Control techniques

The dynamic behaviour of a flexible manipulator may be considered as a combination of rigid-body and flexible dynamics. Accordingly, control strategies devised for such systems are to take account of both rigid-body motion and flexible motion control.

The former corresponds to methods developed within the framework of conventional rigid manipulator control. The latter, on the other hand, corresponds to approaches developed within the framework of vibration control of flexible structures.

Vibration control techniques for flexible structures are generally classified into two categories: passive and active control (Tokhi and Veres, 2002). Active control utilizes the principle of wave interference. This is realised by artificially generating anti-source(s) (actuator(s)) to destructively interfere with the unwanted disturbances and thus result in reduction in the level of vibration. Active control of FMSs can in general be divided into two categories: open-loop and closed-loop control. Open-loop control involves altering the shape of actuator commands by considering the physical and vibration properties of the FMS. The approach may account for changes in the system after the control input is developed. Closed-loop control differs from open-loop control in that it uses measurements of the system state and change the actuator input accordingly to reduce the system response oscillation.

1.3.1 Passive control

Passive control utilizes the absorption property of matter and thus is realised by a fixed change in the physical parameters of the structure, for example, adding viscoelastic material to increase the damping properties of the flexible manipulator. It has been reported that the control of vibration of a flexible manipulator by passive means is not sufficient by itself to eliminate structural deflection (Book et al., 1986). On the other hand, if only active control is used then, owing to actuator and sensor dynamics, destabilisation of modes near the bandwidth of the actuator or sensor may result (Aubrun, 1980). To avoid such destabilisation a certain amount of passive damping will be required to be employed, thus using hybrid control, that is, a combination of active and passive control methods. Combined active/passive control strategies have been proposed previously where low-frequency modes of vibration are controlled by active means and the modes with frequencies just above the actively controlled modes are controlled by passive means (Plunkell and Lee, 1970).

Several methods of passive vibration control of FMSs have been developed over the years. These mainly include methods of implementation of a constrained vis-coelastic damping layer to provide an energy dissipation medium (Kerwin, 1959) and

the utilization of composite materials in the construction of a flexible manipulator to provide higher strength and stiffness-to-weight ratio and larger structural damping than a metallic flexible manipulator Aubrun, 1980; Choi et al., 1988; Thompson and Sung, 1986). Observations have shown that although passive damping provides a sharp increase in damping at higher frequency modes, the lower frequency modes still remain uncontrolled. Moreover, the addition of viscoelastic material and a con-straining layer leads to an increase in the size and dynamic load of the system (Tzou, 1988).

1.3.2 Open-loop control

Open-loop control methods have been considered in vibration control where the con-trol input is developed by considering the physical and vibrational properties of the FMS. Although, the mathematical theory of open-loop control is well established, only few successful applications in the control of distributed parameter systems, including flexible manipulator, have been reported (Dellman et al., 1956; Singh et al., 1989). The method involves the development of suitable forcing functions in order to reduce the vibration at resonance modes. The methods developed include shape command methods, the computed torque technique and bang-bang control.

Shaped command methods attempt to develop forcing functions that minimise vibra-tions and the effect of parameters that affect the resonance modes (Aspinwall, 1980;

Meckl and Seering, 1990; Swigert, 1980; Wang, 1986). Common problems of con-cern encountered in these methods include long move (response) time, instability owing to un-reduced modes and controller robustness in the case of a large change of the manipulator dynamics.

In the computed torque approach, depending on the detailed model of the system and desired output trajectory, the joint torque input is calculated using a model inversion process (Moulin and Bayo, 1991). The technique suffers from several prob-lems, owing to, for instance, model inaccuracy, uncertainty over implementability of the desired trajectory, sensitivity to system parameter variations and response time penalties for a causal input.

Bang-bang control involves the utilization of single and multiple switched bang-bang control functions (Onsay and Akay, 1991). Bang-bang-bang control functions require accurate selection of switching time, depending on the representative dynamic model of the system. A minor modelling error could cause switching error and thus result in a substantial increase in the residual vibrations. Although, utilization of minimum energy inputs has been shown to eliminate the problem of switching times that arise in the bang-bang input (Jayasuriya and Choura, 1991), the total response time, however, becomes longer (Meckl and Seering, 1990; Onsay and Akay, 1991).

1.3.3 Closed-loop control

Effective control of a system always depends on accurate real-time monitoring and the corresponding control effort. Initial discussions on feedback control of a flexible manipulator and the usefulness of optimal regulator as applied to this problem date

6 Flexible robot manipulators

back to the early 1970s. It is known in the conventional approach that compensation can alter the first vibrational mode by either adding some damping or extending the bandwidth of the system (Ogata, 2001). Compensation, however, will limit the performance of the manipulator because inputs with frequency contents above the first flexible mode could still cause vibration. Various modern control designs have been proposed during the last two decades for FMSs with different types of vibration measuring systems.

When free motion of a system consists mainly of a limited number of clearly separable modes, then it is possible to control these modes directly using the so-called independent modal space control (IMSC) method, where the controller is designed for each mode independent of other modes (Baz et al., 1992; Sinha and Kao, 1991).

Modal space control has been used for suppression of flexible motion in a three-link log loading manipulator with which considerable improvement has been achieved over conventional joint-based collocated controller. Although, initial investigations on the use of IMSC lack consideration of the location of the actuator (Meirovitch et al., 1983), later investigations have shown that actuator placement is important for suppression of spillover and, thus, methods for optimal placement of sensors and actuators have been developed (Schulz and Heimbold, 1983).

Variable structure control (VSC) utilizes a viable high-speed switching feedback control law to drive the plant’s state trajectory onto a specified and user-specified surface in the state-space, and to maintain the plant’s state trajectory on this surface for all subsequent times. One of the first studies on the application of VSC to one-link flexible manipulators was reported by Qian and Ma (1992), where they controlled the end-point position in a non-collocated manner. In this study, a sliding surface (a line in the study) is constructed from the end-point position and its derivative is employed in the design. Qian and Ma (1992) claim that if the slope of this line is chosen positive and the system variables are made to stay on this line, these would converge to zero exponentially, thus yielding a stable system in sliding mode. The performance of the controller was evaluated through a series of simulations, followed by an analysis of the designed control system. Thomas and Bandyopadhyay (1997), however, have pointed out that the choice of a positive constant as the slope for this switching line would not guarantee the stability of the system in sliding mode. The switching line is in fact a switching hyper-surface in view of the functional relationship of the tip (end-point) position with the generalized coordinates of the system through mode shape functions. The stability of the system in sliding mode is guaranteed only if the motion on this hyper-surface is asymptotically stable (Young, 1977), whereas the positive value for the slope of the switching line employed by Qian and Ma (1992) will not guarantee this stability. Moreover, a stable VSC controller based on a state transformation has been designed in this study.

The application of VSC to multi-link flexible manipulators is very limited. There are difficulties in both modelling and controller design. Sira-Ramirez et al. (1992) have derived dynamical sliding mode regulators within the context of generalized observability canonical form (GOCF) (Fliess, 1989). The GOCF is obtained by means of a state elimination procedure, carried out on the system of differential equa-tions describing the manipulator dynamics. Therefore the system can be considered

as a linear system. Although simulation examples illustrate the performance of the proposed controller for a robotic manipulator with flexible joint, it is not easy to apply to general multi-link manipulators with flexible links. There are also applications of VSC to other plants similar to flexible manipulators, for example, a spacecraft with flexibility (Karray and Modi, 1995), a flexible structure on the ground (Iwamoto et al., 2002), a disk drive actuator (Supino and Romano, 1997), and so on.

An appreciable amount of work carried out on the control of FMSs involves the utilization of strain gauges, mainly to measure mode shapes (Sakawa et al., 1985).

There are two essential components involved in measuring the modal response using strain gauges. The first is a method of measurement of the modes of vibration of the flexible manipulator. The second is the development of a computational technique for distinguishing different modes in the overall deflection of the flexible manipulator.

Once modal information is available a control loop can be closed for each mode either to damp or to actively drive the manipulator in a manner that reduces the vibration. It appears that the strain gauge measurement is very simple and relatively inexpensive to use. However, the technique may place more stringent requirements on the dynamic modelling and control tasks. Strain gauges have the disadvantage of not giving a direct measurement of manipulator displacement, as they can only provide local information. Thus, displacement measurement by using strain gauges requires more complex and possibly time-consuming computations, which can lead to inaccuracies.

To solve the problem of displacement measurement, as encountered in using strain gauges only, attempts have been made to develop schemes that incorporate end-point measurements as well (Cannon and Schmitz, 1984; Kotnik et al., 1988).

Some researchers have proposed an approach that utilizes local or global measurement of flexible displacement of a manipulator to control system vibration (Harashima and Ueshiba, 1986; Wang et al., 1989). In this method, the deflection of the manipulator is detected (measured), using, for example, a charge coupled device (CCD) camera or laser beam, relative to a rotating reference X–Y frame fixed to the hub of the manipulator. However, as an end-point position control system has a smaller stability margin than collocated control, it is necessary to include a collocated rate feedback (hub-velocity) to obtain acceptable performance of the closed-loop system. By using an end-point sensor, more accurate end-point positioning can be accomplished, but the resulting controller is less robust to plant uncertainties than the corresponding collocated design.

The difficulty in maintaining stability and performance robustness, owing to spillover effects from unmodelled modes that occur when a high-order system is controlled by a low-order controller, is of major concern in the control of flexible systems. To improve robustness it is typically required that the controller bandwidth be sufficiently reduced (Nesline and Zarchan, 1984). Studies have shown that most robust control techniques that ensure stability in the presence of parameter errors can only increase damping by a limited amount (Dorato, 1987). If the inherent damping is very low, this increase may be insufficient to adequately improve the response.

Moreover, the controllers rely on accurate system models. This makes the controller very sensitive to modelling errors, leading to degradation in system performance and,

8 Flexible robot manipulators

in some cases, instability. It is evident that in using either global or local displacement measurement a device is required to be attached on the manipulator. This affects the behaviour of the manipulator (Mace, 1991).

Both feedforward and feedback control structures have been utilized in the control of vibration of FMSs (Shchuka and Goldenberg, 1989; Wells and Schueller, 1990).

These include combined feedforward and feedback methods based on control law partitioning schemes, which use end-point position signal in an outer loop to control the flexible modes and the inner loop to control the rigid-body motion. Although, the pole-zero cancellation property of the feedforward control speeds up the system response, it increases overshoot and oscillation. However, it is found that, in contrast to many high-order compensators, systems with feedforward control incorporating proportional and derivative (PD) feedback are not highly sensitive to plant parameter variations.

In investigations carried out on control of FMSs the only non-collocated sensor/

actuator pairs that have successfully been employed include motor torque with either the manipulator strain or global/local end-point position. However, practical realisa-tion of both methods has associated short- and long-term drawbacks. If a state-space description of the closed-loop dynamics is available, it is possible to use acceleration feedback to stabilise a rigid manipulator (Stadenny and Belanger, 1986). Investiga-tions on the control of a FMS using acceleration feedback to design the compensator and the end-point position feedback using a design based on a full-state feedback observer have shown that the controller using end-point position feedback exhibits a relatively slow and rough response in comparison with an acceleration feedback controller; the difference becoming more noticeable with increasing slewing angle (Kotnik et al., 1988). Moreover, acceleration feedback produces relatively higher overshoot. The use of acceleration feedback appears to have intuitive appeal from an engineering design viewpoint, particularly because of the relative ease of implemen-tation and low cost. Moreover, in sensing acceleration for control implemenimplemen-tation, all sensing and actuation equipment is structure mounted. This implies that issues such as camera positioning or field of view are not of major concern, which is an important consideration, specifically, in large-scale applications such as telerobotics. Further-more, applications to multi-link flexible manipulators (MLFMs) could benefit from such methods to a greater extent. Some researchers have also proposed adaptive con-trol methods to compensate for parameter variations (Feliu et al., 1990; Yang et al., 1991). However, these approaches utilize optical methods of global/local end-point sensing for obtaining the feedback signal.

Many of the controllers have been designed on the basis of various input shaping mechanisms using both open- and closed-loop configurations. Zuo and Wang (1992) designed a closed-loop control mechanism based on shaped input filter, to reduce or eliminate vibrations and to reject external disturbances of a multi-link manipulator. An adaptive input shaping control scheme for vibration suppression in slewing flexible structures with particular application to flexible-link robotic manipulators has been reported by Tzes and Yurkovich (1993). The scheme combines a frequency-domain identification technique, with input shaping, in order to adjust critical parameters of the input shapers in the case of payload variation or other unmodelled dynamics. The

scheme was realised through simulation and experimentation. Hillsley and Yurkovich (1993) have reported a composite control strategy for a two-link flexible robotic arm in conjunction with post-slew feedback scheme. In this work attention has been focused on end-point position control, for point-to-point movements assuming a fixed reference frame for the base with two rotary joints. Khorrami et al. (1994) addressed experimentation on rigid-body-based controllers with input preshaping for a two-link flexible manipulator (TLFM). The scheme is shown to be effective when the plant dynamics are linear and time invariant. It has also been shown that application of an inner-loop non-linear control to cancel some of the non-linearities and to reduce configuration dependence of structural frequencies enhances the performance of the input pre-shaping scheme. Borowiec and Tzes (1996) proposed a frequency-shaped explicit output feedback force control for a TLFM. In this work the frequency shaping dependence has been included to eliminate the undesirable effects associated with control and observation spillover. Magee et al. (1997) developed a control approach, combining command shaping and internal damping, to control a small robot attached to the end of a flexible manipulator. They also verified the proposed control system experimentally using two separate test-beds.

1.3.4 Artificial intelligence control

It is noted that the non-linear dynamics of rigid manipulators are compensated by an inverse-dynamic strategy, and use of such an approach for a flexible manipulator is restricted by non-minimum phase characteristics of the arm when end-point response is taken as output of the system (Talebi et al., 1998b). Several conventional approaches have been proposed as solutions to this problem based on different methods such as non-causal torque, singular perturbation, integral manifold, transmission zero and redefined output (Bayo and Moulin, 1989; De Luca and Siciliano, 1989; Geniele et al., 1992; Hashtrudi-Zaad and Khorasani, 1996; Kwon and Book, 1990; Madhavan and Singh, 1991; Moallem et al., 1997; Schoenwald and Özgüner, 1990; Siciliano and Book, 1988; Wang and Vidaysagar, 1989a,b). However, performance of these control strategies may not be satisfactory in real-applications as it is difficult to accurately model a flexible manipulator.

In many cases, when it is difficult to obtain a model structure for a system with traditional system identification techniques, intelligent techniques are desired that can describe the system in the best possible way (Elanayar and Yung, 1994). Genetic algorithms (GAs) and artificial neural networks (ANNs) are commonly used for modelling dynamic systems. The main advantages of utilizing GAs for system iden-tification are that they simultaneously evaluate many points in the parameter space and converge towards the global solution (Kargupta and Smith, 1991; Kristinsson and Dumont, 1992). The superiority of a GA over recursive least squares (RLS) in modelling a fixed-free flexible beam has been addressed by Hossain et al. (1995). In contrast, neural network (NN) approaches for system identification offer many advan-tages over traditional ones especially in terms of flexibility and hardware realisation (Ljung and Sjöberg, 1992). This technique is quite efficient in modelling non-linear systems or if the system possesses non-linearities to any degree.

10 Flexible robot manipulators

Application of NNs for identification and control of dynamic systems has gained significant momentum in recent years. Narendra and Parthasarathy (1990) addressed system identification using the globally approximating characteristics of NNs. Neuro-modelling with different approaches, involving backpropagation, has been reported by various researchers (Nerrand et al., 1994; Srinivasan et al., 1994). The successful application of radial basis function (RBF) networks for modelling dynamic systems is also widely addressed in the literature (Casdagli, 1989; He and Lapedes, 1993;

Sze, 1995). Chen et al. (1991) proposed orthogonal least square learning algorithm for RBF networks to model non-linear dynamic systems. Elanayar and Yung (1994) have addressed the use of RBF to approximate dynamic and state equations and to estimate state variables of stochastic systems.

A considerable amount of work has been carried out to develop and implement NN-based controllers for flexible manipulators. Cheng and Wen (1993) proposed a neuro-controller to drive a flexible arm to a desired trajectory along with using hub position and velocity measurement techniques for stabilising the system. New-ton and Xu (1993) have addressed the joint tracking control problem for a space manipulator using feedback error learning technique. In this case, end-point posi-tion tracking cannot be guaranteed especially for high-speed desired trajectories.

A considerable amount of work has been carried out to develop and implement NN-based controllers for flexible manipulators. Cheng and Wen (1993) proposed a neuro-controller to drive a flexible arm to a desired trajectory along with using hub position and velocity measurement techniques for stabilising the system. New-ton and Xu (1993) have addressed the joint tracking control problem for a space manipulator using feedback error learning technique. In this case, end-point posi-tion tracking cannot be guaranteed especially for high-speed desired trajectories.