ORIGINAL ARTICLE

O. Sato (*) · N. Takahashi · A. Uekubo · M. KonoDepartment of Computer Science and Systems Engineering, Faculty of Engineering, University of Miyazaki, 1-1 Gakuenkibanadai-Nishi, Miyazaki 889-2192, Japane-mail: sheep@cs.miyazaki-u.ac.jp

A. SatoDepartment of Mechanical Engineering, Miyakonojo National College of Technology, Miyakonojo, Japan

This work was presented in part at the 15th International Symposium on Artifi cial Life and Robotics, Oita, Japan, February 4–6, 2010

Artif Life Robotics (2010) 15:151–155 © ISAROB 2010DOI 10.1007/s10015-010-0785-8

Osamu Sato · Asaji Sato · Nobuya Takahashi Akira Uekubo · Michio Kono

Analysis of a manipulator in consideration of the relative motion between a tray and an object

were easily calculated by iterative dynamic programming, and the dynamic characteristics of the system were analyzed.

In this article, the equations of motion of a manipulator whose mechanism has a passive revolute joint are derived after consideration of the characteristics of DC servomo-tors, and a performance criterion for saving energy is defi ned after consideration of the energy consumption by the driving source. When the manipulator is operated in a verti-cal plane, the system is highly nonlinear due to gravity, and an analytical solution cannot be found. Then a numerical approach is necessary. Considering the relative motion between the tray and the object, the trajectories of velocity for energy saving are calculated by iterative dynamic pro-gramming. The dynamic characteristics of manipulator control based on the above-mentioned trajectory are ana-lyzed theoretically and investigated experimentally.

2 Modeling the manipulator with a passive joint

The dynamic equations of a manipulator with four degrees of freedom, as shown in Fig. 1, which is able to move in a vertical plane, are as follows.

A A A A

A A A A

A A A A

A A A A

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

− +− +− +

��������

θθθθ

τ ττ ττ τ

1

2

3

4

1 2 15

2 3 25

3 4 3

A

A

A 55

4 45τ +

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥A

(1)

where

A a a C a C a C A a C a C a C11 1 6 2 8 23 9 234 12 6 2 8 23 9 234= + + + = + +, ,

A a C a C A a C A a a C a C a C13 8 23 9 234 14 9 234 21 2 6 2 7 3 10 34= + = = + + +, , ,

A a a C a C A a C a C A a C22 2 7 3 10 34 23 7 3 10 34 24 10 34= + + = + =, , ,

A a a C a C a C A a a C a C31 3 7 3 8 23 11 4 32 3 7 3 11 4= + + + = + +, ,

Abstract In this article, equations of motion of a manipula-tor, whose mechanism has a tray installed with a passive revolute joint, are derived after consideration of the char-acteristics of the driving source. Considering the relative motion between the tray and the object, the trajectories of the velocity for saving energy are calculated by iterative dynamic programming. Also, the dynamic characteristics of manipulator control based on the trajectory for saving energy are analyzed theoretically and investigated experimentally.

Key words Manipulator · Trajectory · Dynamic program-ming · DC motor · Minimum energy · Passive joint

1 Introduction

For the purpose of enlarging the work space, it is necessary to study the optimal control of a manipulator whose mecha-nism has a passive joint. In a previous report,1 a casting manipulator was introduced, and it had a large work space compared with its simple mechanism. However, the consid-eration of energy consumption is not enough. In the hori-zontal plane, obstacle avoidance motion planning for a three-axis planar manipulator with a passive joint is inves-tigated,2 but energy consumption is not considered.

In previous reports,3,4 trajectories for saving the energy of a manipulator whose mechanism has two active joints

Received and accepted: April 12, 2010

152

A a a C A a a CA a a C A a a

33 3 11 4 34 5 11 4

43 4 11 4 44 4 5

= + = − += + = +

, ,, ,

A a a C a C a C A a a C a C41 4 9 234 10 34 11 4 42 4 10 34 11 4= + + + = + +, ,

A a S a S

a

15 6 1 2

2

2 8 1 2 3

2

23

9 1 2 3 4

= +( ) + + +( )+ + + +( )� � � � �

� � � �θ θ θ θ θ

θ θ θ θ22

234 12 1S a C− ,

A a S a S

a S

25 621 2 7 1 2 3

2

3

10 1 2 3 4

2

34

= − + + +( )+ + + +( ) −

� � � �

� � � �θ θ θ θ

θ θ θ θ aa C13 12,

A a S a S

a S a C

35 7 1 2

2

3 8 12

23

11 1 2 3 4

2

4 14

= − +( ) −

+ + + +( ) −

� � �

� � � �θ θ θ

θ θ θ θ 1123,

A a S a S

a S a C

45 921 234 10 1 2

2

34

11 1 2 3

2

4 15 1

= − − +( )− + +( ) −

� � �

� � �θ θ θ

θ θ θ 2234,

a I m r m m m la I m r m m l

G

G

1 1 1 12

2 3 4 12

2 2 2 22

3 4 22

= + + + +( )= + + +( )

,,

a I m r m r a m r a I r rG Gb b3 3 3 32

4 32

4 4 42

5 42 2= + + ′ = = ( ), , ,

a m r m m l l a m r m r la m r m r l

6 2 2 3 4 2 1 7 3 3 4 3 2

8 3 3 4 3 1

= + +( ){ } = + ′( )= + ′( )

, ,,

a m l r a m l r a m r ra m r m l m l m l

9 4 1 4 10 4 2 4 11 4 3 4

12 1 1 2 1 3 1 4 1

= = = ′= + + +( )

, , ,gg,

a m r m l m l g a m r m r g a m gr13 2 2 3 2 4 2 14 3 3 4 3 15 4 4= + +( ) = + ′( ) =, , ,

C S C Sj j j j jk j k jk j k= = = +( ) = +( )cos , sin , cos , sin ,θ θ θ θ θ θ

C Sjkn j k n jkn j k n= + +( ) = + +( )cos , sin ,θ θ θ θ θ θ

C Sjknp j k n p jknp j k n p= + + +( ) = + + +( )cos , sin ,θ θ θ θ θ θ θ θ

j k n p= = = =( )1 4 1 4 1 4 1 4∼ ∼ ∼ ∼, , ,

and the relation between φ̇ and θ̇4 is

r rb� �φ θ= − 4 4. (2)

where φ̇ is the rotational speed of the object.The kinetic energy of the mechanism is

Ka a a

a

= + +( ) + + +( )+ + + +

11

2 21 2

2 31 2 3

2

41 2 3

2 2 2

2

� � � � � �

� � � �

θ θ θ θ θ θ

θ θ θ θθ θ

θ θ θ θ θ θ θ θ

4

2 54

2

6 1 1 2 2 7 1 2 1 2 3 3

2( ) +

+ +( ) + +( ) + +( )

a

a C a C

�

� � � � � � � �

++ + +( ) + + + +( )+ +

a C a C

a8 1 1 2 3 23 9 1 1 2 3 4 234

10 1

� � � � � � � � ��

θ θ θ θ θ θ θ θ θθ �� � � � �� � � � � � �

θ θ θ θ θθ θ θ θ θ θ θ

2 1 2 3 4 34

11 1 2 3 1 2 3 4

( ) + + +( )+ + +( ) + + +

C

a (( )C4,

(3)

and the potential energy of the mechanism is

U a S a S a S a S= + + +12 1 13 12 14 123 15 234. (4)

The applied voltage of the servomotor is

e b b b bj j j j j j j j fj j= + + + ( )1 2 3 3� �� �θ θ θτ τ sign (5)

where

b k R k D b R k I b R kj vj aj tj mj j aj tj m j aj tj1 2 3= + ( ) = ( ) =, , ,

and iaj is the electric current of the armature, Raj is the resis-tance of the armature, Imj is the moment of inertia of the armature, and Dmj is the coeffi cient of viscous damping. Then the electric current is

i e k Raj j vj j aj= −( )�θ . (6)

and the energy consumed is

E dte ij ajj

= ⋅( )∫∑=1

3

. (7)

3 Simulation of the pendulum movement

Under the condition that joints 1 and 2 are fi xed, simula-tions of the system are shown in Figs. 2 and 3.

We take the parameters of the system as m3 = 0.109, m4 = 0.122 kg, r4 = 0.075, rb = 0.015 m, and the initial condi-tions are θ3i = −(π/2), θ4i = −(π/5), and θ̇3i = θ̇4i = 0. Figure 2 shows the motion of the system like a pendulum movement, and Fig. 3 shows the response in terms of the conservation of mechanical energy of the pendulum movement.

4 Simulation of the manipulator

We take the parameters of the system as shown in Table 1.Figure 4 shows a fl ow chart for the iterative dynamic

programming method. In frame (A), the trajectory for saving energy is searched by dynamic programming.3 In frame (B), the searching region is shifted to minimize the energy consumed, and the width of the region is smaller.

Figure 5 shows the trajectory for searching, and the initial trajectory for searching is expressed as

� �� � � �

θ θθ θ π θ θ θ θ

j ijfj ij

f

fj ij

f

fj ij

tt

t= +

−⎛⎝⎜

⎞⎠⎟

+−

−+⎛

⎝⎜⎞⎠⎟2 2

sinn .πttf

⎛⎝⎜

⎞⎠⎟

(8)

Fig. 1. Mechanism of the manipulator

153

Fig. 2. Pendulum movement about the tray and the object. a Simulation of pendulum movement about the tray and the object. b Experimental results of pendulum movement (every 0.13 s)

Fig. 3. Mechanical energy of pendulum movement

Fig. 4. Flow chart for the simulation

Table 1. Parameters of the manipulator

Parameter Value Parameter Value

l1 (m) 0.080 IG1 (kgm2) 1.7 × 10−5

l2 (m) 0.115 IG2 (kgm2) 8.4 × 10−5

l3 (m) 0.015 IG3 (kgm2) 1.2 × 10−6

r1 (m) 0.044 IGb (kgm2) 4.8 × 10−7

r2 (m) 0.078 kt1,kt2 (Nm/A) 0.046r3 (m) 0.028 kv1,kv2 (Vs/rad) 0.046r3’ (m) 0.000 Dm1,Dm2 (Nms/rad) 7.9 × 10−5

r4 (m) 0.005 τf1, τf2 (Nm) 0.0013rb (m) 0.010 Ra1, Ra2 (Ω) 3.5m1 (kg) 0.020 m3 (kg) 0.018m2 (kg) 0.047 m4 (kg) 0.012

The tray and the object are shown in Fig. 6. Considering the relative motion between the tray and the object, the performance criterion is changed to

′ = ⋅( ) + − ( ){ }∫∑=

E dt ke ij aj

T

j0

1

3

4 2θ λ . (9)Fig. 5. Trajectory for searching

154

Fig. 6. Tray and object Fig. 7. Locus of the throwing motion

Fig. 8. Throwing motion of the manipulator. a Experimental results. b Theoretical results

The response of the manipulator from the initial position (θ1i = −π/4, θ2i = −π/4, θ3i = 0, θ̇1i = 0, θ̇2i = θ̇3i = 0) to the release position (θ1f = 13π/36, θ2f = −7π/36, θ3f = −7π/24, θ̇1f = 13.9 rad/s, θ̇2f = −1.1 rad/s) is shown in Fig. 7, with a working time of tf = 0.8 (s). In this case, joint 3 is passive, and joints 1 and 2 are actuated. In order to prevent the object falling from the tray at turning point 2, and to avoid preventing the object releasing from the tray at point 3, the center of gravity of the tray is adjusted after consideration of the analysis about the relative motion between the object and the tray.

5 Experimental results

In this section, we show the results of a fundamental experi-ment to examine the effectiveness of modeling for the simulations.

Figure 8a shows the experimental results of the manipu-lator in throwing motion, under the same conditions as in Fig. 7. This apparatus was also used in a previous report.4 The mechanism of the manipulator is a closed type where a tray (link 3) is connected to link 2 by a passive revolute joint. Under the condition that the torque of joint 3 is zero, the angular accelerations of joints 1 and 2 are given by an iterative dynamic programming method, and the accelera-tion of link 3 and the torques of joints 1 and 2 are calculated simultaneously. The parameters of the system are shown in Table 1. Motors 1 and 2 (rated 24 V, 60 W) are on the frame, and the sampling time of the control is 0.002 s. The feedback gain for angular displacement is 50 V/rad, and the feedback gain for angular velocity is 0.5 Vs/rad. Figure 8b shows the theoretical results, which are similar to the experimental results (Fig. 8a).

155

Figure 9a,b shows the experimental responses of the angular velocity of the motor, which are calculated by the angular displacement measured by a rotary encoder. Figure 9c shows the angular displacement of the tray. From the initial position to the release position, the theoretical results (broken line) are similar to the experimental results (solid

Fig. 9. Experimental results

line). From these results, it is considered that the modeling for the simulation is effective.

From the experimental results, it is clear that the holding and releasing of the object are possible by analyzing the relative motion between the object and the tray.

6 Conclusions

The results obtained in this work are summarized as follows.

1. It is considered that the holding and releasing of the object are possible by analyzing the relative motion between the object and the tray.

2. From the experimental results, it is considered that the modeling for the simulation is effective.

References

1. Arisumi H, Yokoi K, Kotoku T, et al (2002) Casting manipulation (experiments on swing and gripper throwing control). Int J JSME 45(1 C):267–274

2. Shiroma N, Lynch KM, Arai H, et al (2000) Motion planning for a three-axis planar manipulator with a passive revolute joint (in Japa-nese). J JSME 66(642):545–552

3. Sato A, Sato O, Takahashi N, et al (2006) Trajectory for saving energy of a direct-drive manipulator in throwing motion. Artif Life Robot-ics 11:61–66

4. Sato A, Sato O, Takahashi N, et al (2008) A study of manipulator with passive revolute joint. Artif Life Robotics 13:31–35