M_{3 x}^{2}}{2 E_{3} I_{3 x}} + \frac{M_{3 z}^{2}}{2 G_{3} I_{3}}) d v_{3} \\ = \frac{L_{3}}{2 E_{3} A_{3}} f_{a 3}^{2} + (\frac{L_{3}}{2 G_{3} A_{3 y}} + \frac{L_{3}^{3}}{6 E_{3} I_{3 x}}) f_{31}^{2} - \frac{τ_{3} \cdot e_{1} L_{3}^{2}}{2 E_{3} I_{3 x}} f_{31} m_{3} \\ + [\frac{{(τ_{3} \cdot e_{1})}^{2} L_{3}}{2 E_{3} I_{3 x}} + \frac{{(τ_{3} \cdot e_{3})}^{2} L_{3}}{2 G_{3} I_{3}}] m_{3}^{2} \end{matrix}

(28)

Thus, the PRPU limb’s flexibility matrix can be obtained as

C_{3} = (\begin{matrix} \frac{L_{3}}{E_{3} A_{3}} & 0 & 0 \\ 0 & \frac{L_{3}}{G_{3} A_{3 y}} + \frac{L_{3}^{3}}{3 E_{3} I_{3 x}} & - \frac{τ_{3} \cdot e_{1} L_{3}^{2}}{2 E_{3} I_{3 x}} \\ 0 & - \frac{τ_{3} \cdot e_{1} L_{3}^{2}}{2 E_{3} I_{3 x}} & \frac{{(τ_{3} \cdot e_{1})}^{2} L_{3}}{E_{3} I_{3 x}} + \frac{{(τ_{3} \cdot e_{3})}^{2} L_{3}}{G_{3} I_{3}} \end{matrix})

(29)

Similarly, the stiffness matrix K ₃ of PRPU limb is

K_{3} = C_{3}^{- 1}

(30)

Overall stiffness matrix of 2-UPR/PRPU KR-PM

According to the constraint wrench system distribution of UPR and PRPU limb, the reaction forces exerting on the MP is accessible as depicted in Figure 7. W = ( F M )^T indicates the external wrenches exerted on the MP’s center. The KR-PM’s overall stiffness matrix is given according to equation (13), and it is

K = J_{1} K_{1} {J_{1}}^{T} + J_{2} K_{2} {J_{2}}^{T} + J_{3} K_{3} {J_{3}}^{T}

(31)

Figure 7.

Forces imposed on the MP.

where $J_{1} = {(\begin{matrix} {S^{T}}_{1} & {S^{r}}_{11} & {S^{r}}_{12} \end{matrix})}^{T}$ , $J_{2} = {(\begin{matrix} {S^{T}}_{2} & {S^{r}}_{21} & {S^{r}}_{22} \end{matrix})}^{T}$ and $J_{3} = {(\begin{matrix} {S^{T}}_{3} & {S^{r}}_{31} & {S^{r}}_{32} \end{matrix})}^{T}$ represent 6 × 3 force/moment matrices.

Equation (31) shows that the elastostatic stiffness performance is related to the KR-PM’s configuration given position and orientation of the MP. The optimal actuating configurations can be obtained using the optimization algorithm according to the best stiffness performance. Besides, according to equations (11) and (14), it can be deduced that the amplitude of the actuating force vector in each limb of this PM is

f_{i} = K_{i} {J_{i}}^{T} K^{- 1} W

(32)

Modeling verification

A simplified model of the 2-UPR/PRPU KR-PM is established using ANSYS software to confirm the accuracy of the theoretical model. The FEM considers the fixed base, MP, and joints to be rigid and assumes that all of the limbs have the same material’s circular cross sections. The limbs are also assumed to have deformable beam elements. The physical parameters of this PM are: l_a = 260 mm, l_b = 390 mm, elastic modulus E₁ = E₂ = E₃ = 210 GPa, shear modulus G₁ = G₂ = G₃ = 80 GPa, the diameter of the limb cross section d₁ = d₂ = d₃ = 100 mm, the distance h = 800 mm denotes the operating height Oo of MP.

To save calculation time, two examples are selected for verification as shown in Table 1. Figure 8 displays the findings of the center point of the MP’s linear and angular deformation. Table 2 also includes comparison findings between the theoretical model and the FEA model. The findings of the two models are found to be largely comparable, and the relative error is shown to be within 1.55%, demonstrating the precision of the theoretical model.

Table 1.

The two examples of 2-UPR/PRPU KR-PM.

Parameters	Unit	Example 1	Example 2
h	mm	800	800
$α$	rad	0	0
$β$	rad	0	0
Fx	N	0	80
Fy	N	0	120
Fz	N	200	160
Mx	N.m	0	80
My	N.m	0	120
Mz	N.m	0	160

Figure 8.

2-UPR/PRPU KR-PM’s deformation results: (a) example 1’s linear deformation, (b) example 1’s angular deformation, (c) example 2’s linear deformation, and (d) example 2’s angular deformation.

Table 2.

Comparison of the 2-UPR/PRPU KR-PM FEA model with the theoretical model.

Example	Approach	$δ_{x}$ mm	$δ_{y}$ mm	$δ_{z}$ mm	$δ θ$ rad	$δ φ$ rad	$δ ψ$ rad
1	Theoretical results	0	1.0346e-7	−0.5044e-7	1.2933e-7	0	0
	FEA	2.1838e-20	1.0354e-7	−0.5048e-7	−1.2943e-7	2.4140e-20	1.2457e-20
	Relative error (%)	–	0.0773	0.0792	0.0773	–	–
2	Theoretical results	4.1552e-5	3.2761e-6	−7.3458e-8	−8.8959e-7	2.6418e-5	4.8613e-5
	FEA	4.1677e-5	3.2855e-6	−7.3516e-8	−8.9452e-7	2.6496e-5	4.8733e-5
	Relative error (%)	1.5510	0.4230	0.0789	0.5511	0.2944	0.2462

Performance analysis

Stiffness index

According to the elastostatic stiffness model, the minimum linear displacement stiffness index³⁶ has been used to assess the PM’s stiffness performance because PMs mainly suffers the external force wrenches when parts processing. Unlike the above stiffness indices,^32–35 the minimum linear displacement stiffness performance index evaluates the mechanism’s minimum linear stiffness when it is subjected to external force wrenches. It is simple to calculate and has a definite physical significance.

Under the external wrenches, the MP’s center point’s minuscule deformation vector is

δ = CW

(33)

where $δ = {(\begin{matrix} \begin{matrix} \begin{matrix} δ_{dx} & δ_{dy} \end{matrix} & δ_{dz} & δ_{φ x} \end{matrix} & δ_{φ y} & δ_{φ z} \end{matrix})}^{T}$ is the deformation vector of the PM’s center point. $δ_{dx}$ , $δ_{dy}$ , and $δ_{dz}$ are the deformations along the x-, y-, and z-axes in terms of linearity, respectively; $δ_{φ x}$ , $δ_{φ y}$ , and $δ_{φ z}$ are the deformations along the x-, y-, and z-axes in terms of angularity, respectively. $W = {(\begin{matrix} \begin{matrix} f_{x} & f_{y} & f_{z} & m_{x} \end{matrix} & m_{y} & m_{z} \end{matrix})}^{T}$ are the external wrenches. $f_{x}$ , $f_{y}$ and $f_{z}$ are the forces along the x-, y-, and z-axes, respectively; $m_{x}$ , $m_{y}$ and $m_{z}$ are the couples along the x-, y-, and z-axes, respectively.

In general, a PM’s overall compliance matrix can be stated as

C = (\begin{matrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{matrix}) = (\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & c_{16} \\ c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & c_{26} \\ c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & c_{36} \\ c_{41} & c_{42} & c_{43} & c_{44} & c_{45} & c_{46} \\ c_{51} & c_{52} & c_{53} & c_{54} & c_{55} & c_{56} \\ c_{61} & c_{62} & c_{63} & c_{64} & c_{65} & c_{66} \end{matrix})

(34)

where $C_{11}$ , $C_{12}$ , $C_{21}$ , and $C_{22}$ are 3 × 3 submatrices. The unit of internal element of $C_{11}$ is m/N, which reflects how a force affects linear deformation; The unit of internal element of $C_{12}$ is m/N.m, which reflects how a couple affects linear deformation; The unit of internal element of $C_{21}$ is rad/N, which reflects how a force affects angular deformation; The unit of internal element of $C_{22}$ is rad/N.m, which reflects how a couple affects angular deformation.

The PM’s maximum compliances under an external force can be expressed as

c_{dfmax} = \sqrt{{| - λ_{df} |}_{max}}

(35)

where $- λ_{df}$ is the eigenvalue of the matrix ${C_{11}}^{T} C_{11}$ .

Taking the reciprocal of equation (35), the minimum linear displacement stiffness performance of the PM under external forces wrenches can be obtained as

k_{dmin} = 1 / c_{dfmax}

(36)

Stiffness optimization algorithm

In general, it is challenging to compute stiffness optimization in real-time. To quickly determine the optimum stiffness’s position given various positions and orientations of the MP, this paper combines PSO and polynomial fitting technology to establish the analytical mapping model of the response surface between the optimal actuating distances and the positions and orientations. The response surface mapping model based on polynomial is defined as

{\begin{matrix} y^{1} (x) = a_{0} + \sum_{i = 1}^{t} b_{i} x_{i} \\ y^{2} (x) = a_{0} + \sum_{i = 1}^{t} b_{i} x_{i} + \sum_{i = 1}^{t} c_{i} x_{i}^{2} + \sum_{i = 1}^{t} \sum_{j = i + 1}^{t} d_{ij} x_{i} x_{j} \\ y^{3} (x) = a_{0} + \sum_{i = 1}^{t} b_{i} x_{i} + \sum_{i = 1}^{t} c_{i} x_{i}^{2} + \sum_{i = 1}^{t} \sum_{j = i + 1}^{t} d_{ij} x_{i} x_{j} + \sum_{i = 1}^{t} e_{i} x_{i}^{3} \\ y^{4} (x) = a_{0} + \sum_{i = 1}^{t} b_{i} x_{i} + \sum_{i = 1}^{t} c_{i} x_{i}^{2} + \sum_{i = 1}^{t} \sum_{j = i + 1}^{t} d_{ij} x_{i} x_{j} + \sum_{i = 1}^{t} e_{i} x_{i}^{3} + \sum_{i = 1}^{t} f_{i} x_{i}^{4} \end{matrix}

(37)

where $y^{i} (x) (i = 1, 2, 3, 4)$ represents the i-th fitting polynomial function; $a_{0}$ , $b_{i}$ , $c_{i}$ , $d_{ij}$ , $e_{i}$ and $f_{i}$ represent the regression coefficient needed to be determined; t represents the variable of design parameter; $x_{i} x_{j}$ represents two parameters coupling, but high-order coupling is not taken into account.

The regression coefficients of the polynomial response surface analytical mapping model can be obtained by decreasing the square of the discrepancy between the observed and estimated the values.

min ε_{i} = \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}

(38)

where $y_{i}$ and ${\hat{y}}_{i}$ are the observed and the estimated values, respectively.

To assess the precision of fitting polynomial, the performance indices of fitting function are used. The most commonly used performance indices are RAAE, RMAE, RMSE, and R², which are indicated as

{\begin{matrix} \begin{matrix} RAAE = \frac{\sum_{i = 1}^{n} | y_{i} - {\hat{y}}_{i} |}{\sum_{i = 1}^{n} | y_{i} - {\bar{y}}_{i} |} & RMAE = \frac{max (| y_{i} - {\hat{y}}_{i} |)}{\sum_{i = 1}^{n} (| y_{i} - {\bar{y}}_{i} | / n)} \end{matrix} \\ \begin{matrix} RMSE = \sqrt{\frac{\sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}{n}} & R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - {\bar{y}}_{i})}^{2}} \end{matrix} \end{matrix}

(39)

Without losing generality, the response surface model (RSM) between the optimal actuating distance q₄ and the MP’s position and orientation is established with the operating height h = 800 mm and the ranges of two rotational angles were set as $(\begin{matrix} α & β \end{matrix}) = (\begin{matrix} - 10 ° & 10 ° \end{matrix})$ . Here are the particular steps: (1) develop the actuating optimization’s mathematical model; (2) based on Latin hypercube design, sampling is conducted in the interval. To ensure the accuracy of the RSM, the center point, midpoint and corner point of the interval should be selected as the sampling points, which are red points as shown in Figure 9; (3) each sampling point is optimized by PSO, and the actuating distance q₄ at the sampling point is obtained; (4) the sampling points are divided into two parts, 70% of which are used for parameter fitting and 30% for external verification.

Figure 9.

Schematic diagram of the sampling points.

Stiffness performance comparison

Comparing with the stiffness performance of non-redundant PM, the physical parameters of 2-UPR/PRPU KR-PM are selected as same as 2UPR-RPU PM³⁹: l_a = 260 mm, l_b = 390 mm, elastic modulus E₁ = E₂ = E₃ = 210 GPa, shear modulus G₁ = G₂ = G₃ = 80 GPa, the diameter of the limb cross section d₁ = d₂ = d₃ = 100 mm, the operating height h = 800 mm. The mathematical model of 2-UPR/PRPU KR-PM is defined as

{\begin{matrix} max k_{dmin} \\ design parameter q_{4} \\ s . t . q_{min} \leq q_{4} \leq q_{max} \end{matrix}

(40)

where $k_{dmin}$ is the minimum line displacement stiffness performance; q_min = 200 mm, q_max = 400 mm.

To increase the fitting precision, the sampling points is 100 in this paper. The accuracy of the polynomial RSM obtained by external validation using the 30% sampling points, and the results as listed in Table 3. Table 3 shows that with the increase of the polynomial degree, the accuracy of the RSM is not significantly improved, but the calculation efficiency decreases with the increase of the unknown amount of the required fitting regression. The acceptance precision of polynomial RSM is that RMSE value is below 0.2 and the R² value is above 0.9.⁴⁰ Considering the accuracy and computational efficiency together, the quartic polynomial is selected to establish the RSM between the optimal actuating distance q₄ and two orientation angles in this paper.

Table 3.

The accuracy of the polynomial RSM.

	Polynomial	RAAE	RMAE	RMSE	R²
q ₄	Linear	0.4662	0.8961	0.0258	0.7851
	Quadratic	0.0998	0.3223	0.0059	0.9887
	Cubic	0.0958	0.2893	0.0056	0.9900
	Quartic	0.0513	0.2158	0.0032	0.9967
	Fifth-degree	0.0476	0.1881	0.0029	0.9972
	Sixth-degree	0.0403	0.1800	0.0027	0.9977
	Seventh-degree	0.0391	0.1660	0.0026	0.9979
	Eighth-degree	0.0385	0.1754	0.0026	0.9979

Figure 10 shows the distribution of the optimal actuating distance q₄ of 2-UPR/PRPU KR-PM. It can be seen from the figure that the optimal actuating distance q₄ changes with the MP’s orientation. The orientation angle $α$ has slightly influence on q₄, but the orientation angle $β$ has considerably influence on q₄. Noteworthy is the fact that the calculation time of the optimal value of q₄ obtained by using the analytical RSM is 0.0004 s, while by using PSO is 25.434 s. Therefore, the RSM can greatly improve the calculation efficiency of the PM. Thus, the actuating distance can be obtained in real-time according to the change of orientation to ensure that the PM performs as stiffly as possible.

Figure 10.

Distribution of the optimal actuating distance q₄ of 2-UPR/PRPU KR-PM.

The distribution of 2-UPR/PRPU KR-PM’s minimum linear displacement stiffness performance is indicated in Figure 11. The graphic shows how the performance of minimum linear displacement stiffness varies slightly depending on the MP’s orientation. With regard to the angle, the stiffness performance is symmetrical, which is compatible with the structural features of 2-UPR/PRPU KR-PM. Additionally, the PRPU limb are arranged in the positive direction of the x-axis, so the positive value of the minimum linear displacement stiffness performance is greater than the negative direction. It is clear that 2-UPR/PRPU KR-PM performs better in terms of minimum linear displacement stiffness in the ranges of $α \in (\begin{matrix} - 10 ° & 10 ° \end{matrix})$ and $β \in (\begin{matrix} 5 ° & 10 ° \end{matrix})$ .

Figure 11.

Distribution of the minimum linear displacement stiffness performance of 2-UPR/PRPU KR-PM.

In order to show the superiority of 2-UPR/PRPU KR-PM’s stiffness performance, the ratio distribution of the minimum linear displacement stiffness performance of 2-UPR/PRPU KR-PM compared with 2-UPR/RPU PM is indicated in Figure 12. The result shows that the ratio $μ$ is greater than 1, which indicates that 2-UPR/PRPU KR-PM’s minimum linear displacement stiffness performance is superior to 2-UPR/RPU PM in the current orientation space. Compared with 2-UPR/RPU PM, the stiffness performance of 2-UPR/PRPU KR-PM is improved by 16% at the highest.

Figure 12.

Comparison of the stiffness performance of 2-UPR/PRPU KR-PM and 2-UPR/RPU PM.

Example 2: 2-PUPR/PRPU KR-PM

Modeling of elastostatic stiffness

Structure description

To further validate the proposed approach, a 2-PUPR/PRPU KR-PM is chosen as another example. This KR-PM is designed according to non-redundant 2-UPR/RPU PM based on Lie group theory as indicated in Figure 13.³⁶ The 2-PUPR/PRPU KR-PM has three kinematically redundant limbs. These limbs take two P joints as the actuators to update the MP’s position and orientation.

Figure 13.

Schematic diagram of 2-PUPR/PRPU KR-PM.

The fixed and moving coordinate frame were set the same as 2-UPR/PRPU KR-PM. Moreover, the 2-PUPR/PRPU KR-PM’s dimension parameters are defined as: OA₁ = q₅, OA₂ = q₆, OA₃ = q₄, oB₁ = oB₂ = oB₃ = l_a.

Inverse kinematics

The inverse position solution of 2-PUPR/PRPU KR-PM is derived as

{\begin{matrix} q_{1} = \pm \sqrt{{(z \tan β - l_{a} \sin β \sin α)}^{2} + {(q_{5} - l_{a} \cos α)}^{2} + {(z - l_{a} \cos β \sin α)}^{2}} \\ q_{2} = \pm \sqrt{{(z \tan β + l_{a} \sin β \sin α)}^{2} + {(- q_{6} + l_{a} \cos α)}^{2} + {(z + l_{a} \cos β \sin α)}^{2}} \\ q_{3} = \pm \sqrt{{(z \tan β - q_{4} + l_{a} \cos β)}^{2} + {(z - l_{a} \sin β)}^{2}} \end{matrix}

(41)

Through equation (41), the input variables q_i (i = 1,2,…,6) of three limbs are not unique given the output parameters ( $α, β, z$ ). Hence, there are infinity input variables, and an optimal input variable can be chosen according to the best stiffness performance.

PUPR limb’s stiffness matrix and overall stiffness matrix

Similarly, the PUPR limb’s constraint wrench system is deduced based on the screw theory as

{\begin{matrix} {S^{T}}_{i} = (\begin{matrix} q_{i} / | q_{i} |; & b_{i} \times q_{i} / | q_{i} | \end{matrix}) \\ {S^{r}}_{i 1} = (\begin{matrix} \cos β & 0 & \sin β; \begin{matrix} 0 & 0 & 0 \end{matrix} \end{matrix}) \\ {S^{r}}_{i 2} = (\begin{matrix} 0 & 0 & 0; \begin{matrix} - \sin β & 0 & \cos β \end{matrix} \end{matrix}) \end{matrix} i = (1, 2)

(42)

Therefore, the overall stiffness matrix of 2-PUPR/PRPU KR-PM can be expressed as

K = J_{1} K_{1} {J_{1}}^{T} + J_{2} K_{2} {J_{2}}^{T} + J_{3} K_{3} {J_{3}}^{T}

(43)

Stiffness performance analysis

The stiffness performance distribution of 2-PUPR/PRPU KR-PM is obtained as shown in Figure 14. The graphic shows how the stiffness performance varies slightly depending on the MP’s orientation. The minimum linear displacement stiffness performance is symmetrical about the angle $α$ , which consistent with the structural features of 2-PUPR/PRPU KR-PM. In addition, the positive value of the minimum linear displacement stiffness performance is greater than its negative direction. It is obvious that 2-PUPR/PRPU KR-PM has better minimum linear displacement stiffness performance in the ranges of $α \in (\begin{matrix} - 10 ° & 10 ° \end{matrix})$ and $β \in (\begin{matrix} 5 ° & 10 ° \end{matrix})$ .

Figure 14.

Distribution of the minimum linear displacement stiffness performance of 2-PUPR/PRPU KR-PM.

Figure 15 shows the distribution of the optimal actuating distances of 2-PUPR/PRPU KR-PM. It can be seen from the figure that the optimal actuating distance q₄ varies greatly within the orientation space, while the optimal actuating distances q₅ and q₆ vary slightly within the orientation space. The findings demonstrate that the actuating distance q₄ has an impact on the minimum linear displacement stiffness performance, while the minimum linear displacement stiffness performance is not sensitive to the actuating distances q₅ and q₆. The main reason is that there is only one limb on the side of PRPU limb and the stiffness performance along this direction that is relatively weak, which can be significantly improved by adjusting q₄. However, the stiffness performance of the other two limbs is relatively strong, which could not be significantly improved by adjusting the actuating distances q₅ and q₆.

Figure 15.

Distribution of the optimal actuating distances of 2-PUPR/PRPU KR-PM: (a) q₄, (b) q₅, and (c) q₆.

Figure 16 shows the distribution of the ratio of the minimum linear displacement stiffness performance of 2-PUPR/PRPU KR-PM compared with 2-UPR/RPU PM. The result shows that 2-PUPR/PRPU KR-PM’s minimum linear displacement stiffness performance is superior to 2-UPR/RPU PM in the current orientation space. Compared with 2-UPR/RPU PM, the stiffness performance of 2-PUPR/PRPU KR-PM is improved by 18% at the highest.

Figure 16.

Comparison of the stiffness performance of 2-PUPR/PRPU KR-PM and 2-UPR/RPU PM.

Conclusions

This work proposes an analytical elastostatic stiffness modeling approach based on the screw theory and the second theorem of Castigliano for KR-PMs. This approach can be employed during the design stage since it has clear modeling methodology, physical meaning and straightforward calculations. The 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs are used as two examples to validate the proposed approach. When the theoretical model is compared to the FEA model, the relative error is under 1.55%. The PM’s stiffness performance is assessed by using the minimum linear displacement stiffness performance index, which is based on the theoretical model. By integrating PSO with polynomial fitting technology, an analytical mapping model of the response surface between the optimal actuating distances and the MP’s given position and orientation is established in order to quickly find the optimal actuating distances of stiffness. It is demonstrated that 2-UPR/PRPU and 2-PUPR/PRPU KR-PMs can significantly enhance the stiffness performance when compared to the 2-UPR/RPU PM’s stiffness performance.

Footnotes

Appendix

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 is supported by the Natural Science Foundation of China (Grant no. 52275036).

ORCID iD

Wei Ye

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Modeling and performance analysis of elastostatic stiffness of kinematically redundant parallel mechanisms

Abstract

Keywords

Introduction

Analytical elastostatic stiffness modeling of KR-PMs

Modeling process

Limb stiffness matrix

Overall stiffness matrix and amplitude of constraint wrench system

Example 1: 2-UPR/PRPU KR-PM

Modeling of analytical elastostatic stiffness

Structure description

Inverse kinematics

Stiffness matrix of UPR limb and PRPU limb

Overall stiffness matrix of 2-UPR/PRPU KR-PM

Modeling verification

Performance analysis

Stiffness index

Stiffness optimization algorithm

Stiffness performance comparison

Example 2: 2-PUPR/PRPU KR-PM

Modeling of elastostatic stiffness

Structure description

Inverse kinematics

PUPR limb’s stiffness matrix and overall stiffness matrix

Stiffness performance analysis

Conclusions

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References