Kinematic Analysis on Land of an Amphibious

Spherical Robot System

Lin Bi1, Jian Guo1 Shuxiang Guo1,2, Zhendong Zhong1 1Tianjin Key Laboratory for Control Theory & Applications in

Complicated Systems and Biomedical Robot Laboratory

Tianjin University of Technology

2IntelligentMechanical Systems Engineering Department

Faculty of Engineering

Kagawa University

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

(bilinsidalin@aliyun.com;jianguo@tjut.edu.cn) guoshuxiang@hotmail.com

Abstract –The amphibious robots have been widely applied

in kinds of fields, such as fishery industry, data collecting,

submarine salvage and military etc. In this paper, we focus on a

novel amphibious spherical robot system. In order to verify the

kinematics characteristics of the amphibious robot, the Jacobian

matrix based on the Denavit-Hartenberg parameters method,

which can reflect the relation between velocity and angular

velocity of the joint, can be obtained. The Jacobian matrix offers

theoretical foundation for the following simulation analysis based

on virtual prototype. We use 3D mechanical modelling software

Solidworks to establish a model and import it into dynamics

simulation software ADAMS (Automatic Dynamic Analysis of

Mechanical Systems). And then the joint simulation experimental

data, such as the position, velocity and acceleration which can

validate the feasibility of the walking gait planning. In the course

of carrying on simulation we can carry on simulation course

observing model altering and result treating. And get the walking

speed of amphibious robot at the same time. Lastly, some

experiments demonstrate that simulation results coincide with

the result of kinematic analysis. The results prove that the

simulation in ADAMS can be accurate to describe the kinematic

characteristics of the robot.

Index Terms - Amphibious Spherical Robot, Walking gait,

Kinematic simulation analysis, ADAMS

I. INTRODUCTION

Over the course of the past 30 years, the robot technology

has developed in a variety of directions, such as rescue [1],

detection, reconnaissance [2], communication system carrier

[3] and fire protection. Nowadays, the research of amphibious

robot development is at a high speed in recent two decades at

home and abroad. Amphibious robot is being used as an

important tool in the exploration of ocean, and at the same

time with the visual feedback system and perceptual system

[4]. The robot can walk on land as well as in rivers, lakes and

oceans with the sensors including the gyroscope, hydraulic

capsule and avoiding blocker sensor, which enlarge the scope

of operation. And the robot can continue working for a long

time by a low-power control device and carry some tiny

underwater robots to complete tasks such as detection [5].

Up to now, a great deal of research is done in amphibious

spherical robot to develop sophisticated systems for

exploration and detection of ocean. In 1991, the University of

Hawaii developed an underwater robot called Odin which is

composed of anodized aluminium [6]. The robot had eight

thrusters, completing with six degrees of freedom underwater

motion, but the volume is too big, not conducive to meet the

miniaturization of system design. Kagawa University (Guo

Lab) developed a spherical robot amphibious which included

several microrobots as son robots[7], and a novel designed

amphibious spherical robot as the mother robot in 2012 [8]. In

the water movement, the servo motors manipulated the

positions of the four propellers actuating the robot. On the

land, servo motors manipulated the four propellers used as

legs to perform walking movement [9]. In 2014, the

researchers of Massachusetts Institute of Technology have

manufactured an underwater robot used for safety inspection.

The main body of the robot was composed of 3D printing

materials. The propulsion system consisted of six pumps,

draining outward through the rubber tube [10]. The

disadvantage of this paper is that some additional sensors such

as pressure sensors and cameras couldn’t be used in the robot.

Since 1970, Chinese scientists and researchers had focused

on the development of underwater robot. Tianmiao Wang

developed a robot, imitating the structure of biological gecko

in 2009 [11]. The study designed the corresponding gait and

realized the motions of walking straight and turning by using

the diagonal gait. Due to the adhesive material, the feet of the

robot was prone to wear. In 2013, Hanxu Sun of Beijing

University of Posts and Telecommunication also carried out a

new amphibious spherical robot which was proposed by using

the characteristic of Fully-closed of the spherical robot with

screw propeller inside [12]. The robot is fully enclosed mobile

platform which will never tip over and its turning radius is

zero, but the mechanical characteristics couldn’t provide

sufficient accuracy in motion control [13].

There are few researches focusing on kinematic modelling

and mechanical simulation in ADAMS, and can’t accurately

reflect the mechanical properties. The innovation points of

this paper lies in investigating the kinematics simulation of

amphibious robot in ADAMS using its planned gait, and a

series of important mechanical characteristics data is obtained

which verifies the accuracy of kinematic model.

This paper is organized as follows. Firstly, we introduce

the robot system structure. In section II, referring to the

mechanical characteristics, the Jacobian matrix of robot is

built by D-H parameters method. We investigate the kinematic

simulation and achieve some important parameters in

ADAMS in section III. In section IV, we carry out some

experiments to control the walking gait of the amphibious

robot which verified the precision of kinematic model and

simulation results. Finally, some conclusions will be given.

II. SYSTEM STRUCTURE

A. The Amphibious Robot System Structure

Fig.1 The prototype of the spherical amphibious robot

The design structure of the amphibious robot is illustrated

in Fig.1. In the future, the lower half spherical shell of bottom

will be processed by 3D printing technology. There are two

active DoFs in the hip joint (Joint 1) and knee joint (Joint 2) of

the robot, which refer to hip flexion and knee flexion.

With adjustable complex actuation methods, the robot can

change the movement mode between quadruped walking and

water-jet propulsion without manual manipulation [14].

Fig.2 The controller system of the spherical amphibious robot

AVR ATMEGA2560 is selected for the microcontroller of

the spherical robot, which can control eight motors and four

water-jet propellers through two L298 (Fig.2). The

microcontroller has eight PWM signals for controlling servo

motor to realize walking motion on land. Another four PWM

signals are used to actuate the water-jet propellers by

regulating duty ratio of PWM signal (Fig.3).

Fig.3 Diagram block of control system

B. Jacobian Matrix of the Robot

In order to obtain the Jacobian matrix which embodies

mapping relation between operating space and joint space, the

kinematic model of spherical robot is built by using DH

parameters method in a given frame. Then, we are going to

consider each of the joint axes, prismatic or revolute, and

analyse its impact on both of the linear velocities and angular

velocities. Therefore, the position and orientation matrix from

the end-actuator to body fixed coordinate system can be

obtained as follow [15]:

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 3 1 3

0

3 0 32

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

2 3 2 23 3 2 4 2

0 0 0 1

b b

s c s c c s s s c l s s l s l a

c c c s s c c s s c c s c

s

c l c s l c l b

s s c s l

c c c s

c l l cT T T

s c

(1)

Where we defineic and

is as cosi

and sin

i ,

respectively. 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

By writing the transform matrix as block matrices, the

position of the end effector can be obtained

0

3 0 1

R PT TT

(2)

1 2 3 1 2 4 1 1

1 2 3 1 2 4 1 1

2 3 2 4 2

x

y

z

P

P P

P

s c l s s l s l a

c c l c s l c l b

s l c l l c

T

T T

T

(3)

Taking the partial derivatives of the rotation angles of the

joint, we can get the Jacobian matrix of the robot.

1 2 3 1 2 4 1 1 1

1 2

1 2

1

2 3 1 2 4

1 2 3 1 2 4 1 1

2

1 2 3 1 2 4

2 2 4

1

3

2

,

0

x x

y y

z z

P P

P P

P P

c c l c s l c l s s l s c l

s c l s s l s l c s l c

T T

T TJ c l

c lT

sT

l

(4)

Link2

Link1

y3 z3

x3

c

2b

Ob

Yb

Zb

Xb

2a

Then, we set the end-effector movement equation as

( )x x q , and take the derivative of time t on both sides.

Finally, the velocity of rotation joint can be acquired [16].

1 2 3 1 21

2

4 1 1 1 2 3 1 2 4

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

2 3 2 40

x

y

z

c c l c s l c l s s l s c l

s c l s s l s l c s l c c l

c l s l

V

V V

V

(5)

In other words, the equation can be simplified as follow:

V J

(6)

The Jacobian matrix reveals the mapping relation between

the velocity of the end-effector and the angular velocity of

each joint. If we got the angular velocity of each joint, we

would calculate the velocity of the end-effector. It offers

theoretical foundation for the following simulation analysis

based on virtual prototype in ADAMS.

III. KINEMATIC SIMULATION

A. Three-dimensional Modeling of Spherical Robot

ADAMS (Automatic Dynamic Analysis of Mechanical

System) is dynamic simulation analysis software about

mechanical system [17]. The software works with thousands

of companies worldwide, in hundreds of industries, to develop

better products faster by using information technology and

services to enhance and automate the product design and

manufacturing process. On account of its lack of three-

dimensional modeling capability, we establish three-

dimensional model of spherical robot through Solidworks

software.

Fig.5 The 3-D model of the robot in Solidworks

Then, we import the model into ADAMS by using

Adams/Exchange module (Fig.6) and add some motion

constraints and force constrains. After the process of

sensitivity kinematic simulation, we refine the parameters of

the robot until get the better result.

Fig.6 The principle of the ADAMS simulation

After importing the simulation model of amphibious robot,

we set the physics parameters according with the requirements

and define the coordinate system of the robot in Fig.7.

Because Solidworks imported into the Adams/View of

simulation model without constrains, we need to add some

corresponding constrains into the mechanism components of

the model. In the robot, each of the hips adds one revolving

(horizontal rotation) and each of the knee joints add one

revolving (longitudinal rotation).

Fig.7 The simulation model of spherical robot in ADAMS platform

The robot movement is driven by motor, now we need to

append drive function to each of the eight servo motors. In the

process of marching, each leg performs a series of actions

“lift-swing-falling-support”. In the support process, the end -

effector will contact with the ground, so that both of them can

produce contact force related to the speed. In order to achieve

a better simulation result, we need to select contact function

and contact friction parameter in setting up contact force in

Fig.8. After parameters setting way is specified, the kinematic

analysis can be carried out by applying the real load.

Fig.8 The definition dialog of contact force

B. Kinematic Simulation in ADAMS/View Environment

The robot gait can be divided into four kinds, crawling

gait, pacing gait, trotting gait, and gallop gait [18]. If the

spherical amphibious robot uses pacing gait or gallop gait to

walk on land, robot can produce serious tilt against the robotic

control. At the same time, considering that pacing gait and

gallop gait are easier to cause wear and tear of the spray pipe,

we select the stable crawling gait.

Fig.9 Crawling motion diagram of the spherical robot

As shown in Fig.9, the robot completes cycle movement

with crawling gait. When the robot is in crawling motion, one

of the legs is in swing phase, the other three legs are in support

phase, holding the whole robot. Then through the post-

processing function of ADAMS, the simulation result about

the kinematic parameters of the robot is displayed in Fig.10.

(a) The angle image of the spherical robot

(b) The angular velocity image of the joint 1

(c) The angular acceleration image of the joint 1

(d) The angle velocity image of the joint 2

(e) The angular acceleration image of the joint 2

(f) The displacement image of the end-effector

Fig.10 The image of the simulation result of the amphibious robot

TABLE I

SYMBOL DEFINITION OF THE SIMULATION

Mark Symbol Specification

L Left

R Right

F Forward

B Back

1 Hip Joint

2 Knee Joint

S Displacement Value Along Y Direction

W Angular Velocity

A Angular Acceleration

MEA Measure

QD Actuate

The symbols in the image of the simulation result are

defined in TABLE I in detail.

Through simulation experiment, the robot can realize

stable walking in accordance with the planned crawling gait

[19]. In the post-processing module, we can see the changes in

the angle velocity and angle acceleration of each joint in the

process of simulation shown in Fig. 10(b)-Fig. 10(e).

According to the previous Jacobian matrix deduced, we can

calculate the trajectory along Y direction, which is displayed

in Fig. 10(f).

From the result, the measure point here is selected as hip

joint, plotted by walking time on the horizontal axis and

distance on the vertical. From Fig. 10(f), we can conclude that

the stride is about 30.0418mm corresponding with the

parameters that we have set, and the walking velocity of robot

is 9.32mm/s

IV. EXPERIMENT AND RESULTS

Amphibious spherical robot can realize the basic

movements such as walking straight, turn and rotary motion.

The implementation of these basic movements has contributed

to performing the tasks, making the research of robot has wide

application prospect and development value.

Fig.11Walking motion

Through controlling the PWM pulse duty ratio [20], we

can control the rotation angle by adjusting the walking speed

and direction of the spherical robot. As shown in Fig.1, the

horizontal servo motor (Joint 1) revolves 30 degrees, and the

vertical servo motor (Joint 2) revolves 20 degree, so that the

robot can walk steadily in the planning direction. In order to

obtain better experiment result, walking experiments have

been carried out on the flat ground in accordance with

crawling gait in Fig.11

Fig.12 The schematic diagram of the robot of on the tile floor

Fig.12 shows a video sequence of the walking motion on

the floor. The displacement of the robot can be measured by

placing a marker ruler on the ground at the corresponding

coordinates and the time can be recorded, and generate curves

using MATLAB [21]. The experimental curve and the fitting

curve are shown respectively in Fig.13. The blue line

represents the true position of the robot, the red line represents

fitting position. The least squares fitting line is given:

9.4302 6.2615y x

Fig.13 The displacement diagram of the robot of on the tile floor

According to the slope in the straight line, we know the

walking speed of the spherical robot is about 9.4302mm/s,

closing enough to the simulation speed (9.32mm/s) by using

ADAMS.

Therefore, the software ADAMS can provide the accurate

kinematic characteristics and theoretic foundation of robot

control research.

0s 3s

6s 9s

12s 15s

IV. CONCLUSION AND FUTURE WORK

The aim of this paper is to provide a novel amphibious

spherical robot to assist the researchers to complete the

underwater complicated missions in limited spaces. In order to

improve the design efficiency and reliability, this thesis

researches into the simulation of spherical robot’s gait of

walking along straight lines with virtual prototypes. In

simulation, through the simulation software ADAMS, the

kinematic characteristic parameters in each joint have been

obtained. In addition this, some experiments have been

conducted to aim at testify the accurate of the simulation

result. From the above, several conclusions are summed up as

follow：

1) A kinematic model established by using Denavit-

Hartenberg parameters method is obtained. Then, we educe

Jacobian matrix to aim at reflecting the relationship between

the linear velocity and angular velocity of each joint. It offers

theoretical foundation for the following simulation analysis

based on virtual prototype.

2) Referencing to the actual geometry parameters and

physical characteristics of the spherical robot, the 3D entity

model of the robot is built in Solidworks and imported into

ADAMS platform. Then the kinematic simulation is

conducted in ADAMS using the planned crawling gait, and a

range of data is collected which provided valuable information

for gait planning of the robot.

3) Some experiments are carried out to validate that the

simulation result in ADAMS is accurate to describe the

kinematic characteristics of the robot.

In the future work, we will add the mechanics analysis of

ANSYS to optimize the structure of leg, which can reduce the

bad effect of the impact between the robot and the land.

Meanwhile, considering the versatility and the integration of

the robot, we can improve the image processing and human-

computer interaction. Besides, we will conduct more

simulations and experiments in a series of gaits and terrains.

ACKNOWLEDGMENT

This research is partly supported by National Natural

Science Foundation of China (61375094), Key Research

Program of the Natural Science Foundation of Tianjin

(13JCZDJC26200), and National High Tech. Research and

Development Program of China (No.2015AA043202)

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