Research on the dynamic coupling of the rigid-flexible manipulator

Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Research on the dynamic coupling of the rigid-flexible manipulator Zhihua Liu, Xiaoqiang Tang n, Liping Wang The State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

art ic l e i nf o

a b s t r a c t

Article history: Received 5 June 2014 Received in revised form 14 September 2014 Accepted 8 October 2014

Because of its large dimension, a rigid-flexible manipulator was designed to serve as the feed support system in a Five-hundred-meter Aperture Spherical radio Telescope (FAST). The rigid-flexible manipulator is composed of a flexible cable-driven parallel manipulator and a rigid Stewart platform. Motion of the Stewart platform may induce vibration of the cable-driven parallel manipulator due to reaction forces. The goal of this study is to investigate the dynamic coupling of the rigid-flexible manipulator. The “Virtual Stewart platform” was introduced to obtain a homogenous matrix to describe how much base motion is produced by a given Stewart platform motion. On this basis, an index was proposed to characterize the dynamic coupling of the rigid-flexible manipulator. Based on the proposed index, the factors influencing the dynamic coupling are investigated. The proposed index can be considered as a performance index in design and control of the system. Influence of the dynamic coupling on the feedback control of the rigidflexible manipulator was also discussed. The simulation results showed that the dynamic performance of the rigid-flexible manipulator is strongly determined by the proposed index. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Rigid-flexible manipulator Dynamic coupling Feedback control FAST

1. Introduction In 1994, Chinese astronomers carried out the conceptual design of the Five-hundred-meter Aperture Spherical radio Telescope (FAST) [1]. Because of the large dimension of the telescope, a rigid-flexible manipulator is designed to serve as the feed support system. The flexible cable-driven parallel manipulator provides the receiver with a wide range of translation and rotation [2]. However, due to the flexibility of the long-span cables, the feed support system bears a concern of possible vibration under wind disturbance in the open air [3]. The rigid Stewart platform is then used to compensate the positioning error for achieving the required accuracy. Compensating control of the rigid mechanism, however, is challenging, especially when the platform's mass and inertia cannot be negligible in comparison with those of the base. Platform motion can induce base motion resulting from dynamic interaction. Furthermore, the base motion will alter the original trajectory of the platform. This dynamic coupling is another cause besides the wind disturbance caused vibration of the rigid-flexible manipulator. However, research on FAST mainly focused on vibration suspension [4–8], dimension optimization [9] and workspace analysis [10,11]. Dynamic coupling of the rigid-flexible manipulator has received very little investigation. This motivates the study in this paper.

n

Corresponding author. E-mail address: [email protected] (X. Tang).

http://dx.doi.org/10.1016/j.rcim.2014.10.001 0736-5845/& 2014 Elsevier Ltd. All rights reserved.

Dynamic coupling problems of free floating robots have been studied by a few groups. Understanding of the dynamic coupling of the system is the first step. Dubowsky and Torres [12] introduced the disturbance mapping to relate the robot joint motion to the base attitude disturbance. Xu [13,14] defined a measure to characterize the degree of the dynamic coupling. The defined measure describes how much base motion is produced by a given robot end-effector motion. To realize zero reaction to the base, dynamic balance control was proposed by Huang et al. [15,16]. In the dynamic balance control, a space robot system consisting of two arms was developed, with the mission arm for accomplishing the capture mission, and the balance arm compensating for the disturbance to the base. To minimize the dynamic disturbance, path planning for space manipulators was also presented. Dubowsky and Torres [12] found paths to minimize the dynamic disturbance with the aid of the enhanced disturbance map. Huang et al. [17] used genetic algorithms to search for the optimal joint inter-knot parameters in order to realize the minimum disturbance. Chung et al. [18] presented new design of gaits for the space manipulator to reduce the dynamic disturbance to a space station. The above methods regarding the free floating robots have certain merit on the rigid-flexible manipulator. However, the defined metric [13,14] does not apply to the base or the end-effector with both translational and rotational freedoms because of non-homogenous dimension in the velocity vectors. Thus, it is necessary to propose a dynamic coupling index with homogenous dimension. The dynamic coupling strongly influences control performance of the rigid-flexible manipulator. Control schemes for the macro/micro manipulators include using the rigid robot to compensate for

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

mL mU mP rB rL rU rP IB IL IU IP B s χ

Nomenclature B

tP RP B_ tP B

ωP tP B εP bi pi Ci Ti mB B

B€

Position of the platform in fBg Rotation matrix of the platform in fBg Linear velocity of the platform in fBg Angular velocity of the platform in fBg Linear acceleration of the platform in fBg Angular acceleration of the platform in fBg Coordinate of bi in fBg Coordinate of pi in fP g Coordinate of C i in fBg Coordinate of T i in fGg Mass of the base

Feed support system

Mass of lower part of the legs Mass of upper part of the legs Mass of the platform Mass center of the base in fBg Mass center of lower part of the legs in fLg Mass center of upper part of the legs in fU g Mass center of the platform in fP g Inertia matrix of the base in fBg Inertia matrix of lower part of the leg in fLg Inertia matrix of upper part of the leg in fU g Inertia matrix of the platform in fP g Unit leg vector in fBg Unit cable vector in fGg

This paper is organized as follows: In Section2, system of the rigid-flexible manipulator is described in details. In Section3, dynamics model of the rigid-flexible manipulator is established. In Section4, dynamic coupling index is proposed based on the concept of “Virtual Stewart platform”. In Section5, influence of the dynamic coupling on the dynamic performance is discussed. In Section6, influential factors of the dynamic coupling are analyzed, including the configuration, the structural parameters and the inertia properties. Finally, Conclusions of this paper are given.

micromanipulator position error while at the same time actively using the micromanipulator to reduce the vibration. Torres et al. [19,20] introduced energy dissipation method to maximizing the energy dissipated by commanding the actuators of the robot to behave as passive linear springs and dampers. George [21,22] developed the inertial damping control to modulate the manipulator actuators to induce the inertial damping forces by sensory feedback of the base vibration. The above control schemes have ability to suppressed high-frequency vibration of the macro manipulator instead of low-frequency vibration of the large span cable-driven parallel manipulator. The major goal of the rigid-flexible manipulator in this paper is to achieve the positioning and orientating precision. The proposed controllers were mainly based on two factors: prediction [23] and feedback [6]. The prediction deals with the base motion caused by the wind disturbance and the feedback aims to construct a closed loop system to achieve the control precision. Duan et al. [5] presented a decoupled tracking and prediction algorithm to predict the position and orientation of the base and designed an upper layer adaptive interaction PID supervisory controller. Ren et al. [4,24] used the predicted inertial motion of the base as the reference input to develop the PD control law for the six actuators. In controlling a rigid-flexible manipulator, the resultant base motion from the platform motion is undesirable and should be restricted within a limited range. If the dynamic coupling is large enough, it may result in ineffectiveness of the prediction and instability of the control system. As a consequence, the rigid-flexible manipulator may vibrate heavily and may even become uncontrollable. Therefore, it is necessary to carry on a research on influence of the dynamic coupling on the control performance.

A3

2. System description In Fig. 1, the FAST is composed of a reflector and a feed support system. The feed support system is a rigid-flexible manipulator consisting of two mechanisms, which are arranged in series: a flexible six-cable-driven parallel manipulator and a rigid Stewart platform. The six-cable-driven parallel manipulator is composed of six towers, six cables and a base. Ai ði ¼ 1; …; 6Þ are the connected points of the cables and the towers. C i ði ¼ 1; …; 6Þ are the connected points of the cables and the base. In Fig. 2, the Stewart platform is composed of a base, a platform and six legs. The leg is connected to the base by a universal joint bi and connected to the platform by a spherical joint pi . The universal joints bi ði ¼ 1; …; 6Þ are distributed in a circle of radius r B and angle of two adjacent points is θB . The spherical joints pi ði ¼ 1; …; 6Þ are distributed in a circle of radius r P and angle of two adjacent points is θP . Coordinates are set up in the rigid-flexible manipulator as follows. Let fGg be the inertial frame, with its origin at the center

A2

A2

A3

A1

A4

73

A1

A4 A5

A6

z

A5

B x

y

Cable

A6

C4 C5

C3

C6

Base

C2 C1

Leg

z Platform

x

G

y

Reflector

Fig. 1. 3-D model of FAST: (a) isometric view, (b) feed support system.

74

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

b4 z b5

x

b6

B

b3 y

b1

z b2

Y

L y

θB

x

b3 p3

b2

p2

rB

b4

θP

p5 p4 p5

x p6

p6

z

z

y p3 P

p1

p2

U

X

p1

rP

p4

b1

y

b5 b6

x

Fig. 2. The Stewart platform: (a) isometric view, (b) top view.

of Ai ði ¼ 1; …; 6Þ, the y-axis crossing the midpoint of A1 A6 , and the z-axis along the opposite of the gravity. Let fBg be the base frame, with its origin at the geometric center of the base, the y-axis crossing the midpoint of b2 b3 and the z-axis perpendicular to the base platform plane. Frame fLg is attached to bi , with its x-axis along the leg and the y-axis along the fixed pivot of the universal joint. Frame fUg is attached to pi , and parallel to the local frame fLg. Let fPg be the platform frame, with its origin at the geometric center of the platform, the y-axis crossing the midpoint of p2 p3 and the z-axis perpendicular to the platform plane. Symbols used in this paper are listed in the nomenclature and have been described at the place of first occurrence.

z

G

y

Ai

x

f c

k

χ Ci

z y

B x

3. Dynamics Fig. 3. Dynamics of the cable.

The linear accelerations of mass center of the base, lower part of the leg, upper part of the leg, and the platform can be given by 
G€ rB ¼ G t€ B þ G εB  G RB rB þ G ωB  G ωB  G R B rB ð1Þ rL ¼ G t€ B þ G εB  G R B B rL þ G ωB 

G€



G

rU ¼ G t€ B þ G εB  G R B B rU þ G ωB 

G€

G

ωB  G R B B rL þ 2G ωB  G R B B r_ L þ G R B B r€ L




ωB  G RB B rU þ 2G ωB  G R B B r_ U þ G RB B r€ U

ð3Þ rP ¼ G t€ B þ G εB  G RB B rP þ G ωB 

G€



G


ωB  G R B B rP þ 2G ωB  G RB B r_ P þ G R B B r€ P

ð4Þ G€

G

where tB , RB , ωB and εB are linear acceleration, orientation, angular velocity, and angular acceleration of the base in fGg; B rL , B_ rL , and B r€ L are position vector, linear velocity, and linear acceleration of lower part of the leg in fBg; B rU , B r_ U , and B r€ U are position vector, linear velocity, and linear acceleration of upper part of the leg in fBg; B rP , B r_ P , and B r€ P are position vector, linear velocity, and linear acceleration of the platform in fBg. Angular velocities of the leg and the platform can be given G

G

G

W ¼ G ωB þ G R B B W

ð5Þ

G

ωP ¼ G ωB þ G RB B ωP

ð6Þ

B

where W and ωP are angular velocity of the leg and the platform in fBg. B

G

E ¼ G εB þ G ω B  G R B B W þ G R B B E

ð7Þ

G

εP ¼ G εB þ G ω B  G R B B ω P þ G R B B εP

ð8Þ




ð2Þ 

Similarly, angular accelerations of the leg and the platform can be given

B

where E and εP are angular accelerations of the leg and the platform in fBg. As shown in Fig. 3, assuming that all the cables are stretched under tension and deflected within their linear elastic range, we model the cable as the spring damper [3,25]. The cable force acting on the base can be calculated in terms of the cable elastic deformation and sliding velocity B

f ¼ kðℓ  ℓ0 Þ  cℓ_

ð9Þ

where ℓ is the cable length, ℓ0 is the initial cable length, ℓ_ is the sliding velocity; k is the elastic coefficient and c is the damping coefficient. According to the Newton–Euler approach, dynamic equations of the rigid-flexible manipulator are obtained as _ B þ mt_ þ mt _ ¼ wg þ we þ JTC f Mt_ B þ Mt where  T tB ¼ G t_ TB G ωTB  t¼

B _T B T tP ωP

T

ð10Þ

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

h f ¼ f1

…

f6

iT

ð11Þ

In the above equations, we is the external force and moment; the detail expressions for JC , M, m, and wg are listed in Appendix A. 4. Dynamic coupling index In the following, a measurement index is proposed to characterize the dynamic coupling. Since the dynamic coupling describes the interaction between parts of the Stewart platform, the gravity, the cable forces, and the external disturbance are set aside in the analysis, therefore, Eq. (10) can be written as _ B þ mt_ þ mt _ ¼0 Mt_ B þ Mt

75

This relationship describes how much base motion is produced by a given platform motion. This relationship will 
T  result in an
eigenvalue problem of the matrix JB M  1 mJ  1 JB M  1 mJ  1 . If the sliding velocity of the leg is of unit vector, the sliding velocity of the virtual leg is bounded within an ellipsoid expanded by JB M  1 mJ  1 , and the axes of the ellipsoid is determined by its singularity value. Therefore, it is rational to define the maximum 
singularity value, σ max JB M  1 mJ  1 , as the Dynamic Coupling Index (DCI for short) because it represents the maximum base motion caused by the dynamic coupling.

5. Dynamic performance

ð12Þ

Integrating Eq. (11) and assuming zero initial conditions, we get MtB þ mt ¼ 0

ð13Þ

Eq. (13) describe relationship between the velocity of the platform and the velocity of the base. However, it is infeasible to measure the dynamic coupling by making use of the above equations, because elements in the velocity vector are not dimensionally homogenous. Then, “Virtual Stewart Platform” is introduced to formulate an equation with homogenous dimension. As illustrated in Fig. 4, it is assumed that a virtual base is mounted on the inertial frame, virtual joint o is at the extension line of the actual leg, virtual leg B connects the virtual joint o and the universal joint b, and virtual leg P connects the virtual joint o and the spherical joint p. At any given time, the actual leg can be expressed in terms of the virtual leg B and virtual leg P L ¼ LP  L B

ð14Þ

According to the deduction in [26], using Jacabian matrix we can obtain sliding velocity vectors of the actual leg and the virtual leg B as L_ ¼ Jt

ð15Þ

L_ B ¼ JB tB

ð16Þ

where the detail expressions for J and JB are listed in Appendix A. Substituting Eq. (15) and Eq. (16) into Eq. (13), we can get L_ B ¼ JB M  1 mJ  1 L_

Motion of the platform can alter trajectory of the base due to the dynamic interaction. Control of the rigid-flexible manipulator is challenging, especially when the platform's mass and inertia cannot be negligible in comparison with the base. If the dynamic coupling is large enough, the control system may result in poor dynamic performance or even become unstable. In the following, we analyze the influence of the DCI on the control system.

5.1. Linearized feedback control As previously mentioned, feedback is an effective way to control the rigid-flexible manipulator. Fig. 5 depicts flow diagram of the feedback control of the rigid-flexible manipulator. The desired trajectory of the platform G tP ; G R P Þ is the reference input. The command strokes li for  the six legs are determined with the pose feedback of the base G tB ; G R B Þ and the desired trajectory of the platform G tP ; G R P Þ, according to the inverse kinematics of the rigid-flexible manipulator. The Laplace transform method is an effective way for analyzing the linear control system. In order to obtain the transfer function of the control system, linearized dynamics of the rigid-flexible manipulator are developed. On taking small variations of both sides of Eq. (10) about a given operation point and, we get _ B þ Mδt _ B þ δmt_ þ mδtþ _ δmt _ þ mδt _ ¼ δwg þ δwe þ δJTC f þ JTC δf Mδt_ B þ δMt_B þ δMt

ð19Þ

ð17Þ

Then, the ratio of the velocity magnitude is determined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 
T 
u_T JB M  1 mJ  1 L_ L JB M  1 mJ  1 ‖L_ B ‖ u ð18Þ ¼t T _ ‖L‖ L̇ L_

If we assume that the mechanical system is in a state of equilibrium _ ¼ 0, m _ ¼ 0. For the sake of convenience, we neglect the such that M _ δm, _ δwg , and δJTC , substituting Eq. (9) into variations of δM, δm, δM, Eq. (19), we obtain the linearized dynamics Mδt_ B þ JTC cJC δtB þ JTC Kδℓ ¼ δwe  mδt_

ð20Þ

where

Virtual base

o lB

 δℓ ¼ δℓ1

…

δℓ6

T

 ; K ¼ k1

…

k6

T

 ; c ¼ c1

…

c6

T

Eq. (20) can be written in the following alternative form by substituting the results from Eq. (15) and Eq. (16). T 1 _ T 1 1 € € δL MJ1 B δL B þ JC cJC JB δL B þ JC KJC JB δL B ¼ δw e mJ

b

lP t P , RP G

G

l

Fig. 4. Virtual base and virtual legs.

t P , G RP

l6

l6

p

ð22Þ

l1

l1 G

Platform

ð21Þ

G

t B , G RB

Fig. 5. Feedback control of the rigid-flexible manipulator.

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Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

δLP

δL

num(s ) den(s )

δL

MJ B−1s 2 + J CT cJ C J B−1s + J CT KJ C J B−1 − mJ −1s 2 MJ B−1s 2 + J CT cJ C J B−1s + J CT KJ C J B−1 − mJ −1s 2

δLB

δwe

MJ B−1s 2 + J CT cJ C J B−1s δLP + J CT KJ C J B−1 − mJ −1s 2

δwe

Fig. 6. Linearized feedback control.

δL

.

+ −

K pp Position loop controller

+ −

EVI K pv (Tiv s + 1) Tiv s

Kt

1 JM

1 s

.

δL

1 s

δL

Velocity loop controller

Fig. 7. Closed-loop control of the leg model.

On taking Laplace transform of Eq. (22) and considering Eq. (14), we get 
T 1 T 1 2 MJ1 δLP ¼ δwe B s þ JC cJC JB s þ JC KJC JB 
T 1 T 1 1 2 2 þ MJ1 s δL B s þJC cJC JB s þ JC KJC JB mJ 

ð23Þ


T 1 T 1 1 2 2 s δLB ¼ δwe  mJ  1 s2 δLP MJ1 B s þ JC cJC JB s þ JC KJC JB  mJ

Table 1 Simulation parameters. Symbol

Physical meaning

Value

JM Kt K pv T iv K pp

Motor inertia Motor torque constant Scale factor of speed loop Integral time constant of speed loop Scale factor of position loop

1.64  10  5 kg m2 0.45 N m/A 0.06 A s/rad 0.001 S 50 1/s

ð24Þ On the basis of Eqs. (23) and (24), linearized feedback control of the rigid-flexible manipulator can be obtained in the following block diagram form. In Fig. 6, δLP is the desired trajectory of the platform, ðsÞ δLB is the feedback pose of the base is expressed, and num is denðsÞ transfer function of position control of the legs. Then, according to Fig. 6, transfer function of the linearized feedback control system is given by

where the detail expressions for A and B are listed in Appendix A. The system is stable provided all the eigenvalues lie on the left half complex plane, i.e., Re ½λi ðAÞ o0

ð28Þ

where

Relationship between the DCI and the maximum real part of the eigenvalues is depicted in Fig. 8. It can be seen that the eigenvalues get close to the image axis as the DCI increases. When the DCI exceeds 0.35, there exist eigenvalues locating on the right half complex plane. In case of divergence of the control system, the DCI should be kept less than 0.35.


1 T JC KJC J1 NðsÞ ¼ I6 s2 þ JB M  1 JTC cJC J1 numðsÞ B s þ JB M B

5.2. Dynamic performance simulation

HðsÞ ¼ NðsÞ  1 DðsÞ

ð25Þ

 JB M  1 mJ  1 s2 numðsÞ 
1 T denðsÞ DðsÞ ¼ I6 s2 þ JB M  1 JTC cJC J1 JC KJC J1 B s þ JB M B JB M  1 mJ  1 s2 numðsÞ

ð26Þ 1

1

It can be seen from Eq. (26) that the matrix JB M mJ has an important effect on pole and zero locations. Roots of 1 T I6 s2 þ JB M  1 JTC cJC J1 JC KJC J1 in Eq. (26) are the natural B s þ JB M B frequencies of the flexible mechanism. If JB M  1 mJ  1 is very close to null matrix, zeros and poles of NðsÞ are composed of the natural frequencies of the flexible mechanism and zeros and poles of the transfer function of the control loop of the legs. For this case, the dynamic coupling is quite weak and good dynamic performance can be achieved. Conversely, the rigid-flexible manipulator will respond poor dynamic performance. In the following, the relationship between the DCI and the stability is analyzed. Simplified position control of the leg is illustrated in Fig. 7 and the detailed parameters are listed in h i T T Table 1. Taking x ¼ δLT δL_ T ETVI δLTB δL_ B as state variables, we express Eq. (25) in state-space as x_ ¼ Ax þBδLP

ð27Þ

In the following, influence of the DCI on the dynamic performance is simulated using simMechanics Toolbox. Simulation model of the feedback control of the rigid-flexible manipulator illustrated in Fig. 5 is established in simMechanics. The cables are modeled as damped springs using the “Body Spring & Damper” block with the spring constant of 3:8524  105 N=m and damping constant of 2000m=s. Parameters of the rigid-flexible manipulator are given in Appendix B in details. In the simulation, the initial pose of the platform is (X¼Y¼0 m, Z¼  2.923 m, ϕ ¼ θ ¼ φ ¼01). The desired trajectory of the platform is along the X-axis, shown in Fig. 9(b) as the dotted line. Mass and inertia matrix of the platform are multiplied by a coefficient ratio. Relationship between the DCI along the trajectory and the ratio is illustrated in Fig. 9(a). The maximum DCI appears at the boundary of the workspace and the greater ratio has the greater DCI. Fig. 9(b) and (c) shows the simulation results. Fig. 9(b) illustrates that the rigidflexible manipulator achieves good tracking performance when the ratio equals to 1. And as shown in Fig. 9(c), the motion of the base induced by the dynamic coupling is bounded within  0.01 m to þ0.02 m. For this case, the DCI is ranging from 0.22 to 0.23 which is relatively little. The maximum displacement of the base becomes greater as the DCI is ranging from 0.37 to 0.38. When the DCI is ranging from 0.50 to 0.51, the maximum displacement of the base

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

reaches 0.075 m, and the tracking performance shown in Fig. 9(b) is undesirable. Influence of the maximum DCI along the trajectory on the maximum position deviation of the platform and the maximum displacement of the base is shown in Fig. 10. Obviously, it can be seen that the maximum position deviation of the platform and the maximum displacement of the base increase with the DCI. The simulation results demonstrate that the DCI has a strong influence on the dynamic performance of the rigid-flexible manipulator. In order to obtain good dynamic performance, the DCI should be kept as small as possible.

Fig. 8. Relation between the DCI and the eigenvalues.

77

6. Analysis of the DCI The DCI provides a rational measure index in the quantification of the dynamic coupling of the rigid-flexible manipulator. In this section, based on the DCI, we present an analysis on the factors influencing the dynamic coupling, including the inertia properties, the structural parameters and the configuration of the Stewart platform. Parameters of the Stewart platform to be studied are given in Appendix B.

Fig. 10. Relationship between the displacement and the DCI.

Fig. 9. Influence of the DCI: (a) DCI along the trajectory, (b) motion of the platform, (c) motion of the base.

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Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

Fig. 11. DCI distribution in the maximum horizontal plane: (a) mesh map, (b) counter map.

Fig. 12. DCI distribution along Z-axis.

Fig. 13. The DCI changes with the rotation angles.

6.1. Distribution of the DCI Distribution of the DCI in the required workspace is draw to analyze the influence of the configuration of the Stewart platform on the dynamic coupling. The required workspace is described as a sphere with a radius of 0.1m and a center of (0, 0, and  2.923 m). The orientation angles ϕ, θ, andφ in the sphere range from  101 to 101. Distribution of the DCI in the maximum plane of the required workspace (Z¼  2.923 m, ϕ ¼ θ ¼ φ ¼01) is shown in Fig. 11. The DCI is minimal in the center of the required workspace, and increases with the platform moves away to the edge. The range of the DCI is from 0.2265 to 0.2281, and is fairly narrow. Distribution of the DCI along the Z-axis (X¼Y¼0 m, ϕ ¼ θ ¼ φ ¼01) is illustrated in Fig. 12. The DCI decreases with the distance of the platform from the base. The DCI varies within relatively wide limits, and is from 0.2185 to 0.2356. For the case where the platform locates at the center of the required workspace (X¼Y¼0 m, Z¼  2.923 m), the relations between the DCI and the rotation angles of the platform are illustrated in Fig. 13. The DCI increases with the rotation angles, but the DCI does not change substantially and the range is from 0.2265 to 0.2322. From the above distribution situation, we deduce that the maximum DCI in the required workspace appears on the surface of the sphere with the orientation angles ϕ ¼ θ ¼ φ ¼101, as shown in Fig. 14. The maximum DCI reaches 0.2460.

Fig. 14. DCI in the boundary of the workspace.

In the whole required workspace, the DCI of the Stewart platform is ranging from 0.2185 to 0.2460. The DCI varies greatly along the Z-axis and appears to become quite large when the platform is situated far from the base. Therefore, for the sake of reducing the dynamic coupling of the rigid-flexible manipulator, the motion of the platform should be kept as close to the base as possible.

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

79

Fig. 15. Influence of the structural parameters: (a) radius of the universal joints, (b) angle of the universal joints, (c) radius of the spherical joints, (d) angle of the spherical joints.

6.2. Structural parameters The dynamic coupling is also closely related to the Structural parameters. Four structural parameters are analyzed: radius of the universal joint r B , angle of the universal joints θB , radius of the spherical joints r P , angle of the spherical joint θP , as shown in Fig. 2. Influence of structural parameters on the distribution of the DCI along the Z-axis is shown in Fig. 15. As shown in Fig. 15(a) and (c), the DCI will decrease as the radiuses of the joints increase. In Fig. 15(b) and (d), on the contrary, the DCI will increase as the angles of the joints increase. The DCI varies greatly with the angles of the joints. In contrast, the radiuses of the joints have weaker influence on the DCI. Therefore, the angles of the joint points are the key elements in reducing the dynamic coupling and should be designed as small as possible. 6.3. Inertia properties In addition, the dynamic coupling is under the influence of the inertia properties of the components. Here, the inertia properties of the base, lower part of the leg, upper part of the leg and the platform are discussed. The mass and the inertia matrix are multiplied by coefficients and influence of the coefficients on the distribution of the DCI when the platform locates at (0, 0, -2.923m) is depicted in Fig. 16. As shown in Fig. 16(a), the DCI decreases as the coefficients of the mass and the inertia matrix of the base increase. In contrast, as shown in Fig. 16(d), the DCI increases as the coefficients of the mass and the inertia matrix of the platform increase. The DCI doesn't change substantially as the coefficients of the mass and the

inertia matrix of the legs increase as shown in Fig. 16(b) and (c). Therefore, for the sake of the dynamic coupling, the inertia of the payload mounted on the platform should not be too large compared to the base. If the payload is already fixed, increasing of the inertia of the base is another way to decrease the dynamic coupling. However, increase of the inertia will leads to decrease of the natural frequencies of the flexible mechanism, which works against system stability according to the analysis in Section 5.2.

7. Conclusions We studied dynamic coupling of the rigid-flexible manipulator. The “Virtual Stewart platform” is introduced to formulate a matrix with homogenous dimension to describe how much base motion is produced by a given Stewart platform motion. The ratio of the velocity magnitude is derived to be an eigenvalue problem of the formulated matrix. Therefore, the maximum singular value of the formulated matrix is defined as the dynamic coupling index of the rigid-flexible manipulator. Furthermore, In order to obtain the transfer function matrix of the control system, linearized dynamics of the rigid-flexible manipulator is developed. It can be seen from the transfer function matrix that pole and zero locations relies heavily on the proposed DCI. Then, we implement a simulation to discuss the influence of the DCI on the dynamic performance. The result shows that the dynamic performance becomes poor or even becomes uncontrollable when the DCI increases. For the sake of the dynamic performance, the DCI should be kept as small as possible. Finally, we conduct a research on the factors influencing the dynamic coupling, including the inertia

80

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

Fig. 16. Influence of the inertia properties: (a) the base, (b) lower part of the leg, (c) upper part of the leg, (d) the platform.

"

properties, the structural parameters and the configuration of the Stewart platform.

#

G

M12 ¼  mB

i¼1

This research is sponsored by the National Natural Science Foundation of China (No. 51475252, 51205224) and the National Natural Science Funds for Distinguished Young Scholars of China (No. 51225503).

G

R B rB 

6

 mL ∑

i¼1

2

cos φ cos θ

6 R ¼ R ðz; φÞ UR ðy; θÞ UR ðx; ϕÞ ¼ 4 sin φ cos θ  sin θ " M¼

M11 ; M12

#

# G

"

"

#" G

R B B rL 

#! G

R B B rL 

# G

6

 mU ∑

i¼1

i



G

R B B rP  "

#" G

cos θ sin ϕ 

i

3

ðmB þ mP þ 6mL þ6mU Þg G

R B r B  mB g þ G R B B r P  mP g 0

1

0

6

6

1#

þ ∑ @G R B B rL  mL gA þ ∑ @G RB B rU  mU gA i¼1

i

i¼1

R B B rU  i




sin φ sin θ cos ϕ  cos φ sin ϕ 7 5 cos θ cos ϕ

sin φ sin θ sin ϕ þ cos φ cos ϕ

#! G

R B B rU 

TðIL þ IU ÞG TT

cos φ sin θ cos ϕ þ sin φ sin ϕ

M21 ; M22

M11 ¼ ðmB þ mP þ 6mL þ6mU ÞI3

i

R B B rP 

i¼1

wg ¼

R B B rL 

#" G

R B r B   mP

þ G R B IB G RTB þ G R B B R P IP B R TP G R TB þ ∑

cos φ sin θ sin ϕ  sin φ cos ϕ

G

i¼1

6

Orientation of the platform based on the Roll-Pitch-Yaw (RPY) Euler angle convention in fBg:

#

"

6

R B B rU 

#" G

R B B rL 

i

"

Appendix A. Expression in detail

#

R B B r P  þ mL ∑

#

G

i¼1

M22 ¼  mB

G

"

þ mU ∑

i¼1

"

R B r B  þ mP 6

# G

R B B rU 

#

"

"

6

R B B r P   mL ∑

#

G

 mU ∑

M21 ¼ mB

# G

"

6

Acknowledgments

"

R B r B   mP

i

Z. Liu et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 72–82

"

G

m¼

#"

RB G

RB

6

m11 ¼ mP I3  ∑

i¼1

m11

m12

m21

m22

#

Universal joints described in fBg:   b1 ; b2 ; b3 ; b4 ; b5 ; b6 2

3:3627 6 ¼ 4 0:5026 0:1700

! !! i1 h i1 6   B B mL B TrL  s þ ∑ mU I3  B TrU  s L L i¼1 h

! i1 i h B s B RP p L i¼1



h h i1 i 6  B B s R P p  ∑ mU I3  B TrU  L i¼1

m12 ¼  mP

h

B

i 6 R P rP  þ ∑

mL

h

B

2

1:4807 6 ¼ 4 1:7337 0:1180





6

B

m22 ¼ B R P IP B RTP  ∑

i¼1

T ðI L þ I U ÞB TT

" G

χ1

⋯

i 1B hB s R P p L

R B C1  χ1

⋯

" J¼

s1

⋯

R P p1  B s1

⋯

B B

0 B JB ¼ B @

"

G

#

RB G

2

RB

" A2 ¼

B



K pv K t J M I6

 I6

0 JB M  1 JTC KJC J1 B

A1   A¼ B 2 0 0 I6 A 1

"  B2 0

B1  0 I6 B1

K pv K t J M T iv I6

0

B

3

2

0:1198 6 IU ¼ 4 0:1093  0:0072

3

0

7 K pv K t K pp 7 7; B 1 ¼ 6 4 J M I6 5 5 K pp I6

# " # 0 B1    0  A2 B2 0 0 I6 B1

0

0

I6



0 ;

#

 2:2418 0:4155 0:1180

 2:2418  0:4155 0:1180

0:7611  2:1492 0:1180

7  0:5026 5  0:1700

3 1:4807  1:7337 7 5 0:1180

 0:0001

0:0000

3

3:5972

0:0000 7 5

0:0000

6:4569

 2:0679 117:1691  0:0099

0:7752

3

7 0:0099 5 117:1755

0:1093

 0:0072

26:8184

0:0003

0:0003

26:8141

3 7 5

Mass, position vector of mass center, and moment of inertia of the platform:  T mP ¼ 11200; rP ¼ 0:5901  0:1238  1:2643 2

1:3850 6 IP ¼ 4 0:0742  0:1325

0:0742 2:1068

3  0:1325  0:0031 7 5

 0:0031

3:0695

References

Appendix B. Parameters of the Stewart parallel manipulator All the parameters in Appendix B are default in SI units. Connected points of the cables and the towers described in fGg:  259:81

0:0000

 150:00 0:0000

 300:00 0:0000

259:81

3

7  150:00 5 0:0000

Connected points of the cables and the base described in fBg: ½C1 ; C2 ; C3 ; C4 ; C5 ; C6  2 3:7500 3:7500  7:5000 6 0:0000 ¼ 4 6:4952 6:4952 0:4850 0:4850 0:4850

3:5455

2

# " # I6 0 ; B2 ¼  JB M  1 JTC cJC J1  JB M  1 mJ  1 B

½A 1 ; A 2 ; A 3 ; A 4 ; A 5 ; A 6  2 259:81 0:0000  259:81 6 150:00 ¼ 4 150:00 300:00 0:0000 0:0000 0:0000

 3:1634  0:1700

3

Mass, position vector of mass center, and moment of inertia of upper part of the legs:  T mU ¼ 56:0959; rU ¼ 0:6961 0:0016  0:0038

s6 7C 7C b6  B s6 5A

0

I6

 2:6609  0:1700

3:3765 6 IL ¼ 4  2:0679  0:7752 31T

⋯

2:6609  0:1700

2

R P p6  B s 6 ⋯

3:1634  0:1700

3:3627

Mass, position vector of mass center, and moment of inertia of lower part of the legs:  T mL ¼ 333:7680; rL ¼ 0:7724  0:0004 0:0007

#T

s6

6 B s1 6 4 b1  B s1

"

B¼

B

 1:2461

6 IB ¼ 105  4  0:0001 0:0000

RB C6  χ6

2

0

6 K pv K t K pp I6  J A1 ¼ 6 M 4  K pp I6

G

 2:1165

0:7611 2:1492 0:1180

2

#T

χ6

 2:1165

Mass, position vector of mass center, and moment of inertia of the base:  T mB ¼ 26585; rB ¼  0:0002  0:0002 0:0595

i 6  i1 i  h h h  mP B rP  B R P rP  þ ∑ mL B rL  B TrL  B s B RP p L i¼1



h i 1 i 6 B   hB B  ∑ mU rU  I3  B TrU  s RP p L i¼1

JC ¼

 1:2461

Spherical joints described in fPg:   p1 ; p2 ; p3 ; p4 ; p5 ; p6

TrL 

i 1  h  B mL B rL  B TrL  s L i¼1





h i1 6 6 B    1 B s þ ∑ mU rU  I3  B TrU  þ ∑ B T ðI L þ I U ÞB TT B s L L i¼1 i¼1

6   m21 ¼ mP B rP   ∑

81

 7:5000 0:0000

3:7500  6:4952

0:4850

0:4850

3 3:7500 7  6:4952 5 0:4850

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