Kinematics and Interactive Simulation System Modeling for Robot

Proceeding of the IEEE International Conference on Information and Automation Yinchuan, China, August 2013

Kinematics and Interactive Simulation System Modeling for Robot Manipulators 1

Xiao Xiao1

2

Department of Electromechanical Engineering University of Macau, Macao SAR, China

School of Mechanical Engineering Tianjin University of Technology Tianjin 300191, China Corresponding Author: E-mail: [email protected]

Language. It is a text based, platform independent, web oriented, and object oriented modeling language [7]. Although OpenGL and VRML are widely used, there are a lot of limitations on their usages. OpenGL only provides limited basic graphic drawing functions, which is not only a big workload, but also inefficient when drawing a complex scene. While VRML only provides four simple geometric modeling nodes and five complex geometric modeling nodes. With the same problem, for some complex mechanical systems, using the provided nodes usually makes modeling too complicated, sometimes even is impossible to accomplish. The 3D modeling software such as SolidWorks and UG can easily construct a complex model. 3D parts or assemblies established by SolidWorks and UG can be easily saved as file format that supported by OpenGL and VRML, which provides a very convenient method to construct complex and vivid virtual scene in OpenGL and VRML. But SolidWorks and UG have not convenient data input/output interface, so they are unsuitable for peripheral device or program to control the models, which has the same problem for some commercial simulation softwares such as VR-Platform and Virtools. In order to solve the above mentioned conflicts, in consideration of interactive simulation system’s potential applications in scientific work and engineering practice, we propose three different kinds of virtual scene modeling methods in terms of virtual scene modeling method based on Visual C++, virtual scene modeling method based on LabVIEW and virtual scene modeling method based on MATLAB. These methods use the advantages of OpenGL, VRML and 3D modeling software, which greatly reduce the difficulty and cost of the system development. The methods are implemented based on Katana450 robot as shown in Fig.1. Forward kinematics and inverse kinematics analyses are performed in Section II. The implementation of every method is discussed in Section III. Experimental results are given in Section IV.

Abstract— In this paper, three convenient, efficient and accurate virtual scene modeling methods are proposed for robot manipulators. 3D modeling softwares in terms of SolidWorks and UG are exploited to construct the 3D models of the virtual scene. The 3D models are established and the data are used and shared by OpenGL and VRML. The virtual scene is displayed on Visual C++, LabVIEW and MATLAB platform, which can be controlled as requested. The forward and inverse kinematics of a robot manipulator named Katana450 are analyzed by using the exponential product method, then the interactive simulation system is established based on the three mentioned platforms, finally the simulation and experimental results are provided base on MATLAB/Simulink. The methods are expected to be widely used in virtual scene modeling of robot manipulators and extended to other complex mechanical systems as well. Index Terms— Interactive simulation system, Paden-Kahan subproblems, OpenGL, VRML.

I. I NTRODUCTION The interactive simulation system is a complex system that related to many technologies. Such as computer graphic technology, display technology, sensor technology and so on. It is visually vivid and highly interactive, which can simulate various physical systems that are used for parameters determination and controlling methods selections. Compared with physical 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 science and 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 interactive virtual assembly path planning [5] and so on. The problem that should be solved first in the process of designing an interactive simulation system is virtual scene modeling. It is one of the key technologies in the interactive simulation system. OpenGL and VRML are commonly used in virtual scene modeling. OpenGL is the widely accepted 2D/3D graphic API in the areas, which is a bottom graphic language pack with a powerful function and convenient usage [6]. VRML is a term referred to Virtual Reality Modeling

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

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

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

Yangmin Li1,2 and Hui Tang1

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Fig. 1.

Substituting 𝜔𝑖 and 𝑟𝑖 into Eq.(2), we get ⎡ ⎤ cos 𝜃1 − sin 𝜃1 0 0 ⎢ sin 𝜃1 cos 𝜃1 0 0 ⎥ ˆ ⎥ 𝑒𝜃1 𝜉1 = ⎢ (6) ⎣ 0 0 1 0 ⎦ 0 0 0 1 ⎡ ⎤ cos 𝜃2 0 sin 𝜃2 0 ⎢ 0 1 0 0 ⎥ ˆ ⎥ (7) 𝑒𝜃2 𝜉2 = ⎢ ⎣ − sin 𝜃2 0 cos 𝜃2 0 ⎦ 0 0 0 1 ⎡ ⎤ cos 𝜃3 0 sin 𝜃3 (1 − cos 𝜃3 ) 𝑙1 ⎢ ⎥ 0 1 0 0 ˆ ⎥ (8) 𝑒𝜃3 𝜉3 = ⎢ ⎣ − sin 𝜃3 0 cos 𝜃3 ⎦ sin 𝜃3 𝑙1 0 0 0 1 ⎡ ⎤ cos 𝜃4 0 sin 𝜃4 (1 − cos 𝜃3 ) (𝑙1 + 𝑙2 ) ⎢ ⎥ 0 1 0 0 ˆ ⎥ 𝑒𝜃4 𝜉4 = ⎢ ⎣ − sin 𝜃4 0 cos 𝜃4 ⎦ sin 𝜃3 (𝑙1 + 𝑙2 ) 0 0 0 1 (9) ⎡ ⎤ 1 0 0 0 ⎢ 0 cos 𝜃5 − sin 𝜃5 0 ⎥ ˆ ⎥ (10) 𝑒𝜃5 𝜉 5 = ⎢ ⎣ 0 sin 𝜃5 cos 𝜃5 0 ⎦ 0 0 0 1

A Katana450 robot.

screw method is much easier than D-H method in a sense. The following section will use exponential product method that based on screw theory to solve the forward and inverse kinematics of the Katana450 robot. A. Forward Kinematics In the screw theory [8]-[9], the motion of the end-effector of the robot can be represented by the motion screws of the joints as ˆ

ˆ

ˆ

𝑔𝑆𝑇 (𝜃) = 𝑒𝜃1 𝜉1 𝑒𝜃2 𝜉2 ⋅ ⋅ ⋅ 𝑒𝜃𝑛 𝜉𝑛 𝑔𝑆𝑇 (0)

(1)

The above formula is called exponential product formula of the robot forward kinematics. 𝑔𝑆𝑇 (0) is the initial configˆ uration of the robot, 𝑒𝜃𝑖 𝜉𝑖 is the matrix exponential form of the motion screw 𝜉, which is defined as ) ] [ 𝜃 𝜔ˆ ( 𝐼 − 𝑒𝜃𝑖 𝜔ˆ 𝑖 (𝜔𝑖 × 𝑣𝑖 ) + 𝜃𝑖 𝜔𝑖 𝜔𝑖𝑇 𝑣𝑖 𝑒𝑖 𝑖 𝜃𝑖 𝜉ˆ𝑖 (2) = 𝑒 0 1

Therefore, the forward kinematics of the Katana450 robot can be obtained by ˆ

Selecting a point on each axis as follows: [ ]𝑇 [ ]𝑇 𝑟1 = 0 0 0 𝑟2 = 0 0 0 [ [ ]𝑇 ]𝑇 𝑟3 = 𝑙1 0 0 𝑟4 = 𝑙1 + 𝑙2 0 0 [ ]𝑇 𝑟5 = 𝑙1 + 𝑙2 + 𝑙3 0 0

ˆ

ˆ

ˆ

(11)

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

Establishing the coordinate systems as shown in Fig. 2, where {𝑆} is the inertial coordinate system, {𝑇 } is the tool coordinate system. 𝜔𝑖 (𝑖 = 1, 2 ⋅ ⋅ ⋅ 5) is the unit vector that represents the direction of the rotational axis, 𝜃𝑖 (𝑖 = 1, 2 ⋅ ⋅ ⋅ 5) is the joint variable, 𝑙𝑖 (𝑖 = 1, 2, 3) is the length of the linkage. As shown in Fig.2, the initial configuration of the robot can be expressed as ⎡ ⎤ 1 0 0 𝑙1 + 𝑙2 + 𝑙3 ⎢ 0 1 0 ⎥ 0 ⎥ 𝑔𝑆𝑇 (0) = ⎢ (3) ⎣ 0 0 1 ⎦ 0 0 0 0 1 The axis of each joint can be represented as [ ]𝑇 [ ]𝑇 𝜔2 = 0 0 1 𝜔1 = 0 0 1 [ ]𝑇 [ ]𝑇 𝜔3 = 0 1 0 𝜔4 = 0 1 0 [ ]𝑇 𝜔5 = 1 0 0

ˆ

𝑔𝑆𝑇 (𝜃) =𝑒𝜃1 𝜉1 𝑒𝜃2 𝜉2 𝑒𝜃3 𝜉3 𝑒𝜃4 𝜉4 𝑒𝜃5 𝜉5 𝑔𝑆𝑇 (0)

B. Inverse Kinematics The inverse kinematics problem of the serial robot is much more complicated than the forward kinematics [10]. Based on Paden-Kahan subproblems [11]-[12], this paper decomposes the inverse kinematics problem of the Katana450 robot into several subproblems which are already known, then the difficulty of the problem can be greatly reduced. According to Eq.(11), we can obtain ˆ

ˆ

ˆ

ˆ

ˆ

𝑒𝜃1 𝜉1 𝑒𝜃2 𝜉2 𝑒𝜃3 𝜉3 𝑒𝜃4 𝜉4 𝑒𝜃5 𝜉5 = 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇 −1 (0)

(12)

The task of the inverse kinematics is to calculate the joint variables when the position and orientation information of the end-effector are given. Let the position and orientation of the end-effector be ⎡ ⎤ 𝑛𝑥 𝑜 𝑥 𝑎 𝑥 𝑝 𝑥 ⎢ 𝑛𝑦 𝑜𝑦 𝑎𝑦 𝑝𝑦 ⎥ ⎥ 𝑔𝑆𝑇 (𝜃) = ⎢ (13) ⎣ 𝑛𝑧 𝑜𝑧 𝑎𝑧 𝑝𝑧 ⎦ 0 0 0 1

(4)

Because joint 1 changes the position of the end-effector without changing the orientation. So we can obtain

(5)

𝜃1 = 𝑎 tan (𝑝𝑦 /𝑝𝑥 )

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(14)

Fig. 2.

Katana 450 robot in its initial configuration.

Axis 𝜉4 and 𝜉5 intersects at 𝑝0 . According to position invariant principle, we can obtain ˆ

ˆ

𝑒𝜃 4 𝜉 4 𝑒 𝜃 5 𝜉 5 𝑝0 = 𝑝 0

Using sub-problem 1 again, we can obtain ( ) 𝜃4 = ± arccos 𝑝𝑔 𝑝𝑇0 ⋅ 𝑝𝑘 𝑝0 / ∣𝑝𝑔 𝑝0 ∣ ⋅ ∣𝑝𝑘 𝑝0 ∣

(15)

Then, we have

Let both side of Eq.(11) act on point 𝑝0 . Hereby, it becomes ˆ

ˆ

ˆ

𝑒𝜃1 𝜉1 𝑒𝜃2 𝜉2 𝑒𝜃3 𝜉3 𝑝0 = 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇 −1 (0) 𝑝0

ˆ

ˆ

ˆ

ˆ

(16)

ˆ

(17)

ˆ

ˆ

ˆ

𝜃3 can be solved by using the sub-problem 2. )) ( ( 𝑙12 + 𝑙22 − 𝑝2𝑥 − 𝑝2𝑦 − 𝑝2𝑧 𝜃3 = ± 𝜋 − 𝑎𝑐𝑟 cos 2𝑙1 𝑙2

(18)

ˆ

ˆ

ˆ

ˆ

𝑒𝜃4 𝜉4 𝑒𝜃5 𝜉5 = 𝑒−𝜃3 𝜉3 𝑒−𝜃2 𝜉2 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇 −1 (0)

(22)

𝑝𝑔 is the center point of the gripper which is on the axis 𝜉5 , according to the position invariant principle, we have ˆ

𝑒 𝜃 5 𝜉 5 𝑝𝑔 = 𝑝𝑔

(23)

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

ˆ

𝑒𝜃4 𝜉4 𝑝𝑔 = 𝑒−𝜃3 𝜉3 𝑒−𝜃2 𝜉2 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇 −1 (0)𝑝𝑔 ˆ

ˆ

ˆ

(29)

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 Exploration is a powerful 2D/3D model file management, browsing and conversion software, which supports 80 kinds of 3D model formats, including Solidworks STL format. It can convert a Solidworks STL format file into a *.CPP file that in accordance with OpenGL language [13]. In this way, it will be very convenient to display and create a variety of complex 3D models and scenes on the Visual C++ platform. By analyzing the *.CPP file transformed from Deep Exploration, we can find that the original 3D model was converted into several triangle patches information, including triangle patch indexes, triangle patch vertexes and triangle patch normal vectors. These information are stored in three arrays in terms of faceindicies[ ][9], vertices[ ][3] and normals[ ][3], respectively. The model will be displayed by drawing the

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

ˆ

ˆ

A. Interactive simulation system based on Visual C++ (20)

Then 𝜃2 can be obtained by using the sub-problem 1 ( ) (21) 𝜃2 = ± arccos 𝑜𝑠 𝑝𝑇1 ⋅ 𝑜𝑠 𝑞0 / ∣𝑜𝑠 𝑝1 ∣ ⋅ ∣𝑜𝑠 𝑞0 ∣

ˆ

ˆ

III. V IRTUAL SCENE MODELING METHODS

𝑒𝜃 2 𝜉 2 𝑝 1 = 𝑞 0

ˆ

ˆ

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

(19)

ˆ

ˆ

ˆ

( ) 𝜃5 = ± arccos 𝑝𝑚 𝑝𝑇0 ⋅ 𝑝𝑛 𝑝0 / ∣𝑝𝑚 𝑝0 ∣ ⋅ ∣𝑝𝑛 𝑝0 ∣

Let 𝑒𝜃3 𝜉3 𝑝0 = 𝑝1 , thus

ˆ

ˆ

𝑒𝜃5 𝜉5 𝑝𝑚 = 𝑒−𝜃4 𝜉4 𝑒−𝜃3 𝜉3 𝑒−𝜃2 𝜉2 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇 −1 (0)𝑝𝑚 (28) Let the right side of Eq.(28) be 𝑝𝑛 . Similarly, using subproblem 1, 𝜃5 can be obtained as

Let 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇 −1 (0) 𝑝0 = 𝑞0 , Thus 𝑒𝜃 2 𝜉 2 𝑒 𝜃 3 𝜉 3 𝑝0 = 𝑞 0

ˆ

𝑒𝜃5 𝜉5 = 𝑒−𝜃4 𝜉4 𝑒−𝜃3 𝜉3 𝑒−𝜃2 𝜉2 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇 −1 (0) (27) [ ]𝑇 Selecting 𝑝𝑚 = 𝑙1 + 𝑙2 1 0 1 on the axis 𝜉4 . Based on Eq.(27), we get

Because 𝜃1 can be obtained by Eq.(14). Thus, Eq.(16) can be rewritten as 𝑒𝜃2 𝜉2 𝑒𝜃3 𝜉3 𝑝0 = 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃) 𝑔𝑆𝑇 −1 (0) 𝑝0

(26)

(24)

ˆ

Let 𝑒−𝜃3 𝜉3 𝑒−𝜃2 𝜉2 𝑒−𝜃1 𝜉1 𝑔𝑆𝑇 (𝜃)𝑔𝑆𝑇 −1 (0)𝑝𝑔 = 𝑝𝑘 , we can get ˆ (25) 𝑒𝜃4 𝜉4 𝑝𝑔 = 𝑝𝑘

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Fig. 3.

3) Adding a Load VRML file vi, which is used to load and display 3D model; 4) Double-clicking the Load VRML file vi and setting the right path of the VRML file in the dialogue box. Indicating the specific name and file format of the loaded VRML file in 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 great influence on the final display effect. It usually needs to set scale, set rotation and set translation before displaying the model in the 3D picture control. The basic block diagram of reading VRML file in LabVIEW is illustrated in Fig.4.

Interactive simulation system based on Visual C++.

triangle patches in series. The function that used for drawing the 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 format file 2) Converting the STL format file into *.CPP file in Deep Exploration 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 subfunction. It is worthwhile to note that the data of the triangle patches will increase with the complexity of the model, especially when the model is made up of several different parts. In this situation, an available method is storing all of the triangle patches 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 by Solidworks in Visual C++.

Fig. 4.

The basic block diagram of reading VRML file in LabVIEW.

In the process of complex virtual scene modeling, the virtual object is usually an assembly that is composed of several different parts. The relative movements between different parts are required as well. In this situation, it needs to separately load each part into the total scene, then assemble them together by using transformation vi and rotation vi. The relative movements between different parts can be realized by changing the parameters of the transformation vi and rotation vi. Fig.5 shows the effect of LabVIEW calling VRML file.

B. Interactive simulation system based on LabVIEW The load file function of 3D picture control in LabVIEW can Load ASE geometry, load VRML file and load STL geometry [14]. This will provide a good solution to construct complex and vivid virtual scene in LabVIEW circumstance. An example of illustrating the process of reading VRML file in LabVIEW is given below: 1) Saving the 3D model constructed by SolidWorks or UG into VRML file format; 2) Adding a 3D picture control on the front panel of LabVIEW, then adding a Create Object vi to the left terminal of the 3D picture control on the block diagram. Right-clicking the scene of the Create Object vi and choose Add Object vi from methods of scene object class, which allows adding an outside 3D model or 3D scene;

Fig. 5.

Interactive simulation system based on LabVIEW.

C. Interactive simulation system based on MATLAB MATLAB has added Virtual Reality Toolbox starting from MATLAB 6 version. With the continuously renewing of the MATLAB, the function of the Virtual Reality Toolbox is constantly improving and refining. It uses the standard VRML technology and combines MATLAB/Simulink with Virtual Reality Technology together, which expands the virtual reality image processing ability of MATLAB and Simulink

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2

circle path 𝑦 2 + (𝑧 + 110) = 1002 , 𝑥 = 150 is given. The path is dispersed and transferred to the inverse kinematics. The joint variables are obtained and the Katana450 robot in the interactive simulation system is controlled to complete a circle path. The angles obtained in the inverse kinematics are calculated in the forward kinematics at the same time. The calculated path and the desired path are compared and analyzed. The results are shown in Fig.7 and Fig.8. The

and provides an effective solution for visual operation and dynamic interaction. Virtual Reality Toolbox can be run under MATLAB interface and Simulink interface. These interfaces provide a method to observe, change and control the virtual scene and object. That is, interacting with the virtual world. The Simulink interface is much more direct and easy to use. The steps of using MATLAB/Simulink to create robot interactive virtual reality scene are listed as follows: 1) Saving the 3D assembly model of the robot as VRML format file, setting the output version to VRML 97, the unit to metre; 2) Using VRML editor V-Realm Builder 2.0 to adjust the subordinate relationship of the Transform nodes according to the kinematic relationships of the robotic joints; 3) Adjusting the field value of the center field in each Transform node, so as to set the right coordinate position of each part; 4) Adding a VR Sink block to a new created Simulink Model. Double-clicking the VR Sink block and loading the saved VRML file into the virtual world. Selecting the fields corresponding 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.

Fig. 7.

Fig. 8.

Interactive simulation system based on MATLAB.

Path tracking result.

Joint variables when tracking the circle path.

maximum position error is less than 3 × 10−12 mm, which indicates the kinematics model obtained in Section II is correct. The joint variables are changing smoothly without singular configuration. Both the path tracking simulation and the kinematics of the Katana450 robot are implemented successfully. The second experiment is a manipulation simulation application. In this experiment, Katana450 robot will pick up a cup on the working table and empty the water in the cup into the basin. To complete this task, we first figure out the position of the cup relative to the basin, then plan a route for the robot and calculate the joint variables needed to complete the task by using the inverse kinematics. Fig.9 to Fig.12 show the

LabVIEW is mainly focused on measurement and control, which has important function that its graphic programming makes program simple and easy. MATLAB has outstanding advantages in numerical calculation and system simulation, each platform has its advantages and can be applied to different applications. IV. E XPERIMENT In order to evaluate the effects of the methods presented in this paper. Two experiments that based on MATLAB platform have been carried out. The first experiment is a path tracking application. Katana450 robot is required to track a desired path in a given orientation. In the experiment, a

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

Fig. 9.

and VRML, then analyzes the solution and introduces the virtual scene modeling method based on Visual C++, LabVIEW and MATLAB in detail, finally proves the feasibility of the interactive simulation system and realizes the function of interactive control by experiments. The main work is focused on the 3D software modeling, instead of complicated programming, it greatly reduces the difficulty of the virtual scene modeling and shortens the development cycle of the interactive simulation system. At the same time, the system based on these methods is easy to maintain and transplant. It will find wide applications in the fields such as military, entertainment, medical and robot. Combining with peripheral devices such as multidimensional mouse and haptic device, the interactive simulation system will be applied to control the real Katana450 robot, more deep research work will be done in the future.

Katana450 robot in its initial configuration.

R EFERENCES

Fig. 10.

Fig. 11.

Fig. 12.

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Katana450 robot is grasping the cup.

Katana450 robot is approaching the basin.

Katana450 robot is emptying the cup.

V. C ONCLUSION Based on OpenGL and VRML, this paper presents three different kinds of interactive simulation system modeling methods. The paper first points out the limitations of OpenGL

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