A Kinematic Modeling of an Amphibious Spherical Robot System

Shuxiang Guo1,2, Yuehui Ji1 Lin Bi1, Xu Ma1 and Yunliang Wang1 1Tianjin Key Laboratory for Control Theory & Applications in

Complicated Systems and Biomedical Robot Laboratory 2Mech. Systems Eng. Depart

Kagawa University Tianjin University of Technology

Binshui Xidao 391, Tianjin, China 2217-20, Hayashi-cho, Takamatsu 761-0396, Japan

jiyuehuitju@163.com guoshuxiang@hotmail.com

Abstract –This paper presents a kinematic model of an amphibious spherical robot through the Denavit-Hartenberg parameters method which will allow us to describe the link to the next or the previous. Then it is going to allow us to precisely define the frames which are going to move with the links, but also allow us to connect with the base through the structure to the end-actuator. Based on the known position and orientation of the robot, a series of kinematics equations are deduced through homogeneous transformation matrix based on the kinematics model and the inverse kinematics and each joint angle can be obtained. The prototype of amphibious robot includes four water-jet propellers and eight servo motors, which can be capable of changing walking mode between water-jet system and quadruped walking system. The walking speed and direction of amphibious robot can be handled by controlling the PWM pulse duty ratio. At last, some gait experiments had been carried out on the flat floor. The results of experiments verified that the model can give important guidance to gait trajectory planning for amphibious spherical robot. Key words: Amphibious Spherical Robot, Kinematic Model, Denavit-Hartenberg Parameters Method, Walking gait

I. INTRODUCTION

The amphibious robot has a strong environmental adaptability and extensive operating range, it also can be easily deployed and recycled. The robot can achieve search, reconnaissance, rescue and communication on land as well as in rivers, lakes and oceans with the sensors including the gyroscope, hydraulic capsule and avoiding blocker sensor, which expanded the practicable range of an amphibious robot. Today, the amphibious robot which can be used in building optical cable at the bottom of the ocean, underwater archeology and other fields becomes a hot topic and it is researched by many scientific research scholars at home and abroad [1]-[5]. The kinematics model describes the relationship between the movement of each leg motion rod and the position and orientation of the end- actuator. So the precise model is one of the key problems of amphibious spherical robot motion control and trajectory planning, so that modeling is a very important basic work [6].

Up to now, a large number of scholars have conducted a number of studies in regard to the gait planning and control algorithm of amphibious spherical robot. In 1996, Finland’s Helsinki University of Technology gave birth to the first spherical robot named Rollo in the world, which was designed

by Aarne Halme, Jussi Suomela et al. The robot used a single wheel mechanism in the body to change the ball center of gravity position to produce drive torque, so that the robot roll forward [7], [8]. In 2000, Bhattacharya improved the driven approach, developed a new type of spherical robot. It is driven by two mutually perpendicular portions [9]. In 2004, University of Western Australia, University of New York and Eindhoven University of Technology in the Netherlands designed a spherical robot which uses the method of positioning output to solve the problem of the stability and tracking[10]. Kagawa University (Guo Lab) developed a spherical robot amphibious that its hemispherical shell inside diameter is 0.25 meters and the outside diameter is 0.26 meters, respectively [11]. Furthermore, the outside of the hemispherical shell is movable, which can be retracted upward or downward swing [12], [13].

Compared to foreign countries, the earlier domestic research on spherical robot started late. Around 2000, the concept of spherical robot was only introduced into China. Nevertheless, domestic scholars in the field of spherical robot had made progress. Xichuan Lin developed a spherical underwater vehicle, which used two water-jet propellers as the actuating device and provided two inlets and two outlets in 2007 [14], [15]. The study used PID control of the underwater vehicle to carry out a series of experiments related to motion control, and achieved certain results. In 2012, Xidian University focused on the detailed research of controllability and non-complete characteristics and proposed an improved particle swarm optimization algorithm applied to the continuous movement control of the spherical robot. In 2013, Hanxu Sun of Beijing University of Posts and Telecommunication also carried out underwater spherical robot related research [16]. The six degrees of freedom spherical underwater robot has two motors, a propeller, apiston-type storage equipment and a central controller inside the spherical shell.

There are few researches focusing on kinematic modeling, although amphibious spherical robot has been a hot research in recent years. A kinematic model has been given in this paper that joint-space and Cartesian-space based on kinematics can be described in the details.

This paper is organized as follows. Firstly, we introduce the Denavit-Hartenberg parameters method and deduce the kinematics model for the amphibious spherical robot on land. Secondly, we carry out some experiments to control the

walking gait of the amphibious robot. Finally, some conclusions will be given.

II. MODELING OF THE AMPHIBIOUS ROBOT

A. Denavit-Hartenberg Parameters Method

Fig.1 The prototype of the spherical amphibious robot

As shown in Fig.1, each leg of an amphibious spherical

robot has a pitching rotary joint and a yawing rotary joint. Each water-jet propeller can be actuated by the two servomotors, and hence the direction of the jetted water can be changed in the horizontal plane and the vertical plane.

The DH parameters or the Denavit-Hartenberg notation allow us to describe the link and its connections to the next or previous link, and also help us to connect the base through the structure to the end- actuator. Each leg has two rotary joints and two links and each of the links is determined by four independent parameters: ia , iα , id , iθ Each parameter is defined as follows in Fig.2[17],[18]:

1) 1ia − : distance of the common normal from 1iz − to iz

2) -1iα : angle about 1ix − from 1iz − to iz

3) id : distance of the common normal from 1ix − to ix

4) iθ : angle about iz from 1ix − to ix

Fig.2 Schematic diagram of DH coordinate frames

Two adjacent coordinates can be transformed by

translation matrix and rotation matrix: (1)Along 1ix − translate the distance 1ia − :

1

1

1 0 0

0 1 0 0( , 0, 0)

0 0 1 0

0 0 0 1

i

i

a

Trans a

−

− =

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(1)

(2)Around 1ix − rotate the angle 1iα − :

1 1

1 1

1 1

1 0 0 0

0 c s 0( , )

0 s 0

0 0 0 1

i i

i i

i i

Rot xc

α αα

α α− −

− −

− −

−=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2)

(3) Along iz translate the distance id :

1 0 0 0

0 1 0 0(0, 0, )

0 0 1

0 0 0 1

i

i

Trans dd

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3)

(4) Around iz rotate the angle iθ :

s 0 0

c 0 0( , )

0 0 1 0

0 0 0 1

i i

i i

i i

c

sRot z

θ θ

θ θθ

−

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(4)

Joint with one degree of freedom can be described in DH parameters. Therefore, to each link, one transformation 1i

iT− is

assigned with: 1

1 1 1( ,0,0) ( , ) (0,0, ) ( , )i

i i i i i i iT Trans a Rot x Trans d Rot zα θ−

− − −=

1

1 1 1 1

1 1 1 1

0

c s

s

0 0 0 1

i i i

i i i i i i i

i i i i i i i

c s a

s c c d s

s s c c d c

θ θ

θ α θ α α α

θ α θ α α α

−

− − − −

− − − −

−

− −=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(5)

Where we define icθ and isθ as cos iθ and sin iθ , respectively.

Fig.3 DH coordinate frames of the single leg

joint1

joint2

Zi+1

Xiai

Zi

Axis i+1

Axis i

θi

aidiai-1

Xi-1

αi-1

Zi-1

Axis i-1

l4

l3

l2

l1

z3

x3

y3

x2

z2

y2x1

z1

y1

x0

y0

z0

As shown in Fig.3, the DH parameters of the single leg have been demonstrated in TABLE I: 1 4l cm= , 2 5l cm= ,

3 4l cm= , 4 =7l cm

TABLE I LINK PARAMETERS OF THE ROBOT

link

1ia

−

1i

α−

i

d i

θ sini

θ cosi

θ 1

sini

α−

1

cosi

α−

1 0 180°

2l 1θ

1 0 0 1

2 1l -90° 0

2θ 0 1 -1 0

3 3l 90°

4l 0° 0 1 1 0

Then, put the TABLE I parameters into the

transformation matrix equations that have been deduced previously:

1 1

1 101

2

s 0 0c 0 0

0 0 10 0 0 1

cs

Tl

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥− −⎢ ⎥⎣ ⎦

2 2 1

12

2 2

s 00 0 1 0

0 00 0 0 1

c l

Ts c

−⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥− −⎢ ⎥⎣ ⎦

3 3 3

423

3 2

s 00 0 1

0 00 0 0 1

c ll

Ts c

−⎡ ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

1 2 1 2 1 11

1 2 1 2 1 11

2 2 2

0 0 12 1 2 0

0 0 0 1

cc cs s clsc ss c sls c l

T T T

− −⎡ ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

= = (6)

0 0 2

3 2 3

1 2 3 1 3 1 2 3 1 2 1 2 1 2 3 1 2 4 1 1

1 2 3 1 3 1 2 3 1 2 1 2 1 2 3 1 2 4 1 1

2 3 2 3 2 2 3 2 4 2

0 0 0 1

T T T

cc c ss cc s sc cs cc l cs l cl

scc cs sc s cc ss sc l ss l sl

s c s s c s l c l l= =

− − − + +

− + + − − − −=

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(7)

Where we define ic and is as cos iθ and sin iθ , respectively.

B. Transformation Matrix Validation

In order to verify the validity of transformation matrix, we set a initial value of a particular joint variable in advance. Here we set the initial value of the joint variable into the equation 0

3T : 1 2 390 , 0 , 0θ θ θ= ° = ° = °

3 103

4 2

0 1 0 01 0 0

0 0 10 0 0 1

l lT

l l

−

− − −=

− − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(8)

The result is consistent with relation between the position an orientation illustrated in Fig.3. So the validity of the model is confirmed. C. Kinematic equation If the parameters of each agency have be known, especially the parameters of leg rods, and the joint angles of the each movement joint has been given, the position and orientation of the end-actuator which is relative to the based coordinate system can be determined [19]. We need to establish proper body fixed coordinate system ( bX , bY , bZ ) in which the origin { bO } is located in the geometric center of the torso and set up the positive direction of bX is the forward direction, the positive direction

of bZ opposite the gravity direction, then the positive

direction of bY can be obtained by right hand rule. The origin of coordinate system {0} relative to the body fixed coordinate system is ( a , b , c ), therefore the transformation matrix from coordinate system {0} to body fixed coordinate system is given:

0

0 1 01 0 0

0 0 10 0 0 1

b

ab

Tc

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

(9)

Where a, b and c are geometry parameters of the torso specified in Fig.4.

Fig.4 The structural model of the amphibious spherical robot

The position and orientation matrix of the end-actuator which relative to body fixed coordinate system is

Link2

Link1

y3 z3

x3

c

2b

ObYb

Zb

Xb

2a

03 0 3b bT T T= =

1 2 3 1 3 1 2 3 1 2 1 2 1 2 3 1 2 4 1 1

1 2 3 1 3 1 2 3 1 2 1 2 1 2 3 1 2 4 1 1

2 3 2 3 2 2 3 2 4 20 0 0 1

s c c c s s c s c c s s s c l s s l s l ac c c s s c c s s c c s c c l c s l c l b

s c s s c s l c l l c

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− − − + + +− + + − − − − +

− − − − +

(10) According to forward kinematics, if we known the joints

variable values of each support leg, we can acquire the position and orientation which is related to the body fixed coordinate system. Furthermore, we can obtain the position and orientation of the torso which relative to the ground. D. Inverse Kinematic equation

Fig.5 The vector coordinate frame of end- actuator

As shown in Fig.5, the position and orientation of end-

actuator can be expressed by three mutually perpendicular unit vectors n ， o and a which n ， o and a constitute right vector product in fixed coordinate system. The position of end-actuator can be used by relative coordinates location

vector { }, ,T

x y zp p p p= . Thus it can be seen that three unit vectors and a location vector constitute the coordinates:

0

3

0 0 0 1

x x x x

y y y y

z z z z

n o a p

n o a pT

n o a p=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(11)

Then, premultiplying transformation matrix 03T with

0 11T

− , we can obtain this:

0 1 0 0 1

1 3 1

0 0 0 1

x x x x

y y y y

z z z z

n o a p

n o a pT T T

n o a p− −=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(12)

0 1 0 1 2 1 2 1

1 1 2 3 2 3 3T T T T T T T−= = =

Due to :

1 1

1 10 1 1

1 0

2

0 0

0 0

0 0 1

0 0 0 1

c s

s cT T

l−

−

− −= =

− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(13)

Which can also be written as

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 2 1

2 3 3

0 0 0 1

x y x y x y x y

x y x y x y x y

z z z z

cn sn co so ca sa c p s p

sn cn so co sa ca s p c pT T T

n o a p

− − − −

− − − − − − − −= =

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(14)

2 3 2 3 2 2 3 2 4 1

3 31

3

2 3 2 3 2 2 3 2 4

0 0

0 0 0 1

c c c s s c l s l l

s cT

s s s s c s l c l

− + +

=− − +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(15)

Due to 1 2 1

2 3 3T T T= , we take equation (14) and (15) on both sides of the d row and column is equal to the corresponding element. So that we can obtain three equations from the above equations:

1 1 0x ys p c p+ = (16)

1 1 2 3 2 4 1x yc p s p c l s l l− = + + (17)

2 3 2 4zp s l c l− = − + (18) The result can be given:

1

1- tan y

x

p

pθ −=

(19)

2

2 atan 2( 1 , ) atan 2( , )zt t p kθ = ± − − (20)

Where ( )2 2 2 2

1 4 3

42

zr l p l l

tl

− + + −= ，

1

2 2

x yk lp p= −+ ,

2 2

x yr p p= +

III. EXPERIMENT

The walking speed and direction of robot are manipulated by controlling the PWM pulse duty ratio. We conduct an experiment on the flat ground, and the robot can walk towards right and walk around the circle. Fig.6 shows that the robot walks towards right on the floor. We recorded the robot trajectory coordinates and generate curves using MATLAB. We recorded the X- and Y-coordinates of its geometric centre in Fig.7. [20].

n=oxa

n

o

a

(a)

(b)

Fig.6 Walking motion to the right.

Fig.7 Experimental trajectory results for walking motion to the right

The robot gait trajectory planning is the process achieving

the locus of discrete points and track regulation error which have a great relationship with the size of the distance between the various discrete points. The existing methods of interpolation have distance interpolation, linear interpolation and the polynomial interpolation. This paper focuses on binomial interpolation. The binomial calibration curve is given: 2236.9564 11.6577 0.115y x x= − + − (21) Where the red line is the experimental curve and the blue line is the calibration curve specified in Fig.8.

Fig.8 Line graph of error between x position

The preset position in the model comes close enough to the experimental position and the error between two position in Fig.8.

Fig.9 shows that the robot walks around the circle on the floor. The circle calibration curve is given:

2 2 2( 23.0933) ( 30.1826) 35.4248x y− + − = (22)

Fig.9 Walking around the circle.

We recorded the robot trajectory coordinates and generate

curves using MATLAB. We recorded the X- and Y-coordinates of its geometric centre in Fig.10. It can be seen that the trajectory is a circle.

Fig.10 Experimental trajectory results for walking around the circle.

Where the blue line is the experimental curve and the red line is the circle calibration curve specified in Fig.10.

IV. CONCLUSION AND FUTURE WORK

An amphibious spherical robot configuration driven by water-jet propellers and servo motors was proposed in the paper. Kinematics for the robot was essential to obtain the position and posture of the joint. It had laid a theoretical foundation of the track planning and movement simulation.1) A kinematic model established by using Denavit-Hartenberg parameters method was obtained. Then, we solved its forward and inverse kinematics equations based on the homogeneous coordinate transformation method and validate it simultaneously.

1) A kinematic model established by using Denavit-Hartenberg parameters method was obtained. Then, we solved its forward and inverse kinematics equations based on the homogeneous coordinate transformation method and validate it simultaneously.

2) Some experiments are designed to validate that the amphibious robot walks towards right and walk around the circle on the flat floor when we change the gait of robot with adjusting the PWM pulse duty ratio and the model of the robot was accurate to describe the system characteristics.

The kinematic model of the amphibious spherical robot described detailed relation between joint-space and Cartesian-space. In future work, we can track the trajectory and simulate the movement in MSC.ADAMS and add more wireless transmitting equipment such as Bluetooth module, Wi-Fi equipment etc. Then do more experiments and simulations with them.

ACKNOWLEDGMENT

This research is supported by Tianjin Key Research Program of Application Foundation and Advanced Technology (13JCZDJC26200).

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