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Abstract

This article presents a derivation of the stiffness matrix of a general redundantly actuated parallel mechanism based on the overall Jacobian. The Jacobian of the constraints and actuations is derived using reciprocal screw theory. Based on the mapping relationship between constraint, actuated and external forces combined with the principle of virtual work, a compatibility equation for the deformation of all of the limbs is achieved, and the stiffness model of the general redundantly actuated parallel mechanism is derived. The 5-UPS/PRPU redundantly actuated parallel machine tool is used to illustrate this method. The parallel machine tool comprehensively reveals the effect of the elastic deformation of active–passive joints and some basic transmission parts. The stiffness model is further validated by experimental data. Moreover, the global stiffness matrix of the general redundantly actuated parallel mechanism can be separated into two parts via matrix decomposition. The first part is the stiffness matrix of the corresponding non-redundant parallel mechanism, and the second part is the stiffness matrix of the redundantly actuated limbs (actuators). The redundantly actuated 5-UPS/PRPU parallel machine tool is also investigated for further analysis. The different stiffness characteristics of the machine tool and its corresponding non-redundant 5-UPS/PRPU parallel machine tool are compared. Actuation redundancy is found to improve the stiffness performance of the machine tool efficiently.

Keywords

Stiffness matrix redundantly actuated parallel mechanism comparison

Introduction

In recent years, redundantly actuated parallel mechanisms have been investigated and developed out of the belief that they have many benefits compared with their non-redundant counterparts. These benefits include the following:

The avoidance of kinematic singularities;^1–3

Enlarged workspace and load capability;⁴

Improved dynamic characteristics.⁵

As one of the most important performance characteristics of parallel mechanisms, stiffness can be understood as the capacity of a mechanical system to maintain loads without any excessive change to the intrinsic geometry of the system. For high-precision mechanical systems such as machine tools⁶, a stiffer structure indicates higher accuracy and higher machining speeds in the end effector. Therefore, determining the stiffness model matrix and investigating its stiffness characteristics over its workspace are necessary and can help programmers direct tool motion planning. Some studies^7–16 have examined the modelling methods and stiffness characteristics of parallel mechanisms. Bi¹⁰ applied the kinetostatic method to develop a stiffness model. Wang et al.¹¹ analysed the stiffness characteristics of a Tricept robot while considering the bend deformation of the constrained passive limb. Joshi and Tsai¹⁷ compared the stiffness characteristics of a Tricept robot with a 3-UPU manipulator while considering the bend deformation of the actuator Shin et al.¹⁸ introduced a stiffness enhancement method by additional support rims. However, fewer efforts have been made to examine the stiffness of general redundantly actuated mechanisms while considering compliance in both actuators and links, and few studies have compared the stiffness performance of redundantly actuated parallel mechanisms with that of their non-redundant counterparts. Wu et al.¹⁹ investigated the stiffness performance of a 3-degree of freedom (DOF) parallel manipulator while considering additional supporting limbs. Zhao et al.⁵ investigated an 8-PSS parallel manipulator that added two extra limbs to the 6-PSS parallel manipulator.

Although some studies have compared redundant parallel manipulators with their non-redundant counterparts, the objects of research remain redundant parallel manipulators with one or more additional limbs. Few studies have compared the stiffness characteristics of redundant parallel manipulators with those of their non-redundant counterparts while replacing passive joints with active joints.

This article is structured as follows. In the ‘Stiffness matrix of a general redundantly actuated parallel mechanism’ section, the stiffness matrix of a general redundantly actuated mechanism is derived. In the ‘Stiffness evaluation of a 5-UPS/PRPU PMT’ section, the 5-UPS/PRPU redundantly actuated parallel machine tool (PMT) is used to illustrate the modelling method, the accuracy of which is verified via experiments. In the ‘Stiffness analysis of a 5-UPS/PRPU PMT’ section, the stiffness characteristics of the 5-UPS/PRPU redundantly actuated PMT are compared with those of its non-redundant counterpart, the 5-UPS/PRPU PMT. The notations of R, U, S and P denote revolute, universal, spherical and prismatic joints, respectively.

Stiffness matrix of a general redundantly actuated parallel mechanism

Description

A general n-DOF parallel mechanism is called a general n-DOF redundantly actuated parallel mechanism when the number of the active joints is m and m > n. In Figure 1, $$_{a, i} (i = 1, 2, \dots, m)$ and $$_{c, j} (j = 1, 2, \dots, 6 - n)$ denote the actuation and constraint wrenches, respectively, and $$_{i} (i = 1, 2, \dots, m)$ denotes the unit screws of the twists of the active joint. The overconstraints²⁰ of the redundantly actuated parallel mechanism are not discussed. Therefore, the number of actuation wrenches applied on the moving platform (MP) is m and the number of constraint wrenches applied on the MP is 6 − n. In addition, imposing more than two actuators/constraint wrenches on the same limb is permitted.

Figure 1.

A general redundantly actuated parallel mechanism.

Jacobian analysis

In Figure 1, the reference coordinate system o-xyz is fixed at the centre o of the MP, and all of the following actuation/constraint wrenches are expressed in the frame o-xyz. The external wrench applied on the MP is denoted as $$_{F}$ , in which the inertia wrench and gravity of the MP are included. To simplify the development of the dynamic equations, all of the supporting limbs are assumed to be massless and all of the passive–active joints frictionless. $$_{P}$ denotes the instantaneous twist of the MP, which can be expressed as

$_{P} = \sum_{k = 1}^{C_{i}} {\overset{\cdot}{θ}}_{i, k} {\hat{$}}_{i, k}

(1)

$_{P} = \sum_{l = 1}^{C_{j}} {\overset{\cdot}{θ}}_{j, l} {\hat{$}}_{j, l}

(2)

Here, the actuated/constraint limb is considered as a serial chain. The MP is connected to the fixed base by C_i /C_j number of 1-DOF joints. $$_{P} = [w^{T} v^{T}]^{T}$ defines the twist of the MP, where ν is the linear velocity of the origin point in the MP and w is the angular velocity.

Based on reciprocal screw theory, the screws that are reciprocal to all of the twists of the jth constraint wrench can form a (6 − C_j ) reciprocal screw set. Let ${\hat{$}}_{c, j}^{T}$ denote the unit screw of the jth constraint wrench by taking the orthogonal product of both sides of equation (2) with every corresponding unit screw of the constraint wrenches

{\hat{$}}_{c, j}^{T} $_{P} = 0

(3)

Thus, according to the definition of the Jacobian of constraints,⁹ the Jacobian of constraints of the general redundantly actuated parallel mechanism can be obtained by assembling each product

J_{c} = [\begin{matrix} {\hat{$}}_{c, 1}^{T} \\ {\hat{$}}_{c, 2}^{T} \\ ⋮ \\ {\hat{$}}_{c, j}^{T} \end{matrix}]

(4)

By locking the active joint of each limb, the screws that are reciprocal to all of the unit screws of the twists, apart from the screws of the actuated joints, that is, the actuation wrenches, form an m reciprocal screw set. Given the orthogonal product of both sides of equation (1), the following equation can be obtained

{\hat{$}}_{a, i}^{T} $_{P} = {\hat{$}}_{a, i}^{T} {\overset{\cdot}{θ}}_{i, k} {\hat{$}}_{i, k}

(5)

By assembling the m reciprocal screws, the following equation can be obtained

J_{x} $_{P} = J_{p} {\overset{\cdot}{q}}_{i}

(6)

where the matrix $J_{x}$ consists of all of the unit screws of the actuation wrenches. These can be expressed as

J_{x} = [\begin{matrix} {\hat{$}}_{a, 1}^{T} \\ {\hat{$}}_{a, 2}^{T} \\ ⋮ \\ {\hat{$}}_{a, m}^{T} \end{matrix}]

(7)

J_{q} = [\begin{matrix} {\hat{$}}_{a, 1}^{T} {\hat{$}}_{p, 1} 0 \dots 0 \\ 0 {\hat{$}}_{a, 2}^{T} {\hat{$}}_{p, 2} \dots 0 \\ ⋮ ⋮ ⋮ \\ 0 0 \dots {\hat{$}}_{a, m}^{T} {\hat{$}}_{p, m} \end{matrix}]

(8)

\overset{\cdot}{q} = {[{\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}, \dots, {\overset{\cdot}{θ}}_{m}]}^{T}

(9)

where ${\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}, \dots, {\overset{\cdot}{θ}}_{m}$ denote the velocity of each actuated joint, and ${\hat{$}}_{p, m}$ denotes the unit screw of the twist of the actuated joints. The matrix $J_{q}$ is an m × m diagonal matrix. Multiplying both sides of equation (6) by $J_{q}^{- 1}$ , the following equation can be obtained

J_{a} $_{p} = \overset{\cdot}{q}

(10)

where

J_{a} = [\begin{matrix} \frac{{\hat{$}}_{a, 1}^{T}}{({\hat{$}}_{a, 1}^{T}, {\hat{$}}_{p, 1})} \\ ⋮ \\ \frac{{\hat{$}}_{a, n}^{T}}{({\hat{$}}_{a, n}^{T}, {\hat{$}}_{p, n})} \\ ⋮ \\ \frac{{\hat{$}}_{a, m}^{T}}{({\hat{$}}_{a, m}^{T}, {\hat{$}}_{p, m})} \end{matrix}]

The Jacobian of non-redundant actuations $J_{a}$ can be obtained by separating the first n rows of $J_{na}$ . The remaining m − n rows are called Jacobian of redundant actuations $J_{ra}$

J_{na} = [\begin{matrix} \frac{{\hat{$}}_{a, 1}^{T}}{({\hat{$}}_{a, 1}^{T}, {\hat{$}}_{p, 1})} \\ ⋮ \\ \frac{{\hat{$}}_{a, n}^{T}}{({\hat{$}}_{a, n}^{T}, {\hat{$}}_{p, n})} \end{matrix}] J_{ra} = [\begin{matrix} \frac{{\hat{$}}_{a, n + 1}^{T}}{({\hat{$}}_{a, n + 1}^{T}, {\hat{$}}_{p, n + 1})} \\ ⋮ \\ \frac{{\hat{$}}_{a, m}^{T}}{({\hat{$}}_{a, m}^{T}, {\hat{$}}_{p, m})} \end{matrix}]

Combining equation (4) with equation (10), we obtain the overall Jacobian of the redundantly actuated parallel mechanism

J = [\begin{matrix} J_{a} \\ J_{c} \end{matrix}]

(11)

Stiffness matrix

In this section, the stiffness matrix of the general redundantly actuated parallel mechanism is obtained using the previously derived Jacobian matrix. The axial deformation of the supporting limbs is considered to determine the actual elastic displacement of the MP. In addition, the fixed base and MP are significantly stiffer mechanical structures and are thus assumed rigid, and the coupling deformation is neglected.

Determination of deformation compatibility equation

It is assumed that the external wrench applied on the MP can be expressed as $w = [f^{T} m^{T}]^{T}$ , where $f = [f_{x} f_{y} f_{z}]^{T}$ denotes a force and $m = [m_{x} m_{y} m_{z}]^{T}$ denotes a torque. In addition, let $τ_{a} = [τ_{na}^{T} τ_{ra}^{T}]^{T}$ and $τ_{c}$ denote the reaction forces/torques of the constraints, $τ_{na}^{T}$ denote the non-redundant actuated wrench and $τ_{ra}^{T}$ denote the redundantly actuated wrench. According to the static equilibrium, the external wrench is balanced with the reaction forces/torques exerted by the actuators and constraints

w = J_{na}^{T} τ_{na} + J_{ra}^{T} τ_{ra} + J_{c}^{T} τ_{c}

(12)

The reaction forces/moments can be represented as

{\begin{matrix} τ_{na} = k_{na} Δ q_{na} \\ τ_{ra} = k_{ra} Δ q_{ra} \\ τ_{c} = k_{c} Δ q_{c} \end{matrix}

(13)

where $Δ q_{na}$ , $Δ q_{ra}$ and $Δ q_{c}$ denote the elastic displacements of the supporting limbs under the action of the non-redundant actuation force/torque, redundant actuation and constraint force/torque, respectively. The corresponding diagonal matrices of stiffness can be expressed as

\begin{matrix} k_{na} = diag [K_{a, 1}, K_{a, 2}, \dots, K_{a, n}] \\ k_{ra} = diag [K_{a, n + 1}, K_{a, n + 2}, \dots, K_{a, m}] \\ k_{c} = diag [K_{a, 1}, K_{a, 2}, \dots, K_{a, 6 - n}] \end{matrix}

(14)

Moreover, let $Δ X = [Δ x Δ y Δ z Δ θ_{x} Δ θ_{y} Δ θ_{z}]^{T}$ be the elastic translation/rotation displacements of the MP, respectively. Based on the principle of virtual work, the following equation can be obtained

w^{T} Δ X = τ_{na}^{T} Δ q_{na} + τ_{ra}^{T} Δ q_{ra} + τ_{c}^{T} Δ q_{c}

(15)

Substituting equations (12) and (13) into equation (15) yields

\begin{matrix} τ_{na}^{T} (J_{na} Δ X - Δ q_{na}) + τ_{ra}^{T} (J_{ra} Δ X - Δ q_{ra}) \\ + τ_{c}^{T} (J_{c} Δ X - Δ q_{c}) = 0 \end{matrix}

(16)

Based on equation (16), the following equations can be obtained

J_{na} Δ X - Δ q_{na} = 0

(17)

J_{ra} Δ X - Δ q_{ra} = 0

(18)

J_{c} Δ X - Δ q_{c} = 0

(19)

Combining equations (17) and (19) into one equation yields

J_{na, c} Δ X - Δ q_{na, c} = 0

(20)

where

J_{na, c} = [\begin{matrix} J_{na} \\ J_{c} \end{matrix}] Δ q_{na, c} = [\begin{matrix} Δ q_{na} \\ Δ q_{c} \end{matrix}]

Based on the assumption made at the beginning of this section, each row of $J_{na, c}$ is linear independent and thus $J_{na, c}$ is an invertible matrix. The deformation compatibility equation can be obtained by substituting equation (20) into equation (18)

J_{ra} J_{na, c}^{- 1} Δ q_{na, c} = Δ q_{ra}

(21)

Determination of stiffness matrix

Substituting equations (13) and (17)–(19) into equation (12) yields

w = J_{na}^{T} k_{na} J_{na} Δ X + J_{ra}^{T} k_{ra} J_{ra} Δ X + J_{c}^{T} k_{c} J_{c} Δ X

(22)

\begin{matrix} w = [J_{na}^{T} J_{ra}^{T} J_{c}^{T}] [\begin{matrix} k_{na} \\ k_{ra} \\ k_{c} \end{matrix}] [\begin{matrix} J_{na} \\ J_{ra} \\ J_{c} \end{matrix}] Δ X \\ w = J^{T} K^{*} J Δ X = K Δ X \end{matrix}

(23)

where $K = J^{T} K^{*} J$ and K is the stiffness matrix of the general redundantly actuated parallel mechanism.

Via linear transformation, the stiffness matrix K can be decomposed into two parts: $K_{n, c}$ and $K_{ra}$

K = [J_{na}^{T} J_{c}^{T}] [\begin{matrix} k_{na} \\ k_{c} \end{matrix}] [\begin{matrix} J_{na} \\ J_{c} \end{matrix}] + J_{ra}^{T} k_{ra} J_{ra} = K_{n, c} + K_{ra}

(24)

where $K_{n, c} = [J_{na}^{T} J_{c}^{T}] [\begin{matrix} k_{na} \\ k_{c} \end{matrix}] [\begin{matrix} J_{na} \\ J_{c} \end{matrix}]$ denotes the stiffness matrix of the corresponding non-redundant parallel mechanism. $K_{ra} = J_{ra}^{T} k_{ra} J_{ra}$ denotes the stiffness matrix of the redundant limbs (or actuators).

There are three types of redundancy. The first is in-branch redundancy, which replaces some of the normally passive joints with the active joints of a parallel manipulator. The second type is actuation redundancy, in which the manipulators are actuated via additional actuated branches that go beyond the minimum necessary. The third type is a hybrid of the first two types. The three types of redundancy are continually modified by corresponding non-redundant mechanisms. Therefore, comparing the stiffness matrix of the redundant mechanism with its corresponding non-redundant mechanism is important when investigating the improvements that result from actuation redundancy. In the next section, a 5-UPS/PRPU redundantly actuated PMT is used as an example and compared with its corresponding non-redundant counterpart.

Stiffness evaluation of a 5-UPS/PRPU PMT

Description

As shown in Figure 2(a), the 5-UPS/PRPU PMT can perform two rotational DOFs and three translational DOFs. The MP and fixed base are connected by five UPS limbs and a PRPU limb. The five UPS limbs are driving limbs that are able to change their length to transform the pose of the MP. The PRPU limb is a passive limb that constrains the rotational DOF of the MP around its normal axis.

Figure 2.

Virtual prototypes of a 5-UPS/PRPU redundant PMT and its non-redundant counterpart. (a) Virtual prototype of the 5-UPS/PRPU PMT: 1. P Joint, 2. R Joint, 3. P Joint, 4. U Joint, 5. S Joint, 6. Spindle, 7. MP, 8. U Joint, 9. Fixed base and 10. Motor. (b) Virtual prototype of the 5-UPS/PRPU redundant PMT.

The outstanding feature of the PMT is that the expression of the forward/inverse dynamic solutions of the constraint limb (PRPU limb) is very concise. Each joint variable of the PRPU limb can be measured by installing sensors on the joint connection, which can be used to calculate the pose of the MP in real time via a direct analytical formula expression of the joint variables. To avoid singularity, improve dynamic performance and realise the closed-loop control of the PMT, a redundant actuator is installed on the PRPU limb (Figure 2(b)). Thus, the PMT can be described as a 5-UPS/PRPU redundantly actuated PMT and a 5-UPS/PRPU non-redundant actuated PMT. Figure 3 shows the improved structure of the 5-DOF redundant PMT and PRPU limb.

Figure 3.

Virtual prototype of the redundantly actuated joint.

Figure 4 presents a schematic diagram of the 5-UPS/PRPU PMT. A base Cartesian coordinate frame designated as the O_A-X_AY_AZ_A frame is fixed at the centre of the fixed base with the x-axis pointing vertically downwards and the y-axis pointing at the universal joint U ₁. A coordinate frame O_B-X_BY_BZ_B is similarly assigned to the centre of the MP, with the x-axis normal to the platform and the z-axis pointing in the opposite direction to the spherical joint S ₁. Following analysis is based on this coordinate system.

Figure 4.

Schematic diagram of the redundant PMT.

U_i (i = 1, 2, red) is the centre of the universal joint distributed on the fixed base (on a circle with a radius of 720 mm). S_i (i = 1, 2, us) is the centre of the spherical joint distributed on the MP (on a circle with a radius r of 200 mm). S ₁ is located on the Z_B -axis. The position coordinate of the universal joint U_i in frame {A} can be expressed as

^{A} P_{Ui} = {[\begin{matrix} ^{A} X_{Ui}^{A} Y_{Ui}^{A} Z_{Ui} \end{matrix}]}^{T} = {[\begin{matrix} H_{i} R_{i} c θ_{i} R_{i} s θ_{i} \end{matrix}]}^{T}

where R_i denotes the distance from O_A to U_i, θ_i denotes the angle between Y_A and R_i and H_i denotes the x coordinate of U_i in the base coordinate system {A}. The position coordinate of the universal joint B_i in frame {B} can be expressed as

^{B} P_{Si} = {[\begin{matrix} ^{B} X_{Si}^{B} Y_{Si}^{B} Z_{Si} \end{matrix}]}^{T} = {[\begin{matrix} h_{i} r_{i} c ϕ_{i} r_{i} s ϕ_{i} \end{matrix}]}^{T}

where R_i denotes the distance from O_A to U_i , $ϕ_{i}$ denotes the angle between Y_A and R_i and h_i is the x coordinate of U_i in the reference coordinate system {B}. Table 1 shows the primary structural parameters of the 5-UPS/PRPU PMT.

Table 1.

Primary structural parameters of the PMT (in mm).

i	R_i	r	θ_i (°)	φ_i (°)
1	780	202	0	−90
2	720	202	45	−18
3	720	202	135	54
4	720	202	225	126
5	720	202	315	198

If $_{B}^{A} T$ denotes the coordinate transformation matrix from {B} to {A}, then the inverse kinematic solution of the mechanism can be expressed as

l_{i} = |^{A} P_{Si} -^{A} P_{Ui} | = |_{B}^{A} T^{B} P_{Si} -^{A} P_{Ui} |, i = 1, 2, \dots, 5

(25)

where $l_{i}$ is the length of each limb.

Jacobian matrix derivation

The twist of the MP can be defined as $$_{P} = [w^{T} v^{T}]^{T}$ , where w and v denote the vectors for the angular and linear velocities, respectively. Thus, the instantaneous twist $$_{P}$ can be expressed as the linear combination of every instantaneous twist

$_{P} = {\overset{\cdot}{d}}_{01} {\hat{$}}_{01} + {\overset{\cdot}{θ}}_{02} {\hat{$}}_{02} + {\overset{\cdot}{d}}_{03} {\hat{$}}_{03} + {\overset{\cdot}{θ}}_{04} {\hat{$}}_{04} + {\overset{\cdot}{θ}}_{05} {\hat{$}}_{05}

(26)

\begin{matrix} $_{P} = {\overset{\cdot}{θ}}_{i 1} {\hat{$}}_{i 1} + {\overset{\cdot}{θ}}_{i 2} {\hat{$}}_{i 2} + {\overset{\cdot}{d}}_{i 3} {\hat{$}}_{i 3} + {\overset{\cdot}{θ}}_{i 2} {\hat{$}}_{i 2} + {\overset{\cdot}{θ}}_{i 4} {\hat{$}}_{i 4} \\ + {\overset{\cdot}{θ}}_{i 5} {\hat{$}}_{i 5} + {\overset{\cdot}{θ}}_{i 6} {\hat{$}}_{i 6} i = 1, 2, \dots, 5 \end{matrix}

(27)

where ${\hat{$}}_{ij}$ denotes the instantaneous twist of the jth joint of the ith limb. When i = 0, it denotes the redundant limb (PRPU limb), and when i = 1,2, …, 5, it denotes the UPS limb. ${\overset{\cdot}{d}}_{ij}$ denotes the intensity of the linear velocities and ${\overset{\cdot}{θ}}_{ij}$ denotes the intensity of the angular velocities.

For the purpose of analysing the kinematic characteristics of the 5-UPS/PRPU PMT, the screw system of the PRPU limb is established and shown in Figure 5.

Figure 5.

Kinematic screw of the PRPU limb.

In Figure 5, $$_{06}$ denotes the normal direction of the plane composed of two rotation axes of the universal joint and $$_{0 B}$ denotes the kinematic screws along the normal direction of the MP. The kinematic screws that correspond to each joint of the PRPU limb can be expressed as

{\begin{matrix} {\hat{$}}_{01} = (0 0 0; 0 0 1) \\ {\hat{$}}_{02} = (0 0 1; 0 0 0) \\ {\hat{$}}_{03} = (0 0 0; l_{2} m_{2} 0) \\ {\hat{$}}_{04} = (0 0 0; p_{4} q_{4} 0) \\ {\hat{$}}_{05} = (l_{5} m_{5} 0; p_{5} q_{5} r_{5}) \end{matrix}

(28)

According to reciprocal screw theory,²⁰ the structure restraint screw of the PRPU limb can be expressed as

$_{r} = (0 0 0; - m_{5} l_{5} 0)

(29)

Equation (29) shows that the motion around the normal axis of the MP is constrained. Given the reciprocal product of both sides of equations (26) and (29), the following can be obtained

J_{c} $_{P} = 0

(30)

where $J_{c}$ is the Jacobian matrix of constraints.

Moreover, by locking the actuators of each UPS limb, the new constraint forces reciprocal to all of the passive joints can be expressed as

{\hat{$}}_{ai} = [\begin{matrix} S_{i} \\ (A_{i} + l_{i}) \times S_{i} \end{matrix}] = [\begin{matrix} S_{i} \\ (A_{i} + B_{i}) \times S_{i} \end{matrix}]

(31)

which points along the axial direction of each limb. In equation (31), $A_{i}$ and $B_{i}$ denote the direction vectors of the ith universal and spherical joints in the base coordinate system.

In the same way, taking the reciprocal product of both sides of equations (27) and (31), the following equations can be derived in matrix form

J_{na} $_{P} = {\overset{\cdot}{q}}_{i 3} i = 1, 2 \dots 5

(32)

where

J_{na} = [\begin{matrix} {[(A_{1} + B_{1}) \times S_{1}]}^{T} S_{1}^{T} \\ {[(A_{2} + B_{2}) \times S_{2}]}^{T} S_{2}^{T} \\ {[(A_{3} + B_{3}) \times S_{3}]}^{T} S_{3}^{T} \\ {[(A_{4} + B_{4}) \times S_{4}]}^{T} S_{4}^{T} \\ {[(A_{5} + B_{5}) \times S_{5}]}^{T} S_{5}^{T} \end{matrix}]

{\overset{\cdot}{q}}_{i 3} = {[\begin{matrix} {\overset{\cdot}{d}}_{13} {\overset{\cdot}{d}}_{23} {\overset{\cdot}{d}}_{33} {\overset{\cdot}{d}}_{43} {\overset{\cdot}{d}}_{53} \end{matrix}]}^{T}

$J_{na}$ is the Jacobian of non-redundant actuations. Each row in $J_{na}$ denotes a unit wrench imposed on the MP by the actuators in each UPS limb.

Likewise, by locking the first prismatic joint of the PRPU limb, the constraint force exerted on the universal joint, which is parallel to the axis of the first prismatic joint, can be obtained

{\hat{$}}_{a 6} = [\begin{matrix} S_{6} \\ r_{6} \times S_{6} \end{matrix}]

(33)

where

J_{ra} $_{P} = {\overset{\cdot}{q}}_{01} J_{ra} = [{\hat{$}}_{a 6}^{T}] {\overset{\cdot}{q}}_{01} = [{\overset{\cdot}{d}}_{01}]

(34)

$J_{ra}$ is the Jacobian of redundant actuations. The row in $J_{ra}$ denotes the unit wrench applied on the MP by the actuators in the PRPU limb.

Stiffness modelling

In this section, the stiffness models of each driving limb of the 5-UPS/PRPU PMT are investigated based on the following assumptions. First, according to the mechanical behaviour of each driving limb, the UPS limbs and parts of the PRPU limb are equivalent to two-force bars. Therefore, only the deformation along its axis direction is considered. Second, the MP, fixed base and small/big slider are significantly stiffer mechanical structures and are thus assumed rigid. As the swing bar (RPU₆ in Figure 5) is not subjected to tension and compression, its axial stiffness is neglected. Third, as shown in Figures 6 and 7, analysis of the actual structure of each limb reveals that the stiffness of each limb is equivalent to a series spring.

Figure 6.

Structure of UPS limb.

Figure 7.

Partial structure of the PRPU limb.

The linear stiffness of the UPS limb can be expressed as

k_{ai} = \sum_{l = 1}^{7} \frac{1}{k_{ail}} i = 1, 2, \dots, 5

(35)

where $k_{ail}$ (l = 1, 2,…, 6) denotes the stiffness of the sleeve, thrust ball bearing, ball nut, U joint, ball screw, inner sleeve and S joint, respectively.

The linear stiffness of the first prismatic joint of the PRPU limb can be expressed as

k_{a 6} = \sum_{l = 1}^{6} \frac{1}{k_{a 6 l}}

(36)

where $k_{a 6 l}$ (l = 1, 2, …, 5) denotes the stiffness of the sleeve gasket 1/2, thrust ball bearing, bearing end plate and ball nut/screw, respectively.

Stiffness of the active joint

The axial stiffness of the general ball screw is correlated with its installation method. There are three methods: one end anti-thrust with the other end free, one end anti-thrust with the other end hinged and both ends anti-thrust. For the first two methods, the axial stiffness can be expressed as

K_{T} = \frac{AE}{l} = \frac{E π d^{2}}{4 l}

(37)

where A is the minimum cross-sectional area of the ball screw, E is the modulus of elasticity, l is the distance between the load acting point and support.

For the third method, the axial stiffness can be expressed as

K_{T} = AE (\frac{1}{l} + \frac{1}{L - l}) = \frac{E π d^{2}}{4} (\frac{1}{l} + \frac{1}{L_{b} - l})

(38)

Stiffness of passive joint

According to the mechanical behaviour of the UPS limb, the linear stiffness of the U joint along the axis direction of each limb should be considered. $k_{u}$ , $k_{v}$ and $k_{w}$ denote the stiffness along the u-, v- and w-axes, respectively, where the w-axis coincides with the axis of the limb. In Figure 8, the local coordinate systems of the inner and outer rings can be expressed as Buvw and Bu′v′w′, respectively. Thus, the three types of linear stiffness along the u-, v- and w-axes of the U joint can be described as

k_{p} = (\begin{matrix} k_{pu} \\ k_{pv} \\ k_{pw} \end{matrix}) {k'}_{p} = (\begin{matrix} {k'}_{pu} \\ {k'}_{pv} \\ {k'}_{pw} \end{matrix})

(39)

Figure 8.

Structure diagram of U joint.

Let $R = (u v w)$ and $R' = (u' v' w')$ represent the transformation matrix, which denotes the coordinate transformation from Buvw and Bu′v′w′ to the global coordinate o-xyz. When the inner and outer rings are connected in series, according to the principle of equivalent potential energy, the stiffness coefficient can be written as

(\begin{matrix} k_{epu}^{- 1} \\ k_{epv}^{- 1} \\ k_{epw}^{- 1} \end{matrix}) = (\begin{matrix} k_{pu}^{- 1} + {(u^{T} R' {k'}_{P} {R'}^{T} u)}^{- 1} \\ k_{pv}^{- 1} + {(v^{T} R' {k'}_{P} {R'}^{T} v)}^{- 1} \\ k_{pv}^{- 1} + {(w^{T} R' {k'}_{P} {R'}^{T} w)}^{- 1} \end{matrix})

(40)

Stiffness of the ball bearing

According to Hertz contact theory, the relationship between the contact load and elastic approach of the two contact bodies can be expressed as

δ = F {(\frac{9 \sum ρ}{2 π^{2} e^{2} E^{2} L_{e}})}^{\frac{1}{3}} Q^{\frac{2}{3}}

(41)

where $2 / E = 1 - μ_{_{1}}^{2} / E_{1} + 1 - μ_{_{2}}^{2} / E_{2}$ , δ is the elastic approach of two contact bodies, F is the elliptic integral of the first kind, $L_{e}$ is the elliptic integral of the second kind, $\sum ρ$ is the sum of the curvature of the contact point on the principal plane, e is the ellipticity parameter, Q is the contact load of the two contact bodies, E is the equivalent modulus of elasticity and $E_{1}$ , $E_{2}$ and $μ_{1}$ , $μ_{2}$ are the modulus of elasticity and Poisson’s ratio, respectively. Hertzian contact stiffness can be derived by taking the derivative of equation (40)

K = 1.5 {(\frac{π eE}{3 F})}^{\frac{2}{3}} {(\frac{2 L}{F \sum ρ})}^{\frac{1}{3}} Q^{\frac{1}{3}}

(42)

With the contact angle and load of the ball and bearing raceway as the known quantity, the contact stiffness of each ball and bearing raceway can be expressed as

\begin{matrix} K_{qs} = 1.5 {(\frac{π e_{qs} E}{3 F_{qs}})}^{\frac{2}{3}} {(\frac{2 L_{qs}}{F_{qs} \sum ρ})}^{\frac{1}{3}} Q_{q}^{\frac{1}{3}} \\ K_{es} = 1.5 {(\frac{π e_{es} E}{3 F_{es}})}^{\frac{2}{3}} {(\frac{2 L_{es}}{F_{es} \sum ρ})}^{\frac{1}{3}} Q_{e}^{\frac{1}{3}} \end{matrix}}

(43)

where the subscripts q, e and s represent inner race, outer race and the sth ball, respectively (Figure 9).

Figure 9.

Contact stiffness between balls and raceways.

The radial and axial stiffness component of the inner/outer race can be expressed as

K_{rqs} = K_{qs} \cos^{2} a_{qs}

(44)

K_{a' qs} = K_{qs} \sin^{2} a_{qs}

(45)

K_{res} = K_{es} \cos^{2} a_{es}

(46)

K_{a' es} = K_{es} \sin^{2} a_{es}

(47)

where $K_{rqs}$ and $K_{res}$ denote the radial stiffness of the inner/outer race between the nth ball, $K_{a' qs}$ and $K_{a' es}$ denote the axial stiffness of the inner/outer race between the nth ball and $a_{qs}$ and $a_{es}$ are the dynamic contact angles, respectively. $K_{es}$ and $K_{qs}$ denote the contact stiffness between the balls and the inner/outer races, respectively.

Based on the series and parallel connection relationships between the stiffness of the Z balls in a ball bearing, the axial stiffness $K_{a'}$ can be derived as

K_{a'} = \sum_{s = 1}^{Z} \frac{K_{a' qs} K_{a' es}}{K_{a' qs} + K_{a' es}}

(48)

Here, the contact angle is 15° and the pre-tightening force is 60 N according to the manufacturer recommendation.

Stiffness value of the components

The specific stiffness value of the related components can be achieved via finite element method (FEM) simulation or experimental test (see Tables 2 –4). The derivation process is not discussed here.

Table 2.

Linear stiffness value of the components of the UPS limb (N µm⁻¹).

Sleeve	Thrust ball bearing	Ball nut/screw	S joint	Inner sleeve
5020	100	107.3	590	165

Table 3.

Linear stiffness value of the components of the UPS limb (N µm⁻¹).

U joint, k_u	U joint, k_v	U joint, k_w	U joint, k_u _′	U joint, k_v _′	U joint, k_w _′
82	135	64	11,800	59,600	655,000

Table 4.

Linear stiffness value of the components of the PRPU limb (N µm⁻¹).

Ball nut/screw	Sleeve gasket 1	Sleeve gasket 2	Thrust ball bearing	Bearing end plate
114	6190	6810	100	33,500

Stiffness matrix description

The stiffness matrix of the 5-UPS/PRPU PMT can be obtained by substituting all of the parameters into equation (23). It can be decomposed into the following form

K = K_{n, c} + K_{ra}

(49)

For the redundantly actuated PMT, $K_{ra}$ denotes the stiffness matrix of the active joint of the PRPU limb and $K_{n, c}$ denotes the stiffness matrix of the non-redundant 5-UPS/PRPU PMT (Figure 2(a)). Thus, the stiffness matrix can be understood as a superposition of the stiffness matrix of the corresponding non-redundant 5-UPS/PRPU PMT, redundantly actuated PMT and redundant PRPU limb. In addition, analysis of the distribution of the elements of the principal diagonal of $K_{ra}$ indicates the influence of actuation redundancy on the PMT.

Experimental verification

Figure 10 shows the initial configuration of the PMT. The coordinates of the central point of the MP in this position is ${(1.019, 0, 0)}^{T}$ m and the orientation is ${(0, 0, 0)}^{T}$ . Thus, the corresponding stiffness matrix can be expressed as

\begin{matrix} K = 10^{7} \times \\ [\begin{matrix} 6.31 - 0.91 0.83 0.14 0.14 - 0.90 \\ - 0.91 1.17 - 0.14 - 0.23 0.001 1.54 \\ 0.83 - 0.14 5.97 - 0.049 0.89 - 0.15 \\ 0.14 - 0.23 - 0.049 8.24 0.11 - 0.30 \\ 0.14 0.001 - 0.89 0.11 1.32 0.014 \\ - 0.90 1.54 - 0.15 - 0.30 0.014 2.05 \end{matrix}] \end{matrix}

(50)

Figure 10.

Stiffness measurement experiment of the PMT.

A hoisting jack is used as an output load to verify the accuracy of the stiffness model. The acting force imposed on the MP can be detected by a six-dimension force/torque sensor, and the micro-displacement of the MP can be measured via AT901-MR laser interferometer. In addition, according to $X = K^{- 1} F$ , the theoretical micro-displacement can be calculated directly. Therefore, by comparing the practical value and theoretical value of the elastic deformation of the MP, the accuracy of the stiffness model can be validated effectively.

To avoid the influence of the deformation caused by gravity, the initial coordinate of the MP is measured without a hoisting jack. Thus, when the hoisting jack imposes a load on the MP, the micro-displacement can be obtained by subtracting the two measured values. Moreover, as the laser interferometer is very sensitive to external disturbance (e.g. noise and vibration), the relative standard deviation of 10 measurements is calculated to guarantee the accuracy of the experiment, which is shown in Table 5.

Table 5 compares the theoretical and actual values of the micro-displacement under different loads. The micro-displacement under a larger load exhibits a more significant difference. When the load is increased to 200 N or higher, the error range is 2%−5%, which is accepted when man-made measurement errors and outside environmental factors are considered. The other five micro-displacements can be obtained in the same way. However, due to space limitations, the experimental process is not discussed here.

Table 5.

Experiment data.

	F_x (N)
Δx (µm)	71.57	118.84	230.68	273.41	337.29	351.26	380.60	430.73
Theoretical value	1.13	1.88	3.65	4.32	5.35	5.57	6.03	6.83
Actual value	1.24	2.01	3.76	4.41	5.48	5.79	6.21	7.11
Error (%)	9.7	6.9	3.01	2.1	2.4	3.9	2.99	4.1

Stiffness analysis of a 5-UPS/PRPU PMT

To investigate the influence of actuation redundancy on the stiffness performance of the PMT, the stiffness distribution of the redundantly actuated 5-UPS/PRPU PMT and its non-redundantly actuated counterpart are compared as shown in Figure 11. Equation (20) notes that entries of the stiffness matrix consist of information about the structural attributes and configuration of the PMT. Therefore, it is of the utmost importance to study the stiffness distribution of the 5-DOF PMT in correspondence with different positions and orientations. The minimum and maximum linear/angle stiffness are calculated and compared under different swing angles (α and β) as follows.

Figure 11.

Stiffness distribution of the redundantly actuated PMT and its corresponding non-redundantly actuated PMT in a plane of x = 1.1 m. (a) Minimum/maximum linear/angular stiffness of the PMT corresponding to the attitude angle α = 0°, β = 0°: (a-1) minimum linear stiffness, (a-2) minimum angular stiffness, (a-3) maximum linear stiffness and (a-4) maximum angular stiffness. (b) Minimum/maximum linear/angular stiffness of the PMT corresponding to the attitude angle α = 10°, β = 0°: (b-1) minimum linear stiffness, (b-2) minimum angular stiffness, (b-3) maximum linear stiffness and (b-4) maximum angular stiffness.

Comparison indicates that the minimum of the linear/angle stiffness are symmetric to the y-axis, which complements the symmetric architecture of the PMT. Moreover, the minimum value of the linear/angle stiffness is distributed on the edge of the workspace, the area in which singularity always appears. In addition, the linear/angle stiffness tends to be the nearby minimum value (z = 0 m, y = −0.13 m). The programmer should try to avoid this area when planning the trajectory of the PMT.

Both linear and angle stiffness are symmetric about the y-axis. Figure 11(a) and (b) compares the influence of actuation redundancy on the stiffness distribution of the 5-UPS/PRPU PMT. In Figure 11(a-1) and (a-2), the minimum stiffness of the redundant PMT is higher than that of the non-redundant PMT. However, actuation redundancy has almost no influence on the maximum stiffness of the PMT when α = 0° and β = 0°. When the angle α is changed, the minimum angular stiffness is not affected by actuation redundancy. Thus, to investigate the relationship between the different attitude angle and the corresponding stiffness distribution of the PMT, the isotropic values of the linear/angular stiffness are calculated as a stiffness performance index that corresponds to different swing angles.

Figure 12(a)–(d) shows that although the variation of attitude angle α can affect the stiffness performance of the PMT, the attitude angle β cannot. In addition, the swing angle β does not affect the isotropy index of the linear stiffness $(λ_{l})$ . More importantly, the isotropy index of the linear stiffness of the redundant PMT is superior to that of its non-redundant counterpart. When the angle α is 10°, the isotropy index of the angle stiffness is almost unaffected by actuation redundancy. Analysis of the PMT stiffness distribution reveals that the anisotropy area comprises the weak workspace. This provides a theoretical reference for the programmer to plan the trajectory of the 5-UPS/PRPU redundantly actuated PMT. According to preceding analysis, actuation redundancy can improve the stiffness isotropy of the PMT. Thus, a more precise mechanical system can be achieved by replacing some of the passive joints with active joints.

Figure 12.

Isotropy index distribution of the redundantly actuated PMT and its corresponding non-redundantly actuated PMT in a plane of x = 1.1 m. (a) Linear/angle stiffness isotropy index distribution corresponding to the attitude angle α = 0°, β = 0° (where $λ_{l}$ denotes the linear stiffness isotropy index and $λ_{a}$ denotes the angle stiffness isotropy index): (a-1) linear stiffness isotropy index and (a-2) angle stiffness isotropy index. (b) Linear/angle stiffness isotropy index distribution corresponding to the attitude angle α = 0°, β = 10°: (b-1) linear stiffness isotropy index and (b-2) angle stiffness isotropy. (c) Linear/angle stiffness isotropy index distribution corresponding to the attitude angle α = 10°, β = 0°: (c-1) linear stiffness isotropy index and (c-2) angle stiffness isotropy index. (d) Linear/angle stiffness isotropy index distribution corresponding to the attitude angle α = 10°, β = 10°: (d-1) linear stiffness isotropy index and (d-2) angle stiffness isotropy index.

Conclusion

The following four conclusions can be drawn:

The stiffness matrix of a general redundantly actuated parallel mechanism is derived based on the Jacobian of actuations and constraints. The stiffness modelling method is validated via a test of the stiffness performance of the 5-UPS/PRPU PMT.

The global stiffness matrix can be separated into two parts: the stiffness matrices of the corresponding non-redundant parallel mechanism and redundant actuators (or limbs). This is significant when comparing the stiffness performance of the redundantly actuated mechanism with that of its corresponding non-redundant mechanism, which can be used to study the influence of actuation redundancy.

The 5-UPS/PRPU redundantly actuated PMT and 5-UPS/PRPU non-redundantly actuated PMT are used to illustrate the influence of actuation redundancy. The redundant actuator can improve the stiffness isotropy of the PMT, which indicates that the load-carrying capacity of the PMT along each direction can be improved.

In addition, this modelling method reveals the stiffness behaviour of the PMT, providing a theoretical reference for the programmer to plan the trajectory. Furthermore, the modelling method is valuable in predicting the stiffness performance of a mechanism over its workspace, assessing whether the stiffness requirement can be satisfied and achieving an optimal design during the design stage. The stiffness modelling method presented in this study is very important for stiffness performance evaluation, static analysis and mechanism design.

Footnotes

Appendix 1 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 research is financially supported by the National Natural Science Foundation of China (Grant No. 51275439).

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Stiffness modelling and comparison of the 5-UPS/PRPU parallel machine tool with its non-redundant counterpart

Abstract

Keywords

Introduction

Stiffness matrix of a general redundantly actuated parallel mechanism

Description

Jacobian analysis

Stiffness matrix

Determination of deformation compatibility equation

Determination of stiffness matrix

Stiffness evaluation of a 5-UPS/PRPU PMT

Description

Jacobian matrix derivation

Stiffness modelling

Stiffness of the active joint

Stiffness of passive joint

Stiffness of the ball bearing

Stiffness value of the components

Stiffness matrix description

Experimental verification

Stiffness analysis of a 5-UPS/PRPU PMT

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References