Kinematics and Interactive Simulation System Modeling forRobot Manipulators

Xiao Xiao11Department of Electromechanical Engineering

University of Macau, Macao SAR, China

Yangmin Li1,2 and Hui Tang12School of Mechanical Engineering

Tianjin University of TechnologyTianjin 300191, China

Corresponding Author: E-mail: ymli@umac.mo

Abstract— In this paper, three convenient, efficient and ac-curate virtual scene modeling methods are proposed for robotmanipulators. 3D modeling softwares in terms of SolidWorksand UG are exploited to construct the 3D models of the virtualscene. The 3D models are established and the data are used andshared by OpenGL and VRML. The virtual scene is displayedon Visual C++, LabVIEW and MATLAB platform, which canbe controlled as requested. The forward and inverse kinematicsof a robot manipulator named Katana450 are analyzed by usingthe exponential product method, then the interactive simulationsystem is established based on the three mentioned platforms,finally the simulation and experimental results are providedbase on MATLAB/Simulink. The methods are expected to bewidely used in virtual scene modeling of robot manipulatorsand extended to other complex mechanical systems as well.

Index Terms— Interactive simulation system, Paden-Kahansubproblems, OpenGL, VRML.

I. INTRODUCTION

The interactive simulation system is a complex systemthat related to many technologies. Such as computer graphictechnology, display technology, sensor technology and so on.It is visually vivid and highly interactive, which can simulatevarious physical systems that are used for parameters deter-mination and controlling methods selections. Compared withphysical experiments, it is lower in cost, higher in reliability,and more suitable for massive exploratory experiments [1].Nowadays it has been applied in many subjects in scienceand technology field. In the field of mechanical engineering,it has found important applications in virtual simulation [2],robot control [3], tele-rehabilitation system [4] and interactivevirtual assembly path planning [5] and so on.

The problem that should be solved first in the process ofdesigning an interactive simulation system is virtual scenemodeling. It is one of the key technologies in the interactivesimulation system. OpenGL and VRML are commonly usedin virtual scene modeling. OpenGL is the widely accepted2D/3D graphic API in the areas, which is a bottom graphiclanguage pack with a powerful function and convenient usage[6]. VRML is a term referred to Virtual Reality Modeling

This work was supported by Research Committee of University of Macauunder Grant No.:MYRG183(Y1-L3)FST11-LYM and MYRG203(Y1-L4)-FST11-LYM).

Language. It is a text based, platform independent, weboriented, and object oriented modeling language [7].

Although OpenGL and VRML are widely used, there are alot of limitations on their usages. OpenGL only provides lim-ited basic graphic drawing functions, which is not only a bigworkload, but also inefficient when drawing a complex scene.While VRML only provides four simple geometric modelingnodes and five complex geometric modeling nodes. With thesame problem, for some complex mechanical systems, usingthe provided nodes usually makes modeling too complicated,sometimes even is impossible to accomplish.

The 3D modeling software such as SolidWorks and UGcan easily construct a complex model. 3D parts or assembliesestablished by SolidWorks and UG can be easily saved asfile format that supported by OpenGL and VRML, whichprovides a very convenient method to construct complex andvivid virtual scene in OpenGL and VRML. But SolidWorksand UG have not convenient data input/output interface,so they are unsuitable for peripheral device or program tocontrol the models, which has the same problem for somecommercial simulation softwares such as VR-Platform andVirtools.

In order to solve the above mentioned conflicts, in consid-eration of interactive simulation system’s potential applica-tions in scientific work and engineering practice, we proposethree different kinds of virtual scene modeling methods interms of virtual scene modeling method based on VisualC++, virtual scene modeling method based on LabVIEW andvirtual scene modeling method based on MATLAB. Thesemethods use the advantages of OpenGL, VRML and 3D mod-eling software, which greatly reduce the difficulty and cost ofthe system development. The methods are implemented basedon Katana450 robot as shown in Fig.1. Forward kinematicsand inverse kinematics analyses are performed in Section II.The implementation of every method is discussed in SectionIII. Experimental results are given in Section IV.

II. KINEMATICS ANALYSIS

The commonly used kinematics analysis methods are D-H method and screw method. Compared with D-H method,screw method only needs to establish two coordinates. So

978-1-4977-1334-3/13/$31.00 ©2013 IEEE

Proceeding of the IEEEInternational Conference on Information and Automation

Yinchuan, China, August 2013

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Fig. 1. A Katana450 robot.

screw method is much easier than D-H method in a sense.The following section will use exponential product methodthat based on screw theory to solve the forward and inversekinematics of the Katana450 robot.

A. Forward Kinematics

In the screw theory [8]-[9], the motion of the end-effectorof the robot can be represented by the motion screws of thejoints as

𝑔𝑆𝑇 (𝜃) = 𝑒𝜃1𝜉1𝑒𝜃2𝜉2 ⋅ ⋅ ⋅ 𝑒𝜃𝑛𝜉𝑛𝑔𝑆𝑇 (0) (1)

The above formula is called exponential product formulaof the robot forward kinematics. 𝑔𝑆𝑇 (0) is the initial config-uration of the robot, 𝑒𝜃𝑖𝜉𝑖 is the matrix exponential form ofthe motion screw 𝜉, which is defined as

𝑒𝜃𝑖𝜉𝑖 =

[𝑒𝜃𝑖�̂�𝑖

(𝐼 − 𝑒𝜃𝑖�̂�𝑖

)(𝜔𝑖 × 𝑣𝑖) + 𝜃𝑖𝜔𝑖𝜔

𝑇𝑖 𝑣𝑖

0 1

](2)

Establishing the coordinate systems as shown in Fig.2, where {𝑆} is the inertial coordinate system, {𝑇} isthe tool coordinate system. 𝜔𝑖 (𝑖 = 1, 2 ⋅ ⋅ ⋅ 5) is the unitvector that represents the direction of the rotational axis,𝜃𝑖 (𝑖 = 1, 2 ⋅ ⋅ ⋅ 5) is the joint variable, 𝑙𝑖 (𝑖 = 1, 2, 3) is thelength of the linkage.

As shown in Fig.2, the initial configuration of the robotcan be expressed as

𝑔𝑆𝑇 (0) =

⎡⎢⎢⎣

1 0 0 𝑙1 + 𝑙2 + 𝑙30 1 0 00 0 1 00 0 0 1

⎤⎥⎥⎦ (3)

The axis of each joint can be represented as

𝜔1 =[0 0 1

]𝑇𝜔2 =

[0 0 1

]𝑇𝜔3 =

[0 1 0

]𝑇𝜔4 =

[0 1 0

]𝑇𝜔5 =

[1 0 0

]𝑇 (4)

Selecting a point on each axis as follows:

𝑟1 =[0 0 0

]𝑇𝑟2 =

[0 0 0

]𝑇𝑟3 =

[𝑙1 0 0

]𝑇𝑟4 =

[𝑙1 + 𝑙2 0 0

]𝑇𝑟5 =

[𝑙1 + 𝑙2 + 𝑙3 0 0

]𝑇 (5)

Substituting 𝜔𝑖 and 𝑟𝑖 into Eq.(2), we get

𝑒𝜃1𝜉1 =

⎡⎢⎢⎣

cos 𝜃1 − sin 𝜃1 0 0sin 𝜃1 cos 𝜃1 0 00 0 1 00 0 0 1

⎤⎥⎥⎦ (6)

𝑒𝜃2𝜉2 =

⎡⎢⎢⎣

cos 𝜃2 0 sin 𝜃2 00 1 0 0

− sin 𝜃2 0 cos 𝜃2 00 0 0 1

⎤⎥⎥⎦ (7)

𝑒𝜃3𝜉3 =

⎡⎢⎢⎣

cos 𝜃3 0 sin 𝜃3 (1− cos 𝜃3) 𝑙10 1 0 0

− sin 𝜃3 0 cos 𝜃3 sin 𝜃3𝑙10 0 0 1

⎤⎥⎥⎦ (8)

𝑒𝜃4𝜉4 =

⎡⎢⎢⎣

cos 𝜃4 0 sin 𝜃4 (1− cos 𝜃3) (𝑙1 + 𝑙2)0 1 0 0

− sin 𝜃4 0 cos 𝜃4 sin 𝜃3 (𝑙1 + 𝑙2)0 0 0 1

⎤⎥⎥⎦(9)

𝑒𝜃5𝜉5 =

⎡⎢⎢⎣

1 0 0 00 cos 𝜃5 − sin 𝜃5 00 sin 𝜃5 cos 𝜃5 00 0 0 1

⎤⎥⎥⎦ (10)

Therefore, the forward kinematics of the Katana450 robotcan be obtained by

𝑔𝑆𝑇 (𝜃)=𝑒𝜃1𝜉1𝑒𝜃2𝜉2𝑒𝜃3𝜉3𝑒𝜃4𝜉4𝑒𝜃5𝜉5𝑔𝑆𝑇 (0) (11)

It is obvious that the position and orientation of the robotcan be easily obtained when the joint variables are provided.

B. Inverse Kinematics

The inverse kinematics problem of the serial robot ismuch more complicated than the forward kinematics [10].Based on Paden-Kahan subproblems [11]-[12], this paperdecomposes the inverse kinematics problem of the Katana450robot into several subproblems which are already known, thenthe difficulty of the problem can be greatly reduced.

According to Eq.(11), we can obtain

𝑒𝜃1𝜉1𝑒𝜃2𝜉2𝑒𝜃3𝜉3𝑒𝜃4𝜉4𝑒𝜃5𝜉5 = 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇−1 (0) (12)

The task of the inverse kinematics is to calculate the jointvariables when the position and orientation information ofthe end-effector are given. Let the position and orientationof the end-effector be

𝑔𝑆𝑇 (𝜃) =

⎡⎢⎢⎣

𝑛𝑥 𝑜𝑥 𝑎𝑥 𝑝𝑥𝑛𝑦 𝑜𝑦 𝑎𝑦 𝑝𝑦𝑛𝑧 𝑜𝑧 𝑎𝑧 𝑝𝑧0 0 0 1

⎤⎥⎥⎦ (13)

Because joint 1 changes the position of the end-effectorwithout changing the orientation. So we can obtain

𝜃1 = 𝑎 tan (𝑝𝑦/𝑝𝑥) (14)

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Fig. 2. Katana 450 robot in its initial configuration.

Axis 𝜉4 and 𝜉5 intersects at 𝑝0. According to positioninvariant principle, we can obtain

𝑒𝜃4𝜉4𝑒𝜃5𝜉5𝑝0 = 𝑝0 (15)

Let both side of Eq.(11) act on point 𝑝0. Hereby, itbecomes

𝑒𝜃1𝜉1𝑒𝜃2𝜉2𝑒𝜃3𝜉3𝑝0 = 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇−1 (0) 𝑝0 (16)

Because 𝜃1 can be obtained by Eq.(14). Thus, Eq.(16) canbe rewritten as

𝑒𝜃2𝜉2𝑒𝜃3𝜉3𝑝0 = 𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇−1 (0) 𝑝0 (17)

Let 𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇−1 (0) 𝑝0 = 𝑞0, Thus

𝑒𝜃2𝜉2𝑒𝜃3𝜉3𝑝0 = 𝑞0 (18)

𝜃3 can be solved by using the sub-problem 2.

𝜃3 = ±(𝜋 − 𝑎𝑐𝑟 cos

(𝑙21 + 𝑙22 − 𝑝2𝑥 − 𝑝2𝑦 − 𝑝2𝑧

2𝑙1𝑙2

))(19)

Let 𝑒𝜃3𝜉3𝑝0 = 𝑝1, thus

𝑒𝜃2𝜉2𝑝1 = 𝑞0 (20)

Then 𝜃2 can be obtained by using the sub-problem 1

𝜃2 = ± arccos(𝑜𝑠𝑝

𝑇1 ⋅ 𝑜𝑠𝑞0/ ∣𝑜𝑠𝑝1∣ ⋅ ∣𝑜𝑠𝑞0∣

)(21)

Substituting 𝜃1, 𝜃2 and 𝜃3 into Eq.(11), we can obtain

𝑒𝜃4𝜉4𝑒𝜃5𝜉5 = 𝑒−𝜃3𝜉3𝑒−𝜃2𝜉2𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇−1(0) (22)

𝑝𝑔 is the center point of the gripper which is on the axis𝜉5, according to the position invariant principle, we have

𝑒𝜃5𝜉5𝑝𝑔 = 𝑝𝑔 (23)

Let both side of Eq.(22) act on point 𝑝𝑔 . That is

𝑒𝜃4𝜉4𝑝𝑔 = 𝑒−𝜃3𝜉3𝑒−𝜃2𝜉2𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇−1(0)𝑝𝑔 (24)

Let 𝑒−𝜃3𝜉3𝑒−𝜃2𝜉2𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇−1(0)𝑝𝑔 = 𝑝𝑘, we can

get𝑒𝜃4𝜉4𝑝𝑔 = 𝑝𝑘 (25)

Using sub-problem 1 again, we can obtain

𝜃4 = ± arccos(𝑝𝑔𝑝

𝑇0 ⋅ 𝑝𝑘𝑝0/ ∣𝑝𝑔𝑝0∣ ⋅ ∣𝑝𝑘𝑝0∣

)(26)

Then, we have

𝑒𝜃5𝜉5 = 𝑒−𝜃4𝜉4𝑒−𝜃3𝜉3𝑒−𝜃2𝜉2𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇−1(0) (27)

Selecting 𝑝𝑚 =[𝑙1 + 𝑙2 1 0 1

]𝑇on the axis 𝜉4.

Based on Eq.(27), we get

𝑒𝜃5𝜉5𝑝𝑚 = 𝑒−𝜃4𝜉4𝑒−𝜃3𝜉3𝑒−𝜃2𝜉2𝑒−𝜃1𝜉1𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇−1(0)𝑝𝑚

(28)Let the right side of Eq.(28) be 𝑝𝑛. Similarly, using sub-

problem 1, 𝜃5 can be obtained as

𝜃5 = ± arccos(𝑝𝑚𝑝𝑇0 ⋅ 𝑝𝑛𝑝0/ ∣𝑝𝑚𝑝0∣ ⋅ ∣𝑝𝑛𝑝0∣

)(29)

Eq.(14), (19), (21), (23) and (29) are the inverse kinematicsequations of Katana450 robot.

III. VIRTUAL SCENE MODELING METHODS

A. Interactive simulation system based on Visual C++

Because Microsoft has included OpenGL in Windows,so OpenGL can closely be associated with Visual C++.In order to make virtual scene modeling in Visual C++simpler, Deep Exploration is introduced. Deep Explorationis a powerful 2D/3D model file management, browsing andconversion software, which supports 80 kinds of 3D modelformats, including Solidworks STL format. It can converta Solidworks STL format file into a *.CPP file that inaccordance with OpenGL language [13]. In this way, it willbe very convenient to display and create a variety of complex3D models and scenes on the Visual C++ platform.

By analyzing the *.CPP file transformed from Deep Explo-ration, we can find that the original 3D model was convertedinto several triangle patches information, including trianglepatch indexes, triangle patch vertexes and triangle patchnormal vectors. These information are stored in three arraysin terms of faceindicies[ ][9], vertices[ ][3] and normals[ ][3],respectively. The model will be displayed by drawing the

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Fig. 3. Interactive simulation system based on Visual C++.

triangle patches in series. The function that used for drawingthe triangle patches is GLint Gen3DObjectList().

The steps of displaying Solidworks models in Visual C++platform are as follows

1) Saving the model created by Solidworks as STL formatfile

2) Converting the STL format file into *.CPP file in DeepExploration

3) Getting faceindicies[ ][9], vertices[ ][3] and normals[][3] arrays from the *.CPP file

4) Getting GLint Gen3DObjectList() sub-function from the*.CPP file

5) Calling the triangle patch information by using the sub-function.

It is worthwhile to note that the data of the triangle patcheswill increase with the complexity of the model, especiallywhen the model is made up of several different parts. In thissituation, an available method is storing all of the trianglepatches information into a *.h file that created beforehand.This can improve the readability of the program codes.Fig.3 shows the effect of displaying the model created bySolidworks in Visual C++.

B. Interactive simulation system based on LabVIEW

The load file function of 3D picture control in LabVIEWcan Load ASE geometry, load VRML file and load STLgeometry [14]. This will provide a good solution to constructcomplex and vivid virtual scene in LabVIEW circumstance.

An example of illustrating the process of reading VRMLfile in LabVIEW is given below:

1) Saving the 3D model constructed by SolidWorks or UGinto VRML file format;

2) Adding a 3D picture control on the front panel ofLabVIEW, then adding a Create Object vi to the left terminalof the 3D picture control on the block diagram. Right-clickingthe scene of the Create Object vi and choose Add Object vifrom methods of scene object class, which allows adding anoutside 3D model or 3D scene;

3) Adding a Load VRML file vi, which is used to loadand display 3D model;

4) Double-clicking the Load VRML file vi and setting theright path of the VRML file in the dialogue box. Indicatingthe specific name and file format of the loaded VRML filein the terminal of the Build Path vi.

Because the settings of the light source, the model scale,and the position of the model in the scene have a greatinfluence on the final display effect. It usually needs to setscale, set rotation and set translation before displaying themodel in the 3D picture control. The basic block diagram ofreading VRML file in LabVIEW is illustrated in Fig.4.

Fig. 4. The basic block diagram of reading VRML file in LabVIEW.

In the process of complex virtual scene modeling, thevirtual object is usually an assembly that is composed ofseveral different parts. The relative movements between dif-ferent parts are required as well. In this situation, it needs toseparately load each part into the total scene, then assemblethem together by using transformation vi and rotation vi. Therelative movements between different parts can be realized bychanging the parameters of the transformation vi and rotationvi. Fig.5 shows the effect of LabVIEW calling VRML file.

Fig. 5. Interactive simulation system based on LabVIEW.

C. Interactive simulation system based on MATLAB

MATLAB has added Virtual Reality Toolbox starting fromMATLAB 6 version. With the continuously renewing ofthe MATLAB, the function of the Virtual Reality Toolboxis constantly improving and refining. It uses the standardVRML technology and combines MATLAB/Simulink withVirtual Reality Technology together, which expands the virtu-al reality image processing ability of MATLAB and Simulink

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and provides an effective solution for visual operation anddynamic interaction.

Virtual Reality Toolbox can be run under MATLAB in-terface and Simulink interface. These interfaces provide amethod to observe, change and control the virtual sceneand object. That is, interacting with the virtual world. TheSimulink interface is much more direct and easy to use.

The steps of using MATLAB/Simulink to create robotinteractive virtual reality scene are listed as follows:

1) Saving the 3D assembly model of the robot as VRMLformat file, setting the output version to VRML 97, the unitto metre;

2) Using VRML editor V-Realm Builder 2.0 to adjust thesubordinate relationship of the Transform nodes according tothe kinematic relationships of the robotic joints;

3) Adjusting the field value of the center field in eachTransform node, so as to set the right coordinate position ofeach part;

4) Adding a VR Sink block to a new created SimulinkModel. Double-clicking the VR Sink block and loading thesaved VRML file into the virtual world. Selecting the fieldscorresponding to the joint varieties and applying;

5) Adding the control parameters.The final robot interactive virtual reality scene is illustrated

in Fig.6. Visual C++ is suitable for engineering design.

Fig. 6. Interactive simulation system based on MATLAB.

LabVIEW is mainly focused on measurement and control,which has important function that its graphic programmingmakes program simple and easy. MATLAB has outstandingadvantages in numerical calculation and system simulation,each platform has its advantages and can be applied todifferent applications.

IV. EXPERIMENT

In order to evaluate the effects of the methods presentedin this paper. Two experiments that based on MATLABplatform have been carried out. The first experiment is a pathtracking application. Katana450 robot is required to tracka desired path in a given orientation. In the experiment, a

circle path 𝑦2 + (𝑧 + 110)2= 1002, 𝑥 = 150 is given. The

path is dispersed and transferred to the inverse kinematics.The joint variables are obtained and the Katana450 robot inthe interactive simulation system is controlled to completea circle path. The angles obtained in the inverse kinematicsare calculated in the forward kinematics at the same time.The calculated path and the desired path are compared andanalyzed. The results are shown in Fig.7 and Fig.8. The

Fig. 7. Path tracking result.

Fig. 8. Joint variables when tracking the circle path.

maximum position error is less than 3× 10−12 mm, whichindicates the kinematics model obtained in Section II iscorrect. The joint variables are changing smoothly withoutsingular configuration. Both the path tracking simulationand the kinematics of the Katana450 robot are implementedsuccessfully.

The second experiment is a manipulation simulation appli-cation. In this experiment, Katana450 robot will pick up a cupon the working table and empty the water in the cup into thebasin. To complete this task, we first figure out the positionof the cup relative to the basin, then plan a route for the robotand calculate the joint variables needed to complete the taskby using the inverse kinematics. Fig.9 to Fig.12 show the

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manipulation process. Tactile sensors can also be installed onthe end-effector of the Katana450 robot [15]-[16], the robotcan sense the gripping forces and better performance can beobtained, which will be our next step work.

Fig. 9. Katana450 robot in its initial configuration.

Fig. 10. Katana450 robot is grasping the cup.

Fig. 11. Katana450 robot is approaching the basin.

Fig. 12. Katana450 robot is emptying the cup.

V. CONCLUSION

Based on OpenGL and VRML, this paper presents threedifferent kinds of interactive simulation system modelingmethods. The paper first points out the limitations of OpenGL

and VRML, then analyzes the solution and introduces thevirtual scene modeling method based on Visual C++, Lab-VIEW and MATLAB in detail, finally proves the feasibilityof the interactive simulation system and realizes the functionof interactive control by experiments.

The main work is focused on the 3D software modeling,instead of complicated programming, it greatly reduces thedifficulty of the virtual scene modeling and shortens thedevelopment cycle of the interactive simulation system. Atthe same time, the system based on these methods is easy tomaintain and transplant. It will find wide applications in thefields such as military, entertainment, medical and robot.

Combining with peripheral devices such as multi-dimensional mouse and haptic device, the interactive sim-ulation system will be applied to control the real Katana450robot, more deep research work will be done in the future.

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