Evaluating sensor configurations for the Extended CTC approach based on sensitivity analysis

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1 Evaluating sensor configurations for the Extended CTC approach based on sensitivity analysis A. Zubizarreta I. Cabanes M. Marcos Ch. Pinto Department of Automatics and System Engineering, University of the Basque Country, Spain ( Department of Mechanical Engineering, University of the Basque Country, Spain ( Abstract: In order to fully exploit the potential of parallel robots advanced control approaches like the Extended CTC approach are required. This approach combines the performance of traditional CTC approaches with the robustness of sensor redundancy. However, to ensure the best performance, a proper sensor configuration has to be implemented. In this paper, a sensitivity analysis based methodology is introduced and used to determine the best sensor configurations in the Gough platform. Results are compared with a statistically significant set of simulations, validating the effectiveness of the approach. Keywords: Robotics, Parallel Robots, Redundant Control, Extended Computed Torque Control, Sensor Redundancy 1. INTRODUCTION Serialrobots have become akey elementinmost industrial processes, as their reprogrammability, flexibility and wide workspace allow these devices to perform a number of tasks. However, inthe need ofincreasing bothproductivity and quality, current industrial tasks demand high speed and accurate operation. These requirements are difficult to be met by serial robots, as their structure is usually bulky and presents joint error accumulation. In the need of suitable robotic structures that fulfill the demands of the industry, Parallel Robots are proposed (Merlet (2006)). These robots are composed by several kinematic chains or limbs that join a mobile platform to a fixed one. This parallel structure provides four major advantages over open chain serial robots: first, higher stiffness than open chain structures; second, errors in one of the kinematic chains can be compensated by the rest; third, in some structures, actuators can be mounted in the base, reducing the moving mass of the mechanism while maintaining its stiffness; and finally, they can handle higher loads than their serial robot counterparts, as the load can be distributed among the several kinematic chains. Due to these features, parallel robots are considered an interesting alternative for precision, pick-and-place and high load handling tasks. However, being a recently rediscovered field, parallel robots still have many unresolved issues that limit their theoretical potential: small workspace, presence of inner This work was supported in part by the MCYT&FEDER under projects DPI , DPI and DPI , the Goverment of the Basque Country under project GIC10/91 and by UPV/EHU under grant GIU07/36. singularities, lack of calibration approaches and complex kinematics and dynamics. In order to give a solution to them, in recent years a great research effort has been made in multiple areas of parallel robots. One of the critical areas in parallel robots is control. A proper and well tuned controller allows to handle the synchronization between the different chains, ensures robustness against disturbances and load changes and provides an adequate dynamic performance even in demanding tasks. In order to fulfill these requirements, researchers have proposed both local control, PID based approaches (Chiacchio et al. (1993); Brecher et al. (2006); Ghorbel et al. (2000)) and model based, multiarticular approaches (Codourey (1998); Davliakos and Papadopoulos (2008); Pietsch et al. (1995); Lee et al. (2003)). However, these approaches are directly imported from serial robots to parallel robots without considering the specific features of the latter. This limits their performance, as effects such as the existence of passive or nonactuated joints are not considered. To solve this issue, in previous works the authors proposed to use extra sensors to measure the motion of passive joints (Zubizarreta et al. (2009); Cabanes et al. (2009)). Using this data, a novel Computed Torque Control based approach, the Extended CTC, was introduced. This approach combines the performance of CTC approach and the robustness of sensor redundancy. This control law was demonstrated to provide better performance than the classical, active joint sensor based, approaches. Although the introduction of extra sensors can improve the dynamic performance even in presence of uncertainties, two questions arise when trying to apply this approach: How many sensors are needed? and where are they lo- Copyright by the International Federation of Automatic Control (IFAC) 1078

2 cated?. Note that the use of extra sensors increases the cost of the prototype, and the integration of them in the structure of the robot is not trivial. Moreover, due to the nonlinear behaviour of parallel robots, the significance of the dataprovidedbythe extrasensorscan varydepending on a) the robot structure itself and its geometry, and b) the task or reference trajectory to be executed. Thus, an appropriate methodology to determine the best location and number of sensors is needed. Optimum extra sensor distribution has already been analyzed in the literature Merlet (1993); Parenti-Castelli and Gregorio (2000). However, these works focused inreducing the computational cost of the direct kinematic problem using extra data. On the other hand, the methodology proposedinthisworkfocuses onprovidingaprocedureto evaluate different sensor configurations from the control point of view, i.e. determining the one that provides the best dynamic behaviour. Thus, the contribution of this paper is a novel, sensitivity analysis based approach, to determine the best sensorconfiguration for the implementation of a redundant sensor based approach, the Extended CTC. The proposed methodology allows to calculate a numerical performance value of each sensor distribution for a given task and robot. Thus, this approach is based on an evaluation of an a-priori determined set of possible or potential configurations, in order to determine the one that presents better robustness and dynamic performance. In order to introduce the approach, it is applied to the 6 degrees of freedom Gough platform. The layout of the paper is as follows. Section 2 introduces the Extended CTC approach. In section 3, the sensitivity analysis based methodology for detecting the best sensor configurations is introduced. In section 4, the proposed approach is applied to the Gough platform and validated using a set of simulation results. Finally, the most important ideas are summarized. 2. EXTENDED COMPUTED TORQUE CONTROL When defining the motion of parallel robots, two sets of variables can be used: the task coordinates x, that define the location of the TCP, and the set of all joint variables q, that include both actuated q a and nonactuated joints q na, so that q = [ q a T q na T ] T. The classical CTC approach, which has been widely analyzed in the literature, uses the data from the actuated joints q a to implement the control law, and estimates the rest of the variables x and q na using the kinematic model of the mechanism. The Extended CTC generalizes the classical CTC control law, allowing to calculate the control action in terms of a set of control coordinates q c, that group both active q a and a set of sensorized nonactuated joints q s, so that q c = [ q a T q s T ] T. The rest of the nonactuated joints that are not sensorized will be known as strictly passive joints q p, so that q na = [ q s T q p T ] T. Thus, the control law defined in the joint space (Fig. 1), τ = D(x,q c,q p )( q cd +K v ė q +K p e q )+ (1) C(x,q p,q c,ẋ, q p, q c ) q c +G(x,q c,q p ) where D, C and G are the Inertia, Coriolis and Gravity matrices of the dynamic model of the robot, K p and K v Fig. 1. Extended CTC in Joint Space, where h = C q a +G are the position and velocity gains, q cd, q cd, q cd are the reference functions of the control coordinates, which are calculated from the references of the task coordinates ẍ d, ẋ d, x d using the Inverse Kinematic Model (IKM) of the robot, e q = q cd q c and ė q = q cd q c are the positioning and velocity error related to the control coordinates. The possibility of introducing extra data in the control law provides two main advantages over the nonredundant, classical CTC approach. First, as some passive joints are sensorized, the controller has more information of the motion of the platform, allowing a more accurate control. And second, if kinematic model uncertainties arise, the redundant data can be used to have a better estimation of the nonmeasured variables required to calculate the control law: the task coordinates x and the strictly passive set of the nonactuated joints q p. 3. EVALUATION OF EXTENDED CTC CONFIGURATIONS THROUGH SENSITIVITY ANALYSIS The Extended CTC approach is defined in a general way, and multiple sensor configurations, i.e. q c configurations, canbe used toimplementit. Although, ithasbeen demonstrated that the use of redundant data provides better dynamic performance (see Zubizarreta et al. (2009)), not all redundant configurations will have the same performance. Thus, for a given structure of parallel robot, a proper sensor configuration has to be determined in order to maximize dynamic performance and robustness while minimizing the number of extra sensors needed. However, this analysis can be a time-consuming one, as multiple simulations have to be done in order to determine, under randomly introduced parameter uncertainties, which configuration is more suitable for a specific task. In order to reduce the computational effort and provide an insight of the dynamic behaviour of the different configurations of the Extended CTC, a sensitivity analysis based approach is proposed. Sensitivity Analysis is a wellknown approach that allows to determine the influence of the variation of a parameter in the output of a dynamic system. So, if all parameters are analyzed, the study determines the most critical parameters based on their relative influence on the output of the system. Thus, as closed loop systems are also dynamic systems, this approach can be used toanalyzethesensitivityofcontrolledsystems, which is a measure of the robustness that the controller provides when parameter uncertainties arise. 1079

3 In the specific case of the Extended CTC approach, the sensitivity of each sensor configuration can be calculated in order to determine the most robust ones (i.e. the ones that present less sensitivity to model parameter variations). For that purpose, sensitivity functions have to be defined for each of the parameters of the kinematic and dynamic model implemented by the controller. These functions depend on a high degree on the reference trajectory and the controller used to execute it, allowing to compare the relative influence of each parameter for each trajectory and controller. In the general case, sensitivity functions are obtained by differentiating the dynamic model to be evaluated with respect to each of the parameters of the model. However, this procedure cannot be used in CTC based approaches. Thus, in Vukobratovic and Filipovic (2000) a different approach is proposed for the sensitivity analysis of the CTC approach applied to serial robots. In this paper, such approach is modified to analyze the sensitivity of the Extended CTC approach in parallel robots. 3.1 Calculation of the Sensitivity functions As in Vukobratovic and Filipovic (2000), the sensitivity function ξθ x k that calculates the deviation δx for the Extended CTC approach along a nominal trajectory P d when the parameter θ k is uncertain is defined by ξ x θ k = δx δx θk (2) where δx is the evolution of the deviation of the task coordinates in the ideal case, i.e., when the dynamic model implemented in the Extended CTC is perfect, and δx θk is the evolution of the deviation of the task coordinates when the parameter θ k of the dynamic model of the Extended CTC varies from the nominal value. The procedure to calculate the evolution of δx θk requires the linearization of the dynamics of the closed loop system. Applying this procedure to the Extended CTC control approach, D q c +h = ˆD( q cd +K v ė q +K p e q )+ĥ (3) where ˆD is the estimated inertia matrix and ĥ = Ĉ q c + Ĝ groups the estimated nonlinear terms (Coriolis and Gravity) of the dynamic model. Note that variations in model parameters only influence these two terms, asd and h represent the real dynamics of the robot. Thus, each term of Eq. (3) must be linearized. If the left term is considered then, δτ = D(x,q p,q cd )δ q c + D q c x δx + D q c q δq+ h x δx + h q δq+ h ẋ δẋ+ h q δ q Note that q c defines the measurable variables. Thus, in order to calculate the linearized model in terms of the (4) variations of the control coordinatesδq c, the velocity Jacobians δx = J p (x,q p,q c )δq c and δq = T q (x,q p,q c )δq c and its derivatives are introduced in Eq. (4). The resulting linearized model in compact form is calculated as, δτ = A m δ q c +B m δ q c +C m δq c (5) where the m subscript denotes the robot dynamics. Note that this equation represents the linearized dynamics of the robot. Therefore it is calculated using the nominal parameters. The linearization of the right term of Eq. (3) follows the same procedure as the one described above, δτ = A c δu+b c δ q c +C c δq c (6) where if small deviations are considered u = K v δ q c K p δq c. In this case,csubscript denotes the robot dynamic model used by the controller, and is calculated using the uncertain parameters. In the sensitivity analysis, these parameters will be modified. Combining Eqs. (5) and (6), the linearized dynamics of the closed loop system is obtained. 0 = A m δ q c +(B m B c +A c K v ) δ q c +(C m C c +A c K p ) δq c (7) that defines an undetermined ODE system as q c can contain redundant data. Thus, the real constraints of the robot are introduced using the constraint Jacobian of the robot, δq c = J qc δx, in order to define an ODE system in terms of δx. If an initial deviation δx 0 is considered, the evolution of the variation of the task coordinates δx can be calculated by integrating the ODE system. Thus, using Eq.(7), δx can be calculated if the nominal parameters are introduced. On the other hand, if a variation in the parameter θ k is introduced, δx θk can be calculated. The sensitivity function evolution is then calculated by substracting both variations as stated in Eq. (2). As proposed by Filipovic and Vukobratovic (2000), the characteristic sensitivity value will be obtained by calculating the maximum absolute value of the sensitivity function for a given parameter variation. 4. APPLICATION TO THE GOUGH PLATFORM 4.1 Robot description The Gough platform, also known as Stewart platform in the literature, is one of the most popular parallel manipulators. Its structure is composed by six extensible UPS serial chains that join a mobile platform to a fixed base. Due to the high stiffness of this architecture, it has been traditionally used in tasks that require heavy load handling (Su and Duan (2000)) or accuracy (Nakadate et al. (2009)). Let O(x,y,z) be the fixed reference frame of the system located in the base platform and P(u,v,w) the mobile reference frame attached to the center of mass of the mobile platform, where the TCP is also located. The motion of the mobile platform is achieved by means of six actuated prismatic joints P. This allows to vary the 1080

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6 IAE (m) IAE (rad) 2.50E E E E E E E E E E Ext. CTC configurations (a) IAE Index. Position Coordinates Ext. CTC configurations (b) IAE Index. Orientation Coordinates. Fig. 5. Average IAE Performance Index. Helicoidal Trajectory 5. CONCLUSION In previous works of the authors, the Extended CTC has been introduced. This approach uses redundant sensor data to increase the dynamic performance of parallel robotsusingactc based controlscheme. However,determining the best sensor configuration is a time consuming task. In this paper, a sensitivity analysis based methodology is introduced to determine the best sensor configuration to implement the redundant sensor based Extended CTC control. The approach uses the contribution of the maximums of the sensitivity functions associated to each model parameter as a reference value for each sensor configuration. Using these values, the relative sensitivity of the proposed configurations can be compared for a given trajectory, allowing to determine the best sensor configurations easily. In order to validate the approach, a set of simulation experiments have been conducted in the Gough platform, demonstrating that the proposed methodology allows to determine the best sensor configurations using less computational time. REFERENCES Brecher, C., Ostermann, T., and Friedrich, D. (2006). Control concept for pkm considering the mechanical coupling between actuator. Proceedings of the 5th Chemnitz Parallel Kinematics Seminar, Cabanes, I., Zubizarreta, A., Marcos, M., and Pinto, C. (2009). Real time distributed control of parallel robots using redundant sensors. Proceedings of the 40th International Symposium on Robotics, Chiacchio, P., Pierrot, F., Sciavicco, L., and Siciliano, B. (1993). Robust design of independent joint controllers with experimentation on a high-speed parallel robot. IEEE Transactions on Industrial Electronics, 40(4), Codourey, A. (1998). Dynamic modeling of parallel robots for computed-torque control implementation. The International Journal of Robotics Research, 17(12), Davliakos, I. and Papadopoulos, E. (2008). Model-based control of a 6-dof electrohydraulic Stewart Gough platform. Mechanism and Machine Theory, 43(11), Filipovic, M. and Vukobratovic, M. (2000). Dynamic accuracy of robotic mechanisms. part 2: Simulation experiments and results discussion. Mechanism and Machine Theory, 35, Ghorbel, F., Chételat, O., Gunawardana, R., and Longchamp, R. (2000). Modeling and set point control of closed-chain mechanisms: Theory and experiment. IEEE Transactions on Control System Technology, 8(5), Lee, S.H., Song, J.B., Choi, W.C., and Hong, D. (2003). Position control of a stewart platform using inverse dynamics control with approximate dynamics. Mechatronics, 13, Merlet, J.P. (1993). Closed form resolution of the direct kinematics of parallel manipulators using extra sensors data. Proceedings IEEE International Conference in Robotics and Automation, Merlet, J.P. (2006). Parallel Robots (Second Edition). Kluwer. Nakadate, R., Uda, H., Hirano, H., Solis, J., Takanishi, A., Minagawa, E., Sugawara, M., and Niki, K. (2009). Development of a robotic carotid blood measurement wta-1rii: Mechanical improvement of gravity compensation mechanism and optimal link position of the parallel manipulator based on ga. IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Parenti-Castelli, V. and Gregorio, R.D. (2000). A new algorithm based on two extra-sensors for real-time computation of the actual configuration of the generalized stewart-gough manipulator. Journal of Mechanical Design, 122(1), Pietsch, I., Krefft, M., Becker, O., Bier, C.C., and Hesselbach, J. (1995). How to reach the dynamic limits of parallel robots? an autonomous control approach. IEEE Transactions on Automation Science and Engineering, 2, Su, Y. and Duan, B. (2000). The application of the stewart platform in large spherical radio telescopes. Journal of Robotic Systems, 17(7), Tsai, L.W. (2000). Solving the inverse dynamics of a stewart-gough manipulator by the principle of virtual work. ASME Journal of Mechanical Design, 122, 3 9. Vukobratovic, M. and Filipovic, M. (2000). Dynamic accuracy of robotic mechanisms. part 1: Parametric sensitivity analysis. Mechanism and Machine Theory, 35, Zubizarreta, A., Cabanes, I., Marcos, M., Pinto, C., and Portillo, E. (2009). Redundant dynamic modelling of the 3rrr parallel robot for control error reduction. Procedings of the 2009 European Control Conference. 1083

SIMULATION ENVIRONMENT PROPOSAL, ANALYSIS AND CONTROL OF A STEWART PLATFORM MANIPULATOR

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