I Advances in Robot Manipulators Advances in Robot Manipulators Edited by Ernest Hall In-Tech intechweb.org Published by In-Teh In-Teh Olajnica 19/2, 32000 Vukovar, Croatia Abstracting and non-profit use of the material is permitted with credit to the source Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher No responsibility is accepted for the accuracy of information contained in the published articles Publisher assumes no responsibility liability for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained inside After this work has been published by the In-Teh, authors have the right to republish it, in whole or part, in any publication of which they are an author or editor, and the make other personal use of the work © 2010 In-teh www.intechweb.org Additional copies can be obtained from: publication@intechweb.org First published April 2010 Printed in India Technical Editor: Sonja Mujacic Cover designed by Dino Smrekar Advances in Robot Manipulators, Edited by Ernest Hall p cm ISBN 978-953-307-070-4 V Preface The purpose of this volume in Advances in Robot Manipulators is to encourage and inspire the continual invention of robot manipulators for science and the good of humanity The concepts of artificial intelligence combined with the engineering and technology of feedback control, have great potential for new, useful and exciting machines The concept of eclecticism for the design, development, simulation and implementation of a real time controller for an intelligent, vision guided robots is now being explored The dream of an eclectic perceptual, creative controller that can select its own tasks and perform autonomous operations with reliability and dependability is starting to evolve We have not yet reached this stage but a careful study of the contents will start one on the exciting journey that could lead to many inventions and successful solutions Editor: Ernest Hall VI VII Contents Preface A Biomimetic steering robot for Minimally invasive surgery application V 001 G Chen, M.T Pham, T Maalej, H Fourati, R Moreau and S Sesmat A Regressor-free Adaptive Control for Flexible-joint Robots based on Function Approximation Technique 027 Ming-Chih Chien and An-Chyau Huang A 9-DoF Wheelchair-Mounted Robotic Arm System: Design, Control, Brain-Computer Interfacing, and Testing 051 Redwan Alqasemi and Rajiv Dubey Advanced Techniques of Industrial Robot Programming 079 Frank Shaopeng Cheng An Open-architecture Robot Controller applied to Interaction Tasks 099 A Oliveira, E De Pieri and U Moreno Collaboration Planning by Task Analysis in Human-Robot Collaborative Manufacturing System 113 Jeffrey Too Chuan Tan, Feng Duan, Ryu Kato and Tamio Arai Collaborative rules operating manipulators 133 José Martins Junior, Luiz Camolesi Jr and Glauco Augusto de Paula Caurin Control of Lightweight Manipulators Based on Sliding Mode Technique 155 Jingxin Shi, Fenglei Ni and Hong Liu Coordinate Transformation Based Contour Following Control for Robotic Systems 183 Chieh-Li Chen and Chao-Chung Peng 10 Design of Adaptive Controllers based on Christoffel Symbols of First Kind 205 Juan Ignacio Mulero-Martínez 11 Development of a New DOF Lightweight Wrist for the Humanoid Robot ARMAR Albert Albers, Jens Ottnad and Christian Sander 237 VIII 12 Development of Tendon Based Dexterous Robot Hand 255 Chung-Hsien Kuo and Chun-Tzu Chen 13 Dimensional Synthesis and Analysis of the 2-UPS-PU Parallel Manipulator 267 Yunfeng Zhao Yanhua Tang Yongsheng Zhao 14 Direct and Indirect Adaptive Fuzzy Control for a Class of MIMO Nonlinear Systems 279 Salim Labiod and Thierry Marie Guerra, 15 Dynamic Trajectory-Tracking Control of an Omnidirectional Mobile Robot Based on a Passive Approach 299 M Velasco-Villa, H Rodríguez-Cortés, I Estrada-Sanchez, H Sira-Ramírez and J A Vázquez 16 Eclectic Theory of Intelligent Robots 315 E L Hall, S M Alhaj Ali, M Ghaffari, X Liao and Ming Cao 17 Enhanced stiffness modeling of serial manipulators with passive joints 331 Anatol Pashkevich, Alexandr Klimchik and Damien Chablat 18 Fast Dynamic Model of a Moving-base 6-DOF Parallel Manipulator 361 António M Lopes 19 Improving the Pose Accuracy of Planar Parallel Robots using Mechanisms of Variable Geometry 381 Jens Kotlarski, Bodo Heimann and Tobias Ortmaier 20 Kinematic Singularities of Robot Manipulators 401 Peter Donelan 21 Motion Control of Industrial Robots in Operational Space: Analysis and Experiments with the PA10 Arm 417 Ricardo Campa, César Ramírez, Karla Camarillo, Víctor Santibáđez and Israel Soto 22 MRI Compatible Robot Systems for Medical Intervention 443 Ming Li, Dumitru Mazilu, Ankur Kapoor and Keith A Horvath 23 On the Optimal Singularity-Free Trajectory Planning of Parallel Robot Manipulators 459 Chun-Ta Chen and Te-Tan Liao 24 Programming-by-Demonstration of Reaching Motions using a Next-State-Planner 479 Alexander Skoglund, Boyko Iliev and Rainer Palm 25 Robot Arms with 3D Vision Capabilities 503 Theodor Borangiu and Alexandru Dumitrache 26 Robot assisted 3D shape acquisition by optical systems Cesare Rossi, Vincenzo Niola, Sergio Savino and Salvatore Strano 515 IX 27 ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 551 Shield B Lin and Sheng-Guo Wang 28 Role of Finite Element Analysis in Designing Multi-axes Positioning for Robotic Manipulators 565 T.T Mon, F.R Mohd Romlay and M.N Tamin 29 Statistical Imitation Learning in Sequential Object Manipulation Tasks 589 Komei Sugiura, Naoto Iwahashi, Hideki Kashioka and Satoshi Nakamura 30 Tangible interfaces for tangible robots 607 Andrew Cyrus Smith 31 Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators 625 Malik Loudini 32 Trajectory Control of RLED Robot Manipulators Using a New Type of Learning Rule Hüseyin Canbolat 651 X 16 Advances in Robot Manipulators Fig 17 Simulation and experimental results of the movement of the endpoint of Colobot Fig 18 Optimization model It is noteworthy that the elongation expression ∆L1 (respectively ∆L2 , ∆L3 ) in (Eq 11) is equivalent to (Eq 3) when the relative pressures P2 and P3 (respectively P1 , P3 and P1 , P2 ) are equal to zero To check this new kinematic model a cross validation is made with three other experiments Three sinus input pressures with amplitude from 0.1 bar to 0.3 bar are applied into three chambers of Colobot The improved kinematic model with the correction coefficient k is used to a straightforward comparison with the sets of data Results shown in Fig 19 and Fig 20 are testimony to the behavior of the proposed model in these two cases A Biomimetic steering robot for Minimally invasive surgery application Fig 19 Verification of corrected model with different pressure inputs (across dead zone) Fig 20 Validation with different pressure inputs 17 18 Advances in Robot Manipulators Guidance control strategy based on proximity multi-sensor system Fig 21 Position of Colobot inside the colon 5.1 Guidance control strategy The objective of sensor-based guidance strategy is to calculate the safe position of the distalend of Colobot compared to the colon wall in real-time based on the measurements of three distance sensors for guidance inside the colon For the sake of simplicity but without loss of generality, it is assumed that a colon is a cylindrical tube and its cross section is an ellipse at the sensor plane Fig 21 illustrates the sensor plane, the distal end of ColoBot and the colon axis With these assumptions, the safe position will be the central axis of the colon To approximate the colon axis, a method based on a circumscribed circle is proposed Since three points D1 , D2 and D3 (Fig 22) of sensor measurements form a triangle, the center of the circumscribed circle of this triangle is chosen as the safe position This approach iterates as following: • Three sensor measurements are collected • Position Pn in the frame Rs (Fig 22) is evaluated with these three measurements • If Pn is a safe position, then it’s necessary to go back to the first step for the next period; otherwise, next the safe position Pn+1 described in the Frame Ru (Fig 22) is calculated through the circumscribed method and is provided to the kinematic control for execution For more details about the guidance control strategy, please find the reference Chen et al (2008) 5.2 Guidance control architecture The control of Colobot is organized in three hierarchical levels, as shown in Fig 23 The first level consists of local pressure control of each Colobot’s chamber through three servovalves A Biomimetic steering robot for Minimally invasive surgery application 19 Fig 22 Computation of the safe position Three independent PI controllers are used to implement the closed-loop pressure control of the chamber The position and orientation of Colobot are controlled at level using an instantaneous inverse Jacobian method This section will describe the implementation detail Level is the sensor-based planning for automatic navigation described in section 5.1 5.3 Formulation of task space control of Colobot After determining the desired trajectory from sensor-based planning, the kinematic control of Colobot will be described in this section It should be noted that two variables are used to represent the position of Colobot inside the colon However, the Colobot has degrees of freedom So this manipulator becomes redundant for the chosen task The velocity kinematic equations are rewritten as following: X = f (Q p ) ∂X ∂(α, φ, L) ∂Q L ˙ ˙ Qp X= ∂(α, φ, L) ∂Q L ∂Q P ˙ ˙ ˙ X = Js Jl J p Q p = J Q p or (12) where X = ( x, y) T , Q L = ( L1 , L2 , L3 ) T , Q p = ( P1 , P2 , P3 ) T and J = Js Jl J p is the Jacobian matrix with relation to the three levels of pressure in the chambers 5.4 Resolution of the inverse kinematics with redundancy In the case of a redundant manipulator with respect to a given task, the inverse kinematic problem admits infinite solutions This suggests that redundancy can be conveniently exploited to meet additional constraints on the kinematic control problem in order to obtain greater manipulability in terms of the manipulator configurations and interaction with the environment A viable solution method is to formulate the problem as a constrained linear 20 Advances in Robot Manipulators Fig 23 sensor-based planning and guidance control procedure optimization problem Work on resolved-rate control Whitney (1969) proposed to use the Moore-Penrose pseudo inversion of the Jacobian matrix as: ˙ ˙ ˙ Q p = J + X = ( J T ( J J T ) −1 ) X (13) In our case, however, there is a mechanical limit range for the elongation of each chamber and the corresponding pressure applied into the chamber of the Colobot The objective function is constructed to be included in the inverse Jacobian algorithms as the second criteria also called the null-space method Hollerbach & Suh (1986); Nakamura (1991) ˙ ˙ ˙ Q p = J + X + µ[ I − J + J ] g (14) where I is the identity matrix, µ is constant and g is a second criterion for the optimization of the solution This objective function evaluates the pressure difference between the applied A Biomimetic steering robot for Minimally invasive surgery application 21 pressure in the chamber and the average pressure applied in the chamber So the cost function w( P) is expressed as follows: w( P) = 3 i∑ =1 Pi − Piave Pimax − Pimin (15) We can then minimize w( p) by choosing: ˙ g = grad (w( P)) = ∂w ∂w ∂w ∂P1 ∂P2 ∂P3 (16) Experimental results This section will present the implementation of the whole system and experimental results of automatic guidance capability of this system in a colon-like tube 6.1 Hardware implementation Fig 24 shows the low-level control system of ColoBot The pressurized air comes through the compressor (1) and the general pressure is adjusted thanks to the device (2) The pressure in the chambers are controlled by three Jet-pipe servovalves (3a, 3b and 3c) Three pressure sensors (4a, 4b and 4c) are connected between the servovalves and the Colobot (5) for the pressure feedback control Suitable drivers and amplifiers in the rack (6) were designed to amplify control signals applied to the actuator A real-time controller is implemented through a DSpace board and coupled with the Real-Time Workshop of Simulink The Simulink block diagram designed for path planning and kinematics algorithms are expressed with Simulink block diagram which will be compiled as real-time executable under the DSP Processor of the DSpace board The system runs at 500 Hz for a real-time control loop Fig 24 The implementation of the whole system 22 Advances in Robot Manipulators 6.2 Experimental results in a colon-like tube A more realistic experiment to test the performance of this semi-autonomous colonoscopy system is to use a colon-like transparent tube to see if Colobot can cross the tube with minimal contact with the tube wall The diameter of the tube is 26 mm and its length is 50 cm (Fig 25) For this guidance experiment, the calibration of the optical fibres was adapted to the transparent tube It is highly probable that results for the distance sensors in a porcine intestine will be similar to those obtained in the human bowel However, the locomotion of the system is manually operated The evolution of the measurements of three optical fibres are represented in the left Fig 26(a) During the entire movement, the distances are never less than 0.8 mm This demonstrates that the colonoscope tip is moving through the tube without touching it For a better representation and visualization, Fig 26(b) shows the extreme positions of the top-end of Colobot as it progresses (with a velocity of cm/s) The position of the Colobot at the centre of the tube is represented by the smallest circle The larger circle represents the tube wall and the line shows the extreme positions of Colobot This experiment demonstrates that Colobot has the capability to guide the exploration of the tube with a sensor-based steering control method Fig 25 Guidance control test in a colon-like tube CONCLUSIONS AND FUTURE WORKS This paper presented a complete robotic system for semi-autonomous colonoscopy It is composed of a microtip, a proximity multi-sensor system and high level real-time control system for guidance control of this robot This system was focused on its guidance ability of endoscope inside the human colon with the fiber optic proximity sensors Colobot is a continuum robot made of silicone rubber It has three DoF with its outer diameter of 17mm and the weight of 20 gram The pneumatic actuators of ColoBot are independently driven through three servovalves The kinematic model of this soft robot was developed based on the geometric deformation and validated its correction A method using a circumscribed circle is utilized to calculate the safe reference position and orientation of the Colobot While kinematic-based orientation control used these reference paths to adjust the position of Colobot inside the colon to achieve guidance Experimental results of guidance control with a transparent tube verified the effectivity of kinematic control and guidance control strategy In the near future, the proposed method will be tested in a vitro environment A Biomimetic steering robot for Minimally invasive surgery application (a) Evolution of three measurements 23 (b) Extreme position projected into the tube plane Fig 26 Guidance control result analysis References (n.d.a) Technical report URL: http://www.rfnorika.com (n.d.b) Technical report URL: http://www.givenimaging.com (n.d.c) Technical report URL: http://www.ascension-tech.com/ Atchley, R (1982) A more reliable electrohydraulic servovalve, Robot VI Conference, Detroit, USA Atchley Controls, Jet Pipe catalogue (n.d.) Bailly, Y & Amirat, Y (2005) Modeling and control of a hybrid continuum active catheter for aortic aneurysm treatment, IEEE International Conference on Robotics and Automation, Barcelona, Spain, pp 924–929 Chen, G., Pham, M & Redace, T (2008) Sensor-based guidance control of a continuum robot for a semi-autonomous colonoscopy, Robotics and autonomous systems 57(6): 712–722 Chen, G., Pham, M T & Redarce, T (2006) Development and kinematic analysis of a siliconerubber bending tip for colonoscopy, IEEE/RSJ Intemational Conference on Intelligent Robots and Systems, Beijing, China, pp 168–173 Chen, G., Pham, M T., Redarce, T., Prelle, C & Lamarque, F (2005) Design and control of an actuator for colonoscopy, 6th International Workshop on Research and Education in Mechatronic, Annecy, France, pp 109–114 Dario, P., Carrozza, M & Pietrabissa, A (1999) Development and in vitro tests of a miniature robotic system for computer-assisted colonoscopy, Jounal of Computer Aided Surgery, 4: 4–14 Dario, P., Paggetti, C., Troisfontaine, N., Papa, E., Ciucci, T., Carrozza, M & Marcacci, M (1997) A miniature steerable end-effector for application in an integrated system for computer-assisted arthroscopy, IEEE International Conference on Robotics and Automation, Albuquerque, USA, pp 1573–1579 24 Advances in Robot Manipulators Fukuda, T., Guo, S., Kosuge, K., Arai, F., Negoro, M & Nakabayashi, K (1994) Micro active catheter system with multi degrees of freedom, Proceedings of the International Conference on Robotics and Automation, San Diego, USA, pp 2290–2295 Glass, P., Cheung, E & Sitti, M (2008) A legged anchoring mechanism for capsule endoscopes using micropatterned adhesives, IEEE Transactions on Biomedical Engineering 55(12): 2759–2767 Gorini, M., Menciassia, A., Pernorio, G., G., S & Dario, P (2006) A novel sma-based actuator for a legged endoscopic capsule, IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomimetics Hollerbach, J & Suh, K (1986) Redundancy resolution of manipulator through torque optimization, A.I.Memo 882, Massachussett Institute of Technology Ikuta, K., Tsukamoto, M & Hirose, S (1988) Shape memory alloy servo actuator system with electric resistance feedback and application for active endoscope, IEEE International Conference on Robotics and Automation, Hitachi City, Japan, pp 427–430 Immega, G & Antonelli, K (1995) The KSI tentacle manipulator, IEEE International Conference on Robotics and Automation, Nagoya, Japan, pp 3149 –3154 Jones, B & Walker, I D (2006) Kinematics for multi-section continuum robots, IEEE Transactions on Robotics 22(1): 43 –55 Kassim, I., Ng, W., Feng, G & Phee, S (2003) Review of locomotion techniques for robotic colonoscopy, Proceedings of the International Conference on Robotics and Automation, Taipei, Taiwan, pp 1086–1091 Kim, B., Lim, H Y., Par, J H & Park, J (2006) Inchworm-like colonoscopic robot with hollow body and steering device, JSME International Journal Series C 49(1): 205–212 Kim, B., sunghak, L., Heong, P J & Jong-oh, P (2005) Design and fabrication of a locomotive mechanism for capsule-type endos using shape-memory alloys (sma), IEEE/ASME Transactions on Mechatronics 10(1): 77–86 Kumar, S., Kassim, I & Asari, V (2000) Design of a vision- guided microrobotic colonoscopy system, Advanced robotics 14(2): 87–114 ´ Lane, D., David, J., Robinson, G., OBrien, D., Sneddon, J., Seaton, E & A., E (1999) The amadeus dextrous subsea hand: Design, modeling, and sensor processing, IEEE Journal of Oceanic engineering 24(1): 96–111 Menciassi, A., J.H., P., Lee, S., Gorini, S., Dario, P & Park, J (2002) Robotic solutions and mechanisms for a semi-autonomous endoscope, Proc of the IEEE-RSJ Int Conf on Intelligent Robots and Systems, Lausane, Switzerland, pp 1379–1384 Menciassi, A., Moglia, A., Gorini, S., Pernorio, G., Stefnini, C & Dario, P (2005) Shape memory alloy clamping devices of a capsule for monitoring tasks in the gastrointestinal tract, Journal of Micromechanics and Microengineering 15(11): 2045–2055 Nakamura, Y (1991) Advanced robotics, Redundancy and Optimization, Addison-Wesley Ohno, H & Hirose, S (2001) Design of slim slime robot and its gait of locomotion, Proc of the IEEE-RSJ Int Conf on Intelligent Robots and Systems, Hawaii, USA, pp 707–715 Phee, S., Ng, W S., Chen, I.-M., Seow-choen, F & Davies, B L (1998) Automation of colonoscopy, part one: Locomotion and steering aspects in automation of colonoscopy, IEEE Engineering in Medecine and Biology Magazine 17(3): 81–89 Piers, J., Reynaerts, D., Van Brussel, H., De Gersem, G & Tang, H T (2003) Design of an advanced tool guiding system for robotic surgery, Proceedings of the International Conference on Robotics and Automation, Taipei, Taiwan, pp 2651–2656 A Biomimetic steering robot for Minimally invasive surgery application 25 Robinson, G & Davies, J (1999) Continuum robots - a state of the art, IEEE International Conference on Robotics and Automation, Detroit Michigan, USA, pp 2849–2853 Sesmat, S (1996) Modélisation, Simulation et Commande dúne Servovalve Electropneumatique (in French), PhD thesis, INSA de Lyon Simaan, N., Taylor, R & Flint, P (2004) A dexterous system for laryngeal surgery- multibackbone bending snake-like slaves for teleoperated dexterous surgical tool manipulation, IEEE International Conference on Robotics and Automation, New Orleans, USA, pp 351–357 Slatkin, A B & Burdick, J (1995) The development of a robot endoscope, Proc of the IEEE-RSJ Int Conf on Intelligent Robots and Systems, Pittsburgh, USA, pp 3315–3320 Sturges, R H (1993) A flexible, tendon-controlled device for endoscopy, The International Journal of Robotics Research 12(2): 121–131 Suzumori, K., Iikura, S & Tannaka, H (1992) Applying a flexible-micro-actuator robotic mechanisms, IEEE control systems 12(1): 21–27 Taylor, R H & Stoianovici, D (2003) Medical robotics in computer-integrated surgery, IEEE Transaction on Robotics and Automation 19(5): 765–781 Wang, X & Meng, M Q.-H (2008) IEEE/RSJ international conference on intelligent robots and systems, Nice, France, pp 1198–1203 Webster, R J I., Romano, J M & Cowan, N J (2009) Mechanics of precurved-tube continuum robots, IEEE Transactions on Robotics 25(1): 67 – 78 Whitney, D (1969) Resolved motion rate control of manipulators and human prostheses, IEEE Transaction on Man-Machine systems 10(2): 47–53 26 Advances in Robot Manipulators A Regressor-free Adaptive Control for Flexible-joint Robots based on Function Approximation Technique 27 X A Regressor-free Adaptive Control for Flexible-joint Robots based on Function Approximation Technique Ming-Chih Chien and An-Chyau Huang Name of the University (Company) Country Abstract An adaptive controller is presented in this paper to control an n-link flexible-joint manipulator with time-varying uncertainties The function approximation technique (FAT) is utilized to represent time-varying uncertainties in some finite combinations of orthogonal basis The tedious computation of the regressor matrix needed in traditional adaptive control is avoided in the new design, and the controller does not require the variation bounds of time-varying uncertainties needed in traditional robust control In addition, the joint acceleration is not needed in the controller realization Via the Lyapunov-like stability theory, adaptive update laws are derived to give convergence of the output tracking error Moreover, the upper bounds of tracking errors in the transient state are also derived A DOF planar manipulator with flexible joints is used in the computer simulation to verify the effectiveness of the proposed controller Keywords: Adaptive control; Flexible-joint robot; FAT INTRODUCTION In practical applications, most controllers for robot manipulators equipped with harmonic devices are based on rigid-body dynamics formulation To achieve high precision tracking performance, the joint flexibility should be carefully considered.1 However, the modeling of flexible-joint robots is far more complex than that of rigid-joint robots Besides, the mathematical model of the robot inevitably contains model inaccuracies such as parametric M C Chien is with the Mechanical and Systems Research Laboratories, Industrial Technology Research Institute, No 195, Sec 4, Chung-Hsing Rd., Chutung, Hsinchu, 310, Taiwan, R.O.C (Tel: +886-3-591-8630/Fax:+886-3-5913607 ,E-mail: D9203401@mail.ntust.edu.tw) A C Huang is with the Department of Mechanical Engineering, National Taiwan University of Science and Technology No 43, Keelung Rd., Sec 4, Taipei, Taiwan, ROC (Tel:+886-2-27376490, Fax: +886-2-37376460, E-mail: achuang@mail.ntust.edu.tw) 28 Advances in Robot Manipulators uncertainties, and unmodeled dynamics Since these inaccuracies may degrade the performance of the closed-loop system, any practical design should consider their effects Under the problems of joint flexibility and model inaccuracies, several strategies based on adaptive control or robust control for flexible-joint robots had been proposed Spong2,3 proposed an adaptive controller for flexible-joint robots by using the singular perturbation formulation Chen and Fu4 presented a two-stage adaptive control scheme for a single-link robot based on a simplified dynamic model Khorasani5 designed an adaptive controller using the concept of integral manifolds for n-link flexible-joint robots Without using the velocity measurements, Lim et al.6 proposed an adaptive integrator backstepping scheme for rigid-link flexible-joint robots Dixon et al.7 designed an adaptive partial state feedback controller by using a nonlinear link velocity filter Yim8 suggested an output feedback adaptive controller based on the backstepping design Kozlowski and Sauer9,10 suggested an adaptive controller under the assumption of bounded disturbances to have semiglobal convergence Tian and Goldenberg11 proposed a robust adaptive controller with joint torque feedback Jain and Khorrami12 suggested a robust adaptive control for a class of flexible-joint robots that are transformable to a special strict feedback form However, like most adaptive control strategies, the uncertainties should be linearly parameterizable into regressor form13 Availability of the regressor matrix is crucial to the derivation of adaptive controllers for robot manipulators This is because traditional adaptive control strategies have a common assumption that the uncertain parameters should be constant or slowly time varying Therefore, the robot dynamics is linearly parameterized into known regressor matrix and an unknown vector with constant parameters In general, derivation of the regressor matrix for a given robot is tedious Once it is obtained, we may find that, for most robots, elements in the unknown vector are simple combinations of system parameters such as link mass, link length and moment of inertia, and these are sometimes relatively easy to measure.13 Huang and Chen14 proposed an adaptive backstepping-like controller based on FAT15-28 for single-link flexible-joint robots with mismatched uncertainties Similar to most backstepping designs, the derivation is too complex to robots with more joints In this paper, we would like to propose a FAT based adaptive controller for n-link flexible-joint robots The tedious computation of the regressor matrix is avoided in the new design Moreover, the novel controller does not require the variation bounds of time-varying uncertainties needed in traditional robust control In addition, the control strategy does not need to feedback joint acceleration Convergence of the output error and the boundedness of all signals are proved using Lyapunov-like direct method with consideration of the effect of the approximation error This paper is organized as follows: in section 2, we derive the proposed adaptive controller in detail; section presents simulation results of a 2-D flexible-joint robot using the proposed controller; finally, some conclusions are given in section MAIN RESULTS The dynamics of an n-rigid link flexible-joint robot can be described by29 D(q)q C(q, q)q g (q) K(θ q) (1) A Regressor-free Adaptive Control for Flexible-joint Robots based on Function Approximation Technique 29 J Bθ K(θ q) u θ where q n (2) θ n is the vector of actuator torques, D(q) is the n n inertia is the vector of link angles, n angles, u is the vector of actuator input matrix, C(q, q)q is an n-vector of centrifugal and Coriolis forces, and g(q) is the gravity vector J , B and K are n n constant diagonal matrices of actuator inertias, damping and joint stiffness, respectively Here, we would like to consider the case when the precise forms of D(q) , C(q, q)q and g(q) are not available and their variation bounds are not given This implies that traditional adaptive control and robust control cannot be applicable In the following, we would like to use the function approximation technique to design an adaptive controller for the flexible-joint robot Moreover, it is well-known that derivation of the regressor matrix for the adaptive control of high DOF rigid robot is generally tedious For the flexible-joint robot in (1) and (2), its dynamics is much more complex than that of its rigid-joint counterpart Therefore, the computation of the regressor matrix becomes extremely difficult Different form the conventional adaptive control schemes for robot manipulators, the proposed FAT-based adaptive controller does not need the computation of the regressor matrix This largely simplifies the implementation of the control loop Define τ K(θ - q) to be the vector of transmission torques, so (1) and (2) becomes11 D(q)q C(q, q)q g (q) τ J t B t τ τ u q (q, q) τ where J t JK 1 , B t BK 1 v q d Λe, and s e Λe and e q qd is the state error, and q (q, q) Jq Bq n (3) (4) Define signal vector q d is the vector of desired Λ diag (1 , , , n ) with i where states, for all i=1, … n Rewrite (3) in the form Ds Cs g Dv Cv τ (5) A Controller Design for Known Robot Suppose D(q) , C(q, q)q and g(q) are known, and we may design a proper control law such that τ follows the trajectory below τ g Dv Cv K d s where Kd (6) is a positive definite matrix Substituting (6) into (5), the closed loop dynamics 30 Advances in Robot Manipulators becomes Ds Cs K d s Define a Lyapunov function candidate as V T s Ds Its time derivative along the trajectory of the closed loop dynamics can be computed as V sT K d s sT (D 2C)s Since D 2C can be proved to be skew-symmetric, the T above equation becomes V s K s It is easy to prove that s is uniformly bounded d s is also uniformly bounded Hence, s as t , or t To make the actual τ converge to the perfect τ in (6), and square integrable, and we may say e as let us consider the reference model Jrr Brτr Kr τr Kr τd Br τd Jrd τ τ where n τ r states Matrices τr τd is the state vector of the reference model and Jr n n , exponentially Define n n (7) τ d n is the desired n n Br and K r are selected such that 1 τ τd (τd , d ) Kr (Br τd Jrd ) , we may rewrite (4) and (7) τ in the state space form as xp Αpxp Bpu Bpq xm Amxm Bm(τd τd ) where T x p τ τ 2n (8) (9) and x m τ r A p 1 J t vectors are augmented I n n n n 1 J t Bt I nn n n 1 Jr Br A m 1 J r K r B p 1 nn J t T τ r 2n are augmented state and system matrices B m 1 nn are augmented input gain J r K r matrices, and the pair ( A m , B m ) is controllable Since all system parameters are assumed to and be available at the present stage, we may select a controller in the form30 u xp τd h(τd , q) where n2 n and nn satisfy A p B p Am (10) and B p Bm , ... P1 < P1max ∆L = 37( P − P 1 1min ) − 54( P1 − P1min ) −9.5( P1 − P1min ) if P2min < P2 < P2max (4) ∆L2 = −9( P2 − P2min )3 − 18 ( P2 − P2min )2 ? ?11 ( P2 − P2min ) ... P1 ) + f ( P3 )) (11 ) ∆L3 = f ( P3 ) + 0.3( f ( P1 ) + f ( P2 )) 16 Advances in Robot Manipulators Fig 17 Simulation and experimental results of the movement of the endpoint of Colobot Fig 18 ... the whole system within a transparent tube * Corresponding author gang.chen@unilever.com Advances in Robot Manipulators Introduction Robotics has increasingly become accepted in the past 20 years