Research ArticleStiffness and Elastic Deformation of 4-DoF Parallel Manipulator with Three Asymmetrical Legs for Supporting Helicopter Rotor

Jialin Song,1 Yang Lu,2 Yongli Wang,3 and Yi Lu 3

1College of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China2Harbin Electric Corporation (Qinhuangdao) Heavy Equipment Company Limited, Qinhuangdao, Hebei 066004, China3College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China

Correspondence should be addressed to Yi Lu; luyi@ysu.edu.cn

Received 21 July 2019; Revised 18 August 2019; Accepted 10 November 2019; Published 1 February 2020

Academic Editor: Gordon R. Pennock

Copyright © 2020 Jialin Song et al. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

�e stiffness and elastic deformation of a 4-DoF parallel manipulator with three asymmetrical legs are studied systematically for supporting helicopter rotor. First, a 4-DoF 2SPS + RRPR type parallel manipulator with two linear SPS type legs and one RRPR type composite leg is constructed and its constraint characteristics are analyzed. Second, the formulas for solving the elastic deformation and the stiffness matrix of the above mentioned three asymmetrical legs are derived. �ird, the formulas for solving the total stiffness matrix and the elastic deformation of this manipulator are derived and analyzed. Finally, its finite element model is constructed and its elastic deformations are solved using both the derived theoretical formulas and the finite element model. �e theoretical solutions of the elastic deformations are verified by that of the finite element model.

1. Introduction

Various less mobility (less than 6-DoF) parallel manipulators (PMs) have been applied widely due to their merits, such as good performances in accuracy, rigidity, ability to manipulate large loads, and they are simple in structure and easy to control [1, 2]. Among them, some less mobility PMs with composite active constrained legs attract more attention because they have larger workspace, better flexibility and fewer legs for avoiding interferences easily; the unnecessary tiny self-move-ment can be eliminated by the composite active constrained legs; more actuators can be installed onto the base for reducing vibration [1, 2]. �erefore, this type of PMs have potential applications for supporting helicopter rotor, airplane opera-tion simulator, parallel machine tools, micro manipulators, sensors, surgical manipulators, tunnel borers, barbettes of war ship, and satellite surveillance platforms. Stiffness is one of the important performances of PMs, because higher stiffness allows larger variable load and higher speeds with higher pre-cision of the end-effector [3]. �erefore, it is significant to analyze the stiffness and to evaluate elastic deformation of this type of PMs in the early design stage. Let (R, P, U, S) be (rev-olute, prismatic, universal, spherical) joint, respectively. In this

aspect, Gosselin and Zhang developed virtual joint method allowed taking into account the links flexibility, which were presented as rigid beams supplemented by linear and torsional springs [3]. Zhang and Lang Sherman [4] established a stiff-ness modeling for PMs with one passive leg. Dong et al. [5] analyzed the stiffness modeling and stiffness distributions of a 5-DOF hybrid robot by considering the component compli-ances associated with the elements of both the PM and the wrist. Li and Xu [6] derived stiffness matrix of a 3-PUU PM based on an overall Jacobian using the screw theory by con-sidering the effect of actuations and constraints. Yang et al. [7] studied elastostatic stiffness modeling of over constrained PMs. Zhou et al. [8] derived the stiffness matrix of a redun-dantly actuated parallel mechanism based on the overall Jacobian. Based on strain energy and Castigliano’s theorem, Enferadi and Tootoonchi [9] obtained mathematical model of the manipulator stiffness matrix. Pashkevich et al. [10] pro-posed a methodology to enhance the stiffness of serial and parallel manipulators with passive joints, the manipulator elements are presented as pseudo-rigid bodies separated by multidimensional virtual springs and perfect passive joints; they [11] also presented a stiffness modeling method for over-constrained PMs with flexible links and compliant actuating

HindawiJournal of RoboticsVolume 2020, Article ID 8571318, 11 pageshttps://doi.org/10.1155/2020/8571318

Journal of Robotics2

joints and the method of FEA-based link stiffness evaluation. Zhao et al. [12] deduced continuous stiffness matrix of a fold-able PM for ship-based and the translation and rotational stiffness along any direction. Pham and Chen [13] established the stiffness model based on the way the flexure members are connected together in serial or parallel combinations. Chen et al. [14] derived a stiffness matrix of 3CPS PM based on the principle of virtual work considering the compliances subject to both actuators and legs. Shan et al. [15] established a overall stiffness model of the 2(3PUS+S) PM through a stiffness mod-eling method of a serial system. Hao and Kong [16] analyzed the mobility of spatial compliant multi-beam modules and derived their compliance matrices using a normalization tech-nique. Wang et al. [17] investigated the stiffness characteristics of a hexaglide parallel loading machine, and derived its total stiffness matrix based on Jacobian matrix and statics. Lu et al. [18, 19] solved stiffness and elastic deformation for some less mobility PMs and serial-parallel manipulators by virtual mechanisms. Others [20–22] studied the stiffness and elastic deformation of PMs using above similar approaches and the virtual experiments in CAD environment. �e above men-tioned approaches for different PMs have their own merits. Since the above mentioned PMs are symmetrical in the struc-ture and the distribution of active legs, the established stiffness matrices are symmetrical, and the elastic deformations of PMs can be solved more easily.

A 2SPS + RRPR type PM is a 4-DoF PM with three asym-metrical legs [23]. When the base of the 2SPS + RRPR type PM is fixed on top of the helicopter, and the rotor and its rotational actuator are installed on the moving platform of the 2SPS + RRPR type PM, the 2SPS + RRPR type PM can be used for the helicopter rotor supporter. Comparing with existing 4-DOF PM, the 2SPS + RRPR type PM has several merits as follows: (1) �e stability and the capability of load bearing can be increased, the force situations can be improved, and the position workspace and the orientation workspace can be

increased largely by rotating a revolute joint which connects the RRPR type composite leg with the base. (2) �e unneces-sary tiny self-movement can be removed and the precision can be increased using the RRPR type composite leg. (3) �e number of oscillating legs is reduced, and the interference can be avoided easily. (4) �e more actuators can be installed onto the base for reducing vibration.

Since the structure of the 2SPS + RRPR type PM is asym-metrical, it is a challenging and a significant issue to study the stiffness and elastic deformation of the 4-DOF PMs with asymmetrical structure by considering its constrained force. �erefore, this paper focuses on the study of the total stiffness and the elastic deformation of the 2SPS + RRPR PM by taking into account the elastic deformation due to constrained wrench. A finite element model of this PM is constructed for verifying the analytic solutions.

2. Kinematics and Statics of 2SPS + RRPR Type PM

A 2SPS + RRPR type PM for supporting helicopter rotor is shown in Figure 1(a). �e 2SPS + RRPR PM is composed of a moving platform �, a fixed base �, and 2 SPS (spherical joint-active prismatic joint-spherical joint) type legs �푟� (�푖 = 1, 3) with the linear actuator, and one RRPR (active revolute joint- revolute joint -active prismatic joint-revolute joint) type composite active leg �2 with a linear actuator and a rotational actuator, see Figure 1(b). Here, � is an equilateral ternary link Δ�푏1�푏2�푏3 with 3 sides �푙� = �푙, 3 vertices ��, and a center point �. � is an equilateral ternary link Δ�퐵1�퐵2�퐵3 with 3 sides �퐿 � = �퐿, 3 vertices ��, and a center point �. Each of �푟�(�푖 = 1, 3) connects � to � by a spherical joint � at ��, a leg �� with active prismatic joint �, and � at ��. �e RRPR-type constrained composite active leg �2 connects � to � by a uni-versal joint � attached to � at �2, a constrained leg �2 with

Motor

Helicopter rotor

(a)

B1, S

r3, P

B2

S, b

b2, R3

S, b3

r1, P r2, P

Z

z

x

ym

B

L3

L2

l2

l3

l1

R2

R2

O

R3

Motor

X Y

o

B3, S

L1

(b)

S

Pin

Pin

(c)

Figure 1: A 2SPS+RRPR PM for supporting helicopter rotor (a), 2SPS+RRPR type PM (b), and a prototype of reconfigurable 3SPS PM (c).

3Journal of Robotics

active prismatic joint �, a revolute joint �3 attached to � at �2. �e universal joint � at �2 is composed of two cross rev-olute joints �1 and �2. Here, �1 is connected with a rotational actuator. �erefore, the moving platform � of the 2SPS + RRPR type PM has 4 DOFs corresponding three rota-tions about �푅1, �푅1, �푅1, and one translation along limb �2. �e degree of freedom of the 2SPS + RRPR type PM has been calculated and verified using its simulation mechanism in [23]. Since each of the SPS-type active legs �푟�(�푖 = 1, 3) only bears the active force along ��, it obviously has relative larger capacity of load bearing and is simple in structure. In addi-tion, the unnecessary tiny self-movement of the 4-DoF 2SPS + RRPR PM can be eliminated effectively and its work-space can be enlarged by the RRPR-type constrained com-posite active leg �2. Comparing with other 4-DOF PMs with four active legs, the 2SPS+RRPR PM with three active legs has merits as follows: (1) �e interference among three active legs and the moving platform can be avoided easily. (2) Its whole mechanism is simplified. (3) Its moving platform pro-vides more room for installing the helicopter rotor, finger mechanisms, tools.

A prototype of the reconfigurable 3SPS experimental model is built, see Figure 1(c). It includes a �, a � and 3 recon-figurable SPS-type legs �푟�(�푖 = 1, 2, 3). Each of �� connects � to � by a spherical joint � at �� a reconfigurable leg �� with active prismatic joint �, and � at bi. Here, � and � are the same as that of the 2SPS+RRPR PM. Each of S joints is composed of three revolute joints �. It can be transformed into a � joint by adding one pin or be transformed into a � joint by adding two pins. �us, the 2SPS + RRPR PM can be constructed easily from the prototype of reconfigurable 3SPS model to transform the upper � joint of �2 into � joint by adding two pins, to trans-form the lower � joint of �2 into � joint by adding one pin, and to add a rotational actuator onto the vertical revolute joint �1 of � joint.

Let ⊥ be a perpendicular constraint, || be a parallel con-straint. Several geometric constraints (R1 being coincident with the axis of motor, �3 being coincident with �푦, �푅1‖�푍, �푅1 ⊥ �퐵, �푅2 ⊥ �푅1, �푅2 ⊥ �푅3, �푅2 ⊥ �푟2, and �푅3 ⊥ �푟2) are satisfied in this PM. Let {�푚} be a coordinate frame �-��� fixed on � at �표, {�퐵} be a coordinate frame �-��� fixed on � at �. Let (�, �, �) be three Euler angles of �푚, �휑 be one of (�훼, �훽, �). Set �푠� = sin�휑, �푐� = cos�휑, and �푡� = tan�휑. �e position vectors �� of �� on � in {�퐵}, the position vectors �b� of �� on � in {�푚}, the position vectors �� of �� on � in {�퐵}, and the position vector � of � on � in {�퐵}, the unit vectors �� of �� and the vector �� of the line �� in {�퐵} can be expressed as follows: [23]

(1)

�� = (�푋���푌���푍��

), ��� = (�푥���푦���푧�� ), �� = (�푋���푌���푍��

),� = (�푋��푌��푍�

), ��� = ( �푥� �푦� �푧��푥� �푦� �푧��푥� �푦� �푧� ),

�� = ������ + �, �푟� = �儨�儨�儨�儨�� − ��

�儨�儨�儨�儨, �� = �� − ���푟� , �� = �� − �.

here, (�푋�, �푌�, ��) are the components of � in {�퐵}; ��� is a rota-tional transformation matrix from {�푚} to {�퐵}; (�푥�, �푥�, �푥�, �푦�,�푦�, �푦�, �푧�, �푧�, �푧�) are nine orientation parameters of �.

�e formulas for solving ��� (�푖 = 1, 2, 3), �� and �� are derived from Equation (1) and represented as follows:

here, � is the distance from �� to �표, �퐸 is the distance from �� to �.Under the geometric constraints of the RRPR-type

constraining active leg �푟2, �퐵�푚� is formed by 3 rotations of

(�푍, �푋1, �푌2), namely, a rotation of � about �-axis i.e., �1, fol-lowed by a rotation of � about �1-axis i.e., �2, and a rotation of � about �2-axis i.e., �3. Here, �1 is formed by � rotating about � by �, and �2 is formed by �1 rotating about �1 by �, see Figure 2. Each of (�푥�, �푥�, �푥�, �푦�, �푦�, �푦�, �푋�, �푌�) can be expressed by (�훼, �훽, �휆) from Equations (1) and (2) as follows:

(2)

�푚��푖 = �푒2(±�푞−10 ),�푚�2 = (0�푒0),��푖 = �퐸2(

±�푞−10 ), �2 = ( 0�퐸0 ),��푖 = 12(

±�푞�푒�푥�푙 − �푒�푦�푙 + 2�푋�표±�푞�푒�푥�푚 − �푒�푦�푚 + 2�푌�표±�푞�푒�푥�푛 − �푒�푦�푛 + 2�푍�표

), �2 = ( �푒�푦�푙 + �푋�표�푒�푦�푚 + �푌�표�푒�푦�푛 + �푍�표

),�푞 = √3; �푖 = 1, ± = +; �푖 = 3, ± = −

(3)

�푥�푙 = �푐�훼�푐�휆 − �푠�훼�푠�훽�푠�휆, �푥�푚 = �푠�훼�푐�휆 + �푐�훼�푠�훽�푠�휆, �푥�푛 = −�푐�훽�푠�휆,�푦�푙 = −�푠�훼�푐�훽, �푦�푚 = �푐�훼�푐�훽, �푦�푛 = �푠�훽,

�푋�표 = −�푦�푙(�푒 + �푍�표�푦�푛)�푦2�푙 + �푦2

�푚= �푠�훼

�푐�훽(�푒 + �푍�표�푠�훽),

�푌�표 = �퐸 − �푦�푚(�푒 + �푍�표�푦�푛)�푦2�푙 + �푦2

�푚= �퐸 − �푐�훼

�푐�훽(�푒 + �푍�표�푠�훽).

c1a3

c2

a1

a2

r3

1

1

3

a

x

β

B1

r3

B3

b1

b3

r1r2

Xo

XYo Y

Z2

z||Z3

y||Y2m

B

Zo

L3

L2

Z3

α

oe1

O

R1

e2R3

b2

B2

R2||X1

r1

r2

β

Y1

Y2

2||Z2

a

αX2, X1

Bo

Z1 Z

λ

c2

X

Figure 2: Kinematics and statics model of 2SPS+RRPR PM.

Journal of Robotics4

��, ��, �, and �� can be expressed by (�훼, �훽, �휆, �푍�) from Equation (1) to Equation (3) as follows:

The force situation of the 2SPS + RRPR PM is shown in Figure 2. The whole workloads can be simplified as a wrench (�, �) applied onto m at o. Here, � is a concentrated force and � is a concentrated torque. (�, �) includes the inertia wrench and the gravity of m, and inertia wrench and the gravity of the active legs and the external working wrench.

A�er solving the kinematics of the general PM and its legs, (�, �) can be solved [23]. (�, �) are balanced by 3 active forces ���(�푖 = 1, 2, 3), an active torque ��, and 2 constrained forces ���(�푗 = 1, 2). Here, each of ��� due to the linear actuators is applied on and along �� at ��, its unit vector �� is the same as that of ��; �� due to the motor 1 is applied on �2 at �2 and coin-cident with �1.

(4)

��푖 =12�푟1

(±�푞�푒(�푐�훼�푐�휆 − �푠�훼�푠�훽�푠�휆) + �푒�푠�훼�푐�훽 − ±�푞�퐸 + 2(�푒 + �푍�표�푠�훽)�푠�훼/�푐�훽±�푞�푒(�푠�훼�푐�휆 + �푐�훼�푠�훽�푠�휆) − �푒�푐�훼�푐�훽 + 3�퐸 − 2(�푒 + �푍�표�푠�훽)�푐�훼/�푐�훽

− ± �푞�푒�푐�훽�푠�휆 − �푒�푠�훽 + 2�푍�표

),

�2 = �2 = (�푐�훼 −�푠�훼 0�푠�훼 �푐�훼 00 0 1

)(1 0 00 �푐�훽 −�푠�훽0 �푠�훽 �푐�훽

)(001)

= (�푠�훼�푠�훽−�푐�훼�푠�훽�푐�훽

),�푞 = √3,�푖 = 1, ± = +,�푖 = 3, ± = −,

��푖 =�푒2(

±�푞(�푐�훼�푐�휆 − �푠�훼�푠�훽�푠�휆) + �푠�훼�푐�훽±�푞(�푠�훼�푐�휆 + �푐�훼�푠�훽�푠�휆) − �푐�훼�푐�훽

−�푞�푐�훽�푐�휆 − �푠�훽),

�2 = �푒(−�푠�훼�푐�훽�푐�훼�푐�훽�푠�훽

), � = ((�푒 + �푍�표�푠�훽)�푠�훼/�푐�훽

�퐸 − (�푒 + �푍�표�푠�훽)�푐�훼/�푐�훽�푍�표

),

�푟2�푖 =�퐷 + 2�푒2 + �퐸�푌�표 ± �푞(�푒�퐷1 − �퐸�푋�표) + �푒�퐸(±�푞�푦�푙 ± �푞�푥�푚 − 3�푥�푙 − �푦�푚)

2 ,

�푟22 = �퐷 − �푒2 − 2�퐸(�푒�푦�푚 + �푌�표), �퐷 = �푋2�표 + �푌2

�표 + �푍2�표 + �퐸2,

�퐷1 = �푥�푙�푋�표 + �푥�푚�푌�표 + �푥�푛�푍�표.

Let �� be the unit vector of ���, �� be the arm vector from ��� to �. Let v and � be the translational and angular velocities. Since ���(�푗 = 1, 2) limits the movement of PMs, based on prin-ciple of virtual work in [23], it is known that ���(�푗 = 1, 2) does not produce any power. �us, there are

�us, the geometric constrains of ���(�푗 = 1, 2) are determined in [23] as follows:

(1) Let v�푟2 be a velocity along prismatic joint � in �푟2, ��푐�푗v�푟2 = 0 i.e., ��푐�푗 ⊥ �2 must be satisfied.(2) Let ��(�푖 = 1, 2, 3) be a unit vector of revolute

joints �� in �2. Let �� × ��� be a torque of ��� about �푅�, �� ⋅ (�� × ���) = 0 must be satisfied. �us, each of ��� must either intersect or be parallel with all the rev-olute joints �� in �2. �us, the geometric constrained conditions {��푐1||�푅2, ��푐1 intersecting with both �1 and �3 at point �푎, ��푐2||�푅3, ��푐2 intersecting with both �1 and �2 at point �2} are satisfied.

From the geometric constrains of ���(�푗 = 1, 2), it leads to

�e general input velocity ��, the general output velocity � in {�퐵}, �퐹��(�푖 = 1, 2, 3), �푇� and �퐹��(�푗 = 1, 2) have been derived based on Equations (4)–(6) as follows:

(5)

�퐹�푐�푗��푗 ⋅ v + (��푗 × �퐹�푐�푗��푗) ⋅ � = 0�푗 = 1, 2 ⇒ ( 00) = ( �1 (�1 × �1)�푇

�2 (�2 × �2)�푇 )(

v

�).

(6)

�1 = �2 = (�푐�훼�푠�훼0), �2 = �3 = (

−�푠�훼�푐�훽�푐�훼�푐�훽�푠�훽

), � = (0�퐸

�푍�+(�푒+�푍��푠�)�푡��푐�

),

� − �2 = �푡�훽(�푒�푠�훽 + �푍�표)(−�푠�훼�푐�훼�푡�훽

), �1 = � − � =(�푒 + �푍�표�푠�훽)

�푐�훽(−�푠�훼�푐�훼�푡�훽

),

�2 = �2 − � = 1�푐�훽(

−�푠�훼(�푒 + �푍�표�푠�훽)�푐�훼(�푒 + �푍�표�푠�훽)

−�푍�표�푐�훽).

dri

S, U S, RA2i I1iI2i A1i

ri

ri–rli r1i aiai

(a)

Tav

R2A12

davI22 I12

A22

r2–r12 r12r2

R3

(b)

R2R3

A22

dc1

I12I22 A12

r2r12r2–r12

c1

(c)

R3R2A12

dc2

I22 I12A22

r2–r12 r12r2c2

(d)

Figure 3: Elastic deformations of SPS-type legs and RRPR-type leg under ���, ���, and ��v.

5Journal of Robotics

here, � is a 6×6 Jacobian matrix of the 2SPS+RRPR PM, �� is an angular velocity of �1 (motor 1).

3. Stiffness Matrix and Elastic Deformation of SPS-Type Legs and RRPR-Type Leg

Suppose that the rigid platform � is elastically suspended and by 3 elastic active legs �� and is constrained by one elastic con-strained leg �2. If only small displacements from its unpreloaded equilibrium position are considered, the overall wrench–deflec-tion relation of the mechanism is linear elasticity. Based on the constructed workspace, each of length of piston/cylinder for active legs and constrained leg can be determined. Let �푟1�푖, �퐴1�푖, �퐼1�푖 and �1�푖 be the length, the section of a piston, the moment of inertia, and the rotational moment of inertia of leg ��, respec-tively. Let (�푟�푖 − �푟1�푖), �퐴2�푖, �퐼2�푖, and �2�푖 be the length, the section, the moment of inertia, and the rotational moment of inertia of a cylinder of ��, respectively. Let ��� and �� be the modulus of elasticity and the rotational modulus of elasticity for leg ��, (�푖 = 1, 2, 3). When each of the active forces ���(�푖 = 1, 2, 3) applies onto the SPS-type active leg �푟� (�푖 = 1, 3) and the RRPR-type constrained active leg �푟�(�푖 = 2) and along ��, the longitudinal elastic differential deformation ��� of leg ��, see Figure 3(a).

�e longitudinal elastic differential deformations of the SPS-type active leg �푟�(�푖 = 1, 3) and the RRPR-type composite active leg �푟�(�푖 = 2) under ��� (a), �e transverse elastic differ-ential deformations of the RRPR-type composite leg r2 under ��� (b, c) and ��v (d).

When each of the active forces ���(�푖 = 1, 2, 3) applies onto the SPS-type active leg �푟�(�푖 = 1, 3) and the RRPR-type con-strained active leg �푟�(�푖 = 2) and along ��, the longitudinal elastic differential deformation ��� of leg �� (see Figure 3(a)) can be solved as below [24]

here, ��� is a longitudinal stiffness of SPS active leg �� and RRPR-type constrained active leg �푟�(�푖 = 2).

(7)

��푟 = �6×6�,� = �−1��푟, �훼�耠 = ( 01×3 ��푇 )�,� = ( v

�), � = ( �푠�훼�푡�훽−�푐�훼�푡�훽1 ),� = (�퐹�푥�퐹�푦�퐹�푧 ), � = (�푇�푥�푇�푦�푇�푧

),��푟 = (

v�푟1v�푟2v�푟3�훼�耠

00),

� = ((((

��푇1 (�1 × �1)�푇��푇2 (�2 × �2)�푇��푇3 (�3 × �3)�푇01×3 ��푇��푇1 (�1 × �1)�푇��푇2 (�2 × �2)�푇

))))

, (�퐹�푎1�퐹�푎2�퐹�푎3�푇�푎�퐹�푐1�퐹�푐2) = −(��푇)−1(��)

(8)�푑�푟�푖 = �퐹�푎�푖�푘�푎�푖 , �퐹�푎�푖 = �푘�푎�푖�푑�푟�푖, �푘�푎�푖 = − �퐸�푖( �푟�푖−�푟1�푖�퐴2�푖

) + ( �푟1�푖�퐴1�푖

) ,

�e active torque �� consists of a component ��� along �2 and a component ��v perpendicular to �2 (see Figure 3(b)). �ey can be expressed as follows:

When ��v is exerted onto leg �2 at universal joint and ��푎v ⊥ �2 is satisfied, the transverse elastic differential deflection ��

v of �2 at its end (Figure 3(b)) can be solved in [24] as follows:

When ��� is exerted onto leg �2 at universal joint, the elastic rotational differential deformation ��1 of leg �2 at its end can be solved based on the elastic deformation formula in [24] as below

here, ��v and ��휃1 are the transverse and rotational stiffness of leg �2 vs. ��v and ���, respectively.

From Equations (10)–(12), it leads to

When ��푐1 is exerted onto leg �2 at point � and ��푐1 �2 is satisfied, the transverse elastic differential deflection ��1 of �2 at its end (see Figure 3(c)) can be solved [24] as follows:

here ��푐1 is a transverse stiffness of r2 vs. ��푐1.When ��푐1 is exerted onto leg �2 at point � and ��푐1 �2 is sat-

isfied, the elastic rotational differential deformation ��2 of leg �2 at its end can be solved in [24] as follows:

Similarly, from Equations (15) and (16), it leads to

(9)��푎 = �푇�푎�, �푇�푎�푢 = ��푎 ⋅ �2 = �푇�푎�푐�훽, �푇�푎v = �푇�푎� ⋅ (�2 × �2) = �푇�푎�푠�훽.

(10)

�푑�푎v= − �푇�푎v2�퐸2

[�푟212�퐼12 + (�푟2 − �푟12)2�퐼22 ] = �푇�푎�푘�푎v ,�푘�푎v = −2�퐸2

[( �푟212�퐼12)�푠�훽 + ((�푟2−�푟12)2�퐼22

)�푠�훽] .

(11)

��1 = −��푎�푢�2

(�12�12+ �2 − �12

�22) = ��푎

��휃1, �푘�휃1 = −�퐺2

(( �푟12�퐽12)�푐�훽 + ( �푟2−�푟12

�퐽22)�푐�훽) ,

(12)

�푑�푎v+ �푑�휃1 = ( 1

�푘�푎v +1�푘�휃1)�푇�푎, → �푇�푎 = �푘1(�푑�푎v + �푑�휃1),

�푘1 = 1( 1�푘�푎v) + ( 1

�푘�휃1).

(13)

�푑�푐1 = �퐹�푐1�푘�푐1 , �푘�푐1 =−3�퐸2

[�푟312/�퐼12] + [(�푟2 − �푟12)(�푟22 + �푟212 + �푟2�푟12)/�퐼22] ,

(14)

�푑�휃2 =�儨�儨�儨�儨�푎 − �푏2�儨�儨�儨�儨�퐹�푐1�퐺2

[(�푟2 − �푟12�퐼22 + �푟12

�퐼12)] = �퐹�푐1�푘�휃2 ,

�푘�휃2 = �퐺2[�儨�儨�儨�儨�푎 − �푏2�儨�儨�儨�儨((�푟2 − �푟12/�퐼22) + (�푟12/�퐼12))] .

Journal of Robotics6

When ��푐2 is exerted onto leg �2 at �2 and ��푐2 �2 is satisfied, the transverse elastic differential deflection ��2 of �2 at its end (see Figure 3(d)) can be solved in [24] as follows:

Since �푑�푎v|�푑�푐1 is satisfied, both ��1 and ��2 are the elastic rota-

tional differential deformations of �2, an equation of force-deformation for the 2SPS + RRPR PM is derived from Equation (8) to Equation (16) as follows:

(15)

�푑�푐1 + �푑�휃2 = ( 1�푘�푐1 + 1

�푘�휃2)�퐹�푐1, → �퐹�푐1 = �푘2(�푑�푐1 + �푑�휃2),�푘2 = 1

( 1�푘�푐1

+ 1�푘�휃2) .

(16)

�푑�푐2 = �퐹�푐2�푘�푐2 , �푘�푐2 =−3�퐸2

[((�푟2 − �푟12)3/�퐼22) + (�푟312 + 3(�푟2 − �푟12)�푟2�푟12/�퐼12)] .

(17)(

��푎1��푎2��푎3��푎��푐1��푐2

) = ��푟(

��1��2��3������2

), (�푑�푟1�푑�푟2�푑�푟3�푑�푐����2

) = �−1�푟 (

��푎1��푎2��푎3��푎��푐1��푐2

),��푟 = (

�푘�푎1 0 0 0 0 00 �푘�푎2 0 0 0 00 0 �푘�푎3 0 0 00 0 0 �푘1 0 00 0 0 0 �푘2 00 0 0 0 0 �푘�푐2)

�푑�푐 = �푑�푎v+ �푑�푐1, �� = ��1 + ��2, �푘1 = 1(1/�푘�푎v)+(1/�푘�휃1) , �푘2 = 1(1/�푘�푐1)+(1/�푘�휃2) .

here, �� is a 6 × 6 symmetric total stiffness matrix of the legs �푟�(�푖 = 1, 2, 3).4. Total Stiffness Matrix and the Elastic Deformation of 2SPS+RRPR PM

Based on principle of virtual work in [22], it is known that when a deformed mechanical system keeps a static balance under all external wrenches, the sum of the work generated by all external wrenches along virtual displacements of the mechanical system and the work produced by all internal wrenches along virtual deformations of the same mechanical system must be zero. �erefore, the sum of the work gener-ated by (��푎1, ��푎2, ��푎3, ��푎, ��푐1, ��푐2) along deformations of the 2SPS + RRPR PM and the work produced by (�,�) along

0 1 2 3

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

r1

r2r3

Exte

nsio

n of

legs

(m)

Time t (s)

(a)

Posit

ion

of m

(m)

0 1 2 3

Yo

Zo

Xo

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Time t (s)

(b)

0 1 2 3

Ta

α

Rota

tiona

l ang

ular

of r

1 (ra

d)To

rque

Ta (

N·m

)

8

7

6

5

4

3

2

1

0

Time t (s)

(c)

0 1 2 3

Fa2

Fc2

Fc1

Fa1

Fa3

50

40

30

20

10

0

–10

–20

–30

Act

ive c

onst

rain

ed fo

rces

(N)

Time t (s)

(d)

0 1 2 3

dr2

dr1 dr3

3

2.5

2

1.5

1

0.5

0

–0.5

–1

–1.5

Def

orm

atio

n of

r i ×

10–4

(mm

)

Time t (s)

(e)

0 1 2 3

do

dXo

dYo

dZo

1

0.5

0

–0.5

–1

Posit

ion

defo

rmat

ion

of o

(mm

)

Time t (s)

(f)

0 1 2 3

dφx

dφz

dφy

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1

–1.2

Def

orm

atio

n of

m ×

10–3

(rad

)

Time t (s)

(g)

0 1 2 3

dc1

dθ2

dav dc2

dθ1

Tran

sver

se d

efor

mat

ion

of r 2

(mm

)Ro

tatio

nal d

efor

mat

ion

of r 2

× 1

0–2 (r

ad)

0.1

0.5

0

–0.5

–0.1

–1.5

Time t (s)

(h)

Figure 4: �eoretical solutions of elastic deformations of 2SPS + RRPR PM.

7Journal of Robotics

Here

� = −[�−1�−1�푟 (��푇)−1]

−1

= −(��푇��푟�)6×6 = (

�푘11 �푘12 ⋅ ⋅ ⋅ �푘16�푘21 �푘22 ⋅ ⋅ ⋅ �푘26...

.

.

....

.

.

.

�푘61 �푘62 ⋅ ⋅ ⋅ �푘66

),

(20)

here � is a 6 × 6 symmetric total stiffness matrix of this manipulator; (�푑�푋�, �푑�푌�, �푑�푍�, �푑�휑�, �푑�휑�, �푑�휑�) are the 6 elastic deformation components of platform. When given (�,�), the elastic differential deformation of this manipulator can be solved from Equation (19).

5. Analytic Solved Example of Elastic Deformation for 2SPS+RRPR PM

In the 2SPS + RRPR type PM, let initial independent pose var-iables vary vs. time � when given pose parameters (�훼, �훽, �휆, �푍�), see Figures 4(a) and 4(b).

Set �퐿 = 1.2m, �푙 = 0.6m; � = −[20 30 60]�N and � = [−0.3 − 0.31]� Nm, �퐸�푖 = 2.11 × 1011 Pa, the diameter of piston and cylinder for active legs �푟�(�푖 = 1, 2, 3) are �퐷�푖1 = �퐷�푖2 = 0.04m, �퐸�푖�퐼�푖1 = �퐸�푖�퐼�푖2 = 26502N ⋅m2, �퐴 �푖1 = �퐴 �푖2 =0.0013m2. By using the relevant theoretical equations and Matlab, the extensions of �� and α are solved, see Figure 4(a). �e position components (�푋�, �푌�, �푍�) of the moving platform � are solved, see Figure 4(b). �ree active forces �퐹��(�푖 = 1, 2, 3), one active torque ��, two constrained forces ��푐1 and ��푐2 are solved, see Figures 4(c) and 4(d). �e longitudinal deformations �푑�푟�(�푖 = 1, 2, 3) of �� are solved, see Figure 4(e). �e position deformations of � at � are solved, see Figure 4(f). �e angular deformations of � are solved, see Figure 4(g). �e transverse deformations �푑�푐�(�푗 = 1, 2) and ��2 + ��

v and the rotational

deformations ��1 and ��2 of �2 are solved, see Figure 4(h).When �1 = 1.8, �푟3 = 1.7, �푟2 = 1.66m. �훼 = 0∘ , � is solved

from Equations (7) and (20) as follows:

6. A FE Model of 2SPS + RRPR PM and Its Solutions

A 3D assembly mechanism of the 2SPS + RRPR PM is con-structed in SolidWorks [25]. Next, its finite element (FE) model is generated in ANSYS, see Figure 5. All relative geom-etry and material parameters of the 3D simulation assembly mechanism are the same as that in Section 5. �e 3 equivalent revolute joints for 3 actuated revolute joints and 4 equivalent spherical joints for 4 actuated spherical joints are constructed, see Figure 5(a). �e applied loads are shown in Figure 5(b). �e boundary condition are explained as follows:

(21)

�퐾 = (0.0923 0.0130 0.0266 0.0038 0.1466 −0.02790.0130 0.2417 0.7799 −0.1465 0.0052 −0.00420.0266 0.7799 4.2941 0.0905 −0.0097 −0.00900.0038 −0.1465 0.0905 0.2721 0.0150 −0.00100.1466 0.0052 −0.0097 0.0150 0.2338 −0.0446−0.0279 −0.0042 −0.0090 −0.0010 −0.0446 0.0084

).

the displacements of point � in {�퐵} must be zero. Let (�푑�푋�, �푑�푌�, �푑�푍�) be 3 translational components of the elastic differential deformation of � at � in {�퐵}; (�푑�휑�, �푑�휑�, �푑�휑�) be 3 rotational components of the elastic differential deforma-tion of � in {�퐵}. �us, based on the theorem of work and energy equal to each other, from Equation (7) to Equation (17), it leads to

�us, from Equations (7), (14), and (15), it leads to

(18)

(

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

)

�푇

(

�퐹�푎1�퐹�푎2�퐹�푎3�푇�푎�퐹�푐1�퐹�푐2

) = −(

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

)

�푇

(��)

⇒ (

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

)

�푇

[−(��푇)−1] = −(

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

)

�푇

,

(

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

)

�푇

(�−1)�푇 = (

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

)

�푇

⇒ �−1(

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

)

= (

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

) ⇒ (

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

) = �(

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

).

(19)

(

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

) = �−1(

�푑�푟1�푑�푟2�푑�푟3�푑�푐�푑�휃�푑�푐2

) ⇒ (

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

)

= �−1�−1�푟 (

�퐹�푎1�퐹�푎2�퐹�푎3�푇�푎�퐹�푐1�퐹�푐2

), (�퐹�푎1�퐹�푎2�퐹�푎3�푇�푎�퐹�푐1�퐹�푐2

)

= −(��푇)−1(��),⇒ ((

�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

))

= �−1(��) ⇒ �−1

= −�−1�−1�푟 (��푇)−1 ⇒ (��) = �(�푑�푋�표�푑�푌�표�푑�푍�표�푑�휑�푥�푑�휑�푦�푑�휑�푧

),

Journal of Robotics8

(a) (b)

(c) (d)

(e) (f) (g)

Figure 5: Simulation solutions of elastic deformations of EF model of the 2SPS + RRPR PM. (a) Equivalent spherical joint � and revolute joints �1 and �2, (b) load condition, (c) FE model of 2SPS + UPR PM and its elastic deformation, (d) elastic deformation of �, (e) elastic deformation ���, (f) elastic deformation ��� and (g) elastic deformation ���.

9Journal of Robotics

7. Analysis of Stiffness and Elastic Deformation of 2SPS+RRPR PM

Several conclusions are obtained from theoretical and simu-lation solutions as follows:

(1) �e solved results of FE model in most cases are approximate numerical results which depend on some key factors such as finite element dimension and type, equivalence between actual joints and simulation joints, selected material parameter, solver, reasonable boundary constraints and connection constraints [23].

(2) It is known from Table 1 that the elastic deformations of FE model of this PM are basically coincident with that of theoretical ones in Section 5.

(3) It is known from Figures 4(e) and 4(h) that the transverse elastic deformations �푑�푐�(�푗 = 1, 2) (0.5→1.2 × 10−4) of �2 due to the constrained forces ��� are greatly larger than the longitudinal elastic defor-mation ��� (0.5→2.5 × 10−7) of � = −[�−1�−1

�푟 (��푇)−1]−1

= −(��푇��푟�)6×6 = (

�푘11 �푘12 ⋅ ⋅ ⋅ �푘16�푘21 �푘22 ⋅ ⋅ ⋅ �푘26...

.

.

....

.

.

.

�푘61 �푘62 ⋅ ⋅ ⋅ �푘66

), due to active forces ���. It implies that the constrained wrench has great influence on the elastic deformation of this PM.

(4) It is known from Figure 4(c), h that the transverse elastic deformations �푑�푐�(�푗 = 1, 2) of �2 due to the constrained forces ��� is larger than the transverse elastic deformation ��

v of �2 due to active torque ��v.

�e transverse elastic deformations and elastic rota-tional deformation of the SPS-type legs �푟�(�푖 = 1, 3) is 0. �erefore, the diameter of piston and cylinder of �2 should be increased.

(5) It is known from Figure 4(h) that the elastic rotational deformation ��1 of �2 due to ��푐1 and the elastic rota-tional deformation ��2 of �2 due to ��푐2 are inversely proportional to each other.

8. Conclusions

A 2SPS + RRPR parallel manipulator with asymmetrical struc-ture is suitable for the helicopter rotor supporting base.

�e formulas for solving the stiffness matrix and the elastic deformation of its three asymmetrical legs are derived. �e formulas for solving its total stiffness matrix and the elastic deformation are derived based on the Jacobian matrix and the stiffness matrix of three asymmetrical legs. Both the stiffness matrix of its three asymmetrical legs and its total stiffness matrix are 6 × 6 symmetric matrices, although this manipula-tor has asymmetrical structure.

�e constrained wrench must be taken into account when establishing its total stiffness matrix and solving its elastic deformation.

�e proposed methodological results can be applied to other less mobility parallel manipulators with asymmetrical structure and active legs for solving the elastic deformations of asymmetrical active legs and the elastic deformations of moving platform.

(1) If no setting is given, all the assembly parts in the FE model may constitute the same elastic body. �erefore, each of the assembly spherical joints in FE model constitutes the same elastic body. �e sim-ulation 3D assembly of spherical joints is used only for varying the pose of PM and the workload applied on � at �.

(2) All the relative geometry parameters of the 3D sim-ulation assembly mechanism are the same as that in Section 5. �e material parameters of �푟�(�푖 = 1, 2, 3) are set as the same as that in Section 5.

(3) Construct 3 equivalent spherical joints for 4 actuated spherical joints, see Figure 5(a). Here, the diameters at the two ends of the SPS-type legs �푟�(�푖 = 1, 3) are reduced sharply.

(4) Construct 3 equivalent revolute joints for the actu-ated revolute joints �푅1, �푅2, �푅3, see Figures 5(a) and 5(b). Here, two holes for each of equivalent revo-lute joints are constructed and kept coincident with each other; the rotational stiffness and the axial stiffness are set as 0 and 1 × 1010 N/mm, respec-tively, according to the requirement for revolute joint in so�ware.

(5) Each of the 3 linear active legs with prismatic joints is formed using the elastic linear rod, which is assigned by the alloy steel. Set SPS leg �푟1 = 1.8, SPS leg �푟3 = 1.7, RRPR leg �푟2 = 1.66m.

(6) A fixed constraint is added onto the base, which is assigned by the alloy steel with rigid body.

(7) �e workload wrench � = −[20 30 60]�N and � = [−0.3 − 0.31]�Nm are applied onto � at �, which is assigned by the alloy steel with the rigid body, see Figure 5(b).

Some solved results of the elastic deformations are shown in Figures 5(c)–5(g) and Table 1.

An existing CAD so�ware provides a function for auto-matically optimal mesh in order to avoid singularity element and to obtain the suitable results of finite element analysis (FEM). �erefore, the 3D assembly mechanism of the 2SPS + RRPR PM is automatically meshed by the function for automatically optimal mesh.

Table 1:  Simulation solved results of elastic deformations of EF model of 2SPS + RRPR PM.

Elastic deformation of �, mm Position of � (m)FE model �eoretical �� �� ��

�� 0.9058 0.8934 FE model position of No. 19319 joint��� −0.8387 −0.8550 −0.049 0.317 1.71��� −0.3434 −0.2547 �eoretical position of ���� 0.0258 0.0534 0 0.3192 1.6540

Journal of Robotics10

��1, ��2: Rotational deformations of �2�푑�훼, �푑�훽, �푑�휆: Differential deformations of 3 Euler angles of ��: One of (�훼, �훽, λ, �훾1, �훾2), �푠� = sin�휑, �푐� = cos�휑, �푡� = tan�휑���: �e length of �푟�, �푗 = 1 for piston, �푗 = 2 for cylinder�퐴 ��, �퐷��: �e cross-section and the diameters of ��||,⊥, |: Perpendicular, parallel collinear constraint.

Data Availability

�e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

�e authors declare that they have no conflicts of interest.

Acknowledgments

�e authors would like to acknowledge Major Research Project (91748125) supported by National Natural Science Foundation of China.

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Nomenclatures

DoF: Degree of freedomPM: Parallel manipulator�, �: Base and moving platform{�푚}: Coordinate frame �-��� fixed on �{�퐵}: Coordinate frame �-��� fixed on ��: �e center point of ��: �e center point of �����: �e vertices of �푚, �퐵(�푗 = 1, 2, 3)�, �: �e side of �푚, �퐵�: �e distances from �� to ��: �e distances from �� to ���: Active leg and its length (�푖 = 1, 2, 3)�1�2�3: �ree revolute joints�, �: Revolute joint, prismatic joint�푈, �푆: Universal joint and spherical joint�푋�, �푌�, ��: Position components of � in {�퐵}�, �, �: �ree Euler angles of ����: Rotational transformation matrix�: Jacobian matrix���: Modulus of elasticity for ����: Rotational modulus of elasticity for ���: �e total complacence matrix of PM��: �e active torque���: Linear inertia moment of �����: Rotational inertia moment of ����: �e general input velocity�: �e general output velocity in {�퐵}��, ��, �푥�, �푦�, ��, �푦�, �푧�, ��, ��: Nine orientation parameters of �

��v��: �e input velocity along active leg ��v, �: �e linear and angular velocities of � at �

in {�퐵}�, �: �e concentrated force and torque

applied on � at ����, ��: �e active forces and their unit vectors��, �: �e constrained forces and their unit

vectors�퐹��, �퐹�v: �e components of ����퐹�푐1, �퐹�푐2: �e components of ��푐, �퐹�푐1||�푟2, �퐹�푐2 �푟2�푇��, �푇�v: �e components of ��푎, �푇�푎�푢||�푟2, �푇�푎v �푟2��: 6 × 6 total stiffness matrix of the legs �푟�(�푖 = 1, 2, 3)���: A longitudinal stiffness of ���푘�푎v , �푘�휃1: �e transverse and rotational stiffness of

leg �2��푐1: Transverse stiffness of �2 in plane with ��푐1�: �e stiffness matrix of PM���: �e longitudinal elastic differential

deformation of leg ���푑�푋�, �푑�푌�, �푑�푍�: 3 deformation components of � at � in {�퐵}�푑�휑�, �푑�휑�, �푑�휑�: 3 rotational deformation components of m in {�퐵}��: Elastic differential deformation of � at �

in {�퐵}��2: Transverse elastic differential deflection of �2 at its end

11Journal of Robotics

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