Robot Manipulators 2011 Part 5 potx

Robot Manipulators 132 where the notations and were introduced in (61), and is defined as It can be shown from Property 7, Property 8, and (Lewis et al., 1996) that can be bounded as (81) where , (i = l,2, ,4) are computable known positive constants. The open-loop robot error system can be obtained by taking the time derivative of ), premultiplying by the robot inertia matrix (), r M x and utilizing (19), (48), and (57) as (82) where the function , contains the uncertain robot and Hunt-Crossley model parameters, and is defined as By representing the function by a NN, the expression in (82) can be written as (83) where the NN input is defined as , are ideal NN weights and denotes the number of hidden layer neurons of the NN. An expression for can be developed to illustrate that the second derivative of the desired trajectory is continuous and does not require acceleration measurements. Based on (83) and the subsequent stability analysis, the robot force control input is designed as (84) where is a constant positive control gain, and are the estimates of the ideal weights, which are designed based on the subsequent stability analysis as (85) where are constant, positive definite, symmetric gain matrices. Substituting (84) into (83) and following a similar approach as in the mass error system in (78)-(80), the closed loop error system for the robot is obtained as (86) Control of Robotic Systems Undergoing a Non-Contact to Contact Transition 133 where is defined as (87) It can be shown from Property 7, Property 8 and (Lewis et al., 1996) that can be bounded as (88( where , (i = 1,2, ,4) are computable known positive constants. 3.2.4 Stability Analysis Theorem: The controller given by (75), (77), (84), and (85) ensures uniformly ultimately bounded regulation of the MSR system in the sense that (89) provided the control gains are chosen sufficiently large (Bhasin et al., 2008). Proof: Let denote a non-negative, radially unbounded function (i.e., a Lyapunov function candidate) defined as (90) It follows directly from the bounds given in (8), Property 8, (64) and (65), that can be upper and lower bounded as (91) where are known positive bounding constants, and is defined as (92) The time derivative of in (90) can be upper bounded (Bhasin et al., 2008) as (93) where are positive constants which can be adjusted through the control gains (Bhasin et al., 2008). Provided the gains are chosen sufficiently large (Bhasin et al., 2008), the definitions in (70) and (92), and the expressions in (90) and (93) can be used to prove that . In a similar approach to the one developed in the first Robot Manipulators 134 section, it can be shown that all other signals remain bounded and the controller given by (75), (77), (84), and (85) is implementable. 4. Conclusion In this chapter, we consider a two link planar robotic system that transitions from free motion to contact with an unactuated mass-spring system. In the first half of the chapter, an adaptive nonlinear Lyapunov-based controller with bounded torque input amplitudes is designed for robotic contact with a stiff environment. The feedback elements for the controller are contained inside of hyperbolic tangent functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from non-contact to contact. The continuous controller in (35) yields semi-global asymptotic regulation of the spring-mass and robot links. Experimental results are provided to illustrate the successful performance of the controller. In the second half of the chapter, a Neural Network controller is designed for a robotic system interacting with an uncertain Hunt-Crossley viscoelastic environment. This result extends our previous result in this area to include a more general contact model, which not only accounts for stiffness but also damping at contact. The use of NN-based estimation in (Bhasin et al., 2008) provides a method to adapt for uncertainties in the robot and impact model. 5. References S. Bhasin, K. Dupree, P. M. Patre, and W. E. Dixon (2008), Neural Network Control of a Robot Interacting with an Uncertain Hunt-Crossley Viscoelastic Environment, ASME Dynamic Systems and Control Conference, Ann Arbor, Michigan, to appear. Z. Cai, M.S. de Queiroz, and D.M. Dawson (2006), A Sufficiently Smooth Projection Operator, IEEE Transactions on Automatic Control, Vol. 51, No. 1, pp. 135-139. D. Chiu and S. Lee (1995), Robust Jump Impact Controller for Manipulators, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Pittsburgh, Pennsylvania, pp. 299-304. N Diolaiti, C Melchiorri, S Stramigioli (2004), Contact impedance estimation for robotic systems, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, Italy, pp. 2538-2543. W. E. Dixon, M. S. de Queiroz, D. M. Dawson, and F. Zhang (1999), Tracking Control of Robot Manipulators with Bounded Torque Inputs, Robotica, Vol. 17, pp. 121-129. W. E. Dixon, E. Zergeroglu, D. M. Dawson, and M. W. Hannan (2000), Global Adaptive Partial State Feedback Tracking Control of Rigid-Link Flexible-Joint Robots, Robotica, Vol. 18. No 3. pp. 325-336. W. E. Dixon, A. Behal, D. M. Dawson, and S. Nagarkatti (2003), Nonlinear Control of Engineering Systems: A Lyapunov-Based Approach, Birkhauser, ISBN 081764265X, Boston. K. Dupree, C. Liang, G. Hu and W. E. Dixon (2006) a, Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing an Impact Collision, International Journal of Robotics and Automation, to appear; see also Proceedings of the IEEE American Control Conference, Minneapolis, MN, pp. 3241-3246. Control of Robotic Systems Undergoing a Non-Contact to Contact Transition 135 K. Dupree, C. Liang, G. Hu and W. E. Dixon (2006) b, Global Adaptive Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing an Impact-Collision, IEEE Transactions on Systems, Man and Cybernetics, to appear; see also Proceedings of the IEEE Conference on Decision and Controls, San Deigo, California, pp. 2039-2044. M. W. Gertz, J. Kim, and P. K. Khosla (1991), Exploiting Redundancy to Reduce Impact Force, IEEE/RSJ International Workshop on Intelligent Robots and Systems IROS, Osaka, Japan, pp. 179-184. G. Gilardi and I. Sharf (2002), Literature survey of contact dynamics modelling, Mechanism and Machine Theory, Volume 37, Issue 10, Pages 1213-1239. N. Hogan (1985), Impedance control: An approach to manipulation: Parts I, II, and III, /. Dynamic Sys. Measurement Control 107:1-24. K.H. Hunt and F.R.E. Crossley (1975), Coefficient of restitution interpreted as damping in vibroimpact, Journal of Applied Mechanics 42, Series E, pp. 440—445. M. Indri and A. Tornambe (2004), Control of Under-Actuated Mechanical Systems Subject to Smooth Impacts, Proceedings of the IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, pp. 1228-1233. S. Jezernik, M Morari (2002), Controlling the human-robot interaction for robotic rehabilitation of locomotion, 7th International Workshop on Advanced Motion Control. H.M. Lankarani and P.E. Nikravesh (1990), A contact force model with hysteresis damping for impact analysis of multi-body systems, Journal of Mechanical Design 112, pp. 369- -376. E. Lee, J. Park, K. A. Loparo, C. B. Schrader, and P. H. Chang (2003), Bang-Bang Impact Control Using Hybrid Impedance Time-Delay Control, IEEE/ASME Transactions on Mechatronics, Vol. 8, No. 2, pp. 272-277. F. L. Lewis, A. Yesildirek, and K. Liu (1996), Multilayer neural-net robot controller: structure and stability proofs, IEEE Transactions. Neural Networks. F. L. Lewis (1999), Nonlinear Network Structures for Feedback Control, Asian Journal of Control, Vol. 1, No. 4, pp. 205-228. F. L. Lewis, J. Campos, and R. Selmic (2002), Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities, SIAM, PA. Z Li, P Hsu, S Sastry (1989), Grasping and Coordinated Manipulation by a Multifingered Robot Hand, The International Journal of Robotics Research, Vol. 8, No. 4,33-50. C. Liang, S. Bhasin, K. Dupree and W. E. Dixon (2007), An Impact Force Limiting Adaptive Controller for a Robotic System Undergoing a Non-Contact to Contact Transition, IEEE Transactions on Control Systems Technology, submitted; see also Proceedings of the IEEE Conference on Decision and Controls, Louisiana, pp. 3555-3560. D.W. Marhefka and D.E. Orin (1999), A compliant contact model with nonlinear damping for simulation of robotic systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, pp. 566—572. A.M. Okamura, N. Smaby, M.R. Cutkosky (2000), An overview of dexterous manipulation, IEEE International Conference on Robotics and Automation. P. R. Pagilla and B. Yu (2001), A Stable Transition Controller for Constrained Robots, IEEE/ASME Transactions on Mechatronics, Vol. 6, No. 1, pp. 65-74. Robot Manipulators 136 P. M. Patre, W. MacKunis, C. Makkar, W. E. Dixon (2008), Asymptotic Tracking for Uncertain Dynamic Systems via a Multilayer NN Feedforward and RISE Feedback Control Structure, IEEE Transactions on Control Systems Technology, Vol. 16, No. 2, pp. 373-379. A. Tornambe (1999), Modeling and Control of Impact in Mechanical Systems: Theory and Experimental Results, IEEE Transactions on Automatic Control, Vol. 44, No. 2, pp. 294-309. I. D. Walker (1990), The Use of Kinematic Redundancy in Reducing Impact and Contact Effects in Manipulation, Proceedings of the IEEE International Conference on Robotics and Automation, Cincinnati, OH, pp. 434-439. I. D. Walker (1994), Impact Configurations and Measures for Kinematically Redundant and Multiple Armed Robot Systems, IEEE Transactions on Robotics and Automation, Vol. 10, No. 5, pp 346-351. K. Youcef-Toumi and D. A. Guts (1989), Impact and Force Control, Proceedings of the IEEE International Conference on Robotics and Automation, AZ, pp. 410-416. 7 Motion Control of a Robot Manipulator in Free Space Based on Model Predictive Control Vincent Duchaine, Samuel Bouchard and Clément Gosselin Université Laval Canada 1. Introduction The majority of existing industrial manipulators are controlled using PD controllers. This type of basically linear control does not represent an optimal solution for the motion control of robots in free space because robots exhibit highly nonlinear kinematics and dynamics. In fact, in order to accommodate configurations in which gravity and inertia terms reach their minimum amplitude, the gain associated with the derivative feedback (D) must be set to a relatively large value, thereby leading to a generally over-damped behaviour that limits the performance. Nevertheless, in most current robotic applications, PD controllers are functional and sufficient due to the high reduction ratio of the transmissions used. However, this assumption is no longer valid for manipulators with low transmission ratios such as human-friendly manipulators or those intended to perform high accelerations like parallel robots. Over the last few decades, a new control approach based on the so-called Model Predictive Control (MPC) algorithm was proposed. Arising from the work of Kalman (Kalman, 1960) in the 1960’s, predictive control can be said to provide the possibility of controlling a system using a proactive rather than reactive scheme. Since this control method is mainly based on the recursive computing of the dynamic model of the process over a certain time horizon, it naturally made its first successful breakthrough in slow linear processes. Common current applications of this approach are typically found in the petroleum and chemical industries. Several attempts were made to adapt this computationally intensive method to the control of robot manipulators. A little more than a decade ago, it was proposed to apply predictive control to nonlinear robotic systems (Berlin & Frank, 1991), (Compas et al., 1994). However, in the latter references, only a restricted form of predictive control was presented and the implementation issues — including the computational burden — were not addressed. Later, predictive control was applied to a broader variety of robotic systems such as a 2-DOF (degree-of-freedom) serial manipulator (Zhang & Wang, 2005), robots with flexible joints (Von Wissel et al., 1994), or electrical motor drives (Kennel et al., 1987). More recently, (Hedjar et al., 2005), (Hedjar & Boucher, 2005) presented simplified approaches using a limited Taylor expansion. Due to their relatively low computation time, the latter approaches open the avenue to real-time implementations. Finally, (Poignet & Gautier, 2000), (Vivas et al., 2003), (Lydoire & Poignet, 2005), experimentally demonstrated Robot Manipulators 138 predictive control on a 4-DOF parallel mechanism using a linear model in the optimization combined with a feedback linearization. Several other control schemes based on the prediction of the torque to be applied at the actuators of a robot manipulator can be found in the literature. Probably the best-known and most commonly used technique is the so-called Computed Torque Method (Anderson, 1989), (Ubel et al., 1992). However, this control scheme has the disadvantage of not being robust to modelling errors. In addition to having the capability of making predictions over a certain time horizon, model predictive control contains a feedback mechanism compensating for prediction errors due to structural mismatch between the model and the process. These two characteristics make predictive control very efficient in terms of optimal control as well as very robust. This chapter aims at providing an introduction to the application of model predictive control to robot manipulators despite their typical nonlinear dynamics and fast servo rate. First, an overview of the theory behind model predictive control is provided. Then, the application of this method to robot control is investigated. After making some assumptions on the robot dynamics, equations for the cost function to be minimized are derived. The solution of these equations leads to an analytic and computationally efficient expression for position and velocity control which are functions of a given prediction time horizon and of the dynamic model of the robot. Finally, several experiments using a 1-DOF pendulum and a 6-DOF cable-driven parallel mechanism are presented in order to illustrate the performance in terms of dynamics as well as the computational efficiency. 2. Overview The very first predictive control schemes appeared around 1980 under the form of Dynamic Matrix Control (DMC) and Generalized Predictive Control (GPC). These two approaches led to a successful breakthrough in the chemical industry and opened the avenue to several new control algorithms later known as the family of model predictive control. A more detailed account of the history of model predictive control can be found in (Morari & Lee, 1999). Model predictive control is based on three main key ideas. These ideas have been well summarized by Camacho and Bordons in their book on the subject (Camacho & Bordons, 2004). This method is in fact based on the explicit use of a model to predict the process output at future time instants horizon. The prediction is done via the calculation of a control sequence minimizing an objective function. It also has the particularity that it is based on a receding strategy, so that at each instant the horizon is displaced toward the future, which involves the application of the first control signal of the sequence calculated at each step. This last particularity partially explains why predictive control is sometime called receding horizon control. As mentioned above, a predictive control scheme required the minimization of a quadratic cost function over a prediction horizon in order to predict the correct control input to be applied to the system. The cost function is composed of two parts, namely, a quadratic function of the deterministic and stochastic components of the process and a quadratic function of the constraints. The latter is one of the main advantages of this control method over many other schemes. It can deal at same time with model regulation and constraints. The constraints can be on the process as well as on the control output. The global function to be minimized can then be written in a general form as: Motion Control of a Robot Manipulator in Free Space Based on Model Predictive Control 139 (1) where Although this function is the key of the effectiveness of the predictive control scheme in terms of optimal control, it is also its weakness in term of computational time. For linear processes, and depending on the constraint function, the optimal control sequence can be found relatively fast. However, for nonlinear model the problem is no longer convex and hence the computation of the function over the prediction horizon becomes computationally intensive and sometime very hard to solve explicitly. 3. Application to manipulators This part aims at showing how Model Predictive Control can be efficiently applied to robot manipulators to suit their fast servo rate. Figure 1 provides a schematic representation of the proposed scheme, where d k represents the error between the output of the system and the output of the model. In the next subsection the different parts of this scheme will be defined for velocity control as well as for position control. Figure 1. MPC applied to manipulator 3.1 Velocity control Velocity control is rarely implemented in conventional industrial manipulators since the majority of the tasks to be performed by robots require precise position tracking. However, over the last few years, several researchers have developed a new generation of robots that Robot Manipulators 140 are capable of working in collaboration with humans (Berbardt et al., 2004), (Peshkin & Colgate, 2001), (Al-Jarrah & Zheng, 1996). For this type of tasks, velocity control seems more appropriate (Duchaine & Gosselin, 2007) due to the fact that the robot is not constrained to given positions but rather has to follow the movement of the human collaborator. Also, velocity control has infinitely spatial equilibrium points which is a very safe intrinsic behaviour in a context where a human being is sharing the workspace of a robot. The predictive control approach presented in this chapter can be useful in this context. 3.1.1 Modelling For velocity control, the reference input is usually relatively constant, especially considering the high servo rates used. Therefore, it is reasonable to assume that the reference velocity remains constant over the prediction horizon. With this assumption, the stochastic predictor of the reference velocity becomes: (2) where stands for the predicted value of r at time step j. The error, d, is obtained by computing the difference between the system's output and the model's output. Taking into account this difference in the cost function will help to increase the robustness of the control to model mismatch. The error can be decomposed in two parts. The first one is the error associated directly with model uncertainties. Often, this component will produce an offset proportional to the mismatch. The error may also include a zero- mean white noise given by the noise of the encoder or some random perturbation that cannot be included in the deterministic model. Since the error term is partially composed of zero-mean white noise, it is difficult to define a good stochastic predictor of the future values. However, in the case considered here, a future error equal to the present one will be simply assumed. This can be expressed as: (3) where is the predicted value of d at time step j. In this chapter, a constraint on the variation of the control input signal (u) over a prediction horizon will be used as a constraint function in the optimization. This is a typical constraint over the control input that helps smoothing the command and tends to maximize the effective life of the actuator, namely: (4) The model of the robot itself is directly linked with its dynamic behaviour. The dynamic equations of a robot manipulator can be expressed as: [...]... haptics 2007, pp 446– 451 Duchaine, V.; Bouchard, S & Gosselin, C (2007) Computationally Efficient Predictive Robot Control, IEEE/ASME Transactions on Mechatronics, Vol.12, pp 57 0 -57 8 Hedjar, R & Boucher, P (20 05) Nonlinear receding-horizon control of rigid link robot manipulators International Journal of Advanced Robotic Systems, pp 15 24 Hedjar, R.; Toumi, R.; Boucher, P & Dumur, D (20 05) Finite horizon... important, it should be noted that the oscillations of elastic modes are now attenuated quickly (compare Fig 5( b,c) and Fig 6(b,c)) 1 0 .52 2 3 4 5 6 8 7 9 10 0 .5 0.48 x(m) 0.46 0.44 0.42 Y0 3 0.4 7 4 0 2 4 6 Time(s) 8 10 1 2 3 4 5 6 8 7 9 1 8 10 12 (a) 0. 45 2 5 9 6 0.38 10 θ1 θ4 0.4 y (m) X0 (b) 0. 35 0.3 0. 25 0 2 4 6 Time (s) 8 10 12 Figure 4 Cartesian trajectory of the manipulator (a) Time evolution of the... industrial robots International Conference on Robotics and Automation, pp 52 8 53 3 Vivas, A., Poignet, P., and Pierrot, F (2003) Predictive functional control for a parallel robot Proceedings of the International Conference on Intelligent Robots and Systems, pp 27 85 2790 Von Wissel, D., Nikoukhah, R., and Campbell, S L (1994) On a new predictive control strategy: Application to a flexible-joint robot Proceedings... Conference on Decision and Control, pp 30 25 3026 Zhang, Z., and Wang, W (20 05) Predictive function control of a two links robot manipulator Proceeding of the Int Conf on Mechatronics and Automation, pp 2004 – 2009 8 Experimental Control of Flexible Robot Manipulators A Sanz and V Etxebarria Universidad del País Vasco Spain 1 Introduction Flexible-link robotic manipulators are mechanical devices whose... on Robotics and automation, pp 1000–10 05 Anderson, R J (1989) Passive computed torque algorithms for robot Proceeding of the 28th Conference on Decision and Control , pp 1638–1644 Berbardt, R.; Lu, D.; & Dong, Y (2004) A novel cobot and control Proceeding of the 5th world congress on intelligent control, pp 46 35 4639 Berlin, F & Frank, P M (1991) Robust predictive robot control Proceeding of the 5th... IEEE Transaction on Robotics and Automation 17, pp 377–390 Poignet, P & Gautier, M (2000) Nonlinear model predictive control of a robot manipulator Proceedings of the 6th International Workshop on Advanced Motion Control, 2000 , pp 401–406 Qin, S & Badgwell, T (1997) An overview of industrial model predictive control technology Chemical Process Control-V, pp 232– 256  154 Robot Manipulators Uebel, M.,... excite the flexible modes to an unwanted extent 164 Robot Manipulators Figure 5 Experimental results for pure rigid LQR control a) Tip trajectory and reference b) Time evolution of the flexible deflections (link 1 deflections) c) Time evolution of the flexible deflections (link 4 deflections) Experimental Control of Flexible Robot Manipulators 1 65 Figure 6 Experimental results for composite (slow-fast)... applied to the flexible manipulator shown in Fig 2, whose geometric sketch is displayed in Fig 3 As seen in the figures, it is a laboratory robot which 158 Robot Manipulators moves in the horizontal plane with a workspace similar to that of common Scara industrial manipulators Links marked 1 and 4 are flexible and the remaining ones are rigid The flexible links' deflections are measured by strain gauges,... − θ 4 )ϕ1 ϕ1 q12 ) + m5 ( l 2 cos(θ1 − θ 4 ) − l 2 sin(θ1 − θ 4 )ϕ4 q4 + l sin(θ1 − θ 4 )ϕ4 q4 2 2 2 1 ′ 2 + l cos(θ1 − θ 4 )ϕ4ϕ4 q4 ) 2 159 Experimental Control of Flexible Robot Manipulators h21 = h12 4 4 1 2 2 2 2 2 ′ ′ h22 = I h 2 + m j 2 (l 2 + l 2ϕ1 2q12 ) + I h3 + I h 4 + m j1 (l 2 + ϕ4 q4 ) + m j 2 (l 2 + ϕ4 q4 ) + 2m2 ( l 2 + l 2ϕ1 2q12 ) + m4l 2 + ς 411q4 3 3 3 2 2 + m5 (l 2 + ϕ4 q4 ) ′ ′... and Robotics Conference Camacho, EF & Bordons, C (2004) Model predictive control, Springer Compas, J M.; Decarreau, P.; Lanquetin, G.; Estival, J & Richalet, J (1994) Industrial application of predictive functionnal control to rolling mill, fast robot, river dam Proceedings of the 3th IEEE Conference on Control Applications , pp 1643–1 655 Duchaine, V & Gosselin, C (2007) General model of human-robot . pp. 355 5- 356 0. 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