PERFORMANCE OF HYBRID LEARNING CONTROL WITH INPUT SHAPING FOR INPUT TRACKING AND VIBRATION SUPPRESSION OF A FLEXIBLE MANIPULATOR

The objective of the work reported in this paper is to investigate the development of an intelligent hybrid iterative learning control scheme with input shaping for input tracking and end-point vibration suppression of a flexible manipulator. The dynamic model of the system is derived using the finite element method. Initially, a collocated proportional-derivative (PD) controller utilizing hub-angle and hub-velocity feedback is developed for control of rigidbody motion of the system. This is then extended to incorporate iterative learning control with genetic algorithm (GA) to optimize the learning parameters and a feedforward controller based on input shaping techniques for control of vibration (flexible motion) of the system. Simulation results of the response of the manipulator with the controllers are presented in time and frequency domains. The performance of the hybrid learning control with input shaping scheme is assessed in terms of input tracking and level of vibration reduction. The effectiveness of the control schemes in handling various payloads is also studied.


INTRODUCTION
Most robot manipulators are designed and built in a manner to maximize stiffness, in an attempt to minimise system vibration and achieve good positional accuracy. High stiffness is achieved by using heavy material. As a consequence, such robots are usually heavy with respect to the operating payload. This, in turn, limits the speed of operation of the robot manipulation, increases the size of actuator, boosts energy consumption and increases the overall cost. Moreover, the payload to robot weight ratio, under such situations, is low. In order to solve these problems, robotic systems are designed to be lightweight and thus possess some level of flexibility.
Flexible manipulators exhibit many advantages over their rigid counterparts: they require less material, are lighter in weight, have higher manipulation speed, lower power consumption, require smaller actuators, are more maneuverable and transportable, are safer to operate due to reduced inertia, have enhanced back-drive ability due to elimination of gearing, have less overall cost and higher payload to robot weight ratio [1]. However, the control of flexible manipulators to maintain accurate positioning is an extremely challenging problem. Due to the flexible nature and distributed characteristics of the system, the dynamics are highly non-linear and complex. Problems arise due to precise positioning requirement, vibration due to system flexibility, difficulty in obtaining accurate model of the system and non-minimum phase characteristics [2,3]. In this respect, a control mechanism that accounts for both the rigid body and flexural motions of the system is required. If the advantages associated with lightness are not to be sacrificed, accurate models and efficient control strategies for flexible robot manipulators have to be developed.
The control strategies for flexible robot manipulator systems can be classified as feedforward (open-loop) and feedback (closed-loop) control schemes. Feed-forward techniques for vibration suppression involve developing the control input through consideration of the physical and vibrational properties of the system, so that system vibrations at response modes are reduced. This method does not require any additional sensors or actuators and does not account for changes in the system once the input is developed. On the other hand, feedbackcontrol techniques use measurement and estimations of the system states to reduce vibration.
Feedback controllers can be designed to be robust to parameter uncertainty. For flexible manipulators, feedforward and feedback control techniques are used for vibration suppression and position control respectively. An acceptable system performance without vibration that accounts for system changes can be achieved by developing a hybrid controller consisting of both control techniques. Thus, a properly designed feedforward controller is required, with which the complexity of the required feedback controller can be reduced. This paper presents investigations into the development of hybrid learning control with input shaping for input tracking and end-point vibration suppression of a flexible manipulator system. A constrained planar single-link flexible manipulator is considered and a simulation environment is developed within Simulink and Matlab for evaluation of performance of the control strategies. In this work, the dynamic model of the flexible manipulator is derived using the finite element (FE) method. Previous simulation and experimental studies have shown that the FE method gives an acceptable dynamic characterization of the actual system [4. Previously, a collocated PD control with a non-collocated PID control has been developed [5]. To demonstrate the effectiveness of the proposed control schemes, initially a joint-based collocated PD control utilising hub-angle and hub-velocity feedback is developed for control of rigid body motion of the manipulator. This is then extended to incorporate an iterative learning control (ILC) scheme, with genetic algorithms (GAs) for optimization of the learning parameters and input shaping for vibration suppression of the manipulator. For noncollocated control, end-point displacement feedback through a PID control configuration is developed whereas in the feedforward scheme, the input shaping technique is utilised as this has been shown to be effective in reducing system vibration [5]. Simulation results of the response of the manipulator with the controllers are presented in time and frequency domains.
The performances of the hybrid learning control with input shaping are assessed in terms of input tracking and level of vibration reduction in comparison to the response with collocated PD and non-collocated PID (PD-PID) control. As the dynamic behaviour of the system changes with different payloads, the effectiveness of the controllers is also studied with a different loading condition. Finally, a comparative assessment of the hybrid learning control scheme in input tracking and vibration suppression of the manipulator is presented.

THE FLEXIBLE MANIPULATOR SYSTEM
A schematic representation of the single-link flexible manipulator system considered in this work is shown in Figure 1, where a control torque ) (t τ is applied at the hub by an actuator motor, , ρ and p M represent Young's modulus, moment of inertia, mass density per unit volume, length, hub inertia and payload of the manipulator respectively. The angular displacement of the link in the POQ co-ordinates is denoted as ) (t θ . w represents the elastic deflection of the manipulator at a distance x from the hub, measured along the OP ' axis. POQ and P ' OQ ' represent the stationary and moving frames respectively. The height (width) of the link is assumed to be much greater than its depth, thus allowing the manipulator to vibrate dominantly in the horizontal direction (POQ plane). To avoid difficulties arising from time varying lengths, the length of the manipulator is assumed to be constant. Moreover, shear deformation, rotary inertia and the effect of axial force are ignored.
For an angular displacement θ and an elastic deflection w , the total displacement ) , ( t x y of a point along the manipulator at a distance x from the hub can be described as a function of both the rigid body motion ) (t θ and elastic deflection ), Thus, the net deflection at x is the sum of a rigid body deflection and an elastic deflection.
Note that by allowing the manipulator to be dominantly flexible in the horizontal direction the elastic deflection of the manipulator can be assumed to be confined to the horizontal plane only. In general, the motion of a manipulator will include elastic deflection in both, the previously been developed using the finite element (FE) method [4]. This is used in this work as a platform for test and evaluation of the proposed control approaches.

CONTROL SCHEMES
In this section, control schemes for rigid-body motion control and vibration suppression of the flexible manipulator are introduced. Initially, a collocated PD control is designed. This is then extended to incorporate an ILC scheme for control of vibration of the system.

Collocated PD control
A common strategy in the control of manipulator systems involves the utilization of PD feedback of collocated sensor signals. Such a strategy is adopted at this stage of the investigation here. A block diagram of the PD controller is shown in Figure 2, where p K and v K are the proportional and derivative gains respectively θ and θ & represent hub angle and hub velocity respectively, f R is the reference hub angle and c A is the gain of the motor amplifier. Here the motor/amplifier set is considered as a linear gain c A . To design the PD controller a linear state-space model of the flexible manipulator was obtained by linearizing the equations of motion of the system. The first two flexible modes of the manipulator were assumed to be dominantly significant. The control signal ) (s u in Figure 2 can thus be written where s is the Laplace variable. The closed-loop transfer function is, therefore, obtained as: is the open-loop transfer function from the input torque to hub angle, given by where Α , B and C are the characteristic matrix, input matrix and output matrix of the system respectively and I is the identity matrix. The closed-loop poles of the system are, thus, given by the closed-loop characteristic equation as: represents the compensator zero which determines the control performance and characterises the shape of root locus of the closed-loop system.
Theoretically any choice of the gains p K and v K assures the stability of the system [6]. In this study, the root locus approach is utilized to design the PD controller. Analyses of the root locus plot of the system show that dominant poles with maximum negative real parts could be achieved with 2 ≈ Z and by setting p K between 0 and 1.2 [7].

Hybrid Collocated PD with Iterative Learning Control
A hybrid collocated PD control structure for control of rigid-body motion of the flexible manipulator with ILC is proposed in this section. In this study, an ILC scheme is developed using PD-type learning algorithm.
Iterative learning control has been an active research area for more than a decade, mainly inspired by the pioneering work of Arimoto et al. [8][9][10]. Learning control begun with the fundamental principle that repeated practice is a common mode of human learning. Given a goal (regulation, tracking, or optimization), learning control, or more specifically, iterative learning control refers to the mechanism by which necessary control can be synthesized by repeated trials. Moore [11] describes ILC as an approach to improving the transient response performance of a system that operates repetitively over a fixed interval. This is especially applicable to a system such as industrial robot which accomplishes most of its tasks repetitively over a period of time. Consider a robot arm in which a number of conditions such as varying the input parameters and disturbances, are imposed. Performance of the arm, e.g.
trajectory control can be evaluated, changed or improved iteratively by means of using the previous response. This is in turn incorporated in the control strategy during the next cycle to improve its performance. In this way, an ILC scheme is established in which unlike conventional adaptive control approaches, the control strategy is changed by changing the command reference signal and not the controller itself. Uchiyama first introduced the concept of iterative learning for generating the optimal input to a system [12]. Arimoto and his coworkers later developed the idea [8][9][10]. and output signal ) (t x k , are stored in memory (some type of memory device is implicitly assumed in the block labeled "learning controller"). By using the desired output of the system ) (t x d and the actual output ) (t x k , the performance error at k th trial can be defined as:

Figure 3: Iterative learning control configuration
The aim of ILC is to iteratively compute a new compensation input signal ) which is stored for use in the next trial. The next input command is chosen in such a way as to guarantee that the performance error will be reduced in the next trial. The important task in the design of a learning controller is to find an algorithm for generating the next input in such a way that the performance error is reduced on successive trials. In other words, the algorithm needs to lead to the convergence of the error to minimum. Another consideration is that it is desirable to have the convergence of the error without or at least with minimal knowledge of the model of the system. Further, the algorithm should be independent of the functional form of the desired response, ) (t x d . Thus, the learning controller would "learn" the best possible control signal for a particular desired output trajectory even if it is newly introduced without the need to reconfigure the algorithm.
In this work a learning algorithm of the following form is considered: Γ Φ, are suitable positive definite constants (or learning parameters) A block diagram of the scheme is shown in Figure 4. It is obvious that the algorithm contains a constant and derivative coefficient of the error. In other words, the expression can be simply called proportional-derivative or PD type learning algorithm. A slightly modified learning algorithm to suit the application is employed here. Instead of using the absolute position tracking error k e , a sum-squared tracking error k e is used. Figure 4 shows a block diagram describing the above expression. This is used with PD collocated control, to realise the hybrid collocated PD with ILC. This is shown in Figure 5.  Figure 4: PD type learning algorithm Figure 5: The collocated PD with iterative learning control structure

GA based hybrid learning control
, N= size of sample , r(t) = reference input, y(t) = measured variable. Thus, the function in equation (8) can be minimized by applying a suitable tuning algorithm as illustrated in the next section or through the application of a genetic algorithm.
The GA used here initializes a random set of population of the two variables Φ and Γ . The algorithm evaluates all members of the population based on the specified performance index.
The algorithm then applies the GA operators such as reproduction, crossover and mutation to generate a new set of population based on the performance of the members of the population [13]. The best member or gene of the population is chosen and saved for next generation. It again applies all operations and selects the best gene among the new population. The best gene of the new population is compared to best gene of previous population. If a predefined termination criterion is not met, a new population is obtained in the same way as above. The termination criterion may be formulated as the magnitude of difference between index value of previous generation and present generation becoming less than a prespecified value. The process continues till the termination criterion is fulfilled.

Hybrid PD and non-collocated control
The use of a non-collocated control, where the end-point of the manipulator is controlled by measuring its position, can be applied to improve the overall performance, as more reliable output measurement can be obtained. The control structure comprises two feedback loops: a) the hub-angle and hub-velocity as inputs to a collocated control law for rigid-body motion control; b) the end-point residual (elastic deformation) as input to a separate non-collocated control law for vibration control. These two loops are then summed together to give a torque input to the system. A block diagram of the control scheme is shown in Figure 6, where α r represents the end-point residual reference input, which is set to zero as the control objective is to have zero vibration during movement of the manipulator. For rigid-body motion control, the PD control strategy developed in the previous section is adopted whereas for the vibration control loop, the end-point residual feedback through a PID control scheme is utilized. The values of proportional (P), derivative (D) and integral (I) gains are adjusted using the Ziegel-Nichols procedure [14]. For the two control loops to work well they have to be decoupled from one another. This can be achieved by using a high-pass filter in the non-collocated control loop.

Flexible Manipulator
System Figure 6: The collocated PD and non-collocated PID control

Hybrid control with input shaping
The method of input shaping involves convolving a desired command with a sequence of impulses. The design objectives are to determine the amplitude and time location of the impulses. A brief derivation is given below. Further details can be found in [15]. where ω is the natural frequency and ξ is the damping ratio of the system. Thus, the impulse response of the system can be obtained as: impulse is selected at time 0. The simplest constraint is zero vibration at expected frequency and damping of vibration using a two-impulse sequence. Hence by setting equation (9) to zero, and solving yields a two-impulse sequence with parameters as: Setting the derivative to zero is equivalent to setting small changes in vibration for changes in the natural frequency. Thus, additional constraints are added into the equation, which after solving yields a three-impulse sequence with parameters as: where K is as in equation (10). The robustness of the input shaper can further be increased by taking and solving the second derivative of the vibration in equation (9). Similarly, this yields a four-impulse sequence with parameters as: 3  2   3   4  3  2   2   3   3  2  2  3  2  1   2  4  2  3  2  1   3  3  1  ,  3  3  1   3   ,  3  3  1   3  ,  3  3  1   1   ,  3  ,  2  , , 0 where K is as in equation (10).
To handle higher vibration modes, an impulse sequence for each vibration mode can be designed independently. Then the impulse sequences can be convoluted together to form a sequence of impulses that attenuates vibration at higher modes. For any vibratory system, the vibration reduction can be accomplished by convolving any desired system input with the impulse sequence. This yields a shaped input that drives the system to a desired location without vibration. Incorporating the input shaping into PD-ILC structure results in the combined PD-ILC and input shaping control structure shown in Figure 7.    Figure 11 (a). The corresponding system response is shown in Figure 11 (b),(c) and (d). The closed-loop parameters with the PD control will subsequently be used to design and evaluate the performance of non-collocated and feedforward control schemes in terms of input tracking capability and level of vibration reduction. The results in Figure 11 for the collocated PD control will be used for comparative assessment of the hybrid control schemes proposed in section 3. The PID controller parameters were tuned using the Ziegel-Nichols method using a closed-loop technique where the proportional gain p k was initially tuned and the integral gain i k and derivative gain d k were then calculated [14]. Accordingly, the PID   Table 1 show the algorithm convergence as a function of generations and the parameter values used in the GA respectively. Figures 15 and 16 show the corresponding responses of the manipulator without payload and with a 15 g payload with PD-ILC. It is noted that the proposed hybrid controller with learning algorithm is capable of reducing the system vibration while resulting in better input tracking performance of the manipulator. The vibration of the system settled within less than 1.5 s, which is much less than that achieved with PD-PID control. The closed-loop system parameters with the PD control will subsequently be used to design and evaluate the performance of ILC and feedforward control schemes in terms of input tracking capability and level of vibration reduction.   For digital implementation of the input shaper, locations of the impulses were selected at the nearest sampling time. In this case, the locations of the second impulse were obtained at 0.042 sec, 0.014 sec and 0.008 sec for the three modes respectively. The developed input shaper was then used to pre-process the input reference shown in Figure 8. Figure  shown that hybrid PD-ILC-IS scheme results in better performance than the PD-PID control in respect of hub-angle response and vibration suppression of the manipulator.