# SIMULATION OF THE KINEMATICALLY REDUNDANT MOBILE MANIPULATOR

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2 to allow the system to perform some tasks autonomously and will also help the operator to solve some difficult situations, e.g., when the whole scene cannot be viewed using the vision system and collisions with obstacles should be avoided. Detailed description of the UTAR system can be found in [2], its current state is shown in Fig. 1b. (a) (b) Figure 1: Idea scheme of the UTAR system (a), and its current state (b) 2 Mathematical Background Mobile manipulator considered in this study consists of the 6-DOF (degrees-of-freedom) arm mounted on the 3-DOF mobile platform. This is kinematically redundant mechanism, since it has more DOF (nine in this case) than required to perform the given task (unique definition of the task in 3D can be realized using six coordinates). Process of calculating the inverse kinematics of the redundant manipulator is referred to redundancy resolution. Kinematic equations that relate locations of the gripper to the corresponding joint angles are mostly highly non-linear and there is no analytical closed-form solution of the inverse kinematics for a general robot structure. This is why the inverse kinematics is solved at the velocity level: q = J + H ẋ + ( I J + H J) ( h(q) q ) T (1) where q is the [n 1] joint velocity vector, J is the [m n] Jacobian matrix, J + H is the inertia weighted Jacobian pseudoinverse defined as J + H = H 1 J T ( J H 1 J T) 1, H is the manipulator inertia matrix, ẋ is the [m 1] gripper velocity vector, I is the identity matrix and h(q) is the scalar function, which becomes large in configurations close to 6-DOF manipulator workspace boundary, singular configurations and is also used for collision avoidance. Detailed explanation of the inverse kinematics is out of the scope of this paper, but can be found in [1]. 3 Simulation Simulation of the redundancy resolution algorithm is performed in Matlab. To speed up the computation no Simulink model is used and everything is programmed in the form of pure Matlab m-files. The key part of the program code is the m-file corresponding to (1). This function is then integrated using an explicit Runge-Kutta integration method (ode23). The input of the system is vector of gripper velocities ẋ, the output is the vector of joint coordinates q. To better understand the behaviour of the control algorithm it is advantageous to run the simulation on-line, rather than off-line. Since the integration algorithm used is a variable-step

3 one, simulation is running in pseudo-real-time only. Gripper velocities may be controlled by standard computer input devices mouse or joystick. Usually, the result of the integration process is not available until the ODE solver finishes the computation. Since for the on-line visualization the actual data are required, the OutputFcn property of the ODE solver is used. Then, the visualization function is called after computing each output point. 4 Visualization For simulations in the field of robotics and mechatronics the visualization of the computed results is crucial. Matlab itself offers powerful tools for visualization both in 2D and 3D. For drawing of robotic systems Matlab patch graphics objects can be used. Thus, such manipulator consists of several polyhedra, cylinders, spheres etc. which are drawn during the initialization of the whole scene. Then, the particular vertices of all patches are translated and rotated accordingly every time the new output values are computed. For this purpose the set command is used, which is changing the properties XData, YData and ZData of the particular patch objects. Although the patch objects make possible to visualize complicated 3D shapes, there are some facts, that complicate the on-line drawing and animation. First of all, the coordinates of all the particular vertices of every graphics object that is moving have to be recomputed at every integration step. Although this can be relatively easy made using the method of homogeneous transformations, all the transformation matrices and other necessary computations must be programmed by the user, which could be difficult for more complicated objects consisting of many patches. Moreover, every change of the model structure needs thorough studying of the corresponding source code and is very inconvenient. Another disadvantage of this approach is the fact that the rendering is slow, especially for large objects. Even if the OpenGL renderer is chosen, animation is too slow and, in addition, the window size is limited. Another possibility how the visualization of various 3D objects can be realized is the use of the Virtual Reality Toolbox, which is extending graphics capabilities of Matlab, [3]. This toolbox is utilizing standard VRML technology, which is an open, text-based, WWW-oriented format, allowing description of 3D scene, sound, internal actions, WWW anchors etc. It can be viewed in any WWW browser with the corresponding VRML plug-in installed. The principle of operation is as follows: First, the VRML model of the object and its environment must be created. This can be accomplished by using of any plain text editor. Nonetheless, writing the VRML code by hand requires deep knowledges of the VRML syntax and is a lengthy process. It is easier to build the scene in any 3D modeler and export it into VRML format or create the model directly using native VRML authoring tool. Once the model is created, the object of class vrworld within Matlab environment is created. Then, it is opened and viewed in the WWW browser. The connection between Matlab and the VRML enabled browser via TCP/IP protocol is established. A flexible Matlab interface to the VRML virtual reality world provided by Virtual Reality Toolbox allows now the easy control of selected model parameters. The above mentioned approach will be now presented on an easy example. First, the model of the mobile manipulator, consisting of the four-wheeled mobile platform with the 6-DOF manipulator arm attached, is created. As the authoring tool the native VRML editor VRealm Builder is used. The screenshot of this editor is shown in Fig. 2. On the left, the hierarchical tree structure of objects nodes is shown. Each node contains a list of fields, which hold values that define parameters for its function. Nodes can be placed in the top level of the tree or as children of other nodes in the tree hierarchy. When a value in a field of certain node is changed, all nodes of its sub-tree will be affected. This allows to define relative positions inside complicated compound objects. The design and creation process of the VRML model is simplified by the friendly graphical user interface. Placing, moving and changing orientation or size of objects can be done using the mouse. Only the basic knowledge of the VRML language principles is necessary.

5 Figure 2: Creation of the VRML model in VRealm Builder Figure 3: Mobile manipulator in virtual environment viewed in the WWW browser

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