Operational Space Control for A Scara Robot

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2 This paper is organized as follows: section 2 is dedicated to model based control represented as join and operational space control; section 3 shows a simply way to calculate the Jacobian matrix; section 4 shows simulation results performed on complex trajectories that can be found in machining tasks on a Scara robot. Finally, section 5 presents the conclusions. II. DESCRIPTION OF THE SCARA ROBOT The SCARA robots is an industrial robot with 4 degrees of freedom, it s built with three revolute joints and one prismatic joint which produce a three dimensional task space. Figure 1 shows a scheme of a SCARA robot. The robot s inverse dynamic model equation which is used in this paper is: Where q and q& are the position vector and velocity vector of the each joint, A(q)is the inertial matrix, Q(q) is the gravitational torque vector, F v q& + F s sign( q& )represent the friction forces of the model (viscous and coulomb), H describes the effects of the gravitational torque and the friction forces and Γ is the applied torque in the joints. Matrix A and H contains the dynamic parameters of the robot. These parameters are the mathematical representation of the robot s dynamical characteristics. The dynamic parameters are: inertia tensor, first moment of the arm, the mass, moment of inertia of the rotor, Coulomb friction and viscous friction. The dynamic parameters are represented in the X vector: The good knowledge of the numerical values of the robot s dynamic parameters is necessary for maximum accuracy of the model. These can be grouped in order to minimize the computational load of the algorithms (Khalil and Dombre, 2001). The resulting parameters are shown in Table 1. TABLE 1: DYNAMIC PARAMETERS OF THE SCARA ROBOT (GROUPED) Figure 1: Robot scheme A. Computed torque control in the joint space When the task requires fast motion of the robot and high dynamic accuracy, it is necessary to improve the performance of the control by taking into account, partially or totally, the dynamic interaction torques. Linealizing and decoupling control is based on canceling the nolinearities in the robot dynamics. Theoretically, it ensures the linearization and de decoupling of the equations of the model, providing a uniform behavior whatever the configuration of the robot (Khalil and Dombre, 2001). Implementing this method requires on-line computation of the inverse dynamic model, as well as knowledge of the numerical values of the inertial parameters and friction parameters, as explined above. Let s assume the joints position and the velocities are measurable. Besides it exists a estimation of A and H matrix in the dynamic model: Then, after substituting (4) into (1) the system converts into a n - linear, decoupled system in the form, q = w( t) which is a linealized system, represented by a double integrator. Where w is the new input control vector (Khalil and Dombre, 2001). It is important to clear up that the industrial consign for this sort of controller is given in the Cartesian space, then it is necessary to convert them to a task space using the inverse geometric model.

3 In this paper, the control vector is: Where K v and K p are the control gain matrix. This control law is shown in Figure 2. Figure 3: Computed torque: block diagram of the tracking control scheme in task space Figure 2: Computed torque: block diagram of the tracking control scheme in joint space B. Computed torque control in the task space In this control type, the movement and the dynamic behavior of the system are completely defined in the task space. By this reason we have to do a conversion of the dynamic behavior from the joint space to the task space. This conversion is possible calculating the jacobian (representation of the linear mapping of the joint space velocities into the task space velocities) (Khalil and Dombre, 2001; Sciavicco and Siciliano, 1996). To obtain q first we use the inverse geometric model (IGM) to compute the joint positions: Now we use the direct kinematic model to calculate the joint velocities (jacobian). Differentiating (7) we obtain the second order direct kinematic model, expressed by: To obtain the second order inverse kinematic model, we clear up q from (8) The dynamic behavior of the robot in the task space is described by the following equation, obtained after substituting (9) into (1): The linelized and decoupled system is described by: For the control of the system in the task space we have (Chevallereau, 1988): Figure 3, shows the diagram of the tracking control with a PD controller. III. JACOBIAN CALCULATION The calculation of the k th term of SCARA robot jacobian matrix is Where k P 4x and k P 4y denote de X an Y components of the position vector respectively. 0 s k, 0 n k, 0 a k are the columns of the orientation matrix. σ k =1 if prismatic joint andσ k =0 if revolute joint. Thus we need to calculate the geometric direct model expressed by: Where T is the transformation matrix of the terminal frame R n relative to frame R 0. This matrix is defined in (14). It is a function of the joint variable vector q (Khalil and Dombre, 2001). IV. SIMULATION RESULTS The simulation of the control schemes is implemented in Matlab/Simulink. The simulation is performed on two complex trajectories, the sampling time is 1ms. The complex trajectories are: a) Circular trajectory with a diameter of 20 mm. This trajectory is achieved in 1s and 3s. b) Linear trajectory with a change of direction of 55. This trajectory is achieved in 1s and 3s. The tuning of CTC controller in joint space and task space for the SCARA robot leads to the gains shown in Table 2. The schemes were shown in Figure 2 and Figure 3, respectively.

4 TABLE 2: GAINS OF THE CONTROLLER Figure 6: Stationary response in circular path, covered in 3s Figure 4: Tracking error for linear path with change of direction, covered in 1s Figure 7: Robot s trajectory using the two control techniques Table 3 indicates the decrease percentage of the maximum tracking error between CTC in task space and CTC in joint space. Figure 5: Transient response in circular path, covered in 3s TABLE 3: COMPARISON BETWEEN CTC IN JOINT SPACE AND CTC IN TASK SPACE V. CONCLUSION This paper compares two control schemes, CTC in joint space and CTC in task space. The comparison is formulated in tracking error. Simulation is performed in Matlab/Simulink environment. The results show an advantage for CTC in task space over CTC in joint space

5 for the tracking error, as shown in Table 3. For the calibrating process of the controllers, the CTC in the task space is better than CTC in joint space because it requires calibration of only three gains. On the other hand the jacobian calculation depends on the number of joints and this increases somewhat the complexity of the process. REFERENCES [1] C. Samson, M. Le Borgne, B. Espinau, 1991, Robot control, Oxford University Press. [2] F. Lewis, C. Abdallah, D. Dawson, 1993, Control of robot manipulators, Macmillan. [3] C. Canudas, B. Siciliano, G. Bastin, 1996, Theory of robot control, Springer Verlag. [4] W. Khalil, E. Dombre, 2002, Modeling identification and control of robots, Hermes Penton Science. [5] I. Cervantes, J. Alvarez-Ramírez, 2001, On the PID tracking control of robot manipulator, Systems & Control Letters, 42, pp [6] M. Spong, 1996, Motion control of robot manipulators, In Handbook of Control. CRC Press, pp [7] L. Sciavicco and B. Siciliano, 1996, Modeling and control of robot manipulators, McGraw-Hill, New York. [8] H. Sage, M. De Mathelin and E. Ostertag, 1999, Robust control of robot manipulators: a survey, International Journal of Control. 72 (6), pp [9] M. Spong, 1992, On the robuts control of robot manipulators, IEEE Transactions on Automatic Control. 37, pp

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